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Corollary 7.31. Suppose \( \left( {M, g}\right) \) is a Riemannian or pseudo-Riemannian manifold of dimension \( n \geq 3 \) . If \( g \) is locally conformally flat, then its Weyl tensor vanishes identically. | Proof. Suppose \( \left( {M, g}\right) \) is locally conformally flat. Then for each \( p \in M \) there exist a neighborhood \( U \) and an embedding \( \varphi : U \rightarrow {\mathbb{R}}^{n} \) such that \( \varphi \) pulls back a flat (Riemannian or pseudo-Riemannian) metric on \( {\mathbb{R}}^{n} \) to a metric o... | Yes |
Proposition 7.32. Suppose \( \left( {M, g}\right) \) is a Riemannian or pseudo-Riemannian manifold of dimension \( n \geq 3 \), and let \( W \) and \( C \) denote its Weyl and Cotton tensors, respectively. Then\n\n\[ \n{\operatorname{tr}}_{g}\left( {\nabla W}\right) = \left( {n - 3}\right) C \n\] \n\nwhere the trace is... | Proof. Writing \( W = {Rm} - P \odot g \) and using the component form of the first contracted Bianchi identity (7.32), we obtain\n\n\[ \n{W}_{ijkl}{;}^{i} = {R}_{{jk};l} - {R}_{{jl};k} - {P}_{{il};}{}^{i}{g}_{jk} - {P}_{{jk};}{}^{i}{g}_{il} + {P}_{{ik};}{}^{i}{g}_{jl} + {P}_{{jl};}{}^{i}{g}_{ik}. \n\] \n\nNote that \(... | Yes |
Proposition 7.34 (Conformal Invariance of the Cotton Tensor in Dimension 3). Suppose \( \left( {M, g}\right) \) is a Riemannian or pseudo-Riemannian 3-manifold, and \( \widetilde{g} = {e}^{2f}g \) for some \( f \in {C}^{\infty }\left( M\right) \) . If \( C \) and \( \widetilde{C} \) denote the Cotton tensors of \( g \)... | Proof. Problem 7-10. | No |
Lemma 7.38. Let \( \\left( {M, g}\\right) \) be a Riemannian or pseudo-Riemannian manifold, and consider the overdetermined system of equations\n\n\[ \n\\nabla \\xi = A\\left( \\xi \\right) \n\]\n\n(7.56)\n\nwhere \( A : {T}^{ * }M \\rightarrow {T}^{2}{T}^{ * }M \) is a smooth map satisfying the following compatibility... | Proof. Let \( p \\in M \) be given. In smooth local coordinates \( \\left( {x}^{i}\\right) \) on a neighborhood of \( p,\\left( {7.56}\\right) \) is equivalent to the overdetermined system\n\n\[ \n\\frac{\\partial {\\xi }_{i}\\left( x\\right) }{\\partial {x}^{j}} = {a}_{ij}\\left( {x,\\xi \\left( x\\right) }\\right) \n... | Yes |
Proposition 8.1 (Properties of the Second Fundamental Form). Suppose \( \left( {M, g}\right) \) is an embedded Riemannian submanifold of a Riemannian or pseudo-Riemannian manifold \( \left( {\widetilde{M},\widetilde{g}}\right) \), and let \( X, Y \in \mathfrak{X}\left( M\right) \). (a) \( \coprod \left( {X, Y}\right) \... | Proof. Choose particular extensions of \( X \) and \( Y \) to a neighborhood of \( M \) in \( \widetilde{M} \), and for simplicity denote the extended vector fields also by \( X \) and \( Y \). We begin by proving that \( \coprod \left( {X, Y}\right) \) is symmetric in \( X \) and \( Y \) when defined in terms of these... | Yes |
Theorem 8.2 (The Gauss Formula). Suppose \( \left( {M, g}\right) \) is an embedded Riemannian submanifold of a Riemannian or pseudo-Riemannian manifold \( \left( {\widetilde{M},\widetilde{g}}\right) \) . If \( X, Y \in \) \( \mathfrak{X}\left( M\right) \) are extended arbitrarily to smooth vector fields on a neighborho... | Proof. Because of the decomposition (8.1) and the definition of the second fundamental form, it suffices to show that \( {\left( {\widetilde{\nabla }}_{X}Y\right) }^{\top } = {\nabla }_{X}Y \) at all points of \( M \) .\n\nDefine a map \( {\nabla }^{\top } : \mathcal{X}\left( M\right) \times \mathcal{X}\left( M\right) ... | Yes |
Corollary 8.3 (The Gauss Formula Along a Curve). Suppose \( \left( {M, g}\right) \) is an embedded Riemannian submanifold of a Riemannian or pseudo-Riemannian manifold \( \left( {\widetilde{M},\widetilde{g}}\right) \), and \( \gamma : I \rightarrow M \) is a smooth curve. If \( X \) is a smooth vector field along \( \g... | Proof. For each \( {t}_{0} \in I \), we can find an adapted orthonormal frame \( \left( {{E}_{1},\ldots ,{E}_{m}}\right) \) in a neighborhood of \( \gamma \left( {t}_{0}\right) \) . (Recall that our default assumption is that \( \dim \widetilde{M} = m \) and \( \dim M = n \) .) In terms of this frame, \( X \) can be wr... | Yes |
Proposition 8.4 (The Weingarten Equation). Suppose \( \left( {M, g}\right) \) is an embedded Riemannian submanifold of a Riemannian or pseudo-Riemannian manifold \( \left( {\widetilde{M},\widetilde{g}}\right) \) . For every \( X \in \mathfrak{X}\left( M\right) \) and \( N \in \Gamma \left( {NM}\right) \), the following... | Proof. Note that at points of \( M \), the covariant derivative \( {\widetilde{\nabla }}_{X}N \) is independent of the choice of extensions of \( X \) and \( N \) by Proposition 4.26. Let \( Y \in \mathfrak{X}\left( M\right) \) be arbitrary, extended to a vector field on an open subset of \( \widetilde{M} \) . Since \(... | Yes |
Theorem 8.5 (The Gauss Equation). Suppose \( \left( {M, g}\right) \) is an embedded Riemannian submanifold of a Riemannian or pseudo-Riemannian manifold \( \left( {\widetilde{M},\widetilde{g}}\right) \). For all \( W, X, Y, Z \in \mathfrak{X}\left( M\right) \), the following equation holds:\n\n\[ \widetilde{Rm}\left( {... | Proof. Let \( W, X, Y, Z \) be extended arbitrarily to an open subset of \( \widetilde{M} \). At points of \( M \), using the definition of the curvature and the Gauss formula, we get\n\n\[ \widetilde{\operatorname{Rm}}\left( {W, X, Y, Z}\right) = \left\langle {{\widetilde{\nabla }}_{W}{\widetilde{\nabla }}_{X}Y - {\wi... | Yes |
Theorem 8.9 (The Codazzi Equation). Suppose \( \left( {M, g}\right) \) is an embedded Riemannian submanifold of a Riemannian or pseudo-Riemannian manifold \( \left( {\widetilde{M},\widetilde{g}}\right) \). For all \( W, X, Y \in \mathfrak{X}\left( M\right) \), the following equation holds:\n\n\[ \n{\left( \widetilde{R}... | Proof. It suffices to show that both sides of (8.6) give the same result when we take their inner products with an arbitrary smooth normal vector field \( N \) along \( M \) :\n\n\[ \n\langle \widetilde{R}\left( {W, X}\right) Y, N\rangle = \left\langle {\left( {{\nabla }_{W}^{F}\mathrm{{II}}}\right) \left( {X, Y}\right... | Yes |
Proposition 8.10 (Geometric Interpretation of II). Suppose \( \left( {M, g}\right) \) is an embedded Riemannian submanifold of a Riemannian or pseudo-Riemannian manifold \( \left( {\widetilde{M},\widetilde{g}}\right), p \in M \), and \( v \in {T}_{p}M \) .\n\n(a) \( \coprod \left( {v, v}\right) \) is the \( \widetilde{... | Proof. Suppose \( \gamma : \left( {-\varepsilon ,\varepsilon }\right) \rightarrow M \) is any regular curve with \( \gamma \left( 0\right) = p \) and \( {\gamma }^{\prime }\left( 0\right) = v \) . Applying the Gauss formula (Corollary 8.3) to the vector field \( {\gamma }^{\prime } \) along \( \gamma \), we obtain\n\n\... | Yes |
Lemma 8.11. Suppose \( V \) is an inner product space, \( W \) is a vector space, and \( B,{B}^{\prime } : V \times V \rightarrow W \) are symmetric and bilinear. If \( B\left( {v, v}\right) = {B}^{\prime }\left( {v, v}\right) \) for every unit vector \( v \in V \), then \( B = {B}^{\prime } \) . | Proof. Every vector \( v \in V \) can be written \( v = \lambda \widehat{v} \) for some unit vector \( \widehat{v} \), so the bilinearity of \( B \) and \( {B}^{\prime } \) implies \( B\left( {v, v}\right) = {B}^{\prime }\left( {v, v}\right) \) for every \( v \), not just unit vectors. The result then follows from the ... | Yes |
THE GAUSS FORMULA FOR A HYPERSURFACE: If \( X, Y \in \mathfrak{X}\left( M\right) \) are extended to an open subset of \( \widetilde{M} \), then\n\n\[ \n{\widetilde{\nabla }}_{X}Y = {\nabla }_{X}Y + h\left( {X, Y}\right) N.\n\] | Parts (a), (b), and (d) follow immediately from substituting (8.10) into the general versions of the Gauss formula and Gauss equation. To prove (c), note first that the general version of the Weingarten equation can be written \( {\left( {\widetilde{\nabla }}_{X}N\right) }^{\top } = - {sX} \) . Since \( \left\langle {{... | No |
Proposition 8.16 (Finite-Dimensional Spectral Theorem). Suppose \( V \) is a finite-dimensional inner product space and \( s : V \rightarrow V \) is a self-adjoint linear endomorphism. Then \( V \) has an orthonormal basis of s-eigenvectors, and all of the eigenvalues are real. | Proof. The proof is by induction on \( n = \dim V \) . The \( n = 1 \) result is easy, so assume that the theorem holds for some \( n \geq 1 \) and suppose \( \dim V = n + 1 \) . Lemma 8.14 shows that \( s \) has a unit eigenvector \( {b}_{0} \) with a real eigenvalue \( {\lambda }_{0} \) . Let \( B \subseteq V \) be t... | Yes |
Proposition 8.17. With notation as above, the components in \( \\left( {{x}^{1},\\ldots ,{x}^{n}}\\right) \) -coordinates of the scalar second fundamental form, the shape operator, and the mean curvature of \( \\left( {{M}_{a},{g}_{a}}\\right) \) (denoted by \( {h}_{a},{s}_{a} \), and \( {H}_{a} \), respectively) with ... | Proof. The normal component of \( {\\widetilde{\\nabla }}_{{\\partial }_{\\alpha }}{\\partial }_{\\beta } \) is \( {\\widetilde{\\Gamma }}_{\\alpha \\beta }^{n + 1}{\\partial }_{v} \), which Corollary 6.42 shows is equal to \( - \\frac{1}{2}{\\partial }_{v}{g}_{\\alpha \\beta }{\\partial }_{r} \) (noting that the roles... | Yes |
Proposition 8.21. Suppose \( \gamma : I \rightarrow {\mathbb{R}}^{m} \) is a unit-speed curve, \( {t}_{0} \in I \), and \( \kappa \left( {t}_{0}\right) \neq 0 \) .\n\n(a) There is a unique unit-speed parametrized circle \( c : \mathbb{R} \rightarrow {\mathbb{R}}^{m} \), called the osculating circle at \( \gamma \left( ... | Proof. An easy geometric argument shows that every circle in \( {\mathbb{R}}^{m} \) with center \( q \) and radius \( R \) has a unit-speed parametrization of the form\n\n\[ c\left( t\right) = q + R\cos \left( \frac{t - {t}_{0}}{R}\right) v + R\sin \left( \frac{t - {t}_{0}}{R}\right) w, \]\n\nwhere \( \left( {v, w}\rig... | No |
Proposition 8.23. Suppose \( M \subseteq {\mathbb{R}}^{n + 1} \) is an embedded hypersurface, \( X : U \rightarrow M \) is a smooth local parametrization of \( M,\left( {{X}_{1},\ldots ,{X}_{n}}\right) \) is the local frame for \( {TM} \) determined by \( X \), and \( N \) is a unit normal field on \( M \) . Then the s... | Proof. Let \( {u}_{0} = \left( {{u}_{0}^{1},\ldots ,{u}_{0}^{n}}\right) \) be an arbitrary point of \( U \) and let \( p = X\left( {u}_{0}\right) \in M \) . For each \( i \in \{ 1,\ldots, n\} \), the curve \( \gamma \left( t\right) = X\left( {{u}_{0}^{1},\ldots ,{u}_{0}^{i} + t,\ldots ,{u}_{0}^{n}}\right) \) is a smoot... | Yes |
Proposition 8.24. Suppose \( M \subseteq {\mathbb{R}}^{n + 1} \) is a Riemannian hypersurface. Let \( p \in M \) , and let \( {\kappa }_{1},\ldots ,{\kappa }_{n} \) denote the principal curvatures of \( M \) at \( p \) with respect to some choice of unit normal. Then there is an isometry \( \varphi : {\mathbb{R}}^{n + ... | Proof. Replacing \( M \) by its image under a translation and a rotation (which are Euclidean isometries), we may assume that \( p \) is the origin and \( {T}_{p}M \) is equal to the span of \( \left( {{\partial }_{1},\ldots ,{\partial }_{n}}\right) \) . Then after reflecting in the \( \left( {{x}^{1},\ldots ,{x}^{n}}\... | Yes |
Example 8.25 (Shape Operators of Spheres). The function \( F : {\mathbb{R}}^{n + 1} \rightarrow \mathbb{R} \) defined by \( F\left( x\right) = {\left| x\right| }^{2} \) is a smooth defining function for each sphere \( {\mathbb{S}}^{n}\left( R\right) \) . The gradient of this function is \( \operatorname{grad}F = 2\math... | \[ N = \frac{1}{R}\mathop{\sum }\limits_{{i = 1}}^{{n + 1}}{x}^{i}{\partial }_{i} \] thus restricts to a unit normal along \( {\mathbb{S}}^{n}\left( R\right) \) . (It is the outward pointing normal.) The shape operator is now easy to compute: \[ {sX} = - \frac{1}{R}\mathop{\sum }\limits_{{i, j = 1}}^{{n + 1}}{X}^{j}\le... | Yes |
Theorem 8.27 (Gauss’s Theorema Egregium). Suppose \( \left( {M, g}\right) \) is an embedded 2-dimensional Riemannian submanifold of \( {\mathbb{R}}^{3} \) . For every \( p \in M \), the Gaussian curvature of \( M \) at \( p \) is equal to one-half the scalar curvature of \( g \) at \( p \), and thus the Gaussian curvat... | Proof. Let \( p \in M \) be arbitrary, and choose an orthonormal basis \( \left( {{b}_{1},{b}_{2}}\right) \) for \( {T}_{p}M \) . In this basis \( g \) is represented by the identity matrix, and the shape operator has the same matrix as the scalar second fundamental form. Thus \( K\left( p\right) = \det \left( {s}_{j}^... | Yes |
Proposition 8.31. Suppose \( {R}_{1} \) and \( {R}_{2} \) are algebraic curvature tensors on a finite-dimensional inner product space \( V \) . Iff for every pair of linearly independent vectors \( v, w \in V \) , \n\n\[ \n\frac{{R}_{1}\left( {v, w, w, v}\right) }{{\left| v \land w\right| }^{2}} = \frac{{R}_{2}\left( {... | Proof. Let \( {R}_{1} \) and \( {R}_{2} \) be tensors satisfying the hypotheses, and set \( D = {R}_{1} - {R}_{2} \) . Then \( D \) is an algebraic curvature tensor, and \( D\left( {v, w, w, v}\right) = 0 \) for all \( v, w \in V \) . (This is true by hypothesis when \( v \) and \( w \) are linearly independent, and it... | Yes |
Proposition 8.32 (Geometric Interpretation of Ricci and Scalar Curvatures).\n\nLet \( \left( {M, g}\right) \) be a Riemannian \( n \) -manifold and \( p \in M \) .\n\n(a) For every unit vector \( v \in {T}_{p}M, R{c}_{p}\left( {v, v}\right) \) is the sum of the sectional curvatures of the 2-planes spanned by \( \left( ... | Proof. Given any unit vector \( v \in {T}_{p}M \), let \( \left( {{b}_{1},\ldots ,{b}_{n}}\right) \) be as in the hypothesis. Then \( R{c}_{p}\left( {v, v}\right) \) is given by\n\n\[ R{c}_{p}\left( {v, v}\right) = {R}_{11}\left( p\right) = {R}_{k11}{}^{k}\left( p\right) = \mathop{\sum }\limits_{{k = 1}}^{n}R{m}_{p}\le... | Yes |
Lemma 8.33. If a Riemannian manifold \( \left( {M, g}\right) \) is frame-homogeneous, then it has constant sectional curvature. | Proof. Frame homogeneity implies, in particular, that given two 2-planes at the same or different points, there is an isometry taking one to the other. The result follows from the isometry invariance of the curvature tensor. | No |
Theorem 8.34 (Sectional Curvatures of the Model Spaces). The following Riemannian manifolds have the indicated constant sectional curvatures:\n\n(a) \( \\left( {{\\mathbb{R}}^{n},\\bar{g}}\\right) \) has constant sectional curvature 0 .\n\n(b) \( \\left( {{\\mathbb{S}}^{n}\\left( R\\right) ,{\\overset{ \\circ }{g}}_{R}... | Proof. First we consider the simplest case: Euclidean space. Since the curvature tensor of \( {\\mathbb{R}}^{n} \) is identically zero, clearly all sectional curvatures are zero. This is also easy to see geometrically, since each plane section is actually a plane, which has zero Gaussian curvature.\n\nNext consider the... | Yes |
Proposition 8.36. A Riemannian metric \( g \) has constant sectional curvature \( c \) if and only if its curvature tensor satisfies\n\n\[ \n{Rm} = \frac{1}{2}{cg} \otimes g.\n\]\n\nIn this case, the Ricci tensor and scalar curvature of \( g \) are given by the formulas\n\n\[ \n{Rc} = \left( {n - 1}\right) {cg};\;S = n... | Proof. Problem 8-29. | No |
Theorem 9.7 (The Gauss-Bonnet Theorem). If \( \left( {M, g}\right) \) is a smoothly triangulated compact Riemannian 2-manifold, then\n\n\[ \n{\int }_{M}{KdA} = {2\pi \chi }\left( M\right) \n\]\n\nwhere \( K \) is the Gaussian curvature of \( g \) and \( {dA} \) is its Riemannian density. | Proof. We may as well assume that \( M \) is connected, because if not we can prove the theorem for each connected component and add up the results.\n\nFirst consider the case in which \( M \) is orientable. In this case, we can choose an orientation for \( M \), and then \( {\int }_{M}{KdA} \) gives the same result wh... | Yes |
Corollary 9.8 Let \( \left( {M, g}\right) \) be a compact Riemannian 2-manifold and let \( K \) be its Gaussian curvature.\n\n(a) If \( M \) is homeomorphic to the sphere or the projective plane, then \( K > 0 \) somewhere.\n\n(b) If \( M \) is homeomorphic to the torus or the Klein bottle, then either \( K \equiv 0 \)... | This corollary has a remarkable converse, proved in the mid-1970s by Jerry Kaz-dan and Frank Warner: If \( K \) is any smooth function on a compact 2-manifold \( M \) satisfying the necessary sign condition of Corollary 9.8, then there exists a Riemannian metric on \( M \) for which \( K \) is the Gaussian curvature. T... | No |
Corollary 9.9 Let \( \left( {M, g}\right) \) be a compact Riemannian 2-manifold and \( K \) its Gaussian curvature.\n\n(a) If \( K > 0 \) everywhere on \( M \), then the universal covering manifold of \( M \) is homeomorphic to \( {\mathbb{S}}^{2} \), and \( {\pi }_{1}\left( M\right) \) is either trivial or isomorphic ... | Proof. Suppose first that \( M \) has positive Gaussian curvature. From the Gauss-Bonnet theorem, \( M \) has positive Euler characteristic. The classification theorem for compact surfaces shows that the only such surfaces are the sphere (with trivial fundamental group) and the projective plane (with fundamental group ... | Yes |
Theorem 10.1 (The Jacobi Equation). Let \( \left( {M, g}\right) \) be a Riemannian or pseudo-Riemannian manifold, let \( \gamma \) be a geodesic in \( M \), and let \( J \) be a vector field along \( \gamma \) . If \( J \) is the variation field of a variation through geodesics, then \( J \) satisfies the following equ... | Proof. Write \( T\left( {s, t}\right) = {\partial }_{t}\Gamma \left( {s, t}\right) \) and \( S\left( {s, t}\right) = {\partial }_{s}\Gamma \left( {s, t}\right) \) as in Chapter 6 . The geodesic equation tells us that\n\n\[ \n{D}_{t}T \equiv 0 \n\]\n\nfor all \( \left( {s, t}\right) \) . We can take the covariant deriva... | Yes |
Proposition 10.2 (Existence and Uniqueness of Jacobi Fields). Let \( \left( {M, g}\right) \) be a Riemannian or pseudo-Riemannian manifold. Suppose \( I \subseteq \mathbb{R} \) is an interval, \( \gamma : I \rightarrow \) \( M \) is a geodesic, \( a \in I \), and \( p = \gamma \left( a\right) \) . For every pair of vec... | Proof. Choose a parallel orthonormal frame \( \left( {E}_{i}\right) \) along \( \gamma \), and write \( v = {v}^{i}{E}_{i}\left( a\right) \) , \( w = {w}^{i}{E}_{i}\left( a\right) \), and \( {\gamma }^{\prime }\left( t\right) = {y}^{i}\left( t\right) {E}_{i}\left( t\right) \) in terms of this frame. Writing an unknown ... | Yes |
Corollary 10.3. Suppose \( \left( {M, g}\right) \) is a Riemannian or pseudo-Riemannian manifold of dimension \( n \), and \( \gamma \) is any geodesic in \( M \) . Then \( \mathcal{J}\left( \gamma \right) \) is a \( {2n} \) -dimensional linear subspace of \( \mathfrak{X}\left( \gamma \right) \) . | Proof. Because the Jacobi equation is linear, \( \mathcal{J}\left( \gamma \right) \) is a linear subspace of \( \mathcal{X}\left( \gamma \right) \) . Let \( p = \gamma \left( a\right) \) be any point on \( \gamma \), and consider the linear map from \( \mathcal{J}\left( \gamma \right) \) to \( {T}_{p}M \oplus {T}_{p}M ... | Yes |
Proposition 10.7. Let \( \left( {M, g}\right) \) be a Riemannian or pseudo-Riemannian manifold. Suppose \( \gamma : I \rightarrow M \) is a geodesic and \( J \) is a Jacobi field along \( \gamma \) . Then the following are equivalent:\n\n(a) \( J \) is a normal Jacobi field.\n\n(b) \( J \) is orthogonal to \( {\gamma }... | Proof. Define a function \( f : I \rightarrow \mathbb{R} \) by \( f\left( t\right) = \left\langle {J\left( t\right) ,{\gamma }^{\prime }\left( t\right) }\right\rangle \), so that \( f\left( t\right) = 0 \) if and only if \( J\left( t\right) \bot {\gamma }^{\prime }\left( t\right) \) . Using compatibility with the metri... | Yes |
Corollary 10.8. Suppose \( \left( {M, g}\right) \) is a Riemannian or pseudo-Riemannian \( n \) -manifold and \( \gamma : I \rightarrow M \) is any nonconstant geodesic. Then \( {\mathcal{J}}^{ \bot }\left( \gamma \right) \) is a \( \left( {{2n} - 2}\right) \) -dimensional subspace of \( \mathcal{J}\left( \gamma \right... | Proof. As we noted in the proof of Corollary 10.3, for every \( a \in I \), the map from \( \mathcal{J}\left( \gamma \right) \) to \( {T}_{\gamma \left( a\right) }M \oplus {T}_{\gamma \left( a\right) }M \) given by \( J \mapsto \left( {J\left( a\right) ,{D}_{t}J\left( a\right) }\right) \) is an isomorphism, and Proposi... | Yes |
Lemma 10.9. Let \( \left( {M, g}\right) \) be a Riemannian or pseudo-Riemannian manifold, \( I \subseteq \mathbb{R} \) an interval containing 0, and \( \gamma : I \rightarrow M \) a geodesic. Suppose \( J : I \rightarrow M \) is a Jacobi field such that \( J\left( 0\right) = 0 \) . If \( M \) is geodesically complete o... | Proof. The proof of Proposition 10.4 showed that \( J \) is the variation field of a variation \( \Gamma \) of the form (10.2), with \( \sigma \) any smooth curve satisfying \( \sigma \left( 0\right) = p \) and \( {\sigma }^{\prime }\left( 0\right) = 0 \), and \( V \) a smooth vector field along \( \sigma \) with \( V\... | Yes |
Proposition 10.10 (Jacobi Fields Vanishing at a Point). Let \( \left( {M, g}\right) \) be a Riemannian or pseudo-Riemannian n-manifold and \( p \in M \) . Suppose \( \gamma : I \rightarrow M \) is a geodesic such that \( 0 \in I \) and \( \gamma \left( 0\right) = p \) . For every \( w \in {T}_{p}M \), the Jacobi field ... | Proof. Under the given hypotheses, Lemma 10.9 showed that the restriction of \( J \) to any compact interval containing 0 is the variation field of a variation \( \Gamma \) through geodesics of the form (10.5). Using the chain rule to compute \( J\left( t\right) = {\partial }_{s}\Gamma \left( {0, t}\right) \) , we arri... | Yes |
Corollary 10.11. Suppose \( \left( {M, g}\right) \) is a Riemannian or pseudo-Riemannian manifold and \( U \) is a normal neighborhood of \( p \in M \) . For each \( q \in U \smallsetminus \{ p\} \), every vector in \( {T}_{q}M \) is the value of a Jacobi field \( J \) along a radial geodesic such that \( J \) vanishes... | Proof. Let \( \left( {x}^{i}\right) \) be normal coordinates on \( U \) . Given \( q = \left( {{q}^{1},\ldots ,{q}^{n}}\right) \in U \smallsetminus \{ p\} \) and \( w = {\left. {w}^{i}{\partial }_{i}\right| }_{q} \in {T}_{q}M \), the curve \( \gamma \left( t\right) = \left( {t{q}^{1},\ldots, t{q}^{n}}\right) \) is a ra... | Yes |
Proposition 10.12 (Jacobi Fields in Constant Curvature). Suppose \( \left( {M, g}\right) \) is a Riemannian manifold with constant sectional curvature \( c \), and \( \gamma \) is a unit-speed geodesic in \( M \) . The normal Jacobi fields along \( \gamma \) vanishing at \( t = 0 \) are the vector fields of the form\n\... | Proof. Since \( g \) has constant curvature, its curvature endomorphism is given by the formula of Proposition 8.36:\n\n\[ R\left( {v, w}\right) x = c\left( {\langle w, x\rangle v-\langle v, x\rangle w}\right) . \]\n\nSubstituting this into the Jacobi equation, we find that a normal Jacobi field \( J \) satisfies\n\n\[... | Yes |
Lemma 10.13. On \( {\mathbb{R}}^{n} \smallsetminus \{ 0\} \), the metric \( \widehat{g} \) defined by (10.14) and the Euclidean metric \( \bar{g} \) are related by\n\n\[ \bar{g} = d{r}^{2} + {r}^{2}\widehat{g} \]\n\nwhere \( r\left( x\right) = \left| x\right| \) is the Euclidean distance from the origin. | Proof. Example 2.24 observed that the map \( \Phi : {\mathbb{R}}^{ + }{ \times }_{\rho }{\mathbb{S}}^{n - 1} \rightarrow {\mathbb{R}}^{n} \smallsetminus \{ 0\} \) given by\n\n\[ \Phi \left( {\rho ,\omega }\right) = {\rho \omega } \]\n\n(10.16)\n\nis an isometry when \( {\mathbb{R}}^{ + }{ \times }_{\rho }{\mathbb{S}}^{... | Yes |
Theorem 10.14 (Constant-Curvature Metrics in Normal Coordinates). Suppose \( \left( {M, g}\right) \) is a Riemannian manifold with constant sectional curvature \( c \) . Given \( p \in M \) , let \( \left( {x}^{i}\right) \) be normal coordinates on a normal neighborhood \( U \) of \( p \) ; let \( r \) be the radial di... | Proof. Let \( \bar{g} \) denote the Euclidean metric in \( x \) -coordinates, and let \( {g}_{c} \) denote the metric defined by the formula on the right-hand side of (10.17). By the properties of normal coordinates, at points of \( U \smallsetminus \{ p\} \), all three metrics \( g,\bar{g} \), and \( {g}_{c} \) make t... | Yes |
Corollary 10.15 (Local Uniqueness of Constant-Curvature Metrics). Let \( \left( {M, g}\right) \) and \( \left( {\widetilde{M},\widetilde{g}}\right) \) be Riemannian manifolds of the same dimension with constant sectional curvature \( c \) . For all points \( p \in M,\widetilde{p} \in \widetilde{M} \), there exist neigh... | Proof. Choose \( p \in M \) and \( \widetilde{p} \in \widetilde{M} \), and let \( U \) and \( \widetilde{U} \) be geodesic balls of small radius \( \varepsilon \) around \( p \) and \( \widetilde{p} \), respectively. Riemannian normal coordinates give maps \( \psi : U \rightarrow {B}_{\varepsilon }\left( 0\right) \subs... | Yes |
Corollary 10.16 (Constant-Curvature Metrics as Warped Products). Suppose \( \left( {M, g}\right) \) is a Riemannian manifold with constant sectional curvature \( c \), and \( U \) is a geodesic ball of radius \( b \) centered at \( p \in M \) . Then \( U \smallsetminus \{ p\} \) is isometric to a warped product of the ... | Proof. By virtue of Theorem 10.14, we may consider \( g \) to be a metric on the ball of radius \( b \) in \( {\mathbb{R}}^{n} \) given by formula (10.17). Let \( \Phi : \left( {0, b}\right) \times {\mathbb{S}}^{n - 1} \rightarrow U \smallsetminus \{ p\} \) and \( \pi : {\mathbb{R}}^{n} \smallsetminus \{ 0\} \rightarro... | Yes |
Corollary 10.17 (Polar Decomposition of Integrals). Suppose \( \\left( {M, g}\\right) \) is a Riemannian manifold with constant sectional curvature \( c \), and \( U \) is an open or closed geodesic ball of radius \( b \) around a point \( p \\in M \) . If \( f : U \\rightarrow \\mathbb{R} \) is any bounded integrable ... | Proof. Because every geodesic ball is orientable, we might as well choose an orientation on \( U \) and interpret \( d{V}_{g} \) as a differential form. Since the boundary of a geodesic ball has measure zero, it does not matter whether \( U \) is open or closed. Similarly, integrating over \( U \\smallsetminus \\{ p\\}... | Yes |
Theorem 10.19 (Characterization of Locally Symmetric Spaces). A Riemannian manifold is a locally symmetric space if and only if its curvature tensor is parallel. | Proof. One direction is taken care of by Problem 7-3. To prove the converse, suppose \( \left( {M, g}\right) \) is a Riemannian manifold with \( \nabla {Rm} \equiv 0 \) . Let \( p \in M \) be arbitrary, and let \( U \) be a geodesic ball centered at \( p \) . The linear map \( A = - \operatorname{Id:}{T}_{p}M \rightarr... | Yes |
Proposition 10.20. Suppose \( \left( {M, g}\right) \) is a Riemannian or pseudo-Riemannian manifold, \( p \in M \), and \( v \in {\mathcal{E}}_{p} \subseteq {T}_{p}M \) . Let \( \gamma = {\gamma }_{v} : \left\lbrack {0,1}\right\rbrack \rightarrow M \) be the geodesic segment \( \gamma \left( t\right) = {\exp }_{p}\left... | Proof. Suppose first that \( v \) is a critical point of \( {\exp }_{p} \) . Then there is a nonzero vector \( w \in {T}_{v}\left( {{T}_{p}M}\right) \) such that \( d{\left( {\exp }_{p}\right) }_{v}\left( w\right) = 0 \) . Because \( {T}_{p}M \) is a vector space, we can identify \( {T}_{v}\left( {{T}_{p}M}\right) \) w... | Yes |
Proposition 10.24. Let \( \\left( {M, g}\\right) \) be a Riemannian manifold and let \( \\gamma : \\left\\lbrack {a, b}\\right\\rbrack \\rightarrow M \) be a geodesic segment. For every pair of piecewise smooth normal vector fields \( V, W \) along \( \\gamma \) ,\n\n\[ I\\left( {V, W}\\right) = - {\\int }_{a}^{b}\\lef... | Proof. On every subinterval \( \\left\\lbrack {{a}_{i - 1},{a}_{i}}\\right\\rbrack \) where \( V \) and \( W \) are smooth,\n\n\[ \\frac{d}{dt}\\left\\langle {{D}_{t}V, W}\\right\\rangle = \\left\\langle {{D}_{t}^{2}V, W}\\right\\rangle + \\left\\langle {{D}_{t}V,{D}_{t}W}\\right\\rangle .\]\n\nThus, by the fundamental... | Yes |
If \( \gamma \) is a geodesic segment and \( V \) is a proper normal piecewise smooth vector field along \( \gamma \), then \( I\left( {V, W}\right) = 0 \) for every proper normal piecewise smooth vector field \( W \) along \( \gamma \) if and only if \( V \) is a Jacobi field. | Problem 10-11. | No |
Theorem 10.26. Let \( \left( {M, g}\right) \) be a Riemannian manifold and \( p, q \in M \) . If \( \gamma \) is a unit-speed geodesic segment from \( p \) to \( q \) that has an interior conjugate point, then there exists a proper normal vector field \( X \) along \( \gamma \) such that \( I\left( {X, X}\right) < 0 \)... | Proof. Suppose \( \gamma : \left\lbrack {a, c}\right\rbrack \rightarrow M \) is a unit-speed geodesic segment, and \( \gamma \left( b\right) \) is conjugate to \( \gamma \left( a\right) \) along \( \gamma \) for some \( a < b < c \) . This means that there is a nontrivial normal Jacobi field \( J \) along \( \gamma \) ... | Yes |
Lemma 10.27. Let \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow M \) be a geodesic segment, and suppose \( {J}_{1} \) and \( {J}_{2} \) are Jacobi fields along \( \gamma \) . Then \( \left\langle {{D}_{t}{J}_{1}\left( t\right) ,{J}_{2}\left( t\right) }\right\rangle - \left\langle {{J}_{1}\left( t\right) ,{D}_... | Proof. Let \( f\left( t\right) = \left\langle {{D}_{t}{J}_{1}\left( t\right) ,{J}_{2}\left( t\right) }\right\rangle - \left\langle {{J}_{1}\left( t\right) ,{D}_{t}{J}_{2}\left( t\right) }\right\rangle \) . Using the Jacobi equation, we compute\n\n\[ \n{f}^{\prime }\left( t\right) = \left\langle {{D}_{t}^{2}{J}_{1}\left... | Yes |
Theorem 10.34. Let \( \left( {M, g}\right) \) be a complete, connected Riemannian manifold and \( p \in M \) .\n\n(a) The cut locus of \( p \) is a closed subset of \( M \) of measure zero.\n\n(b) The restriction of \( {\exp }_{p} \) to \( \overline{\operatorname{ID}\left( p\right) } \) is surjective.\n\n(c) The restri... | Proof. To prove that the cut locus is closed, suppose \( \left( {q}_{i}\right) \) is a sequence of points in \( \operatorname{Cut}\left( p\right) \) converging to some \( q \in M \) . Write \( {q}_{i} = {\exp }_{p}\left( {{t}_{\text{cut }}\left( {p,{v}_{i}}\right) {v}_{i}}\right) \) for unit vectors \( {v}_{i} \) . By ... | Yes |
Every compact, connected, smooth n-manifold is homeomorphic to a quotient space of \( {\overline{\mathbb{B}}}^{n} \) by an equivalence relation that identifies only points on the boundary. | Let \( M \) be a compact, connected, smooth \( n \) -manifold, let \( p \) be any point of \( M \), and let \( g \) be any Riemannian metric on \( M \) . Because a compact metric space has finite diameter, every unit vector in \( {T}_{p}M \) has a finite cut time, no greater than the diameter of \( M \) . Let \( {\bar{... | Yes |
Proposition 10.36. Let \( \left( {M, g}\right) \) be a complete, connected Riemannian manifold. For each \( p \in M \), the injectivity radius at \( p \) is the distance from \( p \) to its cut locus if the cut locus is nonempty, and infinite otherwise. | Proof. Given \( p \in M \), let \( d \) denote the distance from \( p \) to its cut locus, with the convention that \( d = \infty \) if the cut locus is empty. Let \( a \in (0,\infty \rbrack \) be arbitrary, and let \( {B}_{a} \subseteq {T}_{p}M \) denote the set of vectors \( v \in {T}_{p}M \) with \( {\left| v\right|... | Yes |
Lemma 11.1. Let \( r,{\partial }_{r} \), and \( {\mathcal{H}}_{r} \) be defined as above.\n\n(a) \( {\mathcal{H}}_{r} \) is self-adjoint.\n\n(b) \( {\mathcal{H}}_{r}\left( {\partial }_{r}\right) \equiv 0 \) .\n\n(c) The restriction of \( {\mathcal{H}}_{r} \) to vectors tangent to a level set of \( r \) is equal to the ... | Proof. Since the covariant Hessian \( {\nabla }^{2}r \) is symmetric, equation (11.2) shows that the Hessian operator is self-adjoint. Part (b) follows immediately from the fact that \( {\mathcal{H}}_{r}\left( {\partial }_{r}\right) = {\nabla }_{{\partial }_{r}}{\partial }_{r} = 0 \) because the integral curves of \( {... | Yes |
Theorem 11.4 (The Riccati Equation). Let \( \left( {M, g}\right) \) be a Riemannian manifold; let \( U \) be a normal neighborhood of a point \( p \in M \) ; let \( r : U \rightarrow \mathbb{R} \) be the radial distance function; and let \( \gamma : \left\lbrack {0, b}\right\rbrack \rightarrow U \) be a unit-speed radi... | Proof. Let \( {t}_{0} \in (0, b\rbrack \) and \( w \in {T}_{\gamma \left( {t}_{0}\right) }M \) be arbitrary. We can decompose \( w \) as \( w = \) \( y + z \), where \( y \) is a multiple of \( {\partial }_{r} \) and \( z \) is tangent to a level set of \( r \) . Since (11.7) is an equation between linear operators, we... | Yes |
Theorem 11.5 (Riccati Comparison Theorem). Suppose \( \left( {M, g}\right) \) is a Riemannian manifold and \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow M \) is a unit-speed geodesic segment. Suppose \( \eta ,\widetilde{\eta } \) are self-adjoint endomorphism fields along \( {\left. \gamma \right| }_{(a, b\r... | To prove this theorem, we will express the endomorphism fields \( \eta ,\widetilde{\eta },\sigma \), and \( \widetilde{\sigma } \) in terms of a parallel orthonormal frame along \( \gamma \) . In this frame, they become symmetric matrix-valued functions, and then the Riccati equations for \( \eta \) and \( \widetilde{\... | Yes |
Corollary 11.8 (Principal Curvature Comparison). Suppose \( \left( {M, g}\right) \) is a Riemannian \( n \) -manifold, \( p \in M, U \) is a normal neighborhood of \( p, r \) is the radial distance function on \( U \), and \( {s}_{c} \) and \( {\pi }_{r} \) are defined as in Proposition 11.3.\n\n(a) If all sectional cu... | Proof. This follows immediately from the fact that the shape operator of each \( r \) - level set is the restriction of \( {\mathcal{H}}_{r} \) by Lemma 11.1(c). | No |
Theorem 11.9 (Jacobi Field Comparison). Suppose \( \left( {M, g}\right) \) is a Riemannian manifold, \( \gamma : \left\lbrack {0, b}\right\rbrack \rightarrow M \) is a unit-speed geodesic segment, and \( J \) is any normal Jacobi field along \( \gamma \) such that \( J\left( 0\right) = 0 \) . For each \( c \in \mathbb{... | Proof. If \( {D}_{t}J\left( 0\right) = 0 \), then \( J \) vanishes identically, so we may as well assume that \( {D}_{t}J\left( 0\right) \neq 0 \) . Let \( {b}_{0} \) be the largest time in \( (0, b\rbrack \) such that \( \gamma \) has no conjugate points in \( \left( {0,{b}_{0}}\right) \) and \( {s}_{c}\left( t\right)... | Yes |
Theorem 11.10 (Metric Comparison). Let \( \left( {M, g}\right) \) be a Riemannian manifold, and let \( \left( {U,\left( {x}^{i}\right) }\right) \) be any normal coordinate chart for \( g \) centered at \( p \in M \) . For each \( c \in \mathbb{R} \), let \( {g}_{c} \) denote the constant-curvature metric on \( U \small... | Proof. Let \( q \in U \smallsetminus \{ p\} \), satisfying the restriction \( {d}_{g}\left( {p, q}\right) < {\pi R} \) if we are in case (a) and \( c = 1/{R}^{2} > 0 \), but otherwise arbitrary; and let \( b = {d}_{g}\left( {p, q}\right) \) . Given \( w \in {T}_{q}M \) , we can decompose \( w \) as a sum \( w = y + z \... | Yes |
Theorem 11.11 (Laplacian Comparison I). Suppose \( \left( {M, g}\right) \) is a Riemannian \( n \) - manifold whose sectional curvatures are all bounded above by a constant \( c \) . Suppose \( p \in M, U \) is a normal neighborhood of \( p, r \) is the radial distance function on \( U \) , and \( {s}_{c} \) is defined... | Proof. By the result of Problem 5-14, \( {\Delta r} = {\operatorname{tr}}_{g}\left( {{\nabla }^{2}r}\right) = \operatorname{tr}\left( {\mathcal{H}}_{r}\right) \) . The result then follows from the Hessian comparison theorem, using the fact that \( \operatorname{tr}\left( {\pi }_{r}\right) = n - 1 \) , which can be veri... | Yes |
Theorem 11.12 (Conjugate Point Comparison I). Suppose \( \left( {M, g}\right) \) is a Riemannian \( n \) -manifold whose sectional curvatures are all bounded above by a constant \( c \) . If \( c \leq 0 \), then no point of \( M \) has conjugate points along any geodesic. If \( c = 1/{R}^{2} > 0 \) , then there is no c... | Proof. The case \( c \leq 0 \) is covered by Problem 10-7, so assume \( c = 1/{R}^{2} > 0 \) . Let \( \gamma : \left\lbrack {0, b}\right\rbrack \rightarrow M \) be a unit-speed geodesic segment, and suppose \( J \) is a nontrivial normal Jacobi field along \( \gamma \) that vanishes at \( t = 0 \) . The Jacobi field co... | Yes |
Lemma 11.13. Suppose \( \left( {M, g}\right) \) is a Riemannian manifold and \( \left( {x}^{i}\right) \) are Riemannian normal coordinates on a normal neighborhood \( U \) of \( p \in M \) . Let \( \det g \) denote the determinant of the matrix \( \left( {g}_{ij}\right) \) in these coordinates, let \( r \) be the radia... | Proof. Corollary 6.10 to the Gauss lemma shows that the vector fields grad \( r \) and \( {\partial }_{r} \) are equal on \( U \smallsetminus \{ p\} \) . Comparing the components of these two vector fields in normal coordinates, we conclude (using the summation convention as usual) that\n\n\[ \n{g}^{ij}{\partial }_{j}r... | Yes |
Theorem 11.14 (Günther’s Volume Comparison). Suppose \( \\left( {M, g}\\right) \) is a connected Riemannian n-manifold whose sectional curvatures are all bounded above by a constant \( c \) . Given \( p \\in M \), let \( {\\delta }_{0} = \\operatorname{inj}\\left( p\\right) \) if \( c \\leq 0 \), and \( {\\delta }_{0} ... | Proof. The volume estimate (11.24) follows easily from the metric comparison theorem, which implies that the determinants of the metrics \( g \) and \( {g}_{c} \) in normal coordinates satisfy \( \\sqrt{\\det g} \\geq \\sqrt{\\det {g}_{c}} \) . If that were all we needed, we could stop here; but to prove the other stat... | Yes |
Theorem 11.16 (Conjugate Point Comparison II). Let \( \left( {M, g}\right) \) be a Riemannian \( n \) - manifold, and suppose there is a positive constant \( c = 1/{R}^{2} \) such that the Ricci curvature of \( M \) satisfies \( {Rc}\left( {v, v}\right) \geq \left( {n - 1}\right) c \) for all unit vectors \( v \) . The... | Proof. Let \( U \) be a normal neighborhood of an arbitrary point \( p \in M \) . The second Laplacian comparison theorem (Thm. 11.15) combined with Lemma 11.13 shows\n\nthat\n\[{\partial }_{r}\log \left( {{r}^{n - 1}\sqrt{\det g}}\right) = {\Delta r} \leq \left( {n - 1}\right) \frac{{s}_{c}^{\prime }\left( r\right) }{... | Yes |
Corollary 11.17 (Injectivity Radius Comparison). Let \( \left( {M, g}\right) \) be a Riemannian \( n \) -manifold, and suppose there is a positive constant \( c = 1/{R}^{2} \) such that the Ricci curvature of \( M \) satisfies \( {Rc}\left( {v, v}\right) \geq \left( {n - 1}\right) c \) for all unit vectors \( v \) . Th... | Proof. Every radial geodesic segment in a geodesic ball is minimizing, but the preceding theorem shows that no geodesic segment of length \( {\pi R} \) or greater is minimizing. Thus no geodesic ball has radius greater than \( {\pi R} \) . | Yes |
Corollary 11.18 (Diameter Comparison). Let \( \left( {M, g}\right) \) be a complete, connected Riemannian n-manifold, and suppose there is a positive constant \( c = 1/{R}^{2} \) such that the Ricci curvature of \( M \) satisfies \( {Rc}\left( {v, v}\right) \geq \left( {n - 1}\right) c \) for all unit vectors \( v \) .... | Proof. This follows from the fact that any two points of \( M \) can be connected by a minimizing geodesic segment, and the conjugate point comparison theorem implies that no such segment can have length greater than \( {\pi R} \) . | Yes |
Theorem 11.19 (Bishop-Gromov Volume Comparison). Let \( \left( {M, g}\right) \) be a connected Riemannian n-manifold, and suppose there is a constant \( c \) such that the Ricci curvature of \( M \) satisfies \( \operatorname{Rc}\left( {v, v}\right) \geq \left( {n - 1}\right) c \) for all unit vectors \( v \) . Let \( ... | Proof. First consider \( \delta \leq \operatorname{inj}\left( p\right) \), in which case a metric ball of radius \( \delta \) in \( M \) is actually a geodesic ball. With the exception of the first and last paragraphs, the proof of Theorem 11.14 goes through with all of the inequalities reversed, and with the first Lap... | Yes |
Lemma 12.1 (Uniqueness of Analytic Continuation). With \( \left( {M, g}\right) ,\left( {\widehat{M},\widehat{g}}\right) \), and \( \varphi \) as above, if \( \left\{ \left( {{U}_{t},{\varphi }_{t}}\right) \right\} \) and \( \left\{ \left( {{U}_{t}^{\prime },{\varphi }_{t}^{\prime }}\right) \right\} \) are two analytic ... | Proof. Let \( \mathcal{T} \) be the set of all \( t \in \left\lbrack {0,1}\right\rbrack \) such that \( {\varphi }_{t} = {\varphi }_{t}^{\prime } \) on a neighborhood of \( \gamma \left( t\right) \) . Then \( 0 \in \mathcal{T} \), because both \( {\varphi }_{0} \) and \( {\varphi }_{0}^{\prime } \) agree with \( \varph... | Yes |
Theorem 12.2 (Monodromy Theorem for Local Isometries). Let \( \left( {M, g}\right) \) and \( \left( {\widehat{M},\widehat{g}}\right) \) be connected Riemannian manifolds. Suppose \( U \) is a neighborhood of a point \( p \in M \) and \( \varphi : U \rightarrow \widehat{M} \) is a local isometry that can be analytically... | Proof. Suppose \( {\gamma }_{0} \) and \( {\gamma }_{1} \) are path-homotopic, and let \( H : \left\lbrack {0,1}\right\rbrack \times \left\lbrack {0,1}\right\rbrack \rightarrow M \) be a path homotopy from \( {\gamma }_{0} \) to \( {\gamma }_{1} \) . Write \( {H}_{s}\left( t\right) = H\left( {t, s}\right) \), so that \... | Yes |
Corollary 12.3. Let \( \left( {M, g}\right) \) and \( \left( {\widehat{M},\widehat{g}}\right) \) be simply connected, complete Riemannian manifolds. Suppose \( U \) is a connected neighborhood of a point \( p \in M \) and \( \varphi : U \rightarrow \) \( \widehat{M} \) is a local isometry that can be analytically conti... | Proof. Let \( q \in M \) be arbitrary. We wish to define \( \Phi \left( q\right) \) to be the value of an analytic continuation of \( \varphi \) along a path from \( p \) to \( q \) . The fact that \( M \) is simply connected means that all paths from \( p \) to \( q \) are path-homotopic, so the monodromy theorem impl... | Yes |
Corollary 12.5 (Characterization of Constant-Curvature Manifolds). The complete, connected, n-dimensional Riemannian manifolds of constant sectional curvature are, up to isometry, exactly the Riemannian quotients of the form \( \widetilde{M}/\Gamma \), where \( \widetilde{M} \) is one of the constant-curvature model sp... | Proof. First suppose \( \left( {M, g}\right) \) is a complete, connected Riemannian \( n \) -manifold with constant sectional curvature, and let \( \pi : \widetilde{M} \rightarrow M \) be its universal covering manifold with the pullback metric \( \widetilde{g} = {\pi }^{ * }g \) . The preceding theorem shows that \( \... | Yes |
Theorem 12.6. Suppose \( \left( {M, g}\right) \) is a complete, simply connected Riemannian manifold with parallel Riemann curvature tensor. Then \( M \) is a symmetric space. | Proof. Let \( p \) be an arbitrary point of \( M \) . Theorem 10.19 shows that \( M \) is locally symmetric, so there is a point reflection \( \varphi : U \rightarrow U \) defined on some connected neighborhood \( U \) of \( p \) . If we can show that \( \varphi \) can be analytically continued along every path startin... | Yes |
Corollary 12.7. Suppose \( \left( {M, g}\right) \) is a complete, connected Riemannian manifold. The following are equivalent:\n\n(a) \( M \) has parallel curvature tensor.\n\n(b) \( M \) is a locally symmetric space.\n\n(c) \( M \) is isometric to a Riemannian quotient \( \widetilde{M}/\Gamma \), where \( \widetilde{M... | Proof. Theorem 10.19 shows that (a) \( \Leftrightarrow \) (b). To show that (a) \( \Rightarrow \) (c), assume that \( M \) has parallel curvature tensor, and let \( \left( {\widetilde{M},\widetilde{g}}\right) \) be its universal covering manifold with the pullback metric. Since the covering map \( \pi : \widetilde{M} \... | Yes |
Theorem 12.8 (Cartan-Hadamard). If \( \left( {M, g}\right) \) is a complete, connected Riemannian manifold with nonpositive sectional curvature, then for every point \( p \in M \), the map \( {\exp }_{p} : {T}_{p}M \rightarrow M \) is a smooth covering map. Thus the universal covering space of \( M \) is diffeomorphic ... | Proof. By Theorem 11.12, the assumption of nonpositive curvature guarantees that \( p \) has no conjugate points along any geodesic. Therefore, by Proposition 10.20, \( {\exp }_{p} \) is a local diffeomorphism on all of \( {T}_{p}M \) . Let \( \widetilde{g} \) be the (variable-coefficient) 2-tensor field \( {\exp }_{p}... | Yes |
Proposition 12.10. Suppose \( \bigtriangleup {ABC} \) is a geodesic triangle in a Cartan-Hadamard manifold \( \left( {M, g}\right) \), and let \( a, b, c \) denote the lengths of the segments opposite the vertices \( A, B \), and \( C \), respectively. The following inequalities hold:\n\n(a) \( {c}^{2} \geq {a}^{2} + {... | Proof. Problem 12-7. | No |
Corollary 12.12. Suppose \( M \) and \( N \) are positive-dimensional compact, connected smooth manifolds, at least one of which is simply connected. Then \( M \times N \) does not admit any Riemannian metric of nonpositive sectional curvature. | Proof. Suppose for the sake of contradiction that \( M \) is simply connected and \( g \) is a metric on \( M \times N \) with nonpositive sectional curvature. If \( \widetilde{N} \) is the universal covering manifold of \( N \), then there is a universal covering map \( \pi : M \times \widetilde{N} \rightarrow M \time... | Yes |
Corollary 12.14. If \( M \) is a smooth manifold that admits a metric of nonpositive sectional curvature, then \( M \) is aspherical. | Proof. Suppose \( M \) admits a nonpositively curved metric, so its universal covering space is diffeomorphic to \( {\mathbb{R}}^{n} \) by Cartan-Hadamard. Since \( {\mathbb{R}}^{n} \) is contractible, it is aspherical. Since covering maps induce isomorphisms on \( {\pi }_{k} \) for \( k > 1 \) (see [Hat02, Prop. 4.1])... | Yes |
Lemma 12.15. Suppose \( \left( {M, g}\right) \) is a Cartan-Hadamard manifold. Given \( q \in M \), let \( f : M \rightarrow \lbrack 0,\infty ) \) be the function \( f\left( x\right) = \frac{1}{2}{d}_{g}{\left( x, q\right) }^{2} \). Then \( f \) is strictly geodesically convex, in the sense that for every geodesic segm... | Proof. We can write \( f\left( x\right) = \frac{1}{2}r{\left( x\right) }^{2} \), where \( r \) is the radial distance function from \( q \) with respect to any normal coordinates, and Proposition 12.9 shows that \( f \) is smooth on all of \( M \). The Hessian comparison theorem implies that \( {\mathcal{H}}_{r} \geq \... | Yes |
Lemma 12.16. Suppose \( \left( {M, g}\right) \) is a Cartan-Hadamard manifold and \( S \) is a compact subset of \( M \) containing more than one point. Then there is a unique closed ball of minimum radius containing \( S \) . | Proof. Because \( S \) is compact, it is bounded. Let \( \delta = \operatorname{diam}\left( S\right) \), and let \( {q}_{0} \) be any point of \( S \), so that \( S \subseteq {\bar{B}}_{\delta }\left( {q}_{0}\right) \) . Let \( {c}_{0} \) be the infimum of the radii of closed metric balls containing \( S \), and let \(... | Yes |
Theorem 12.17 (Cartan’s Fixed-Point Theorem). Suppose \( \left( {M, g}\right) \) is a Cartan-Hadamard manifold and \( G \) is a compact Lie group acting smoothly and isometrically on \( M \) . Then \( G \) has a fixed point in \( M \), that is, a point \( {p}_{0} \in M \) such that \( \varphi \cdot {p}_{0} = {p}_{0} \)... | Proof. Let \( {q}_{0} \in M \) be arbitrary, and let \( S = G \cdot {q}_{0} \) be the orbit of \( {q}_{0} \) . If \( S \) contains only the point \( {q}_{0} \), then \( {q}_{0} \) is a fixed point, so we may assume that the orbit contains at least two points. Because \( S \) is the image of the continuous map from \( G... | Yes |
Corollary 12.18 (Cartan’s Torsion Theorem). Suppose \( \left( {M, g}\right) \) is a complete, connected Riemannian manifold with nonpositive sectional curvature. Then \( {\pi }_{1}\left( M\right) \) is torsion-free. | Proof. Let \( \left( {\widetilde{M},\widetilde{g}}\right) \) be the universal covering manifold of \( M \) with the pullback metric. Then \( \left( {\widetilde{M},\widetilde{g}}\right) \) is a Cartan-Hadamard manifold, and \( M \) is isometric to a Riemannian quotient \( \widetilde{M}/\Gamma \), where \( \Gamma \) is s... | Yes |
Lemma 12.21. Suppose \( \left( {M, g}\right) \) is a compact, connected Riemannian manifold, and \( \pi : \widetilde{M} \rightarrow M \) is its universal covering manifold endowed with the metric \( \widetilde{g} = {\pi }^{ * }g \) . Then every covering automorphism of \( \pi \) has an axis, which restricts to a lift o... | Proof. Let \( \varphi \) be any nontrivial covering automorphism of \( \pi \), so it is also an isometry of \( \left( {\widetilde{M},\widetilde{g}}\right) \) . Every continuous path \( \widetilde{\sigma } : \left\lbrack {0,1}\right\rbrack \rightarrow \widetilde{M} \) from a point \( \widetilde{x} \in \widetilde{M} \) t... | No |
Lemma 12.22. Suppose \( \left( {M, g}\right) \) is a Cartan-Hadamard manifold with strictly negative sectional curvature. If \( \varphi : M \rightarrow M \) is an axial isometry, then its axis is unique up to reparametrization. | Proof. Let \( \varphi : M \rightarrow M \) be an axial isometry, and suppose \( {\gamma }_{1},{\gamma }_{2} : \mathbb{R} \rightarrow M \) are both axes for \( \varphi \) . After reparametrizing the geodesics if necessary, we can assume that \( \varphi \) translates both geodesics by time 1 .\n\nSuppose first that the t... | Yes |
Corollary 12.23. No product of positive-dimensional connected compact manifolds admits a metric of strictly negative sectional curvature. | Proof. Problem 12-8. | No |
Theorem 12.24 (Myers). Let \( \left( {M, g}\right) \) be a complete, connected Riemannian \( n \) - manifold, and suppose there is a positive constant \( R \) such that the Ricci curvature of \( M \) satisfies \( \operatorname{Rc}\left( {v, v}\right) \geq \left( {n - 1}\right) /{R}^{2} \) for all unit vectors \( v \) .... | Proof. The diameter comparison theorem (Cor. 11.18) shows that the diameter of \( M \) is no greater than \( {\pi R} \) . To show that \( M \) is compact, we just choose a base point \( p \) and note that every point in \( M \) can be connected to \( p \) by a geodesic segment of length at most \( {\pi R} \) . Therefor... | Yes |
Corollary 12.25. Suppose \( \left( {M, g}\right) \) is a compact, connected Riemannian \( n \) -manifold whose Ricci tensor is positive definite everywhere. Then \( M \) has finite fundamental group. | Proof. Because the unit tangent bundle of \( M \) is compact by Proposition 2.9, the hypothesis implies that there is a positive constant \( c \) such that \( {Rc}\left( {v, v}\right) \geq c \) for all unit tangent vectors \( v \) . Thus the hypotheses of Myers’s theorem are satisfied with \( \left( {n - 1}\right) /{R}... | Yes |
Corollary 12.27. If \( \left( {M, g}\right) \) is a complete Einstein manifold with positive scalar curvature, then \( M \) is compact. | Proof. If \( g \) is Einstein with positive scalar curvature, then \( {Rc} = \frac{1}{n}{Sg} \) satisfies the hypotheses of Myers’s theorem with \( \left( {n - 1}\right) /{R}^{2} = \frac{1}{n}S \) . | Yes |
Theorem 12.29 (Milnor). Suppose \( \left( {M, g}\right) \) is a complete, connected Riemannian \( n \) -manifold with nonnegative Ricci curvature. Every finitely generated subgroup of \( {\pi }_{1}\left( M\right) \) has polynomial growth of order at most \( n \) . | Proof. Problem 12-11. | No |
Corollary 12.33. Suppose \( \left( {M, g}\right) \) is a compact, connected, even-dimensional Riemannian manifold with strictly positive sectional curvature. If \( M \) is orientable, then \( {\pi }_{1}\left( M\right) \) is trivial, and if not, then \( {\pi }_{1}\left( M\right) \cong \mathbb{Z}/2 \) . | Proof. The orientable case is part of Synge’s theorem. If \( M \) is nonorientable, it has a two-sheeted orientable covering manifold \( \widetilde{M} \) ; Synge’s theorem shows that \( \widetilde{M} \) is simply connected, so it is the universal covering space of \( M \) . Because each fiber of the universal covering ... | Yes |
Theorem 12.36 (The Splitting Theorem). If (M, g) is a complete, connected, non-compact Riemannian manifold with nonnegative Ricci curvature, and \( M \) contains a line, then there is a Riemannian manifold \( \left( {N, h}\right) \) with nonnegative Ricci curvature such that \( M \) is isometric to the Riemannian produ... | The proofs of all three of the above theorems, which can be found, for example, in [Pet16], are elaborate applications of comparison theory. | No |
Theorem 12.39 (Böhm-Wilking). Every simply connected compact Riemannian manifold with positive curvature operator is diffeomorphic to a sphere. | The method of proof of these three sphere theorems was to start with an initial metric \( {g}_{0} \), and then define a one-parameter family of metrics \( \left\{ {{g}_{t} : t \geq 0}\right\} \) evolving according to the following partial differential equation, called the Ricci flow:\n\n\[ \frac{\partial }{\partial t}{... | Yes |
Theorem 12.40 (Differentiable Sphere Theorem). Let \( \left( {M, g}\right) \) be a compact, simply connected Riemannian manifold of dimension \( n \geq 4 \) . If \( M \) is strictly pointwise \( \frac{1}{4} \) -pinched, then it is diffeomorphic to \( {\mathbb{S}}^{n} \) with its standard smooth structure. | For a nice exposition of the proof, see the recent book [Bre10]. | No |
Theorem 1.2 (Topological Invariance of Dimension). A nonempty n-dimensional topological manifold cannot be homeomorphic to an \( m \) -dimensional manifold unless \( m = n \) . | For the proof, see Theorem 17.26. In Chapter 2, we will also prove a related but weaker theorem (diffeomorphism invariance of dimension, Theorem 2.17). See also [LeeTM, Chap. 13] for a different proof of Theorem 1.2 using singular homology theory. | No |
Let \( U \subseteq {\mathbb{R}}^{n} \) be an open subset, and let \( f : U \rightarrow {\mathbb{R}}^{k} \) be a continuous function. The graph of \( f \) is the subset of \( {\mathbb{R}}^{n} \times {\mathbb{R}}^{k} \) defined by\n\n\[ \Gamma \left( f\right) = \left\{ {\left( {x, y}\right) \in {\mathbb{R}}^{n} \times {\... | Because \( \varphi \) is the restriction of a continuous map, it is continuous; and it is a homeomorphism because it has a continuous inverse given by \( {\varphi }^{-1}\left( x\right) = \left( {x, f\left( x\right) }\right) \) . Thus \( \Gamma \left( f\right) \) is a topological manifold of dimension \( n \) . In fact,... | Yes |
For each integer \( n \geq 0 \), the unit \( n \)-sphere \( {\mathbb{S}}^{n} \) is Hausdorff and second-countable because it is a topological subspace of \( {\mathbb{R}}^{n + 1} \). To show that it is locally Euclidean, for each index \( i = 1,\ldots, n + 1 \) let \( {U}_{i}^{ + } \) denote the subset of \( {\mathbb{R}... | Let \( f : {\mathbb{B}}^{n} \rightarrow \mathbb{R} \) be the continuous function\n\n\[ f\left( u\right) = \sqrt{1 - {\left| u\right| }^{2}} \]\n\nThen for each \( i = 1,\ldots, n + 1 \), it is easy to check that \( {U}_{i}^{ + } \cap {\mathbb{S}}^{n} \) is the graph of the function\n\n\[ {x}^{i} = f\left( {{x}^{1},\ldo... | Yes |
The \( n \)-dimensional real projective space, denoted by \( {\mathbb{{RP}}}^{n} \) (or sometimes just \( {\mathbb{P}}^{n} \) ), is defined as the set of 1-dimensional linear subspaces of \( {\mathbb{R}}^{n + 1} \), with the quotient topology determined by the natural map \( \pi : {\mathbb{R}}^{n + 1} \smallsetminus \{... | For each \( i = 1,\ldots, n + 1 \), let \( {\widetilde{U}}_{i} \subseteq {\mathbb{R}}^{n + 1} \smallsetminus \{ 0\} \) be the set where \( {x}^{i} \neq 0 \) , and let \( {U}_{i} = \pi \left( {\widetilde{U}}_{i}\right) \subseteq {\mathbb{{RP}}}^{n} \) . Since \( {\widetilde{U}}_{i} \) is a saturated open subset, \( {U}_... | Yes |
Example 1.8 (Product Manifolds). Suppose \( {M}_{1},\ldots ,{M}_{k} \) are topological manifolds of dimensions \( {n}_{1},\ldots ,{n}_{k} \), respectively. The product space \( {M}_{1} \times \cdots \times {M}_{k} \) is shown to be a topological manifold of dimension \( {n}_{1} + \cdots + {n}_{k} \) as follows. | It is Hausdorff and second-countable by Propositions A. 17 and A.23, so only the locally Euclidean property needs to be checked. Given any point \( \left( {{p}_{1},\ldots ,{p}_{k}}\right) \in \) \( {M}_{1} \times \cdots \times {M}_{k} \), we can choose a coordinate chart \( \left( {{U}_{i},{\varphi }_{i}}\right) \) for... | Yes |
Lemma 1.10. Every topological manifold has a countable basis of precompact coordinate balls. | Proof. Let \( M \) be a topological \( n \) -manifold. First we consider the special case in which \( M \) can be covered by a single chart. Suppose \( \varphi : M \rightarrow \widehat{U} \subseteq {\mathbb{R}}^{n} \) is a global coordinate map, and let \( \mathcal{B} \) be the collection of all open balls \( {B}_{r}\l... | Yes |
Proposition 1.11. Let \( M \) be a topological manifold.\n\n(a) \( M \) is locally path-connected.\n\n(b) \( M \) is connected if and only if it is path-connected.\n\n(c) The components of \( M \) are the same as its path components.\n\n(d) \( M \) has countably many components, each of which is an open subset of \( M ... | Proof. Since each coordinate ball is path-connected, (a) follows from the fact that \( M \) has a basis of coordinate balls. Parts (b) and (c) are immediate consequences of (a) and Proposition A.43. To prove (d), note that each component is open in \( M \) by Proposition A.43, so the collection of components is an open... | Yes |
Proposition 1.12 (Manifolds Are Locally Compact). Every topological manifold is locally compact. | Proof. Lemma 1.10 showed that every manifold has a basis of precompact open subsets. | No |
Lemma 1.13. Suppose \( X \) is a locally finite collection of subsets of a topological space \( M \) .\n\n(a) The collection \( \{ \bar{X} : X \in \mathcal{X}\} \) is also locally finite.\n\n(b) \( \overline{\mathop{\bigcup }\limits_{{X \in X}}X} = \mathop{\bigcup }\limits_{{X \in X}}\bar{X} \) . | Exercise 1.14. Prove the preceding lemma. | No |
Theorem 1.15 (Manifolds Are Paracompact). Every topological manifold is paracompact. In fact, given a topological manifold \( M \), an open cover \( X \) of \( M \) , and any basis \( \mathcal{B} \) for the topology of \( M \), there exists a countable, locally finite open refinement of \( X \) consisting of elements o... | Proof. Given \( M, X \), and \( \mathcal{B} \) as in the hypothesis of the theorem, let \( {\left( {K}_{j}\right) }_{j = 1}^{\infty } \) be an exhaustion of \( M \) by compact sets (Proposition A.60). For each \( j \), let \( {V}_{j} \doteq {\bar{K}}_{j + 1} \smallsetminus \) Int \( {K}_{j} \) and \( {W}_{j} = \operato... | Yes |
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