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0000000000000000000000000000000000000000..8b000aeb65b68964641815f577e7a89cdf8ab308 --- /dev/null +++ b/09E2T4oBgHgl3EQfNAa-/content/tmp_files/2301.03732v1.pdf.txt @@ -0,0 +1,545 @@ +arXiv:2301.03732v1 [math.DG] 10 Jan 2023 +A SCHUR’S THEOREM VIA A MONOTONICITY AND THE +EXPANSION MODULE +LEI NI +Abstract. In this paper we present a monotonicity which extends a classical theorem +of A. Schur comparing the chord length of a convex plane curve with a space curve of +smaller curvature. We also prove a Schur’s Theorem for spherical curves, which extends +the Cauchy’s Arm Lemma. +1. Introduction +For a convex curve c(s) : [0, L] → R2 and a smooth curve in ˜c(s) : [0, L] → R3 of the same +length (both parametrized by the arc-length), A. Schur’s theorem [7] (Theorem A page 31, +see also [5]) asserts that if both curves are embedded, and the curvature of the space curve +˜k(s) := | ˜T ′|(s), where ˜T(s) = ˜c′(s) is the tangent vector, is not greater than the curvature +k(s) of the convex curve, then d(˜c(0), ˜c(L)) ≥ d(c(0), c(L)). From the proof of [7] it is easy +to see R3 can be replaced by Rn with n ≥ 2. +The theorem can be proven for curves whose tangents have finite discontinuous jumps, +and to the situation that the curvature of the smaller curve is a curve in Rn+1 for n ≥ 1. +In terms of the generalization to curves with finite discontinuous points for the tangent, it +assumes that there exists {sj}0≤j≤N such that 0 = s0 < s1 < · · · < sk < · · · < sN = L such +that both c(s) and ˜c(s) are regular embedded curves for s ∈ (sj−1, sj) for all 1 ≤ j ≤ N +satisfying k(s) ≥ ˜k(s), and for each 1 ≤ j ≤ N − 1 at the point c(sj) and ˜c(sj), the oriented +turning angles, which are measured by signed distance αj := dSn(c′(sj−), c′(sj+)) > 0 and +˜αj = dSn(˜c′(sj−), ˜c′(sj+)), satisfy that αj ≥ ˜αj for all 1 ≤ j ≤ N − 1. The convexity of +c(s) and the simpleness assumption imply that +N +� +j=1 +� sj +sj−1 +k(s) ds + +N−1 +� +j=1 +αj ≤ 2π. +(1.1) +This extension, together with some ingenious applications of the hinge’s theorem, allows +one to prove the famous Cauchy’s Arm Lemma for geodesic arms in the unit sphere (consist- +ing of continuous broken great/geodesic arcs with finite jumps of the tangents) in Lemma +II on the pages 37–38 of [7]. The Lemma became famous due to that it had an incom- +plete/false proof by Cauchy originally [4]. The corrected proof appeared in [1, 11]. This +spherical Cauchy’s Arm Lemma can also be proved by an induction argument [12], whose +idea in fact in part resembles the proof of the smooth case to some degree. Note that this +lemma of Cauchy plays a crucial role in the rigidity of convex polyhedra in R3, which finally +was vastly generalized to convex surfaces (convex bodies enclosed) as the famous Pogorelov +monotypy theorem (cf. [3] Section 21). +1 + +2 +LEI NI +The Schur’s theorem also can be applied to prove the four-vertex theorem for convex plane +curves, besides implying a Theorem of H. A. Schwartz which asserts: For any curve c of +length L with curvature k(s) ≤ 1/r, let C be the circle passing c(0) and c(L) of radius r, +then L is either not greater than the length of the lesser circular arc, or not less than the +length of the greater circular arc of C. +High dimensional (intrinsic) analogues of A. Schur’s +theorem include the Rauch’s comparison theorem and the Toponogov comparison theorem. +The later however has the limit of requiring that the manifold with less curvature must be +a space form of constant sectional curvature. +First we have the following slight more general version of Schur’s theorem in terms of a +monotonicity. +Theorem 1.1. Let c : [0, L] → R2 be an embedded convex plane curve with curvature +k(s) ≥ 0. Let ˜c : [0, L] → Rn+1 (n ≥ 1) be a curve such that ˜k(s) ≤ k(s). Then for any +0 ≤ s′ < s′′ ≤ L there exist an isometric inclusion ιs′,s′′ : R2 → Rn+1 with ιs′,s′′(0) = 0 +such that +I(s) := ⟨˜c(s) − ιs′,s′′(c(s)), ιs′,s′′(c(s′′) − c(s′))⟩ +is monotone non-decreasing for s ∈ [s′, s′′], or equivalently +⟨ ˜T(s) − ιs′,s′′(T (s)), ιs′,s′′(c(s′′) − c(s′))⟩ ≥ 0, +∀ s ∈ [s′, s′′]. +(1.2) +As s′ → s′′, the inclusion ιs′,s′′ converges to an inclusion identifying T (s) with ˜T(s). +Corollary 1.2. Under the same assumption as in the theorem, for any s′ ≤ s′ +∗ < s′′ +∗ ≤ s′′, +⟨c(s′′ +∗) − c(s′ +∗), c(s′′) − c(s′)⟩ ≤ ⟨˜c(s′′ +∗) − ˜c(s′ +∗), ιs′,s′′(c(s′′) − c(s′))⟩. +(1.3) +When s′ = s′ +∗ and s′′ = s′′ +∗ we have that +|c(s′′) − c(s′)|2 ≤ ⟨˜c(s′′) − ˜c(s′), ιs′,s′′(c(s′′) − c(s′))⟩. +(1.4) +The estimate (1.4) implies Schur’s theorem by the Cauchy-Schwarz inequality applied to +the right hand side of (1.4): +|c(s′′) − c(s′)| ≤ |˜c(s′′) − ˜c(s′)|, +∀ 0 ≤ s′ < s′′ ≤ L. +This extension allows one to rephrase the result in terms of the concept of the expansion +module [2, 9] of vector fields. If X : Ω ⊂ Rn+1 → Rn+1 is a vector field defined on a convex +domain, then the expansion module is a function of one variable ψ(t) such that +⟨X(y) − X(x), y − x +|y − x|⟩ ≥ 2ψ +�|x − y| +2 +� +. +Since ˜c(s) and c(s) are related via the parameter s, one may view ˜c as a related vector field +defined over ιs′,s′′(c(s)) ∈ Rn+1. Now the estimate in Theorem 1.1 simply asserts that the +related vector fields ˜c(s) has an expansion module function ψ(t) = t with respect to the +associated vector ιs′,s′′(c(s)). +From the above connection between the concept of curvature and the expansion module +it is our hope that a high dimensional Schur’s theorem could be discovered through the +consideration involving the expansion module. +Given that Schur’s theorem implies the Cauchy’s Arm Lemma for the arms of great arcs +in the unit sphere, a natural question is that if the spherical analogue of Schur’s theorem +still holds. Namely, given two embedded spherical curves c(s) and ˜c(s) in the unit sphere +S2 ⊂ R3 parametrized by the arc-length s ∈ [0, L] with L ≤ π. Assume that c(s) is convex + +AN EXTENSION OF SCHUR’S THEOREM +3 +with geodesic curvature k(s) > 0 and that the geodesic curvature of ˜c satisfies |˜k|(s) ≤ k(s). +Does it still hold that |c(0) − c(L)| ≤ |˜c(0) − ˜c(L)|? One could also allow the tangent of +curves to have same amount of finite many jumps at {sj}. In that case, the oriented angles +αj and ˜αj are assumed to satisfy that αj ≥ ˜αj as in the case of Schur’s theorem. The +Cauchy’s Arm Lemma in the sphere answers the question affirmatively in the special case +where both curves have zero geodesic curvature for the smooth parts. Here we confirm this +conjecture by proving +Theorem 1.3. The Schur’s theorem holds for two curves in S2 ⊂ R3 under the above +configurations similar to that of Theorem 1.1. +The proof is via construction of auxiliary curves with one of them being a convex plane +curve and appealing to the original Schur’s theorem. This is part of the reason we present +the proof of Theorem 1.1 with care and details. Note that this result generalizes the spherical +Cauchy’s Arm Lemma. It would be interesting to see if it plays any role in the proof of +Pogorelov’s monotype theorem. There were extensions of A. Schur’s theorem in hyperbolic +spaces [6] and in the Minkowski plane [8] earlier. It is plausible that the method of this +paper can be used to simplify the argument in the former work via Theorem 2.1. +2. Proof of Theorem 1.1 +We prove theorem and its corollary together. After an inclusion ι : R2 → Rn+1, which +shall be specified later, we may consider the tangent T (s) and ˜T(s) as two curves in Sn. For +the proof we need to choose a point N ∈ image(T (s)). For the situation when the tangent +T (s) has a jump at sj, the minimizing arc jointing T (sj−) and T (sj+) is also considered +to be part of the image. They together form a part of a great circle which is denoted by +Image(T (s)). +Consider the two curves T (s) and ˜T (s) inside Sn(1). The first one is a plane curve, hence +is part of a great arc. We first find a s∗ ∈ [s′, s′′] (and then let N := T (s∗)), such that T (s∗) +is parallel to c(s′′) − c(s′) using the convexity of the cone over image(T (s)). +Since T (s), s ∈ [s′, s′′] forms a part of a great circle, letting j1 be the smallest j with sj ≥ s′ +and j2 being the greatest j with sj ≤ s′′, by the mean value theorem +c(s′′) − c(s′) += +� sj1 +s′ +T (s) ds + +� +j1≤j≤j2−1 +� sj+1 +sj +T (s) ds + +� s′′ +sj2 +T (s) ds += +(sj1 − s′)T ((s∗)j1) + +j2−1 +� +j1 +(sj+1 − sj)T ((s∗)j+1) + (s′′ − sj2)T ((s∗)j2+1). +Since +1 +s′′−s′ multiple of the right hand above lies inside the cone over the image of T (s) for s ∈ +[s′, s′′], it implies that there exists (a unique) s∗ ∈ [s′, s′′] such that T (s∗) = λ(c(s′′) − c(s′)) +with λ = +1 +|c(s′′)−c(s′)|. +Now consider the two products Pi defined as (with N = T (s∗)) +P1 := ⟨c(s′′) − c(s′), N⟩ = +� s′′ +s′ ⟨T (s), N⟩ ds, +P2 := ⟨˜c(s′′) − ˜c(s′), N⟩ = +� s′′ +s′ ⟨ ˜T(s), N⟩ ds. +From the choice of s∗, ⟨c(s′′) − c(s′), N⟩ = |c(s′′) − c(s′)|. Now P1 = |c(s′′) − c(s′)|, and +P2 = ⟨˜c(s′′) − ˜c(s′), c(s′′) − c(s′)⟩/|c(s′′) − c(s′)|. + +4 +LEI NI +The claimed estimate (1.4) amounts to showing that the second product is bounded from +below by the first after a proper inclusion. Let j3 be the biggest j with sj ≤ s∗. Observe +the convexity of c(s) implies that +αj0 + +� +j1≤j≤j3 +αj + +� sj1 +s′ +k(s) ds + +� +j1≤j≤j3−1 +� sj+1 +sj +k(s) ds + +� s∗ +sj3 +k(s) ds = π, +� +j3+1≤j≤j2 +αj + αj4 + +� sj3+1 +s∗ +k(s) ds + +� +j3+1≤j≤j2−1 +� sj+1 +sj +k(s) ds + +� s′′ +sj2 +k(s) ds = π, +with αj0 being the angle from −N to T (s′) and αj4 being the angle from T (s′′) and −N. +This implies that the images of T ([s′, s∗]) (denoted as curve Γ1(s)) and T ([s∗, s′′]) (denoted +as the spherical curve Γ2(s)) are two minimizing arcs of the great circle (formed by the +intersection of the plane in which c(s) lies and Sn(1)). We also denote the spherical curves +corresponding to ˜T by �Γi. Hence +max{Length(Γ1), Length(Γ2)} ≤ π. +(2.1) +On the other hand, by rotation we may arrange the inclusion ι such that ˜T(s∗) = N = T (s∗). +Now we estimate +π +≥ +dSn(T (s′), N) = dSn(T (s′), T (s∗)) = Length(Γ1) += +� +j1≤j≤j3 +αj + +� sj1 +s′ +k(s) ds + +� +j1≤j≤j3−1 +� sj+1 +sj +k(s) ds + +� s∗ +sj3 +k(s) ds +≥ +� +j1≤j≤j3 +˜αj + +� sj1 +s′ +˜k(s) ds + +� +j1≤j≤j3−1 +� sj+1 +sj +˜k(s) ds + +� s∗ +sj3 +˜k(s) ds += +� +j1≤j≤j3 +˜αj + +� sj1 +s′ +| ˜T ′|(s) ds + +� +j1≤j≤j3−1 +� sj+1 +sj +| ˜T ′|(s) ds + +� s′′ +sj3 +| ˜T ′|(s) ds +≥ +dSn( ˜T(s′), ˜T(s∗)). +The second line above follows from the definition of the curvature measuring the rotating +angle of the tangent for a curve [10] (cf. page 49) and that for a convex curve k(s) ≥ 0. The +third line uses the assumption, and the last line follows from the definition of the (spherical) +distance between two points being the infimum of the length of all possible connecting curves +in Sn. The same argument also implies that for any s ∈ [s′, s∗] +π ≥ dSn(T (s), N) = Length(Γ1|[s,s∗]) ≥ dSn( ˜T(s), N). +This implies that +⟨T (s), N⟩ = cos(dSn(T (s), T (s∗)) ≤ cos(dSn( ˜T(s), ˜T (s∗)) = cos(dSn( ˜T(s), N)). +(2.2) +Rewriting the above estimate we have that +⟨ ˜T (s) − T (s), N⟩ ≥ 0 +for s ∈ [s′, s∗], which implies (1.2). A similar argument proves that the same inequality +holds also for s ∈ [s∗, s′′]. Putting them together we have (1.2). The above proof also works +for any N = T (s∗) such that (2.1) holds, while (1.1) implies that one can always choose a +s∗ ∈ [0, L] independent of s′ and s′′. (However such s∗ is far from being unique.) + +AN EXTENSION OF SCHUR’S THEOREM +5 +Now we compare the two products Pi by writing +P1 = +� s′′ +s′ ⟨T (s), N⟩ ds = +�� s∗ +s′ ++ +� s′′ +s∗ +� +cos (dSn(T (s), T (s∗)) ds. +We express P2 accordingly. The above estimate (2.2) implies that +� s∗ +s′ +cos(dSn(T (s), T (s∗)) ds ≤ +� s∗ +s′ +cos(dSn( ˜T (s), ˜T(s∗)) ds. +(2.3) +Similarly, we have +� s′′ +s∗ +cos(dSn(T (s), T (s∗)) ds ≤ +� s′′ +s∗ +cos(dSn( ˜T (s), ˜T(s∗)) ds. +(2.4) +From (2.3) and (2.4) we have that P1 ≤ P2, namely the desired claim (1.4). +From the proof we have the following more general monotonicity. +Proposition 2.1. Let c : [0, L] → R2 be an embedded convex plane curve with curvature +k(s) ≥ 0. Let ˜c : [0, L] → Rn+1 (n ≥ 1) be a curve such that ˜k(s) ≤ k(s). Then there exists +s∗ ∈ [0, L] and an inclusion ι : R2 → Rn+1 with ι(0) = 0 and ι(T (s∗)) = ˜T(s∗), such that +I′ +1(s) = ⟨ ˜T (s) − ι(T (s)), ˜T(s∗)⟩ ≥ 0, +∀ s ∈ [0, L], where I1(s) = ⟨˜c(s) − ι(c(s)), ˜T (s∗)⟩. +(2.5) +Here the choices of s∗ and ι are more flexible than in Theorem 1.1, where they are essentially +unique. +The proof can be easily adopted to show a comparison between a time-like curves in a +Minkowski plane L2 +1 and another time-like curve in the three dimensional Minkowski space +L3 +1 with signature (+, −, −). In fact in terms of the monotonicity one may choose s∗ freely. +Following the convention of the physics a vector u is called time-like if ⟨u, u⟩ > 0. For a +time-like curve c(s), |T (s)|2 = |c′(s)|2 = 1. Hence T (s) can be viewed as a point in the +hyperbolic line/plane defined as x2 +1 − x2 +2 = 1 (or x2 +1 − x2 +2 − x2 +3 = 1). It can be checked easily +that −1 multiple of the restricted metric on the surface is the standard hyperbolic metric. +The angle between two tangents T (s1) and T (s2) is given by ⟨T (s1), T (s2)⟩ = cosh ϕ(s2, s1). +A simple computation shows that ϕ(s2, s1) equals to the hyperbolic distance between T (s1) +and T (s2). +For space-like curves the length of the vector u is defined to be +� +−⟨u, u⟩. +Equipped with the above basics, a similar consideration as the above gives the following +result. +Theorem 2.1. Let c(s) : [0, L] be a time-like convex curve in L2 +1 parametrized by the arc- +length, and let ˜c(s) : [0, L] be a similarly parametrized regular time-like curve in L3 +1. Assume +that k(s) ≥ |˜k|(s). Then for any s∗ ∈ [0, L] and an isometric inclusion of ι : L2 +1 → L3 +1, which +identifies T (s∗) with ˜T(s∗), we have that +I′ +2(s) = ⟨ι(T (s)) − ˜T(s), ˜T(s∗)⟩ ≥ 0, where I2(s) = ⟨ι(c(s)) − ˜c(s), ˜T (s∗)⟩. +(2.6) +In particular, |c(L) − c(0)| ≥ |˜c(L) − ˜c(0)|. +The last statement of (ii) generalizes the result of [8] by allowing the second curve ˜c(s) a +space curve in L3 +1. Note that for the curves in two Minkowski planes, the result for space-like +curves is the same as that for the time-like curves. To prove the last conclusion we first +integrate (2.6) with s∗ so chosen that T (s∗) is proportional to c(L) − c(0), and then apply +the reserved Cauchy-Schwarz inequality (which holds for two time-like vectors). + +6 +LEI NI +3. Proof of Theorem 1.3 +We start with some basics on spherical (smooth) curves. +Let c(s) be a curve in S2 +parametrized by the arc-length. Let T (s) be its tangent, which is orthogonal to c(s). Let +V (s) = c(s) × T (s) be the cross product of c(s) and T (s) in R3, which is a normal of c(s) +in Tc(s)S2. The triple {c(s), T (s), V (s)} forms an orthonormal moving frame (of R3) along +c(s). Since the geodesic curvature of a curve in the sphere (in a surface) is the changing +rate of the tangential great circles (tangential geodesics in general, by (8-3) of page 157 of +[13]), and that V (s) provides a natural parametrization of the tangential great circles, the +derivative of V (s) yields the geodesic curvature of c(s). This can also be formulated in terms +of the following result. +Proposition 3.1. Let k(s) be the geodesic curvature of c(s) (with respect to S2). Then the +following holds for {c(s), T (s), V (s)}. +c′(s) += +T (s), +(3.1) +T ′(s) += +k(s)V (s) − c(s), +(3.2) +V ′(s) += +−k(s)T (s). +(3.3) +Proof. The first equation is definition. Also by definition k(s) = ⟨T ′(s), V (s)⟩. Hence from +0 = d2 +ds2 +� +|c|2(s) +� += 2⟨T (s), T (s)⟩ + 2⟨c(s), T ′(s)⟩ = 2 + 2⟨c(s), T ′(s)⟩ +we deduce the second equation. Now by the second equation +V ′(s) = T (s) × T (s) + c(s) × T ′(s) = k(s) c(s) × V (s) = −k(s)T (s). +This prove the third one, hence completes the proof of the proposition. +□ +The local convexity of c(s) is equivalent to k(s) ≥ 0. The basic construction is to look +at the cone C(c(s)) over the spherical curve c(s) centered at the origin, and obtain a plane +curve by taking the intersection of C(c(s)) with a plane P not passing the origin to obtain +a plane curve Pc(s). This curve can be expressed as R(s)c(s) with R(s) being the distance +of Pc(s) to the origin. We need the following formula for the curvature of the space curve +in R3 applied to Pc(s). +Proposition 3.2. If c(s) is a convex curve in S2, Pc(s) is a convex curve in P. +The +curvature k(s) of Pc(s) (as a space curve of R3) is given by +k2(s) = |P′ +c(s) × P′′ +c (s)|2 +|P′c(s)|3 +. +(3.4) +Proof. From the geometric definition of the convexity we know that c(s) lies in a signed +semi-sphere cut out by any tangent great circle obtained by a plane passing the origin. +Then it is clear that Pc(s) lies on the corresponding half plane cut out by the corresponding +tangent line of Pc(s) in P. This proves the convexity of Pc(s). The formula for the curvature +of a space curve is well known and computational. See for example page 51 of [10]. Of course +the formula applies to the case that the curve happens to be a plane curve. +□ +Now let τ be the arc-length parameter of Pc(s). Direct calculation shows that +τ(s) = +� s +0 +� +(R′(s))2 + R2(s) ds. +(3.5) + +AN EXTENSION OF SCHUR’S THEOREM +7 +Now we construct a space curve ˜P˜c(s) corresponding to ˜c(s) by defining it as R(s)˜c(s). In +general, this is not a plane curve. The key observation is that the arc-length parameter for +˜P˜c(s) is the same as that of Pc(s), namely it is given by (3.5) as well, since |˜c|(s) = 1 = |c(s)| +and |˜c′|(s) = 1 = |c′(s)|. Moreover its curvature ˜k(s) (as a curve in R3) can be expressed +similarly as +˜k2(s) = | ˜P′ +˜c(s) × ˜P′′ +˜c (s)|2 +| ˜P′ +˜c(s)|3 +. +(3.6) +Namely the second part of Proposition 3.2 applies to ˜P˜c(s) as well since it holds for any +space curve in R3. The key step is the following comparison. +Proposition 3.3. Under the assumption that the geodesic curvature k(s) of c(s) and the +geodesic curvature ˜k(s) of ˜c(s) satisfy k(s) ≥ |˜k(s)| ≥ 0, the curvatures of Pc(s) and ˜P˜c(s) +satisfies k(s) ≥ 0 and k(s) ≥ |˜k|(s). +Proof. Since Pc(s) is convex, we have that k(s) ≥ 0, it suffices to show that k2(s) ≥ ˜k2(s). +First we observe that |P′ +c(s)|2 = R2(s) + (R′(s))2 = | ˜P′ +˜c(s)|2. +This reduces the desired +estimate to +|P′ +c(s) × P′′ +c (s)|2 ≥ | ˜P′ +˜c(s) × ˜P′′ +˜c (s)|2. +(3.7) +Using the fact that {c(s), T (s), V (s)} forms an oriented orthonormal moving frame, a direct +calculation, using Proposition 3.1, shows that +P′ +c(s) × P′′ +c (s) += +(R′(s)c(s) + R(s)T (s)) × (R′′(s)c(s) + 2R′(s)T (s) + R(s)T ′(s)) += +(2(R′(s))2 − R(s)R′′(s))V (s) − R′(s)R(s)k(s) T (s) +−R(s)R′′(s) V (s) + R2(s)k(s)c(s) + R2(s)V (s) += +R2(s)k(s)c(s) − R′(s)R(s)k(s) T (s) ++(2(R′(s))2 − 2R(s)R′′(s) + R2(s))V (s). +Hence we have that +|P′ +c(s) × P′′ +c (s)|2 += +(R4(s) + (R′(s)R(s))2)k2(s) ++(2(R′(s))2 − 2R(s)R′′(s) + R2(s))2. +(3.8) +A similar calculation shows that +| ˜P′ +˜c(s) × ˜P′′ +˜c (s)|2 += +(R4(s) + (R′(s)R(s))2)˜k2(s) ++(2(R′(s))2 − 2R(s)R′′(s) + R2(s))2. +(3.9) +From (3.8) and (3.9), the assumption k(s) ≥ |˜k|(s) implies (3.7), hence the desired estimate +of the proposition. +□ +Now Proposition 3.3 and (3.5) implies that Pc(τ) and ˜P˜c(τ) are two curves satisfying the +assumption of Theorem 1.1. Hence we have that +d(Pc(0), Pc(τ(L))) ≤ d( ˜P˜c(0), ˜P˜c(τ(L))). +Theorem 1.3 for the smooth curves now follows from the hinge theorem of Euclidean geom- +etry. + +8 +LEI NI +For the general case when the tangents of c(s) and ˜c(s) have finite jumps at {sj}, if we +denote the turning angles at Pc(sj) and ˜P˜c(sj) by θj and ˜θj, then +cos θj = +R′(sj−)R′(sj+) + R2(sj) cos αj +� +((R′(sj−))2 + R2(sj)) ((R′(sj+))2 + R2(sj)) +. +By a similar formula for cos ˜θj we deduce that θj ≥ ˜θj if αj ≥ ˜αj. Hence Theorem 1.3 +follows from the general case of Theorem 1.1. +Acknowledgments +The author would like to thank Burkhard Wilking for helpful discussions, Paul Bryant, +Jon Wolfson, H. Wu and Fangyang Zheng for their interests to the problem considered. +References +[1] A. D. Alexandrow, Konvexe Polyeder. German translation from Russian; Akademie-Verlag, Berlin, 1958. +[2] B. Andrews and J. Clutterbuck, Proof of the fundamental gap conjecture. J. Amer. Math. Soc. 24 +(2011), no. 3, 899–916. +[3] H. Busemann, Convex surfaces. Interscience Tracts in Pure and Applied Mathematics, no. 6. Interscience +Publishers, Inc., New York; Interscience Publishers Ltd., London 1958 ix+196 pp. +[4] A. Cauchy, Sur les polygones et les poly`edres. Second M´emoire, Oeuvres Compl`etes, IIe S´erie, vol. 1; +Paris, 1905. +[5] S. S. Chern, +Curves and surfaces in Euclidean space. 1967 Studies in Global Geometry and Analysis +pp. 16–56 Math. Assoc. America, Buffalo, N.Y.; distributed by Prentice-Hall, Englewood Cliffs, N.J. +[6] C. L. Epstein, The theorem of A Schur in hyperbolic spaces. Preprint 46 pages, 1985. +[7] H. Hopf, Differential Geometry in the Large. Notes taken by Peter Lax and John Gray. With a preface +by S. S. Chern. Second edition. With a preface by K. Voss. Lecture Notes in Mathematics, 1000. +Springer-Verlag, Berlin, 1989. viii+184 pp. +[8] R. L´opez, The theorem of Schur in the Minkowski plane. Jour. Geom. Phys. 61 (2011), 342–346. +[9] L. Ni, Estimates on the modulus of expansion for vector fields solving nonlinear equations. J. Math. +Pures Appl. (9) 99 (2013), no. 1, 1–16. +[10] A. V. Pogorelov, Differential Geometry. P. Noodhoff N. V. 1960. +[11] E. Steinitz and H. Rademacher, Vorlesungen ¨uber die Th´eorie der Polyeder. Springer-Verlag, Berlin, +1934. +[12] I. J. Schoenberg and S. C. Zaremba, +On Cauchy’s lemma concerning convex polygons. Canadian J. +Math. 19 (1967), 1062–1071. +[13] D. J. Struik, Lectures on Classical Differential Geometry. 2nd Edition, Dover, 1988. +Lei Ni. Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, +USA +Email address: leni@ucsd.edu + diff --git a/09E2T4oBgHgl3EQfNAa-/content/tmp_files/load_file.txt b/09E2T4oBgHgl3EQfNAa-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d1f569cf30b04a09cbd595949328a979c87437ad --- /dev/null +++ b/09E2T4oBgHgl3EQfNAa-/content/tmp_files/load_file.txt @@ -0,0 +1,335 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf,len=334 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='03732v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='DG] 10 Jan 2023 A SCHUR’S THEOREM VIA A MONOTONICITY AND THE EXPANSION MODULE LEI NI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' In this paper we present a monotonicity which extends a classical theorem of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Schur comparing the chord length of a convex plane curve with a space curve of smaller curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' We also prove a Schur’s Theorem for spherical curves, which extends the Cauchy’s Arm Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Introduction For a convex curve c(s) : [0, L] → R2 and a smooth curve in ˜c(s) : [0, L] → R3 of the same length (both parametrized by the arc-length), A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Schur’s theorem [7] (Theorem A page 31, see also [5]) asserts that if both curves are embedded, and the curvature of the space curve ˜k(s) := | ˜T ′|(s), where ˜T(s) = ˜c′(s) is the tangent vector, is not greater than the curvature k(s) of the convex curve, then d(˜c(0), ˜c(L)) ≥ d(c(0), c(L)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' From the proof of [7] it is easy to see R3 can be replaced by Rn with n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' The theorem can be proven for curves whose tangents have finite discontinuous jumps, and to the situation that the curvature of the smaller curve is a curve in Rn+1 for n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' In terms of the generalization to curves with finite discontinuous points for the tangent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' it assumes that there exists {sj}0≤j≤N such that 0 = s0 < s1 < · · · < sk < · · · < sN = L such that both c(s) and ˜c(s) are regular embedded curves for s ∈ (sj−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' sj) for all 1 ≤ j ≤ N satisfying k(s) ≥ ˜k(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' and for each 1 ≤ j ≤ N − 1 at the point c(sj) and ˜c(sj),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' the oriented turning angles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' which are measured by signed distance αj := dSn(c′(sj−),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' c′(sj+)) > 0 and ˜αj = dSn(˜c′(sj−),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' ˜c′(sj+)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' satisfy that αj ≥ ˜αj for all 1 ≤ j ≤ N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' The convexity of c(s) and the simpleness assumption imply that N � j=1 � sj sj−1 k(s) ds + N−1 � j=1 αj ≤ 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='1) This extension, together with some ingenious applications of the hinge’s theorem, allows one to prove the famous Cauchy’s Arm Lemma for geodesic arms in the unit sphere (consist- ing of continuous broken great/geodesic arcs with finite jumps of the tangents) in Lemma II on the pages 37–38 of [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' The Lemma became famous due to that it had an incom- plete/false proof by Cauchy originally [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' The corrected proof appeared in [1, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' This spherical Cauchy’s Arm Lemma can also be proved by an induction argument [12], whose idea in fact in part resembles the proof of the smooth case to some degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Note that this lemma of Cauchy plays a crucial role in the rigidity of convex polyhedra in R3, which finally was vastly generalized to convex surfaces (convex bodies enclosed) as the famous Pogorelov monotypy theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' [3] Section 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' 1 2 LEI NI The Schur’s theorem also can be applied to prove the four-vertex theorem for convex plane curves, besides implying a Theorem of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Schwartz which asserts: For any curve c of length L with curvature k(s) ≤ 1/r, let C be the circle passing c(0) and c(L) of radius r, then L is either not greater than the length of the lesser circular arc, or not less than the length of the greater circular arc of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' High dimensional (intrinsic) analogues of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Schur’s theorem include the Rauch’s comparison theorem and the Toponogov comparison theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' The later however has the limit of requiring that the manifold with less curvature must be a space form of constant sectional curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' First we have the following slight more general version of Schur’s theorem in terms of a monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Let c : [0, L] → R2 be an embedded convex plane curve with curvature k(s) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Let ˜c : [0, L] → Rn+1 (n ≥ 1) be a curve such that ˜k(s) ≤ k(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Then for any 0 ≤ s′ < s′′ ≤ L there exist an isometric inclusion ιs′,s′′ : R2 → Rn+1 with ιs′,s′′(0) = 0 such that I(s) := ⟨˜c(s) − ιs′,s′′(c(s)), ιs′,s′′(c(s′′) − c(s′))⟩ is monotone non-decreasing for s ∈ [s′, s′′], or equivalently ⟨ ˜T(s) − ιs′,s′′(T (s)), ιs′,s′′(c(s′′) − c(s′))⟩ ≥ 0, ∀ s ∈ [s′, s′′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='2) As s′ → s′′, the inclusion ιs′,s′′ converges to an inclusion identifying T (s) with ˜T(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Under the same assumption as in the theorem, for any s′ ≤ s′ ∗ < s′′ ∗ ≤ s′′, ⟨c(s′′ ∗) − c(s′ ∗), c(s′′) − c(s′)⟩ ≤ ⟨˜c(s′′ ∗) − ˜c(s′ ∗), ιs′,s′′(c(s′′) − c(s′))⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='3) When s′ = s′ ∗ and s′′ = s′′ ∗ we have that |c(s′′) − c(s′)|2 ≤ ⟨˜c(s′′) − ˜c(s′), ιs′,s′′(c(s′′) − c(s′))⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='4) The estimate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='4) implies Schur’s theorem by the Cauchy-Schwarz inequality applied to the right hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='4): |c(s′′) − c(s′)| ≤ |˜c(s′′) − ˜c(s′)|, ∀ 0 ≤ s′ < s′′ ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' This extension allows one to rephrase the result in terms of the concept of the expansion module [2, 9] of vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' If X : Ω ⊂ Rn+1 → Rn+1 is a vector field defined on a convex domain, then the expansion module is a function of one variable ψ(t) such that ⟨X(y) − X(x), y − x |y − x|⟩ ≥ 2ψ �|x − y| 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Since ˜c(s) and c(s) are related via the parameter s, one may view ˜c as a related vector field defined over ιs′,s′′(c(s)) ∈ Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Now the estimate in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='1 simply asserts that the related vector fields ˜c(s) has an expansion module function ψ(t) = t with respect to the associated vector ιs′,s′′(c(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' From the above connection between the concept of curvature and the expansion module it is our hope that a high dimensional Schur’s theorem could be discovered through the consideration involving the expansion module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Given that Schur’s theorem implies the Cauchy’s Arm Lemma for the arms of great arcs in the unit sphere, a natural question is that if the spherical analogue of Schur’s theorem still holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Namely, given two embedded spherical curves c(s) and ˜c(s) in the unit sphere S2 ⊂ R3 parametrized by the arc-length s ∈ [0, L] with L ≤ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Assume that c(s) is convex AN EXTENSION OF SCHUR’S THEOREM 3 with geodesic curvature k(s) > 0 and that the geodesic curvature of ˜c satisfies |˜k|(s) ≤ k(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Does it still hold that |c(0) − c(L)| ≤ |˜c(0) − ˜c(L)|?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' One could also allow the tangent of curves to have same amount of finite many jumps at {sj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' In that case, the oriented angles αj and ˜αj are assumed to satisfy that αj ≥ ˜αj as in the case of Schur’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' The Cauchy’s Arm Lemma in the sphere answers the question affirmatively in the special case where both curves have zero geodesic curvature for the smooth parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Here we confirm this conjecture by proving Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' The Schur’s theorem holds for two curves in S2 ⊂ R3 under the above configurations similar to that of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' The proof is via construction of auxiliary curves with one of them being a convex plane curve and appealing to the original Schur’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' This is part of the reason we present the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='1 with care and details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Note that this result generalizes the spherical Cauchy’s Arm Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' It would be interesting to see if it plays any role in the proof of Pogorelov’s monotype theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' There were extensions of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Schur’s theorem in hyperbolic spaces [6] and in the Minkowski plane [8] earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' It is plausible that the method of this paper can be used to simplify the argument in the former work via Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='1 We prove theorem and its corollary together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' After an inclusion ι : R2 → Rn+1, which shall be specified later, we may consider the tangent T (s) and ˜T(s) as two curves in Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' For the proof we need to choose a point N ∈ image(T (s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' For the situation when the tangent T (s) has a jump at sj, the minimizing arc jointing T (sj−) and T (sj+) is also considered to be part of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' They together form a part of a great circle which is denoted by Image(T (s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Consider the two curves T (s) and ˜T (s) inside Sn(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' The first one is a plane curve, hence is part of a great arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' We first find a s∗ ∈ [s′, s′′] (and then let N := T (s∗)), such that T (s∗) is parallel to c(s′′) − c(s′) using the convexity of the cone over image(T (s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Since T (s), s ∈ [s′, s′′] forms a part of a great circle, letting j1 be the smallest j with sj ≥ s′ and j2 being the greatest j with sj ≤ s′′, by the mean value theorem c(s′′) − c(s′) = � sj1 s′ T (s) ds + � j1≤j≤j2−1 � sj+1 sj T (s) ds + � s′′ sj2 T (s) ds = (sj1 − s′)T ((s∗)j1) + j2−1 � j1 (sj+1 − sj)T ((s∗)j+1) + (s′′ − sj2)T ((s∗)j2+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Since 1 s′′−s′ multiple of the right hand above lies inside the cone over the image of T (s) for s ∈ [s′, s′′], it implies that there exists (a unique) s∗ ∈ [s′, s′′] such that T (s∗) = λ(c(s′′) − c(s′)) with λ = 1 |c(s′′)−c(s′)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Now consider the two products Pi defined as (with N = T (s∗)) P1 := ⟨c(s′′) − c(s′), N⟩ = � s′′ s′ ⟨T (s), N⟩ ds, P2 := ⟨˜c(s′′) − ˜c(s′), N⟩ = � s′′ s′ ⟨ ˜T(s), N⟩ ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' From the choice of s∗, ⟨c(s′′) − c(s′), N⟩ = |c(s′′) − c(s′)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Now P1 = |c(s′′) − c(s′)|, and P2 = ⟨˜c(s′′) − ˜c(s′), c(s′′) − c(s′)⟩/|c(s′′) − c(s′)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' 4 LEI NI The claimed estimate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='4) amounts to showing that the second product is bounded from below by the first after a proper inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Let j3 be the biggest j with sj ≤ s∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Observe the convexity of c(s) implies that αj0 + � j1≤j≤j3 αj + � sj1 s′ k(s) ds + � j1≤j≤j3−1 � sj+1 sj k(s) ds + � s∗ sj3 k(s) ds = π, � j3+1≤j≤j2 αj + αj4 + � sj3+1 s∗ k(s) ds + � j3+1≤j≤j2−1 � sj+1 sj k(s) ds + � s′′ sj2 k(s) ds = π, with αj0 being the angle from −N to T (s′) and αj4 being the angle from T (s′′) and −N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' This implies that the images of T ([s′, s∗]) (denoted as curve Γ1(s)) and T ([s∗, s′′]) (denoted as the spherical curve Γ2(s)) are two minimizing arcs of the great circle (formed by the intersection of the plane in which c(s) lies and Sn(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' We also denote the spherical curves corresponding to ˜T by �Γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Hence max{Length(Γ1), Length(Γ2)} ≤ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='1) On the other hand, by rotation we may arrange the inclusion ι such that ˜T(s∗) = N = T (s∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Now we estimate π ≥ dSn(T (s′), N) = dSn(T (s′), T (s∗)) = Length(Γ1) = � j1≤j≤j3 αj + � sj1 s′ k(s) ds + � j1≤j≤j3−1 � sj+1 sj k(s) ds + � s∗ sj3 k(s) ds ≥ � j1≤j≤j3 ˜αj + � sj1 s′ ˜k(s) ds + � j1≤j≤j3−1 � sj+1 sj ˜k(s) ds + � s∗ sj3 ˜k(s) ds = � j1≤j≤j3 ˜αj + � sj1 s′ | ˜T ′|(s) ds + � j1≤j≤j3−1 � sj+1 sj | ˜T ′|(s) ds + � s′′ sj3 | ˜T ′|(s) ds ≥ dSn( ˜T(s′), ˜T(s∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' The second line above follows from the definition of the curvature measuring the rotating angle of the tangent for a curve [10] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' page 49) and that for a convex curve k(s) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' The third line uses the assumption, and the last line follows from the definition of the (spherical) distance between two points being the infimum of the length of all possible connecting curves in Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' The same argument also implies that for any s ∈ [s′, s∗] π ≥ dSn(T (s), N) = Length(Γ1|[s,s∗]) ≥ dSn( ˜T(s), N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' This implies that ⟨T (s), N⟩ = cos(dSn(T (s), T (s∗)) ≤ cos(dSn( ˜T(s), ˜T (s∗)) = cos(dSn( ˜T(s), N)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='2) Rewriting the above estimate we have that ⟨ ˜T (s) − T (s), N⟩ ≥ 0 for s ∈ [s′, s∗], which implies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' A similar argument proves that the same inequality holds also for s ∈ [s∗, s′′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Putting them together we have (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' The above proof also works for any N = T (s∗) such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='1) holds, while (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='1) implies that one can always choose a s∗ ∈ [0, L] independent of s′ and s′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' (However such s∗ is far from being unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=') AN EXTENSION OF SCHUR’S THEOREM 5 Now we compare the two products Pi by writing P1 = � s′′ s′ ⟨T (s), N⟩ ds = �� s∗ s′ + � s′′ s∗ � cos (dSn(T (s), T (s∗)) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' We express P2 accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' The above estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='2) implies that � s∗ s′ cos(dSn(T (s), T (s∗)) ds ≤ � s∗ s′ cos(dSn( ˜T (s), ˜T(s∗)) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='3) Similarly, we have � s′′ s∗ cos(dSn(T (s), T (s∗)) ds ≤ � s′′ s∗ cos(dSn( ˜T (s), ˜T(s∗)) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='4) From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='4) we have that P1 ≤ P2, namely the desired claim (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' From the proof we have the following more general monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Let c : [0, L] → R2 be an embedded convex plane curve with curvature k(s) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Let ˜c : [0, L] → Rn+1 (n ≥ 1) be a curve such that ˜k(s) ≤ k(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Then there exists s∗ ∈ [0, L] and an inclusion ι : R2 → Rn+1 with ι(0) = 0 and ι(T (s∗)) = ˜T(s∗), such that I′ 1(s) = ⟨ ˜T (s) − ι(T (s)), ˜T(s∗)⟩ ≥ 0, ∀ s ∈ [0, L], where I1(s) = ⟨˜c(s) − ι(c(s)), ˜T (s∗)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='5) Here the choices of s∗ and ι are more flexible than in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='1, where they are essentially unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' The proof can be easily adopted to show a comparison between a time-like curves in a Minkowski plane L2 1 and another time-like curve in the three dimensional Minkowski space L3 1 with signature (+, −, −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' In fact in terms of the monotonicity one may choose s∗ freely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Following the convention of the physics a vector u is called time-like if ⟨u, u⟩ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' For a time-like curve c(s), |T (s)|2 = |c′(s)|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Hence T (s) can be viewed as a point in the hyperbolic line/plane defined as x2 1 − x2 2 = 1 (or x2 1 − x2 2 − x2 3 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' It can be checked easily that −1 multiple of the restricted metric on the surface is the standard hyperbolic metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' The angle between two tangents T (s1) and T (s2) is given by ⟨T (s1), T (s2)⟩ = cosh ϕ(s2, s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' A simple computation shows that ϕ(s2, s1) equals to the hyperbolic distance between T (s1) and T (s2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' For space-like curves the length of the vector u is defined to be � −⟨u, u⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Equipped with the above basics, a similar consideration as the above gives the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Let c(s) : [0, L] be a time-like convex curve in L2 1 parametrized by the arc- length, and let ˜c(s) : [0, L] be a similarly parametrized regular time-like curve in L3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Assume that k(s) ≥ |˜k|(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Then for any s∗ ∈ [0, L] and an isometric inclusion of ι : L2 1 → L3 1, which identifies T (s∗) with ˜T(s∗), we have that I′ 2(s) = ⟨ι(T (s)) − ˜T(s), ˜T(s∗)⟩ ≥ 0, where I2(s) = ⟨ι(c(s)) − ˜c(s), ˜T (s∗)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='6) In particular, |c(L) − c(0)| ≥ |˜c(L) − ˜c(0)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' The last statement of (ii) generalizes the result of [8] by allowing the second curve ˜c(s) a space curve in L3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Note that for the curves in two Minkowski planes, the result for space-like curves is the same as that for the time-like curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' To prove the last conclusion we first integrate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='6) with s∗ so chosen that T (s∗) is proportional to c(L) − c(0), and then apply the reserved Cauchy-Schwarz inequality (which holds for two time-like vectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' 6 LEI NI 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='3 We start with some basics on spherical (smooth) curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Let c(s) be a curve in S2 parametrized by the arc-length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Let T (s) be its tangent, which is orthogonal to c(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Let V (s) = c(s) × T (s) be the cross product of c(s) and T (s) in R3, which is a normal of c(s) in Tc(s)S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' The triple {c(s), T (s), V (s)} forms an orthonormal moving frame (of R3) along c(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Since the geodesic curvature of a curve in the sphere (in a surface) is the changing rate of the tangential great circles (tangential geodesics in general, by (8-3) of page 157 of [13]), and that V (s) provides a natural parametrization of the tangential great circles, the derivative of V (s) yields the geodesic curvature of c(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' This can also be formulated in terms of the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Let k(s) be the geodesic curvature of c(s) (with respect to S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Then the following holds for {c(s), T (s), V (s)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' c′(s) = T (s), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='1) T ′(s) = k(s)V (s) − c(s), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='2) V ′(s) = −k(s)T (s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='3) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' The first equation is definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Also by definition k(s) = ⟨T ′(s), V (s)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Hence from 0 = d2 ds2 � |c|2(s) � = 2⟨T (s), T (s)⟩ + 2⟨c(s), T ′(s)⟩ = 2 + 2⟨c(s), T ′(s)⟩ we deduce the second equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Now by the second equation V ′(s) = T (s) × T (s) + c(s) × T ′(s) = k(s) c(s) × V (s) = −k(s)T (s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' This prove the third one, hence completes the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' □ The local convexity of c(s) is equivalent to k(s) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' The basic construction is to look at the cone C(c(s)) over the spherical curve c(s) centered at the origin, and obtain a plane curve by taking the intersection of C(c(s)) with a plane P not passing the origin to obtain a plane curve Pc(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' This curve can be expressed as R(s)c(s) with R(s) being the distance of Pc(s) to the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' We need the following formula for the curvature of the space curve in R3 applied to Pc(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' If c(s) is a convex curve in S2, Pc(s) is a convex curve in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' The curvature k(s) of Pc(s) (as a space curve of R3) is given by k2(s) = |P′ c(s) × P′′ c (s)|2 |P′c(s)|3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='4) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' From the geometric definition of the convexity we know that c(s) lies in a signed semi-sphere cut out by any tangent great circle obtained by a plane passing the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Then it is clear that Pc(s) lies on the corresponding half plane cut out by the corresponding tangent line of Pc(s) in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' This proves the convexity of Pc(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' The formula for the curvature of a space curve is well known and computational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' See for example page 51 of [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Of course the formula applies to the case that the curve happens to be a plane curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' □ Now let τ be the arc-length parameter of Pc(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Direct calculation shows that τ(s) = � s 0 � (R′(s))2 + R2(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='5) AN EXTENSION OF SCHUR’S THEOREM 7 Now we construct a space curve ˜P˜c(s) corresponding to ˜c(s) by defining it as R(s)˜c(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' In general, this is not a plane curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' The key observation is that the arc-length parameter for ˜P˜c(s) is the same as that of Pc(s), namely it is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='5) as well, since |˜c|(s) = 1 = |c(s)| and |˜c′|(s) = 1 = |c′(s)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Moreover its curvature ˜k(s) (as a curve in R3) can be expressed similarly as ˜k2(s) = | ˜P′ ˜c(s) × ˜P′′ ˜c (s)|2 | ˜P′ ˜c(s)|3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='6) Namely the second part of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='2 applies to ˜P˜c(s) as well since it holds for any space curve in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' The key step is the following comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Under the assumption that the geodesic curvature k(s) of c(s) and the geodesic curvature ˜k(s) of ˜c(s) satisfy k(s) ≥ |˜k(s)| ≥ 0, the curvatures of Pc(s) and ˜P˜c(s) satisfies k(s) ≥ 0 and k(s) ≥ |˜k|(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Since Pc(s) is convex, we have that k(s) ≥ 0, it suffices to show that k2(s) ≥ ˜k2(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' First we observe that |P′ c(s)|2 = R2(s) + (R′(s))2 = | ˜P′ ˜c(s)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' This reduces the desired estimate to |P′ c(s) × P′′ c (s)|2 ≥ | ˜P′ ˜c(s) × ˜P′′ ˜c (s)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='7) Using the fact that {c(s), T (s), V (s)} forms an oriented orthonormal moving frame, a direct calculation, using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='1, shows that P′ c(s) × P′′ c (s) = (R′(s)c(s) + R(s)T (s)) × (R′′(s)c(s) + 2R′(s)T (s) + R(s)T ′(s)) = (2(R′(s))2 − R(s)R′′(s))V (s) − R′(s)R(s)k(s) T (s) −R(s)R′′(s) V (s) + R2(s)k(s)c(s) + R2(s)V (s) = R2(s)k(s)c(s) − R′(s)R(s)k(s) T (s) +(2(R′(s))2 − 2R(s)R′′(s) + R2(s))V (s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Hence we have that |P′ c(s) × P′′ c (s)|2 = (R4(s) + (R′(s)R(s))2)k2(s) +(2(R′(s))2 − 2R(s)R′′(s) + R2(s))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='8) A similar calculation shows that | ˜P′ ˜c(s) × ˜P′′ ˜c (s)|2 = (R4(s) + (R′(s)R(s))2)˜k2(s) +(2(R′(s))2 − 2R(s)R′′(s) + R2(s))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='9) From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='8) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='9), the assumption k(s) ≥ |˜k|(s) implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='7), hence the desired estimate of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' □ Now Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='3 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='5) implies that Pc(τ) and ˜P˜c(τ) are two curves satisfying the assumption of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Hence we have that d(Pc(0), Pc(τ(L))) ≤ d( ˜P˜c(0), ˜P˜c(τ(L))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='3 for the smooth curves now follows from the hinge theorem of Euclidean geom- etry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' 8 LEI NI For the general case when the tangents of c(s) and ˜c(s) have finite jumps at {sj}, if we denote the turning angles at Pc(sj) and ˜P˜c(sj) by θj and ˜θj, then cos θj = R′(sj−)R′(sj+) + R2(sj) cos αj � ((R′(sj−))2 + R2(sj)) ((R′(sj+))2 + R2(sj)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' By a similar formula for cos ˜θj we deduce that θj ≥ ˜θj if αj ≥ ˜αj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Hence Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='3 follows from the general case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Acknowledgments The author would like to thank Burkhard Wilking for helpful discussions, Paul Bryant, Jon Wolfson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'} +page_content=' Wu and Fangyang Zheng for their interests to the problem considered.' metadata={'source': 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a/0NE0T4oBgHgl3EQfuAEL/content/2301.02598v1.pdf b/0NE0T4oBgHgl3EQfuAEL/content/2301.02598v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6c90d7d8f43b62f838bc05dcafd7f0d51cd8fc8a --- /dev/null +++ b/0NE0T4oBgHgl3EQfuAEL/content/2301.02598v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:15cd73033707ef7802fcac91cf6432c4a79c47a0acaa42d6e84a560f5d3f51fc +size 13483434 diff --git a/0tAyT4oBgHgl3EQfPPbs/content/tmp_files/2301.00023v1.pdf.txt b/0tAyT4oBgHgl3EQfPPbs/content/tmp_files/2301.00023v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..61c45f085ec7d5a5faf635e33badfb24c0f15126 --- /dev/null +++ b/0tAyT4oBgHgl3EQfPPbs/content/tmp_files/2301.00023v1.pdf.txt @@ -0,0 +1,1316 @@ +Imitator: Personalized Speech-driven 3D Facial Animation +Balamurugan Thambiraja1 +Ikhsanul Habibie2 +Sadegh Aliakbarian3 +Darren Cosker3 +Christian Theobalt2 +Justus Thies1 +1 Max Planck Institute for Intelligent Systems, T¨ubingen, Germany +2 Max Planck Institute for Informatics, Saarland, Germany +3 Microsoft Mixed Reality & AI Lab, Cambridge, UK +Figure 1. Imitator is a novel method for personalized speech-driven 3D facial animation. Given an audio sequence and a personalized +style-embedding as input, we generate person-specific motion sequences with accurate lip closures for bilabial consonants (’m’,’b’,’p’). +The style-embedding of a subject can be computed by a short reference video (e.g., 5s). +Abstract +Speech-driven 3D facial animation has been widely ex- +plored, with applications in gaming, character animation, +virtual reality, and telepresence systems. State-of-the-art +methods deform the face topology of the target actor to sync +the input audio without considering the identity-specific +speaking style and facial idiosyncrasies of the target ac- +tor, thus, resulting in unrealistic and inaccurate lip move- +ments. To address this, we present Imitator, a speech-driven +facial expression synthesis method, which learns identity- +specific details from a short input video and produces novel +facial expressions matching the identity-specific speaking +style and facial idiosyncrasies of the target actor. Specif- +ically, we train a style-agnostic transformer on a large fa- +cial expression dataset which we use as a prior for audio- +driven facial expressions. Based on this prior, we optimize +for identity-specific speaking style based on a short refer- +ence video. To train the prior, we introduce a novel loss +function based on detected bilabial consonants to ensure +plausible lip closures and consequently improve the realism +of the generated expressions. Through detailed experiments +and a user study, we show that our approach produces tem- +porally coherent facial expressions from input audio while +preserving the speaking style of the target actors. Please +check out the project page for the supplemental video and +more results. +1. Introduction +3D digital humans raised a lot of attention in the past +few years as they aim to replicate the appearance and mo- +tion of real humans for immersive applications, like telep- +resence in AR or VR, character animation and creation for +entertainment (movies and games), and virtual mirrors for +e-commerce. +Especially, with the introduction of neural +rendering [27, 28], we see immense progress in the photo- +1 +arXiv:2301.00023v1 [cs.CV] 30 Dec 2022 + +Imitator +Personalized Style-Embedding +Audio +Bilabial Consonants +Personalized 3D Facial Animation +Timerealistic synthesis of such digital doubles [11,20,38]. These +avatars can be controlled via visual tracking to mirror the fa- +cial expressions of a real human. However, we need to con- +trol the facial avatars with text or audio inputs for a series +of applications. For example, AI-driven digital assistants +rely on motion synthesis instead of motion cloning. Even +telepresence applications might need to work with audio in- +puts only, when the face of the person is occluded or cannot +be tracked, since a face capture device is not available. To +this end, we analyze motion synthesis for facial animations +from audio inputs; note that text-to-speech approaches can +be used to generate such audio. Humans are generally sen- +sitive towards faces, especially facial motions, as they are +crucial for communication (e.g., micro-expressions). With- +out full expressiveness and proper lip closures, the gener- +ated animation will be perceived as unnatural and implausi- +ble. Especially if the person is known, the facial animations +must match the subject’s idiosyncrasies. +Recent methods for speech-driven 3D facial anima- +tion [5, 10, 16, 21] are data-driven. +They are trained on +high-quality motion capture data and leverage pretrained +speech models [13,23] to extract an intermediate audio rep- +resentation. We can classify these data-driven methods into +two categories, generalized [5,10,21] and personalized an- +imation generation methods [16]. In contrast to those ap- +proaches, we aim at a personalized 3D facial animation syn- +thesis that can adapt to a new user while only relying on in- +put RGB videos captured with commodity cameras. Specif- +ically, we propose a transformer-based auto-regressive mo- +tion synthesis method that predicts a generalized motion +representation. This intermediate representation is decoded +by a motion decoder which is adaptable to new users. A +speaker embedding is adjusted for a new user, and a new +motion basis for the motion decoder is computed. +Our +method is trained on the VOCA dataset [5] and can be ap- +plied to new subjects captured in a short monocular RGB +video. As lip closures are of paramount importance for bi- +labial consonants (’m’,’b’,’p’), we introduce a novel loss +based on the detection of bilabials to ensure that the lips +are closed properly. We take inspiration from the locomo- +tion synthesis field [14,18], where similar losses are used to +enforce foot contact with the ground and transfer it to our +scenario of physically plausible lip motions. +In a series of experiments and ablation studies, we +demonstrate that our method is able to synthesize facial ex- +pressions that match the target subject’s motions in terms of +style and expressiveness. Our method outperforms state-of- +the-art methods in our metrical evaluation and user study. +Please refer to our supplemental video for a detailed qual- +itative comparison. In a user study, we confirm that per- +sonalized facial expressions are important for the perceived +realism. +The contributions of our work Imitator are as follows: +• a novel auto-regressive motion synthesis architec- +ture that allows for adaption to new users by disen- +tangling generalized viseme generation and person- +specific motion decoding, +• and a lip contact loss formulation for improved lip clo- +sures based on physiological cues of bilabial conso- +nants (’m’,’b’,’p’). +2. Related Work +Our work focuses on speech-driven 3D facial animation +related to talking head methods that create photo-realistic +video sequences from audio inputs. +Talking Head Videos: Several prior works on speech- +driven generation focus on the synthesis of 2D talking head +videos. Suwajanakorn et al. [25] train an LSTM network on +19h video material of Obama to predict his person-specific +2D lip landmarks from speech inputs, which is then used for +image generation. Vougioukas et al. [33] propose a method +to generate facial animation from a single RGB image +by leveraging a temporal generative adversarial network. +Chung et al. [4] introduce a real-time approach to gener- +ate an RGB video of a talking face by directly mapping the +audio input to the video output space. This method can re- +dub a new target identity not seen during training. Instead of +performing direct mapping, Zhou et al. [39] disentangles the +speech information in terms of speaker identity and content, +allowing speech-driven generation that can be applied to +various types of realistic and hand-drawn head portraits. A +series of work [24,29,36,37] uses an intermediate 3D Mor- +phable Model (3DMM) [2,8] to guide the 2D neural render- +ing of talking heads from audio. Wang et al. [34] extend this +work also to model the head movements of the speaker. Lip- +sync3d [17] proposes data-efficient learning of personalized +talking heads focusing on pose and lighting normalization. +Based on dynamic neural radiance fields [11], Ad-nerf [12] +and DFA-NeRF [35] learn personalized talking head mod- +els that can be rendered under novel views, while being +controlled by audio inputs. In contrast to these methods, +our work focuses on predicting 3D facial animations from +speech that can be used to drive 3D digital avatars with- +out requiring retraining of the entire model to capture the +person-specific motion style. +Speech-Driven 3D Facial Animation: Speech-driven 3d +facial animation is a vivid field of research. Earlier meth- +ods [6, 7, 9, 15, 32] focus on animating a predefined facial +rig using procedural rules. HMM-based models generate +visemes from input text or audio, and the facial anima- +tions are generated using viseme-dependent co-articulation +models [6, 7] or by blending facial templates [15]. With +recent advances in machine learning, data-driven meth- +ods [3, 5, 10, 16, 21, 26, 29] have demonstrated their capa- +bility to learn viseme patterns from data. These methods +2 + +Figure 2. Our architecture takes audio as input which is encoded by a pre-trained Wav2Vec2.0 model [1]. This audio embedding ˆa1:T is +interpreted by an auto-regressive viseme decoder which generates a generalized motion feature ˆv1:T . A style-adaptable motion decoder +maps these motion features to person-specific facial expressions ˆy1:T in terms of vertex displacements on top of a template mesh. +are based on pretrained speech models [1, 13, 23] to gen- +erate an abstract and generalized representation of the in- +put audio, which is then interpreted by a CNN or auto- +regressive model to map to either a 3DMM space or directly +to 3D meshes. Karras et al. [16] learn a 3D facial animation +model from 3-5 minutes of high-quality actor specific 3D +data. VOCA [5] is trained on 3D data of multiple subjects +and can animate the corresponding set of identities from in- +put audio by providing a one-hot encoding during inference +that indicates the subject. MeshTalk [21] is a generalized +method that learns a categorical representation for facial +expressions and auto-regressively samples from this cate- +gorical space to animate a given 3D facial template mesh +of a subject from audio inputs. FaceFormer [10] uses a +pretrained Wav2Vec [1] audio representation and applies a +transformer-based decoder to regress displacements on top +of a template mesh. Like VOCA, FaceFormer provides a +speaker identification code to the decoder, allowing one to +choose from the training set talking styles. In contrast, we +aim at a method that can adapt to new users, capturing their +talking style and expressiveness. +3. Method +Our goal is to model person-specific speaking style and +the facial idiosyncrasies of an actor, to generate 3D facial +animations of the subject from novel audio inputs. As in- +put, we assume a short video sequence of the subject which +we leverage to compute the identity-specific speaking style. +To enable fast adaptation to novel users without significant +training sequences, we learn a generalized style-agnostic +transformer on VOCAset [5]. This transformer provides +generic motion features from audio inputs that are inter- +pretable by a person-specific motion decoder. +The mo- +tion decoder is pre-trained and adaptable to new users via +speaking style optimization and refinement of the motion +basis. To further improve synthesis results, we introduce a +novel lip contact loss based on physiological cues of the bi- +labial consonants [7]. In the following, we will detail our +model architecture and the training objectives and describe +the style adaptation. +3.1. Model Architecture +Our architecture consists of three main components (see +Figure 2): an audio encoder, a generalized auto-regressive +viseme decoder, and an adaptable motion decoder. +Audio Encoder: Following state-of-the-art motion synthe- +sis models [5, 10], we use a generalized speech model to +encode the audio inputs A. Specifically, we leverage the +Wav2Vec 2.0 model [1]. The original Wav2Vec is based on +a CNN architecture designed to produce a meaningful la- +tent representation of human speech. To this end, the model +is trained in a self-supervised and semi-supervised manner +to predict the immediate future values of the current input +speech by using a contrastive loss, allowing the model to +learn from a large amount of unlabeled data. Wav2Vec 2.0 +extends this idea by quantizing the latent representation and +incorporating a Transformer-based architecture [31]. We re- +sample the Wav2Vec 2.0 output with a linear interpolation +layer to match the sampling frequency of the motion (30fps +for the VOCAset, with 16kHz audio), resulting in a con- +textual representation {ˆa}T +t=1 of the audio sequence for T +motion frames. +Auto-regressive Viseme Decoder: The decoder Fv takes +the contextual representation of the audio sequence as input +and produces style agnostic viseme features ˆvt in an auto- +regressive manner. These viseme features describe how the +lip should deform given the context audio and the previ- +ous viseme features. In contrast to Faceformer [10], we +propose to use of a classical transformer architecture [31] +as viseme decoder, which learns the mapping from audio- +3 + +Start +11 +Linear Deformation Basis +Cross-Modal MH Attention +Multi-Head Self-Attention + Speaker Identity +Token +Positional Encoding +Feed Forward +12 +Wav2Vec +Linear +Linear + Relu +Linear + Relu +Linear + Relu +Style Linear +Linear + Relu +03 +Si +y3 +DT +A +