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Theorem 22.9. The density topology in \( R \) is regular. | Proof. Let \( x \) be a point of a set \( A \in \mathcal{T} \) . Then \( A \) has density 1 at \( x \) . For each positive integer \( n \), let \( {F}_{n} \) be an ordinary-closed subset of \( (x - 1/{2n} \) , \( x + 1/{2n}) \cap A \) such that \( m\left( {F}_{n}\right) > \left( {1 - 1/n}\right) m\left\lbrack {\left( {... | Yes |
Theorem 1.2.1. Suppose a real-valued function \( H \) on \( {\mathbb{R}}^{n} \) is differentiable away from the origin of \( {\mathbb{R}}^{n} \) . Then the following two statements are equivalent:\n\n- \( H \) is positively homogeneous of degree \( r \) . That is,\n\n\[ H\left( {\lambda y}\right) = {\lambda }^{r}H\left... | Proof.\n\n* Suppose \( H \) satisfies \( H\left( {\lambda y}\right) = {\lambda }^{r}H\left( y\right) \) for all positive \( \lambda \) . Fix \( y \) . Differentiating this equation with respect to the parameter \( \lambda \) gives\n\n\[ {y}^{i}{H}_{{y}^{i}}\left( {\lambda y}\right) = r{\lambda }^{r - 1}H\left( y\right)... | Yes |
Theorem 1.2.2. Let \( F \) be a nonnegative real-valued function on \( {\mathbb{R}}^{n} \) with the properties:\n\n* \( F \) is \( {C}^{\infty } \) on the punctured space \( {\mathbb{R}}^{n} \smallsetminus 0 \) .\n\n* \( F\left( {\lambda y}\right) = {\lambda F}\left( y\right) \) for all \( \lambda > 0 \) .\n\n* The \( ... | ## Proof of the theorem.\n\n## (i) Positivity:\n\nConsider (1.2.5); namely, \( {g}_{{ij}\left( y\right) }{y}^{i}{y}^{j} = {F}^{2}\left( y\right) \) . The hypothesized strong convexity of \( F \) says that the left-hand side is positive whenever \( y \neq 0 \), thus \( F \) is strictly positive on \( {\mathbb{R}}^{n} \s... | Yes |
Theorem 1.4.1 (Deicke) [D]. Let \( F \) be a Minkowski norm on \( {\mathbb{R}}^{n} \). The following three statements are equivalent:\n\n(a) \( F \) is Euclidean. That is, it arises from an inner product.\n\n(b) \( {A}_{ijk} = 0 \) for all \( i, j, k \).\n\n(c) \( {A}_{k} \mathrel{\text{:=}} {g}^{ij}{A}_{ijk} = 0 \) fo... | ## Remarks:\n\n* The equivalence between the first two statements comes directly from (1.4.2). Namely, \( {A}_{ijk} \) is proportional to the vertical derivative of \( {g}_{ij} \). Thus this derivative vanishes if and only if \( {g}_{ij} \) has no \( y \) - dependence (which means that \( F \) comes from an inner produ... | No |
Proposition 4.1.1 (Rund) [R]. Every unit speed parametrization \( y\left( t\right) \) of the indicatrix \( \left( {S, h}\right) \) must satisfy the following ODE:\n\n(4.1.7a)\n\n\[ \ddot{y} + I\dot{y} + y = 0 \]\n\nThat is,\n\n(4.1.7b)\n\n\[ {\ddot{y}}^{i} + I{\dot{y}}^{i} + {y}^{i} = 0\;\text{ for }i = 1,2. \] | Proof. Differentiating (4.1.4) once with respect to \( t \) yields\n\n(4.1.8)\n\n\[ {g}_{{ij}\left( y\right) }{\dot{y}}^{i}{y}^{j} = 0 \]\n\nwhich says that the position vector \( y \) and the velocity vector \( \dot{y} \) are \( \widehat{g} \) - orthogonal. Note that in (4.1.4), differentiating \( {g}_{ij} \) produces... | Yes |
Corollary 4.1.2. Let \( F \) be a Minkowski norm on \( {\mathbb{R}}^{2} \). (It is smooth and strongly convex on \( {\mathbb{R}}^{2} \smallsetminus 0 \).) Then the average value of the Cartan scalar, over the indicatrix \( S \), is zero. In particular, for any unit speed parametrization of \( S \), we have\n\n\[ \n{\in... | Proof. As before, let \( y\left( t\right) = \left( {{y}^{1}\left( t\right) ,{y}^{2}\left( t\right) }\right) ,0 \leq t \leq L \) be a unit speed parametrization of \( \mathrm{S} \). Since the indicatrix is a closed curve in \( {\mathbb{R}}^{2} \smallsetminus 0 \), the functions \( {y}^{i}\left( t\right) \) are periodic ... | Yes |
Proposition 4.2.1. Suppose the unit speed parametrizations \( y\left( t\right) ,\widetilde{y}\left( t\right) \) trace out the indicatrices \( S,\widetilde{S} \) in a positive manner with respect to the canonical orientation of \( {\mathbb{R}}^{2} \) . Then the following two statements are equivalent:\n\n- The two Minko... | Proof.\n\nThat (a) \( \Rightarrow \) (b) :\n\nSuppose the two Minkowski planes are equivalent in the sense (a). Then \( L \) maps \( \mathrm{S} \) into \( \widetilde{\mathrm{S}} \) and therefore \( L\left\lbrack {y\left( t\right) }\right\rbrack \) parametrizes the indicatrix \( \widetilde{\mathrm{S}} \) . Next, at all ... | Yes |
Proposition 5.1.1. Let \( \sigma \left( t\right) ,0 \leq t \leq r \) be a regular piecewise \( {C}^{\infty } \) curve in a Finsler manifold \( \left( {M, F}\right) \) . The following two statements are equivalent:\n\n(a) \( {L}^{\prime }\left( 0\right) = 0 \) for all piecewise \( {C}^{\infty } \) variations of \( \sigm... | Proof of Proposition 5.1.1. Consider only variations in which all the \( t \) -curves share the same endpoints. For these, the variation vector field vanishes at \( {t}_{o} = 0 \) and \( {t}_{k} = r \) . After restricting (5.1.16) to the base curve, we get:\n\n(*)\n\n\[ \n{L}^{\prime }\left( 0\right) = - \mathop{\sum }... | Yes |
Lemma 6.1.1 (The Gauss Lemma).\n\n- Fix \( y \in {T}_{x}M \smallsetminus 0 \) and set \( r \mathrel{\text{:=}} F\left( {x, y}\right) \) . Suppose \( {\exp }_{x}y \) is defined.\n\n- Let \( T \) denote the velocity field of the geodesic \( {\exp }_{x}\left( {ty}\right) ,0 \leq t \leq 1 \) emanating from \( x \) . Call t... | ## For the technical statement:\n\nFix an instant \( \tau \in \left\lbrack {0,1}\right\rbrack \) . Then \( {\tau y} \in {\mathrm{S}}_{x}\left( {\tau r}\right) \) . Let \( c\left( u\right) , - \epsilon < u < + \epsilon \) be any curve on \( {\mathrm{S}}_{x}\left( {\tau r}\right) \) that passes through the point \( {\tau... | No |
Lemma 6.2.1. Let \( \\left( {M, F}\\right) \) be a Finsler manifold. At every point \( p \\in M \) , there exists a local coordinate system \( \\varphi : \\bar{U} \\rightarrow {\\mathbb{R}}^{n} \) that has the following properties:\n\n- The closure of \( U \) is compact, \( \\varphi \\left( p\\right) = 0 \), and \( \\v... | Proof. Take any local coordinate system \( \\varphi : W \\rightarrow {\\mathbb{R}}^{n} \) at \( p \) with \( \\varphi \\left( p\\right) = 0 \). Let \( {\\mathbb{B}}^{n}\\left( r\\right) \) denote the standard open ball \( \\left\{ {\\left( {v}^{i}\\right) \\in {\\mathbb{R}}^{n} : \\sqrt{{\\delta }_{ij}{v}^{i}{v}^{j}} <... | No |
Proposition 6.5.1. Let \( p \) be any point in a Finsler manifold \( \left( {M, F}\right) \) , where \( F \) is positively (but perhaps not absolutely) homogeneous of degree one. Suppose\n\n* \( M \) is connected, and\n\n* \( {\exp }_{p} \) is defined on all of \( {T}_{p}M \) .\n\nThen:\n\n- For any \( q \in M \), ther... | Proof. Fix a sufficiently small \( r \) . Then (6.3.1) assures us that\n\n\[{\exp }_{p}\left\lbrack {{\mathrm{\;B}}_{p}\left( r\right) }\right\rbrack = {\mathcal{B}}_{p}^{ + }\left( r\right) \text{ and }{\exp }_{p}\left\lbrack {{\mathrm{\;S}}_{p}\left( r\right) }\right\rbrack = {\mathcal{S}}_{p}^{ + }\left( r\right) .\... | Yes |
Proposition 7.1.1. Let \( \sigma \left( t\right) = {\exp }_{p}\left( {tT}\right) ,0 \leq t \leq r \) be a constant speed geodesic from \( p = \sigma \left( 0\right) \) to \( q = \sigma \left( r\right) \) . The following five statements are mutually equivalent:\n\n(a) The point \( q \) is not conjugate to \( p \) along ... | ## Proof. By definition, (b) and (a) are equivalent.\n\n## That (b) and (c) are equivalent:\n\nSuppose (b) holds. Consider any variation whose \( t \) -curves are all geodesics. According to \( §{5.4} \), its variation vector field \( U \) is a Jacobi field. Thus, if \( U \) is zero at \( p \) and \( q \) ,(b) will for... | No |
Theorem 7.7.1 (Bonnet-Myers). Let \( \left( {M, F}\right) \) be a forward geodesically complete connected Finsler manifold of dimension \( n \) . Suppose its Ricci scalar has the following uniform positive lower bound\n\n\[ \operatorname{Ric} \geq \left( {n - 1}\right) \lambda > 0. \]\n\nEquivalently, suppose its Ricci... | - Every geodesic of length \( \geq \frac{\pi }{\sqrt{\lambda }} \) must contain conjugate points:\n\nConsider any unit speed geodesic \( \sigma \left( t\right) ,0 \leq t \leq L \) with velocity field \( T = T\left( t\right) \) . As usual, abbreviate \( {g}_{\left( \sigma, T\right) } \) by \( {g}_{T} \) . Use parallel t... | Yes |
Proposition 8.4.1. Let \( \left( {M, F}\right) \) be a forward geodesically complete Finsler manifold. Let \( S \) be the indicatrix bundle.\n\n(1) The function \( \left( {x, y}\right) \mapsto {c}_{y} \) is lower semicontinuous from \( S \) into \( (0,\infty \rbrack \) . That is,\n\n\[ \mathop{\liminf }\limits_{{\left(... | The lower semicontinuity of \( \left( {x, y}\right) \mapsto {c}_{y} \) :\n\nAccording to Exercise 8.3.2, for \( \left( {\widetilde{x},\widetilde{y}}\right) \) close to \( \left( {x, y}\right) \) and any fixed \( r \) in \( \left( {0,{c}_{y}}\right) \), we have \( {c}_{\widetilde{y}} \geq r + \epsilon \) for some positi... | No |
Proposition 8.5.2. Fix a point \( x \) in a forward geodesically complete Finsler manifold \( \left( {M, F}\right) \) and let \( {D}_{x},{\mathcal{D}}_{x} \) be as defined above. Let \( \mathcal{C}u{t}_{x} \) denote the cut locus of \( x \) . Then:\n\n(1) \( {\exp }_{x} \) maps the connected open set \( {D}_{x} \) diff... | ## Proof.\n\nFor (1), it suffices to check that \( {\exp }_{x} \) is injective on \( {D}_{x} \) . Suppose not; then we have\n\n(8.5.3a)\n\n\[ \n{\exp }_{x}\left( {{t}_{1}{y}_{1}}\right) = {\exp }_{x}\left( {{t}_{2}{y}_{2}}\right) = : \text{ some }\widetilde{x}\text{, say,} \n\]\n\nfor certain unit vectors \( {y}_{1},{y... | Yes |
Lemma 8.6.1. Let \( \left( {M, F}\right) \) be a forward geodesically complete Finsler manifold.\n\n(1) If \( M \) is compact, every geodesic contains a cut point.\n\n(2) If \( M \) is connected and if every geodesic emanating from some particular \( x \) contains a cut point, then \( M \) is compact. | Proof. Actually,(1) has already been addressed in Exercise 8.1.2. If \( M \) is compact, it must have finite diameter \( D \) . No geodesic longer than \( D \) can remain minimizing. Thus (1) holds.\n\nConsider (2). Our special \( x \) here guarantees that each \( {i}_{y} \) is finite, hence \( {\mathcal{D}}_{x} \) [se... | No |
Proposition 8.6.2. Fix a point \( x \) in a compact Finsler manifold \( \left( {M, F}\right) \) . Let \( \mathcal{C}u{t}_{x} \) denote its cut locus. Then:\n\n(1) \( \mathcal{C}u{t}_{x} \) is a compact and connected subset of \( M \) .\n\n(2) The distance \( d\left( {x,\mathcal{C}u{t}_{x}}\right) \) is attained at some... | Proof. Let \( {\widehat{x}}_{y} \) abbreviate the cut point \( {\exp }_{x}\left( {{i}_{y}y}\right) \) of \( x \) in the direction of \( y \) . Since \( M \) is compact, the map \( y \mapsto {\widehat{x}}_{y} \) is defined everywhere on the indicatrix \( {\mathrm{S}}_{x} \) . This map is continuous in view of Corollary ... | Yes |
Theorem 8.7.1. Let \( \left( {M, F}\right) \) be a connected Finsler manifold.\n\n(1) Suppose \( \left( {M, F}\right) \) is forward geodesically complete.\n\n- Fix any two points \( p \) and \( q \) in \( M \) . Then every homotopy class of paths from \( p \) to \( q \) contains a shortest (but perhaps not uniquely so)... | Consider a specific homotopy class, say \( \alpha \), of piecewise smooth curves. [In case (1), these would be piecewise smooth curves from the fixed \( p \) to the fixed \( q \) .] The geometric length of this class is defined as\n\n\[ \left| \alpha \right| \mathrel{\text{:=}} \mathop{\inf }\limits_{{c \in \alpha }}L\... | No |
Proposition 9.1.2. On any Finsler manifold \( \left( {M, F}\right) \) of nonpositive flag curvature, no geodesic can contain any conjugate points. | Proof. This follows from Theorem 9.1.1 by setting \( \lambda = 0 \) there. Indeed, let \( \sigma \left( t\right) ,0 \leq t \leq L \) be any geodesic with velocity \( T \) . Suppose there is a nonzero Jacobi field \( J \) along \( \sigma \) such that \( J\left( 0\right) = 0 = J\left( r\right) \), with \( r \in \left\lbr... | Yes |
Theorem 9.2.1. Let \( \left( {\widetilde{M},\widetilde{F}}\right) \) and \( \left( {M, F}\right) \) be two Finsler manifolds of the same dimension. Here, the Finsler structures are positively (but perhaps not absolutely) homogeneous of degree one. Suppose:\n\n* Both Finsler manifolds are connected, and the domain \( \l... | Proof. The completeness of the target and the surjectivity of \( \varphi \) both follow from a precise understanding of how short geodesics in the two spaces are related to each other. We relegate the proof to Exercise 9.2.1.\n\nIt remains to deduce that \( \varphi \) is a covering projection. We accomplish this by che... | No |
Proposition 9.2.2. Let \( \left( {M, F}\right) \) be any forward geodesically complete, connected Finsler manifold of nonpositive flag curvature. For any \( p \in M \) , the exponential map \( {\exp }_{p} : {T}_{p}M \rightarrow M \) is a covering projection. | Proof. Given the hypotheses, our \( {\exp }_{p} : {T}_{p}M \rightarrow M \) is a surjective \( {C}^{1} \) local diffeomorphism. If we had set \( \widetilde{M} \mathrel{\text{:=}} {T}_{p}M,\varphi \mathrel{\text{:=}} {\exp }_{p},\widetilde{F} \mathrel{\text{:=}} {\varphi }^{ * }F = F \circ {\varphi }_{ * } \) , and then... | Yes |
Lemma 9.3.2. Let \( \left( {M, F}\right) \) be a Finsler manifold.\n\n- Suppose that at some \( p \in M \), the exponential map \( {\exp }_{p} : {T}_{p}M \rightarrow M \) is a covering projection.\n\n- Let \( {\sigma }_{0}\left( t\right) \mathrel{\text{:=}} {\exp }_{p}\left( {t{T}_{0}}\right) \) and \( {\sigma }_{1}\le... | Proof. The contrapositive of the first conclusion encompasses the second conclusion. So it suffices to establish the first one.\n\nSuppose \( {\sigma }_{0} \) is homotopic to \( {\sigma }_{1} \), through a homotopy \( h\left( {t, u}\right) ,0 \leq t \leq L \) , \( 0 \leq u \leq 1 \) with fixed endpoints \( p \) and \( ... | Yes |
Theorem 9.3.3. Let \( \left( {M, F}\right) \) be any forward geodesically complete, connected Finsler manifold of nonpositive flag curvature.\n\n- Fix \( p, q \in M \) . Then, within each homotopy class of paths from \( p \) to \( q \), there exists a unique shortest smooth geodesic within that class.\n\n- In particula... | Proof. The existence has been ascertained in Theorem 8.7.1; it only requires forward geodesic completeness and connectedness.\n\nWe establish uniqueness here. Since \( \left( {M, F}\right) \) has, by hypothesis, nonpositive flag curvature, the exponential map \( {\exp }_{p} : {T}_{p}M \rightarrow M \) is a covering pro... | Yes |
Theorem 9.4.1 (Cartan-Hadamard). Let \( \\left( {M, F}\\right) \) be any forward geo-desically complete, connected Finsler manifold of nonpositive flag curvature. Then:\n\n(1) Geodesics in \( \\left( {M, F}\\right) \) do not contain conjugate points.\n\n(2) For any fixed \( p \\in M \), the exponential map \( {\\exp }_... | Proof. The first two conclusions have already been established in Propositions 9.1.2 and 9.2.2.\n\nIt remains to check that the covering projection \( {\\exp }_{p} \) is injective whenever the manifold \( M \) is simply connected. We give two separate and independent arguments. The first one depends on the Covering Hom... | No |
Theorem 9.5.2. Let \( \sigma \left( t\right), a \leq t \leq b \) be a unit speed geodesic, with velocity field \( T \), in a Finsler manifold of dimension \( n \) . Let \( K \) abbreviate the collection of flag curvatures \( \left\{ {\left. {K\left( {T, W}\right) : }\right| \;W \in {T}_{\sigma \left( t\right) }M,\;a \l... | Proof. In the following proof, when we speak of piecewise \( {C}^{\infty } \) vector fields along a geodesic, we mean those that vanish at the endpoints of that geodesic. This omission effects a less cumbersome prose. Note that \( W \) vanishes at the endpoints of \( \sigma \) if and only if its transplant \( \widetild... | No |
Corollary 9.8.2. Let \( \\left( {M, F}\\right) \) be an \( n \) -dimensional Finsler manifold. Let \( \\sigma \\left( t\\right) ,0 \\leq t \\leq L \) be a unit speed geodesic in \( M \), with velocity field \( T\\left( t\\right) \). Suppose:\n\n- The flag curvature \( K\\left( {T, W}\\right) \\geq \\lambda \) for any \... | ## Exercises\n\nExercise 9.8.1: Establish Corollaries 9.8.1 and 9.8.2 using the Rauch comparison theorem (Theorem 9.6.1), together with \( §{9.7} \).\n\n(a) For Corollary 9.8.1: Let the comparison space \( \\left( {{M}_{o},{F}_{o}}\\right) \) be an \( n \) - dimensional complete Riemannian manifold of constant sectiona... | No |
Proposition 10.1.1 (Ichijy \( \overline{\mathrm{o}} \) ). Let \( \left( {M, F}\right) \) be a Berwald space. Then:\n\n- Given any parallel vector field \( W \) along a curve \( \sigma \) in \( M \), its Finslerian norm \( F\left( W\right) \) is necessarily constant along \( \sigma \) .\n\n- Whenever \( M \) is connecte... | Proof. By (1.2.5), we can express the Finslerian norm of \( W \) as\n\n\[ F\left( W\right) = \sqrt{{g}_{W}\left( {W, W}\right) } \]\n\nwhere \( {g}_{W} \mathrel{\text{:=}} {g}_{{ij}\left( {\sigma, W}\right) }d{x}^{i} \otimes d{x}^{j} \) . Note that even though \( {\Gamma }^{i}{}_{jk} \) has no \( y \) dependence, the s... | Yes |
Proposition 10.2.1. Let \( \left( {M, F}\right) \) be a Finsler manifold. Then the following five criteria are equivalent:\n\n(a) The hv-part of the Chern curvature vanishes identically: \( {P}_{j}{}^{i}{}_{kl} = 0 \) .\n\n(b) The Cartan tensor is covariantly constant along all horizontal directions on the slit tangent... | Proof. Let us make a few preliminary observations:\n\n- The equivalence between (a) and (b) was established near the end of §3.4. See (3.4.13). The key consists of constitutive formulas that come from first Bianchi identities. These formulas express the curvature \( P \) entirely in terms of the horizontal covariant de... | Yes |
Lemma 10.3.1. Given any Finsler surface \( \left( {M, F}\right) \) which is either \( y \) -local or \( y \) -global, the following two statements are equivalent:\n\n(a) The Finsler structure \( F \) is of Berwald type.\n\n(b) Its Cartan scalar \( I \) is horizontally constant; that is, \( {I}_{1} = 0 = {I}_{2} \) . | Proof. In Exercise 4.4.7, we enumerated the only two a priori nonvanishing components of the \( {hv} \) -curvature \( \mathrm{P} \), relative to the Berwald frame \( \left\{ {{e}_{1},{e}_{2}}\right\} \) of the pulled-back bundle \( {p}^{ * }{TM} \) (which sits over \( {SM} \) ). They are \( {P}_{2111} \) and \( {P}_{11... | No |
Proposition 10.4.1. Let \( D \) be a torsion-free linear connection on a finite dimensional manifold \( M \) . Let \( p \) be any point in \( M \) . If the curvature of \( D \) vanishes in a neighborhood of \( p \), then there is a local coordinate system \( \left( {x}^{i}\right) \) about \( p \) in which all the conne... | Proof. We sketch an argument from Volume II of Spivak's book [Sp2]. Specifically, we focus on his \ | No |
Proposition 10.5.1. Let \( \left( {M, F}\right) \) be a Finsler manifold. Let \( {R}_{ikl}^{i} \) and \( {P}_{j}{}^{i}{}_{kl} \) be, respectively, the hh- and hv-curvatures of the Chern connection. Then the following three conditions are equivalent:\n\n(a) \( \left( {M, F}\right) \) is locally Minkowskian.\n\n(b) \( {R... | Proof. Let us make some preliminary observations.\n\n* Our discussions at the end of \( §{2.4} \) and \( §{3.3} \) show that (a) \( \Rightarrow \) (b).\n\n* It is apparent that \( \left( \mathrm{b}\right) \Rightarrow \left( \mathrm{c}\right) \) .\n\nThus it remains to check that \( \left( \mathrm{c}\right) \Rightarrow ... | Yes |
Proposition 10.6.1. Let \( \\left( {M, F}\\right) \) be a Landsberg surface for which the Finsler structure \( F \) is smooth on \( {TM} \\smallsetminus 0 \) . Then the value of \( K \) at any point \( y\\left( t\\right) \) of the indicatrix \( {S}_{x}M \) is determined by the Cartan scalar \( I \) according to the fol... | Proof. Since \( J = 0 \) on a Landsberg surface,(4.4.8) reduces to\n\n\[ {K}_{3} + {IK} = 0.\]\n\nRestricting this to \( {\\mathrm{S}}_{x}M \) gives \( \\dot{K}\\left( t\\right) + I\\left( t\\right) K\\left( t\\right) = 0 \) . Our hypothesis on \( F \) implies that \( I\\left( t\\right) \) is continuous; hence the solu... | Yes |
Theorem 10.6.2 (Szabó) [Sz]. Let \( \\left( {M, F}\\right) \) be a connected Berwald surface for which the Finsler structure \( F \) is smooth and strongly convex on all of \( {TM} \\smallsetminus 0 \) .\n\n- If \( K \) vanishes identically, then \( F \) is locally Minkowskian everywhere.\n\n- If \( K \) is not identic... | Proof. According to Exercise 4.4.7, \( K \) vanishing identically implies the same of the \( {hh} \) -Chern curvature \( {R}_{j}{}^{i}{}_{kl} \) . And being Berwald is synonymous with \( {P}_{j}{}^{i}{}_{kl} = 0 \) . Proposition 10.5.1 now tells us that \( \\left( {M, F}\\right) \) must be locally Minkowskian. This tak... | No |
11.0 The Importance of Randers Spaces | In 1941, G. Randers [Ra] studied a very interesting type of Finsler structures. These are called Randers metrics, and we first encountered them in \( §{1.3} \). Randers metrics are important for six reasons.\n\n- They occur naturally in physical applications, most notably in electron optics. According to Ingarden's acc... | Yes |
If \( {\widetilde{b}}_{j \mid k} = 0 \), namely, the 1-form \( \widetilde{b} \) is parallel with respect to \( \widetilde{a} \), then our Randers space \( \left( {M, F}\right) \) is of Berwald type. | - \( {\widetilde{b}}_{j \mid k} = 0 \) as a sufficient condition:\n\nGiven that \( \widetilde{b} \) is parallel with respect to \( \widetilde{a} \), formula (11.3.11) reduces to \( {G}^{i} = {\widetilde{\gamma }}_{jk}^{i}{y}^{j}{y}^{k} \) . Taking two \( y \) derivatives of \( {G}^{i} \) then gives \( 2{\widetilde{\gam... | Yes |
Proposition 12.1.1. Let \( \left( {M, F}\right) \) be a Finsler manifold. Let \( R \) be the \( {hh} \) - curvature of the Chern connection, for the portion of \( {\pi }^{ * }{TM} \) over \( {TM} \smallsetminus 0 \) . The five statements listed below are mutually equivalent: | And, given any of them, we have\n\n(12.1.1)\n\n\[{\ddot{A}}_{ijk} + \lambda {A}_{ijk} = 0\]\n\nwhere\n\n\[{\ddot{A}}_{ijk} \mathrel{\text{:=}} {\dot{A}}_{{ijk} \mid s}{\ell }^{s} = {\left( {A}_{{ijk} \mid r}{\ell }^{r}\right) }_{\mid s}{\ell }^{s} = {A}_{{ijk}\left| r\right| s}{\ell }^{r}{\ell }^{s}.\n\nThe above two f... | No |
Theorem 13.4.1 (Hopf). Let \( \left( {M, g}\right) \) be a complete connected Riemannian manifold of constant sectional curvature \( \lambda \) . Denote by \( \left( {\widetilde{M},\widetilde{g}}\right) \) the standard model of a simply connected complete Riemannian space with constant sectional curvature \( \lambda \)... | Proof. Here, both \( \widetilde{M} \) and \( M \) are complete and connected. So, according to Theorem 9.2.1, every smooth local isometry \( \varphi : \widetilde{M} \rightarrow M \) must be an onto map and a covering projection.\n\nAlso,(2) follows from (1). Indeed, \( \widetilde{M} \) is simply connected and \( M \) ,... | Yes |
Lemma 13.5.1. Let \( V \) be any globally defined vector field on a compact Riemannian manifold \( \left( {M, g}\right) \) without boundary. Denote the volume form in natural coordinates by \( \sqrt{g}{dx} \). Then\n\n\[ \n{\int }_{M}\left( {{\nabla }_{i}{V}^{i}}\right) \sqrt{g}{dx} = 0 \n\] | Proof. Because \( M \) is compact, we can decompose it as a finite union of closed subsets \( {\bar{U}}_{\alpha } \) whose interiors \( {U}_{\alpha } \) are mutually disjoint from each other. Furthermore, we may assume that each \( {\bar{U}}_{\alpha } \) lies inside some coordinate neighborhood.\n\nOver each \( {\bar{U... | Yes |
Theorem 13.5.2 (Hopf). Let \( \\left( {M, g}\\right) \) be a compact connected Riemannian manifold without boundary. Then every globally defined function \( f \) , with \( {\\Delta f} \\geq 0 \) everywhere or \( {\\Delta f} \\leq 0 \) everywhere, must be constant. In particular, there are no nonconstant globally define... | Proof. By replacing \( f \) with \( - f \) if necessary, we may assume without loss of generality that \( {\\Delta f} \\geq 0 \) everywhere.\n\nIn view of the divergence lemma and (13.5.4), we have\n\n\[ \n{\\int }_{M}{\\Delta f}\\sqrt{g}{dx} = 0 \n\]\n\nThis integral makes sense because \( M \) is compact. Since \( {\... | Yes |
Theorem 13.6.1 (Bochner). Let \( \theta \) be a globally defined 1-form on a compact boundaryless Riemannian manifold \( \left( {M, g}\right) \). * Suppose the quadratic form defined by the Ricci tensor of \( g \) is positive-definite. That is, \( {\operatorname{Ric}}_{ij}{V}^{i}{V}^{j} > 0 \) at \( x \) whenever \( V ... | Proof. Take the inner product of (13.6.4) with \( \theta \). All the integrals make sense because \( M \) is compact and \( \theta \) is globally defined. Carry out an integration-by-parts on the term involving the double covariant derivative. Since \( M \) is boundaryless, there is no subsequent boundary integral beca... | Yes |
Theorem 14.4.1 (Deicke). Let \( F \) be a Minkowski norm on \( {\mathbb{R}}^{n} \), smooth and strongly convex at all \( y \neq 0 \) . Let\n\n\[ \n{g}_{ij} \mathrel{\text{:=}} {\left( \frac{1}{2}{F}^{2}\right) }_{{y}^{i}{y}^{j}},\;{A}_{ijk} \mathrel{\text{:=}} \frac{F}{2}{\left( \frac{1}{2}{F}^{2}\right) }_{{y}^{i}{y}^... | * Statements (1) and (2) are equivalent, and is local. This can be seen directly from the definition of the Cartan tensor.\n\n* Statements (3) and (4) are equivalent, and is local. Note that \( \det \left( {g}_{ij}\right) \) is constant if and only if \( \sqrt{g} \) is constant. And, we have\n\n\[ \n{\left( \sqrt{g}\ri... | No |
Proposition 14.6.1. Let \( \\left( {{\\mathbb{R}}^{n}, F}\\right) \) be any Minkowski space of dimension \( n \\geq 3 \) . Here, \( F \) is typically only positively homogeneous of degree one. Then the following statements are equivalent:\n\n(a) The metric \( \\widehat{g} \\mathrel{\\text{:=}} {g}_{ij}\\left( y\\right)... | Proof. The Gauss equation (14.6.6) holds only when \( n \\geq 3 \) . It tells us that (a) \\Rightarrow (b). Also,(b) \\Rightarrow (c) is logically immediate.\n\nIt remains to show that (c) \\Rightarrow (a).\n\n* Given (c), we conclude from (14.6.6) that all sectional curvatures \( \\widehat{K} \) must vanish on the lev... | No |
Proposition 14.8.1. Let \( F \) be a Minkowski norm on \( {\mathbb{R}}^{n} \) that is possibly only positively homogeneous. Let \( {F}^{ * } \) denote the dual norm, in the functional analysis sense, on \( {\mathbb{R}}^{n * } \) . Then:\n\n- The Legendre transformation \( y \mapsto {y}^{b} \), namely,\n\n\[ \n{y}^{j} \... | - The Minkowski norm is by hypothesis smooth at all nonzero \( y \) . Hence the same holds for the fundamental tensor \( {g}_{ij}\left( y\right) \) . This now gives the smoothness of the map \( {y}^{j} \mapsto {y}_{j} \mathrel{\text{:=}} {g}_{ij}\left( y\right) {y}^{i} \) .\n\n- According to (14.8.2), the said map is a... | Yes |
Proposition 14.9.1. Let \( \left( {{\mathbb{R}}^{n}, F}\right) \) be a Minkowski space. Suppose the Minkowski norm \( F \) is absolutely homogeneous of degree one. Then\n\n\[ \n{\operatorname{vol}}_{\widehat{g}}\left( \bar{B}\right) \leq \operatorname{vol}\left( \overline{\mathbb{B}}\right) .\n\]\n\nEquivalently,\n\n\[... | Proof. Formula (14.9.4) says that \( {\operatorname{vol}}_{\widehat{g}}\left( \overline{\mathrm{B}}\right) = \frac{1}{n}{\operatorname{vol}}_{\dot{g}}\left( \mathrm{\;S}\right) \) . Specializing it to Euclidean space gives \( \operatorname{vol}\left( \overline{\mathbb{B}}\right) = \frac{1}{n}\operatorname{vol}\left( \m... | Yes |
Theorem 14.9.2 (Brickell). Suppose:\n\n- \( \\left( {{\\mathbb{R}}^{n}, F}\\right) \) is a Minkowski space, with \( F \) smooth and strongly convex on all of \( {\\mathbb{R}}^{n} \\smallsetminus 0 \) .\n\n- \( n \\geq 3 \) .\n\n- \( F \) is absolutely homogeneous of degree one.\n\n- The curvature tensor of \( \\widehat... | Proof. The argument we describe involves three major ingredients:\n\n* The Gauss equation (14.6.6).\n\n* Hopf’s classification of Riemannian space forms (Theorem 13.4.1).\n\n* The mixed-volume inequality (Proposition 14.9.1). This in turn comes from the Blaschke-Santaló inequality (14.7.8) and a detailed understanding ... | Yes |
Theorem 9.1. (Canonical local heights) Let \( V/k \) be a smooth variety defined over a number field, let \( D \in \operatorname{Div}\left( V\right) \), and let \( \phi : V \rightarrow V \) be a morphism. Suppose that\n\n\[{\phi }^{ * }D = {\alpha D} + \operatorname{div}\left( f\right)\]\n\nfor some number \( \alpha > ... | Proof. See Call-Silverman [1, Theorem 2.1]. | No |
Theorem 9.3. (Néron [2]) Let \( A/k \) be an abelian variety defined over a number field. For each divisor \( D \in \operatorname{Div}\left( A\right) \) there is a local height function\n\n\[ \n{\widehat{\lambda }}_{D} : \mathop{\coprod }\limits_{{v \in {M}_{k}}}{A}_{D}\left( {k}_{v}\right) \rightarrow \mathbb{R} \n\]\... | Proof. See Lang [6, Chapter 11]. | No |
Proposition 2.19 (Local Criterion for Continuity). A map \( f : X \rightarrow Y \) between topological spaces is continuous if and only if each point of \( X \) has a neighborhood on which (the restriction of) \( f \) is continuous. | Proof. If \( f \) is continuous, we may simply take each neighborhood to be \( X \) itself. Conversely, suppose \( f \) is continuous in a neighborhood of each point, and let \( U \subseteq Y \) be any open subset; we have to show that \( {f}^{-1}\left( U\right) \) is open. Any point \( x \in {f}^{-1}\left( U\right) \)... | Yes |
Let \( {\mathbb{B}}^{n} \subseteq {\mathbb{R}}^{n} \) be the unit ball, and define a map \( F : {\mathbb{B}}^{n} \rightarrow {\mathbb{R}}^{n} \) by\n\n\[ F\left( x\right) = \frac{x}{1 - \left| x\right| }.\] | Direct computation shows that the map \( G : {\mathbb{R}}^{n} \rightarrow {\mathbb{B}}^{n} \) defined by\n\n\[ G\left( y\right) = \frac{y}{1 + \left| y\right| } \]\n\nis an inverse for \( F \) . Thus \( F \) is bijective, and since \( F \) and \( {F}^{-1} = G \) are both continuous, \( F \) is a homeomorphism. It follo... | Yes |
Show that the map \( \varphi : C \rightarrow {\mathbb{S}}^{2} \) is a homeomorphism by showing that its inverse can be written \[ {\varphi }^{-1}\left( {x, y, z}\right) = \frac{\left( x, y, z\right) }{\max \{ \left| x\right| ,\left| y\right| ,\left| z\right| \} }.\] | \[ {\varphi }^{-1}\left( {x, y, z}\right) = \frac{\left( x, y, z\right) }{\max \{ \left| x\right| ,\left| y\right| ,\left| z\right| \} }.\] | No |
Proposition 2.30. Suppose \( X \) and \( Y \) are topological spaces, and \( f : X \rightarrow Y \) is any map.\n\n(a) \( f \) is continuous if and only if \( f\left( \bar{A}\right) \subseteq \overline{f\left( A\right) } \) for all \( A \subseteq X \) . | Proof. Problem 2-6. | No |
Proposition 2.37. Let \( X \) be a Hausdorff space.\n\n(a) Every finite subset of \( X \) is closed.\n\n(b) If a sequence \( \left( {p}_{i}\right) \) in \( X \) converges to a limit \( p \in X \), the limit is unique. | Proof. For part (a), consider first a set \( \left\{ {p}_{0}\right\} \) containing only one point. Given \( p \neq \) \( {p}_{0} \), the Hausdorff property says that there exist disjoint neighborhoods \( U \) of \( p \) and \( V \) of \( {p}_{0} \) . In particular, \( U \) is a neighborhood of \( p \) contained in \( X... | Yes |
Proposition 2.37. Let \( X \) be a Hausdorff space.\n\n(a) Every finite subset of \( X \) is closed.\n\n(b) If a sequence \( \left( {p}_{i}\right) \) in \( X \) converges to a limit \( p \in X \), the limit is unique. | Proof. For part (a), consider first a set \( \left\{ {p}_{0}\right\} \) containing only one point. Given \( p \neq \) \( {p}_{0} \), the Hausdorff property says that there exist disjoint neighborhoods \( U \) of \( p \) and \( V \) of \( {p}_{0} \) . In particular, \( U \) is a neighborhood of \( p \) contained in \( X... | Yes |
Proposition 2.43. Let \( X \) and \( Y \) be topological spaces and let \( \mathcal{B} \) be a basis for \( Y \) . A map \( f : X \rightarrow Y \) is continuous if and only if for every basis subset \( B \in \mathcal{B} \), the subset \( {f}^{-1}\left( B\right) \) is open in \( X \) . | Proof. One direction is easy: if \( f \) is continuous, the preimage of every open subset, and thus certainly every basis subset, is open. Conversely, suppose \( {f}^{-1}\left( B\right) \) is open for every \( B \in \mathcal{B} \) . Let \( U \subseteq Y \) be open, and let \( x \in {f}^{-1}\left( U\right) \) . Because ... | Yes |
Proposition 2.44. Let \( X \) be a set, and suppose \( \mathcal{B} \) is a collection of subsets of \( X \) . Then \( \mathcal{B} \) is a basis for some topology on \( X \) if and only if it satisfies the following two conditions:\n\n(i) \( \mathop{\bigcup }\limits_{{B \in \mathcal{B}}}B = X \) .\n\n(ii) If \( {B}_{1},... | Proof. If \( \mathcal{B} \) is a basis for some topology, the proof that it satisfies (i) and (ii) is left as an easy exercise. Conversely, suppose that \( \mathcal{B} \) satisfies (i) and (ii), and let \( \mathcal{T} \) be the collection of all unions of elements of \( \mathcal{B} \) .\n\n\n\nBecause the identity is always continuous, the characteristic property implies that \( {\iota }_{S} \) is also continuous. | No |
Corollary 3.10. Let \( X \) and \( Y \) be topological spaces, and suppose \( f : X \rightarrow Y \) is continuous.\n\n(a) RESTRICTING THE DOMAIN: The restriction of \( f \) to any subspace \( S \subseteq X \) is continuous.\n\n(b) RESTRICTING THE CODOMAIN: If \( T \) is a subspace of \( Y \) that contains \( f\left( X... | Proof. Part (a) follows from Corollary 3.9, because \( {\left. f\right| }_{S} = f \circ {\iota }_{S} \) . Part (b) follows from the characteristic property applied to the subspace \( T \) of \( Y \), and part (c) follows by composing \( f \) with the inclusion \( Y \hookrightarrow Z \) . | Yes |
Proposition 3.11 (Other Properties of the Subspace Topology). Suppose \( S \) is a subspace of the topological space \( X \) .\n\n(a) If \( R \subseteq S \) is a subspace of \( S \), then \( R \) is a subspace of \( X \) ; in other words, the subspace topologies that \( R \) inherits from \( S \) and from \( X \) agree... | Proof. For part (a), let \( U \subseteq R \) be any subset. Assume first that \( U \) is open in the subspace topology that \( R \) inherits from \( S \). This means \( U = R \cap V \) for some open subset \( V \subseteq S \) (Fig. 3.3). The fact that \( V \) is open in \( S \) means that \( V = W \cap S \) for some op... | Yes |
Proposition 3.16. A continuous injective map that is either open or closed is a topological embedding. | Proof. Suppose \( X \) and \( Y \) are topological spaces and \( f : X \rightarrow Y \) is a continuous injective map. Note that \( f \) defines a bijective map from \( X \) to \( f\left( X\right) \), which is continuous by Corollary 3.10(b). To show that this map is a homeomorphism, it suffices by Exercise 2.29 to sho... | No |
Proposition 3.16. A continuous injective map that is either open or closed is a topological embedding. | Proof. Suppose \( X \) and \( Y \) are topological spaces and \( f : X \rightarrow Y \) is a continuous injective map. Note that \( f \) defines a bijective map from \( X \) to \( f\left( X\right) \), which is continuous by Corollary 3.10(b). To show that this map is a homeomorphism, it suffices by Exercise 2.29 to sho... | No |
If \( U \subseteq {\mathbb{R}}^{n} \) is an open subset and \( f : U \rightarrow {\mathbb{R}}^{k} \) is any continuous map, the graph of \( f \) (Fig. 3.5) is the subset \( \Gamma \left( f\right) \subseteq {\mathbb{R}}^{n + k} \) defined by\n\n\[ \Gamma \left( f\right) = \left\{ {\left( {x, y}\right) = \left( {{x}_{1},... | Let \( {\Phi }_{f} : U \rightarrow {\mathbb{R}}^{n + k} \) be the continuous injective map\n\n\[ {\Phi }_{f}\left( x\right) = \left( {x, f\left( x\right) }\right) . \]\n\nJust as in Example 3.14, \( {\Phi }_{f} \) defines a continuous bijection from \( U \) onto \( \Gamma \left( f\right) \), and the restriction to \( \... | Yes |
To see that \( {\mathbb{S}}^{n} \) is a manifold, we need to show that each point has a Euclidean neighborhood. | For each \( i = 1,\ldots, n + 1 \), let \( {U}_{i}^{ + } \) denote the open subset of \( {\mathbb{R}}^{n + 1} \) consisting of points with \( {x}_{i} > 0 \), and \( {U}_{i}^{ - } \) the set of points with \( {x}_{i} < 0 \) . If \( x \) is any point in \( {\mathbb{S}}^{n} \), some coordinate \( {x}_{i} \) must be nonzer... | Yes |
Finally, consider the doughnut surface, which is the surface of revolution \( D \subseteq {\mathbb{R}}^{3} \) obtained by starting with the circle \( {\left( x - 2\right) }^{2} + {z}^{2} = 1 \) in the \( {xz} \) - plane (called the generating circle), and revolving it around the \( z \) -axis (Fig. 3.7). It is characte... | \[ F\left( {u, v}\right) = \left( {\left( {2 + \cos {2\pi u}}\right) \cos {2\pi v},\left( {2 + \cos {2\pi u}}\right) \sin {2\pi v},\sin {2\pi u}}\right) . \] This maps the plane onto \( D \) . It is not one-to-one, because \( F\left( {u + k, v + l}\right) = F\left( {u, v}\right) \) for any pair of integers \( \left( {k... | Yes |
Lemma 3.23 (Gluing Lemma). Let \( X \) and \( Y \) be topological spaces, and let \( \left\{ {A}_{i}\right\} \) be either an arbitrary open cover of \( X \) or a finite closed cover of \( X \) . Suppose that we are given continuous maps \( {f}_{i} : {A}_{i} \rightarrow Y \) that agree on overlaps: \( {\left. {f}_{i}\ri... | Proof. In either case, it follows from elementary set theory that there exists a unique map \( f \) such that \( {\left. f\right| }_{{A}_{i}} = {f}_{i} \) for each \( i \) . If the sets \( {A}_{i} \) are open, the continuity of \( f \) follows immediately from the local criterion for continuity (Proposition 2.19). On t... | Yes |
Theorem 3.24 (Uniqueness of the Subspace Topology). Suppose \( S \) is a subset of a topological space \( X \) . The subspace topology on \( S \) is the unique topology for which the characteristic property holds. | Proof. Suppose we are given an arbitrary topology on \( S \) that is known to satisfy the characteristic property. For this proof, let \( {S}_{g} \) denote the set \( S \) with the given topology, and let \( {S}_{s} \) denote \( S \) with the subspace topology. To show that the given topology is equal to the subspace t... | Yes |
Theorem 3.27 (Characteristic Property of the Product Topology). Suppose \( {X}_{1} \times \cdots \times {X}_{n} \) is a product space. For any topological space \( Y \), a map \( f : Y \rightarrow \) \( {X}_{1} \times \cdots \times {X}_{n} \) is continuous if and only if each of its component functions \( {f}_{i} = {\p... | Proof. Suppose each \( {f}_{i} \) is continuous. To prove that \( f \) is continuous, it suffices to show that the preimage of each basis subset \( {U}_{1} \times \cdots \times {U}_{k} \) is open. A point \( y \in Y \) is in \( {f}^{-1}\left( {{U}_{1} \times \cdots \times {U}_{k}}\right) \) if and only if \( {f}_{i}\le... | Yes |
Corollary 3.28. If \( {X}_{1},\ldots ,{X}_{n} \) are topological spaces, each canonical projection \( {\pi }_{i} : {X}_{1} \times \cdots \times {X}_{n} \rightarrow {X}_{i} \) is continuous. | Proof. Suppose that \( {X}_{1} \times \cdots \times {X}_{n} \) is endowed with some topology that satisfies the characteristic property. Since the proof of Corollary 3.28 uses only the characteristic property, it follows that the canonical projections \( {\pi }_{i} \) are continuous with respect to both the product top... | No |
Theorem 3.30 (Uniqueness of the Product Topology). Let \( {X}_{1},\ldots ,{X}_{n} \) be topological spaces. The product topology on \( {X}_{1} \times \cdots \times {X}_{n} \) is the unique topology that satisfies the characteristic property. | Proof. Suppose that \( {X}_{1} \times \cdots \times {X}_{n} \) is endowed with some topology that satisfies the characteristic property. Since the proof of Corollary 3.28 uses only the characteristic property, it follows that the canonical projections \( {\pi }_{i} \) are continuous with respect to both the product top... | Yes |
Proposition 3.33. A product of continuous maps is continuous, and a product of homeomorphisms is a homeomorphism. | Proof. Because a map is continuous provided that the preimages of basis open subsets are open, the first claim follows from the fact that \( {\left( {f}_{1} \times \cdots \times {f}_{k}\right) }^{-1}\left( {{U}_{1} \times \cdots \times }\right. \) \( \left. {U}_{k}\right) \) is just the product of the open subsets \( {... | Yes |
Proposition 3.35. If \( {M}_{1},\ldots ,{M}_{k} \) are manifolds of dimensions \( {n}_{1},\ldots ,{n}_{k} \), respectively, the product space \( {M}_{1} \times \cdots \times {M}_{k} \) is a manifold of dimension \( {n}_{1} + \cdots + {n}_{k} \) . | Proof. Proposition 3.31 shows that the product space is Hausdorff and second countable, so only the locally Euclidean property needs to be checked. Given any point \( p = \left( {{p}_{1},\ldots ,{p}_{k}}\right) \in {M}_{1} \times \cdots \times {M}_{k} \), for each \( i \) there exists a neighborhood \( {U}_{i} \) of \(... | Yes |
Proposition 3.36. The torus \( {\mathbb{T}}^{2} \) is homeomorphic to the doughnut surface \( D \) of Example 3.22. | Proof. The key geometric idea is that both surfaces are parametrized by two angles. For \( D \), the angles are \( \varphi = {2\pi u} \) and \( \theta = {2\pi v} \) as in (3.3); for \( {\mathbb{T}}^{2} \), they are the angles in the two circles. Although one must be cautious using angle functions because they cannot be... | Yes |
Corollary 3.39. If \( {\left( {X}_{\alpha }\right) }_{\alpha \in A} \) is an indexed family of nonempty topological spaces with infinitely many indices such that \( {X}_{\alpha } \) is not a trivial space, then the box topology on \( \mathop{\prod }\limits_{{\alpha \in A}}{X}_{\alpha } \) does not satisfy the character... | Proof. As we remarked above, under the hypotheses of the corollary, the box topology is not equal to the product topology. The result follows from the uniqueness statement of Theorem 3.37. | No |
Theorem 3.41 (Characteristic Property of Disjoint Union Spaces). Suppose that \( {\left( {X}_{\alpha }\right) }_{\alpha \in A} \) is an indexed family of topological spaces, and \( Y \) is any topological space. A map \( f : \mathop{\coprod }\limits_{{\alpha \in A}}{X}_{\alpha } \rightarrow Y \) is continuous if and on... | Proof. Problem 3-10. | No |
Define an equivalence relation on the square \( I \times I \) by setting \( \left( {x,0}\right) \sim \) \( \left( {x,1}\right) \) for all \( x \in I \), and \( \left( {0, y}\right) \sim \left( {1, y}\right) \) for all \( y \in I \) (Fig. 3.9). | This can be visualized as the space obtained by pasting the top boundary segment of the square to the bottom to form a cylinder, and then pasting the left-hand boundary circle of the resulting cylinder to the right-hand one. Later we will prove that the resulting quotient space is homeomorphic to the torus (see Example... | No |
Define \( {\mathbb{P}}^{n} \), the real projective space of dimension \( \mathbf{n} \), to be the set of 1-dimensional linear subspaces (lines through the origin) in \( {\mathbb{R}}^{n + 1} \) . There is a natural map \( q : {\mathbb{R}}^{n + 1} \smallsetminus \{ 0\} \rightarrow {\mathbb{P}}^{n} \) defined by sending a... | Projective space can also be viewed in another way. If we define an equivalence relation on \( {\mathbb{R}}^{n + 1} \smallsetminus \{ 0\} \) by declaring two points \( x, y \) to be equivalent if \( x = {\lambda y} \) for some nonzero real number \( \lambda \), then there is an obvious identification between \( {\mathb... | Yes |
Proposition 3.56. Suppose \( P \) is a second countable space and \( M \) is a quotient space of \( P \) . If \( M \) is locally Euclidean, then it is second countable. Thus if \( M \) is locally Euclidean and Hausdorff, it is a manifold. | Proof. Let \( q : P \rightarrow M \) denote the quotient map, and let \( \mathcal{U} \) be a cover of \( M \) by coordinate balls. The collection \( \left\{ {{q}^{-1}\left( U\right) : U \in \mathcal{U}}\right\} \) is an open cover of \( P \), which has a countable subcover by Theorem 2.50. If we let \( {\mathcal{U}}^{\... | Yes |
Proposition 3.57. Suppose \( q : X \rightarrow Y \) is an open quotient map. Then \( Y \) is Hausdorff if and only if the set \( \mathcal{R} = \left\{ {\left( {{x}_{1},{x}_{2}}\right) : q\left( {x}_{1}\right) = q\left( {x}_{2}\right) }\right\} \) is closed in \( X \times X \) . | Proof. First assume \( Y \) is Hausdorff. If \( \left( {{x}_{1},{x}_{2}}\right) \notin \mathcal{R} \), then there are disjoint neighborhoods \( {V}_{1} \) of \( q\left( {x}_{1}\right) \) and \( {V}_{2} \) of \( q\left( {x}_{2}\right) \), and it follows that \( {q}^{-1}\left( {V}_{1}\right) \times {q}^{-1}\left( {V}_{2}... | Yes |
Example 3.66. Consider the map \( \omega : I \rightarrow {\mathbb{S}}^{1} \) that wraps the interval once around the circle at constant speed, given (in complex notation) by \( \omega \left( s\right) = {e}^{2\pi is} \) . This map is continuous and surjective. To show that it is a quotient map, let \( U \subseteq {\math... | If \( U \) is open, then \( {\omega }^{-1}\left( U\right) \) is open by continuity. Conversely, suppose \( {\omega }^{-1}\left( U\right) \) is open, and let \( z \) be a point in \( U \) . If \( z \neq 1 \), then \( z = \omega \left( {s}_{0}\right) \) for a unique \( {s}_{0} \in \left( {0,1}\right) \), and there is som... | Yes |
Consider the map \( \omega : I \rightarrow {\mathbb{S}}^{1} \) that wraps the interval once around the circle at constant speed, given (in complex notation) by \( \omega \left( s\right) = {e}^{2\pi is} \). This map is continuous and surjective. To show that it is a quotient map, let \( U \subseteq {\mathbb{S}}^{1} \) b... | If \( U \) is open, then \( {\omega }^{-1}\left( U\right) \) is open by continuity. Conversely, suppose \( {\omega }^{-1}\left( U\right) \) is open, and let \( z \) be a point in \( U \). If \( z \neq 1 \), then \( z = \omega \left( {s}_{0}\right) \) for a unique \( {s}_{0} \in \left( {0,1}\right) \), and there is some... | Yes |
Proposition 3.67. If \( q : X \rightarrow Y \) is a surjective continuous map that is also an open or closed map, then it is a quotient map. | Proof. If \( q \) is open, it takes saturated open subsets to open subsets (because it takes all open subsets to open subsets). If \( q \) is closed, it takes saturated closed subsets to closed subsets. In either case, it is a quotient map by Proposition 3.60. | Yes |
Theorem 3.70 (Characteristic Property of the Quotient Topology). Suppose \( X \) and \( Y \) are topological spaces and \( q : X \rightarrow Y \) is a quotient map. For any topological space \( Z \), a map \( f : Y \rightarrow Z \) is continuous if and only if the composite map \( f \circ q \) is continuous: | Proof. This result follows immediately from the fact that for any open subset \( U \subseteq \) \( Z,{f}^{-1}\left( U\right) \) is open in \( Y \) if and only if \( {q}^{-1}\left( {{f}^{-1}\left( U\right) }\right) = {\left( f \circ q\right) }^{-1}\left( U\right) \) is open in \( X \) . | Yes |
Theorem 3.73 (Passing to the Quotient). Suppose \( q : X \rightarrow Y \) is a quotient map, \( Z \) is a topological space, and \( f : X \rightarrow Z \) is any continuous map that is constant on the fibers of \( q \) (i.e., if \( q\left( x\right) = q\left( {x}^{\prime }\right) \), then \( f\left( x\right) = f\left( {... | Proof. The existence and uniqueness of \( \widetilde{f} \) follow from elementary set theory: given \( y \in Y \), there is some \( x \in X \) such that \( q\left( x\right) = y \), and we can set \( \widetilde{f}\left( y\right) = f\left( x\right) \) for any such \( x \). The hypothesis on \( f \) guarantees that \( \wi... | Yes |
Theorem 3.75 (Uniqueness of Quotient Spaces). Suppose \( {q}_{1} : X \rightarrow {Y}_{1} \) and \( {q}_{2} : X \rightarrow {Y}_{2} \) are quotient maps that make the same identifications (i.e., \( {q}_{1}\left( x\right) = \) \( \left. {{q}_{1}\left( {x}^{\prime }\right) \text{if and only if}{q}_{2}\left( x\right) = {q}... | Proof. By Theorem 3.73, both \( {q}_{1} \) and \( {q}_{2} \) pass uniquely to the quotient as in the following diagrams:\n\n\n\n\n\nSince both of these diagrams commute, it follows that\n\n\[ \n{\widetilde{q}}_{1} \cir... | Yes |
To show that the quotient space \( I/ \sim \) of Example 3.47 is homeomorphic to the circle, all we need to do is exhibit a quotient map \( \omega : I \rightarrow {\mathbb{S}}^{1} \) that makes the same identifications as \( \sim \) . | The map described in Example 3.66 is such a map. | No |
Proposition 3.77 (Properties of Adjunction Spaces). Let \( X{ \cup }_{f}Y \) be an adjunction space, and let \( q : X \coprod Y \rightarrow X{ \cup }_{f}Y \) be the associated quotient map.\n\n(a) The restriction of \( q \) to \( X \) is a topological embedding, whose image set \( q\left( X\right) \) is a closed subspa... | Proof. We begin by showing that \( {\left. q\right| }_{X} \) is a closed map. Suppose that \( B \) is a closed subset of \( X \) . To show that \( q\left( B\right) \) is closed in the quotient space, we need to show that \( {q}^{-1}\left( {q\left( B\right) }\right) \) is closed in \( X \coprod Y \), which is equivalent... | Yes |
Example 3.86 (More Topological Groups). In view of Proposition 3.84, each of the following is a topological group, with the product topology or subspace topology as appropriate:\n\n- Euclidean space \( {\mathbb{R}}^{n} = \mathbb{R} \times \cdots \times \mathbb{R} \) as a group under vector addition\n\n- the group \( {\... | If \( G \) is a topological group and \( g \in G \), left translation by \( g \) is the map \( {L}_{g} : G \rightarrow \) \( G \) defined by \( {L}_{g}\left( {g}^{\prime }\right) = g{g}^{\prime } \) . It is continuous, because it is equal to the composition\n\n\[ G\overset{{i}_{g}}{ \rightarrow }G \times G\overset{m}{ ... | Yes |
Proposition 3.87. Suppose \( G \) is a topological group acting on a topological space \( X \) .\n\n(a) If the action is continuous, then it is an action by homeomorphisms.\n\n(b) If \( G \) has the discrete topology, then the action is continuous if and only if it is an action by homeomorphisms. | Proof. First suppose the action is continuous. This means, in particular, that for each \( g \in G \) the map \( x \mapsto g \cdot x \) is continuous from \( X \) to itself, because it is the composition \( x \mapsto \left( {g, x}\right) \mapsto g \cdot x \) . Each such map is a homeomorphism, because the definition of... | Yes |
As an application, let us consider the coset space \( \mathbb{R}/\mathbb{Z} \) . Because \( \mathbb{Z} \) is a subgroup of the topological group \( \mathbb{R} \), there is a natural free continuous action of \( \mathbb{Z} \) on \( \mathbb{R} \) by translation: \( n \cdot x = n + x \) . (Because \( \mathbb{R} \) is abel... | Consider also the map \( \varepsilon : \mathbb{R} \rightarrow {\mathbb{S}}^{1} \) defined (in complex notation) by \[ \varepsilon \left( r\right) = {e}^{2\pi ir}. \] It is straightforward to check that this is a local homeomorphism and thus an open map, so it is a quotient map. Because it makes the same identifications... | Yes |
A topological space \( X \) is connected if and only if the only subsets of \( X \) that are both open and closed in \( X \) are \( \varnothing \) and \( X \) itself. | Assume first that \( X \) is connected, and suppose that \( U \subseteq X \) is open and closed. Then \( V = X \smallsetminus U \) is also open and closed. If both \( U \) and \( V \) were nonempty, then they would disconnect \( X \) ; therefore, either \( V \) is empty, which means that \( U = X \), or \( U \) is empt... | Yes |
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