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Proposition 3.12. Let \( G \) be a Lie group and \( \mathfrak{g} \) its Lie algebra. Suppose \( g \) is a left-invariant Riemannian metric on \( G \), and let \( \langle \cdot , \cdot \rangle \) denote the corresponding inner product on \( \mathfrak{g} \) as in Lemma 3.10. Then \( g \) is bi-invariant if and only if \(... | Proof. We begin the proof with some preliminary computations. Suppose \( g \) is left-invariant and \( \langle \cdot , \cdot \rangle \) is the associated inner product on \( \mathfrak{g} \) . Let \( \varphi \in G \) be arbitrary, and note that \( {C}_{\varphi } \) is the composition of left multiplication by \( \varphi... | Yes |
Lemma 3.13. Suppose \( V \) is a finite-dimensional real vector space and \( H \) is a subgroup of \( \mathrm{{GL}}\left( V\right) \) . There exists an \( H \) -invariant inner product on \( V \) if and only if \( H \) has compact closure in \( \mathrm{{GL}}\left( V\right) \) . | Proof. Assume first that there exists an \( H \) -invariant inner product \( \langle \cdot , \cdot \rangle \) on \( V \) . This implies that \( H \) is contained in the subgroup \( \mathrm{O}\left( V\right) \subseteq \mathrm{{GL}}\left( V\right) \) consisting of linear isomorphisms of \( V \) that are orthogonal with r... | Yes |
Theorem 3.14 (Existence of Bi-Invariant Metrics). Let \( G \) be a Lie group and \( \mathfrak{g} \) its Lie algebra. Then \( G \) admits a bi-invariant metric if and only if \( \operatorname{Ad}\left( G\right) \) has compact closure in \( \mathrm{{GL}}\left( \mathrm{g}\right) \) . | Proof. Proposition 3.12 shows that there is a bi-invariant metric on \( G \) if and only if there is an \( \operatorname{Ad}\left( G\right) \) -invariant inner product on \( \mathfrak{g} \), and Lemma 3.13 in turn shows that the latter is true if and only if \( \operatorname{Ad}\left( G\right) \) has compact closure in... | Yes |
Corollary 3.15. Every compact Lie group admits a bi-invariant Riemannian metric. | Proof. If \( G \) is compact, then \( \operatorname{Ad}\left( G\right) \) is a compact subgroup of \( \mathrm{{GL}}\left( \mathrm{g}\right) \) because Ad: \( G \rightarrow \mathrm{{GL}}\left( \mathrm{g}\right) \) is continuous. | Yes |
Theorem 3.22 (Uniformization of Compact Surfaces). Every compact, connected, smooth 2-manifold admits a locally frame-homogeneous Riemannian metric, and a Riemannian covering by the Euclidean plane, hyperbolic plane, or round unit sphere. | Sketch of proof. The proof relies on the topological classification of compact surfaces (see, for example, [LeeTM, Thms. 6.15 and 10.22]), which says that every connected compact surface is homeomorphic to a sphere, a connected sum of one or more tori, or a connected sum of one or more projective planes. The crux of th... | No |
Lemma 3.23. If \( \left( {M, g}\right) \) is a connected homogeneous Riemannian manifold that possesses a point reflection at one point, then it is symmetric. | Proof. Suppose \( \left( {M, g}\right) \) satisfies the hypothesis, and let \( \varphi : M \rightarrow M \) be a point reflection at \( p \in M \) . Given any other point \( q \in M \), by homogeneity there is an isometry \( \psi : M \rightarrow M \) satisfying \( \psi \left( p\right) = q \) . Then \( \widetilde{\varph... | Yes |
Theorem 3.25. For all \( r, s \), and \( R \) as above, \( {\mathbb{S}}^{r, s}\left( R\right) \) and \( {\mathbb{H}}^{r, s}\left( R\right) \) are pseudo-Riemannian manifolds of signature \( \left( {r, s}\right) \) . | ## Proof. Problem 3-22. | No |
Theorem 3.26. All pseudo-Euclidean spaces, pseudospheres, and pseudohyperbolic spaces are frame-homogeneous. | Proof. Problem 3-23. | No |
Lemma 4.1 (Locality). Suppose \( \nabla \) is a connection in a smooth vector bundle \( E \rightarrow \) \( M \) . For every \( X \in \mathfrak{X}\left( M\right), Y \in \Gamma \left( E\right) \), and \( p \in M \), the covariant derivative \( {\left. {\nabla }_{X}Y\right| }_{p} \) depends only on the values of \( X \) ... | Proof. First consider \( Y \) . Replacing \( Y \) by \( Y - \widetilde{Y} \) shows that it suffices to prove \( {\left. {\nabla }_{X}Y\right| }_{p} = 0 \) if \( Y \) vanishes on a neighborhood of \( p \) .\n\nThus suppose \( Y \) is a smooth section of \( E \) that is identically zero on a neighborhood \( U \) of \( p ... | Yes |
Proposition 4.3 (Restriction of a Connection). Suppose \( \nabla \) is a connection in a smooth vector bundle \( E \rightarrow M \). For every open subset \( U \subseteq M \), there is a unique connection \( {\nabla }^{U} \) on the restricted bundle \( {\left. E\right| }_{U} \) that satisfies the following relation for... | Proof. First we prove uniqueness. Suppose \( {\nabla }^{U} \) is any such connection and \( X \in \) \( \mathcal{X}\left( U\right) \) and \( Y \in \Gamma \left( {\left. E\right| }_{U}\right) \) are arbitrary. Given \( p \in U \), we can use a bump function to construct a smooth vector field \( \widetilde{X} \in \mathca... | No |
Proposition 4.5. Under the hypotheses of Lemma 4.1, \( {\left. {\nabla }_{X}Y\right| }_{p} \) depends only on the values of \( Y \) in a neighborhood of \( p \) and the value of \( X \) at \( p \) . | Proof. The claim about \( Y \) was proved in Lemma 4.1. To prove the claim about \( X \) , it suffices by linearity to assume that \( {X}_{p} = 0 \) and show that \( {\left. {\nabla }_{X}Y\right| }_{p} = 0 \) . Choose a coordinate neighborhood \( U \) of \( p \), and write \( X = {X}^{i}{\partial }_{i} \) in coordinate... | Yes |
Proposition 4.6. Let \( M \) be a smooth manifold with or without boundary, and let \( \nabla \) be a connection in \( {TM} \). Suppose \( \left( {E}_{i}\right) \) is a smooth local frame over an open subset \( U \subseteq M \), and let \( \left\{ {\Gamma }_{ij}^{k}\right\} \) be the connection coefficients of \( \nabl... | Proof. Just use the defining properties of a connection and compute: \[ {\nabla }_{X}Y = {\nabla }_{X}\left( {{Y}^{j}{E}_{j}}\right) \] \[ = X\left( {Y}^{j}\right) {E}_{j} + {Y}^{j}{\nabla }_{{X}^{i}{E}_{i}}{E}_{j} \] \[ = X\left( {Y}^{j}\right) {E}_{j} + {X}^{i}{Y}^{j}{\nabla }_{{E}_{i}}{E}_{j} \] \[ = X\left( {Y}^{j}... | Yes |
Proposition 4.7 (Transformation Law for Connection Coefficients). Let \( M \) be a smooth manifold with or without boundary, and let \( \nabla \) be a connection in TM. Suppose we are given two smooth local frames \( \left( {E}_{i}\right) \) and \( \left( {\widetilde{E}}_{j}\right) \) for TM on an open subset \( U \sub... | Proof. Problem 4-3. | No |
Example 4.9 (The Tangential Connection on a Submanifold of \( {\mathbb{R}}^{n} \) ). Let \( M \subseteq {\mathbb{R}}^{n} \) be an embedded submanifold. Define a connection \( {\nabla }^{\top } \) on \( {TM} \), called the tangential connection, by setting\n\n\[ \n{\nabla }_{X}^{\top }Y = {\pi }^{\top }\left( {\left. {\... | It is immediate from the definition that \( {\nabla }_{X}^{\top }Y \) is linear over \( {C}^{\infty }\left( M\right) \) in \( X \) and over \( \mathbb{R} \) in \( Y \), so to show that \( {\nabla }^{\top } \) is a connection, only the product rule needs to be checked. Let \( f \in {C}^{\infty }\left( M\right) \), and l... | Yes |
Lemma 4.10. Suppose \( M \) is a smooth \( n \) -manifold with or without boundary, and \( M \) admits a global frame \( \left( {E}_{i}\right) \) . Formula (4.9) gives a one-to-one correspondence between connections in TM and choices of \( {n}^{3} \) smooth real-valued functions \( \left\{ {\Gamma }_{ij}^{k}\right\} \)... | Proof. Every connection determines functions \( \left\{ {\Gamma }_{ij}^{k}\right\} \) by (4.8), and Proposition 4.6 shows that those functions satisfy (4.9). On the other hand, given \( \left\{ {\Gamma }_{ij}^{k}\right\} \), we can define \( {\nabla }_{X}Y \) by (4.9); it is easy to see that the resulting expression is... | No |
Proposition 4.12. The tangent bundle of every smooth manifold with or without boundary admits a connection. | Proof. Let \( M \) be a smooth manifold with or without boundary, and cover \( M \) with coordinate charts \( \left\{ {U}_{\alpha }\right\} \) ; the preceding lemma guarantees the existence of a connection \( {\nabla }^{\alpha } \) on each \( {U}_{\alpha } \) . Choose a partition of unity \( \left\{ {\varphi }_{\alpha ... | Yes |
Proposition 4.13 (The Difference Tensor). Let \( M \) be a smooth manifold with or without boundary. For any two connections \( {\nabla }^{0} \) and \( {\nabla }^{1} \) in TM, define a map \( D : \mathcal{X}\left( M\right) \times \mathcal{X}\left( M\right) \rightarrow \mathcal{X}\left( M\right) \) by\n\n\[ D\left( {X, ... | Proof. It is immediate from the definition that \( D \) is linear over \( {C}^{\infty }\left( M\right) \) in its first argument, because both \( {\nabla }^{0} \) and \( {\nabla }^{1} \) are. To show that it is linear over \( {C}^{\infty }\left( M\right) \) in the second argument, expand \( D\left( {X,{fY}}\right) \) us... | Yes |
Theorem 4.14. Let \( M \) be a smooth manifold with or without boundary, and let \( {\nabla }^{0} \) be any connection in TM. Then the set \( \mathcal{A}\left( {TM}\right) \) of all connections in \( {TM} \) is equal to the following affine space:\n\n\[ \mathcal{A}\left( {TM}\right) = \left\{ {{\nabla }^{0} + D : D \in... | Proof. Problem 4-4. | No |
Proposition 4.16. Let \( M \) be a smooth manifold with or without boundary, and let \( \nabla \) be a connection in \( {TM} \). Suppose \( \left( {E}_{i}\right) \) is a local frame for \( M,\left( {\varepsilon }^{j}\right) \) is its dual coframe, and \( \left\{ {\Gamma }_{ij}^{k}\right\} \) are the connection coeffici... | Proof. Problem 4-5. | No |
Proposition 4.17 (The Total Covariant Derivative). Let \( M \) be a smooth manifold with or without boundary and let \( \nabla \) be a connection in TM. For every \( F \in \) \( \Gamma \left( {{T}^{\left( k, l\right) }{TM}}\right) \), the map \[ \nabla F : \underset{k\text{ copies }}{\underbrace{{\Omega }^{1}\left( M\r... | Proof. This follows immediately from the tensor characterization lemma (Lemma B.6): \( {\nabla }_{X}F \) is a tensor field, so it is multilinear over \( {C}^{\infty }\left( M\right) \) in its \( k + l \) arguments; and it is linear over \( {C}^{\infty }\left( M\right) \) in \( X \) by definition of a connection. | Yes |
Let \( M \) be a smooth manifold with or without boundary and let \( \nabla \) be a connection in TM. For every smooth vector field or tensor field \( F \) , \[ {\nabla }_{X, Y}^{2}F = {\nabla }_{X}\left( {{\nabla }_{Y}F}\right) - {\nabla }_{\left( {\nabla }_{X}Y\right) }F. \] | A covariant derivative \( {\nabla }_{Y}F \) can be expressed as the trace of \( \nabla F \otimes Y \) on its last two indices: \[ {\nabla }_{Y}F = \operatorname{tr}\left( {\nabla F \otimes Y}\right) \] as you can verify by noting that both expressions have the same component formula, \( {F}_{{j}_{1}\ldots {j}_{l};m}^{{... | Yes |
Let \( u \) be a smooth function on \( M \) . Then \( \nabla u \in \Gamma \left( {{T}^{\left( 0,1\right) }{TM}}\right) = {\Omega }^{1}\left( M\right) \) is just the 1 -form \( {du} \), because both tensors have the same action on vectors: \( \nabla u\left( X\right) = {\nabla }_{X}u = {Xu} = {du}\left( X\right) \) . The... | Proposition 4.21 shows that its action on smooth vector fields \( X, Y \) can be computed by the following formula:\n\n\[ \n{\nabla }^{2}u\left( {Y, X}\right) = {\nabla }_{X, Y}^{2}u = {\nabla }_{X}\left( {{\nabla }_{Y}u}\right) - {\nabla }_{\left( {\nabla }_{X}Y\right) }u = Y\left( {Xu}\right) - \left( {{\nabla }_{Y}X... | Yes |
Theorem 4.24 (Covariant Derivative Along a Curve). Let \( M \) be a smooth manifold with or without boundary and let \( \nabla \) be a connection in TM. For each smooth curve \( \gamma : I \rightarrow M \), the connection determines a unique operator\n\n\[ \n{D}_{t} : \mathfrak{X}\left( \gamma \right) \rightarrow \math... | Proof. For simplicity, we prove the theorem for the case of vector fields along \( \gamma \) ; the proof for arbitrary tensor fields is essentially identical except for notation.\n\nFirst we show uniqueness. Suppose \( {D}_{t} \) is such an operator, and let \( {t}_{0} \in I \) be arbitrary. An argument similar to that... | No |
Proposition 4.26. Let \( M \) be a smooth manifold with or without boundary, let \( \nabla \) be a connection in \( {TM} \), and let \( p \in M \) and \( v \in {T}_{p}M \) . Suppose \( Y \) and \( \widetilde{Y} \) are two smooth vector fields that agree at points in the image of some smooth curve \( \gamma : I \rightar... | Proof. We can define a smooth vector field \( Z \) along \( \gamma \) by \( Z\left( t\right) = {Y}_{\gamma \left( t\right) } = {\widetilde{Y}}_{\gamma \left( t\right) } \) . Since both \( Y \) and \( \widetilde{Y} \) are extensions of \( Z \), it follows from condition (iii) in Theorem 4.24 that both \( {\nabla }_{v}Y ... | Yes |
Corollary 4.28. Let \( M \) be a smooth manifold and let \( \nabla \) be a connection in TM. For each \( p \in M \) and \( v \in {T}_{p}M \), there is a unique maximal geodesic \( \gamma : I \rightarrow M \) with \( \gamma \left( 0\right) = p \) and \( {\gamma }^{\prime }\left( 0\right) = v \), defined on some open int... | Proof. Given \( p \in M \) and \( v \in {T}_{p}M \), let \( I \) be the union of all open intervals containing 0 on which there is a geodesic with the given initial conditions. By Theorem 4.27, all such geodesics agree where they overlap, so they define a geodesic \( \gamma : I \rightarrow M \), which is obviously the ... | Yes |
Theorem 4.32 (Existence and Uniqueness of Parallel Transport). Suppose \( M \) is a smooth manifold with or without boundary, and \( \nabla \) is a connection in TM. Given a smooth curve \( \gamma : I \rightarrow M,{t}_{0} \in I \), and a vector \( v \in {T}_{\gamma \left( {t}_{0}\right) }M \) or tensor \( v \in \) \( ... | Proof. As in the proof of Theorem 4.24, we carry out the proof for vector fields. The case of tensor fields differs only in notation.\n\nFirst suppose \( \gamma \left( I\right) \) is contained in a single coordinate chart. Then \( V \) is parallel along \( \gamma \) if and only if its components satisfy the linear syst... | Yes |
Corollary 4.33 (Parallel Transport Along Piecewise Smooth Curves). Suppose \( M \) is a smooth manifold with or without boundary, and \( \nabla \) is a connection in \( {TM} \). Given an admissible curve \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow M \) and a vector \( v \in {T}_{\gamma \left( {t}_{0}\right... | Proof. Let \( \left( {{a}_{0},\ldots ,{a}_{k}}\right) \) be an admissible partition for \( \gamma \) . First define \( {\left. V\right| }_{\left\lbrack {a}_{0},{a}_{1}\right\rbrack } \) to be the parallel transport of \( v \) along the first smooth segment \( {\left. \gamma \right| }_{\left\lbrack {a}_{0},{a}_{1}\right... | Yes |
Theorem 4.34 (Parallel Transport Determines Covariant Differentiation). Let \( M \) be a smooth manifold with or without boundary, and let \( \nabla \) be a connection in TM. Suppose \( \gamma : I \rightarrow M \) is a smooth curve and \( V \) is a smooth vector field along \( \gamma \) . For each \( {t}_{0} \in I \) , | Proof. Let \( \left( {E}_{i}\right) \) be a parallel frame along \( \gamma \), and write \( V\left( t\right) = {V}^{i}\left( t\right) {E}_{i}\left( t\right) \) for \( t \in I \) . On the one hand,(4.23) shows that \( {D}_{t}V\left( {t}_{0}\right) = {\dot{V}}^{i}\left( {t}_{0}\right) {E}_{i}\left( {t}_{0}\right) \) . On... | Yes |
Corollary 4.35 (Parallel Transport Determines the Connection). Let \( M \) be a smooth manifold with or without boundary, and let \( \nabla \) be a connection in TM. Suppose \( X \) and \( Y \) are smooth vector fields on \( M \) . For every \( p \in M \) , \[ {\left. {\nabla }_{X}Y\right| }_{p} = \mathop{\lim }\limits... | Proof. Given \( p \in M \) and a smooth curve \( \gamma \) such that \( \gamma \left( 0\right) = p \) and \( {\gamma }^{\prime }\left( 0\right) = {X}_{p} \), let \( V\left( t\right) \) denote the vector field along \( \gamma \) determined by \( Y \), so \( V\left( t\right) = {Y}_{\gamma \left( t\right) } \) . By proper... | Yes |
Proposition 4.36. Suppose \( M \) is a smooth manifold with or without boundary, \( \nabla \) is a connection in \( {TM} \), and \( A \) is a smooth vector or tensor field on \( M \) . Then \( A \) is parallel on \( M \) if and only if \( \nabla A \equiv 0 \) . | Proof. Problem 4-12. | No |
Lemma 4.37 (Pullback Connections). Suppose \( M \) and \( \widetilde{M} \) are smooth manifolds with or without boundary. If \( \widetilde{\nabla } \) is a connection in \( T\widetilde{M} \) and \( \varphi : M \rightarrow \widetilde{M} \) is a diffeomorphism, then the map \( {\varphi }^{ * }\widetilde{\nabla } : \mathf... | Proof. It is immediate from the definition that \( {\left( {\varphi }^{ * }\widetilde{\nabla }\right) }_{X}Y \) is linear over \( \mathbb{R} \) in \( Y \) . To see that it is linear over \( {C}^{\infty }\left( M\right) \) in \( X \), let \( f \in {C}^{\infty }\left( M\right) \), and let \( \widetilde{f} = f \circ {\var... | No |
Proposition 4.38 (Properties of Pullback Connections). Suppose \( M \) and \( \widetilde{M} \) are smooth manifolds with or without boundary, and \( \varphi : M \rightarrow \widetilde{M} \) is a diffeomorphism. Let \( \widetilde{\nabla } \) be a connection in \( T\widetilde{M} \) and let \( \nabla = {\varphi }^{ * }\wi... | Proof. Problem 4-13. | No |
Proposition 5.1. Let \( M \subseteq {\mathbb{R}}^{n} \) be an embedded submanifold, \( \gamma : I \rightarrow M \) a smooth curve in \( M \), and \( V \) a smooth vector field along \( \gamma \) that takes its values in \( {TM} \) . Then for each \( t \in I \) , \[ {D}_{t}^{\top }V\left( t\right) = {\pi }^{\top }\left(... | Proof. Let \( {t}_{0} \in I \) be arbitrary. By Proposition 2.14, on some neighborhood \( U \) of \( \gamma \left( {t}_{0}\right) \) in \( {\mathbb{R}}^{n} \) there is an adapted orthonormal frame for \( {TM} \), that is, a local orthonormal frame \( \left( {{E}_{1},\ldots ,{E}_{n}}\right) \) for \( T{\mathbb{R}}^{n} \... | Yes |
Corollary 5.2. Suppose \( M \subseteq {\mathbb{R}}^{n} \) is an embedded submanifold. A smooth curve \( \gamma : I \rightarrow M \) is a geodesic with respect to the tangential connection on \( M \) if and only if its ordinary acceleration \( {\gamma }^{\prime \prime }\left( t\right) \) is orthogonal to \( {T}_{\gamma ... | Proof. As noted in Example 4.8, the connection coefficients of the Euclidean connection on \( {\mathbb{R}}^{n} \) are all zero. Thus it follows from (4.15) that the Euclidean covariant derivative of \( {\gamma }^{\prime } \) along \( \gamma \) is just its ordinary acceleration: \( {\bar{D}}_{t}{\gamma }^{\prime }\left(... | Yes |
Proposition 5.3. Suppose \( M \) is an embedded Riemannian or pseudo-Riemannian submanifold of the pseudo-Euclidean space \( {\mathbb{R}}^{r, s} \) . A smooth curve \( \gamma : I \rightarrow M \) is a geodesic with respect to \( {\nabla }^{\top } \) if and only if \( {\gamma }^{\prime \prime }\left( t\right) \) is \( {... | - Exercise 5.4. Prove the preceding proposition. | No |
Proposition 5.5 (Characterizations of Metric Connections). Let \( \left( {M, g}\right) \) be a Riemannian or pseudo-Riemannian manifold (with or without boundary), and let \( \nabla \) be a connection on TM. The following conditions are equivalent:\n\n(a) \( \nabla \) is compatible with \( g : {\nabla }_{X}\langle Y, Z... | Proof. First we prove (a) \( \Leftrightarrow \) (b). By (4.14) and (4.12), the total covariant derivative of the symmetric 2-tensor \( g \) is given by\n\n\[ \left( {\nabla g}\right) \left( {Y, Z, X}\right) = \left( {{\nabla }_{X}g}\right) \left( {Y, Z}\right) = X\left( {g\left( {Y, Z}\right) }\right) - g\left( {{\nabl... | Yes |
Proposition 5.8. If \( M \) is an embedded Riemannian or pseudo-Riemannian subman-ifold of \( {\mathbb{R}}^{n} \) or \( {\mathbb{R}}^{r, s} \), the tangential connection on \( M \) is compatible with the induced Riemannian or pseudo-Riemannian metric. | Proof. We will show that \( {\nabla }^{\top } \) satisfies (5.1). Suppose \( X, Y, Z \in \mathfrak{X}\left( M\right) \), and let \( \widetilde{X},\widetilde{Y},\widetilde{Z} \) be smooth extensions of them to an open subset of \( {\mathbb{R}}^{n} \) or \( {\mathbb{R}}^{r, s} \) . At points of \( M \), we have\n\n\[{\na... | Yes |
Proposition 5.9. If \( M \) is an embedded (pseudo-)Riemannian submanifold of a (pseudo-)Euclidean space, then the tangential connection on \( M \) is symmetric. | Proof. Let \( M \) be an embedded Riemannian or pseudo-Riemannian submanifold of \( {\mathbb{R}}^{n} \), where \( {\mathbb{R}}^{n} \) is endowed either with the Euclidean metric or with a pseudo-Euclidean metric \( {\bar{q}}^{\left( r, s\right) }, r + s = n \) . Let \( X, Y \in \mathfrak{X}\left( M\right) \), and let \... | Yes |
Theorem 5.10 (Fundamental Theorem of Riemannian Geometry). Let \( \left( {M, g}\right) \) be a Riemannian or pseudo-Riemannian manifold (with or without boundary). There exists a unique connection \( \nabla \) on TM that is compatible with \( g \) and symmetric. It is called the Levi-Civita connection of \( g \) (or al... | Proof. We prove uniqueness first, by deriving a formula for \( \nabla \) . Suppose, therefore, that \( \nabla \) is such a connection, and let \( X, Y, Z \in \mathfrak{X}\left( M\right) \) . Writing the compatibility equation three times with \( X, Y, Z \) cyclically permuted, we obtain\n\n\[ X\langle Y, Z\rangle = \le... | Yes |
Corollary 5.11 (Formulas for the Levi-Civita Connection). Let \( \left( {M, g}\right) \) be a Riemannian or pseudo-Riemannian manifold (with or without boundary), and let \( \nabla \) be its Levi-Civita connection.\n\n(a) IN TERMS OF VECTOR FIELDS: If \( X, Y, Z \) are smooth vector fields on \( M \), then\n\n\[ \left\... | Proof. We derived (5.9) and (5.10) in the proof of Theorem 5.10. To prove (5.12), apply formula (5.9) with \( X = {E}_{i}, Y = {E}_{j} \), and \( Z = {E}_{l} \), to obtain\n\n\[ {\Gamma }_{ij}^{q}{g}_{ql} = \left\langle {{\nabla }_{{E}_{i}}{E}_{j},{E}_{l}}\right\rangle \]\n\n\[ = \frac{1}{2}\left( {{E}_{i}{g}_{jl} + {E... | Yes |
Proposition 5.13 (Naturality of the Levi-Civita Connection). Suppose \( \left( {M, g}\right) \) and \( \left( {\widetilde{M},\widetilde{g}}\right) \) are Riemannian or pseudo-Riemannian manifolds with or without boundary, and let \( \nabla \) denote the Levi-Civita connection of \( g \) and \( \widetilde{\nabla } \) th... | Proof. By uniqueness of the Levi-Civita connection, it suffices to show that the pullback connection \( {\varphi }^{ * }\widetilde{\nabla } \) is symmetric and compatible with \( g \) . The fact that \( \varphi \) is an isometry means that for any \( X, Y \in \mathfrak{X}\left( M\right) \) and \( p \in M \) ,\n\n\[ \le... | Yes |
Corollary 5.14 (Naturality of Geodesics). Suppose \( \left( {M, g}\right) \) and \( \left( {\widetilde{M},\widetilde{g}}\right) \) are Riemannian or pseudo-Riemannian manifolds with or without boundary, and \( \varphi : M \rightarrow \) \( \widetilde{M} \) is a local isometry. If \( \gamma \) is a geodesic in \( M \), ... | Proof. This is an immediate consequence of Proposition 4.38, together with the fact that being a geodesic is a local property. | No |
Proposition 5.15. Suppose \( \left( {M, g}\right) \) is a Riemannian or pseudo-Riemannian manifold. The connection induced on each tensor bundle by the Levi-Civita connection is compatible with the induced inner product on tensors, in the sense that \( X\langle F, G\rangle = \left\langle {{\nabla }_{X}F, G}\right\rangl... | Proof. Since every tensor field can be written as a sum of tensor products of vector and/or covector fields, it suffices to consider the case in which \( F = {\alpha }_{1} \otimes \cdots \otimes {\alpha }_{k + l} \) and \( G = {\beta }_{1} \otimes \cdots \otimes {\beta }_{k + l} \), where \( {\alpha }_{i} \) and \( {\b... | Yes |
Proposition 5.16. Let \( \left( {M, g}\right) \) be an oriented Riemannian manifold. The Riemannian volume form of \( g \) is parallel with respect to the Levi-Civita connection. | Proof. Let \( p \in M \) and \( v \in {T}_{p}M \) be arbitrary, and let \( \gamma : \left( {-\varepsilon ,\varepsilon }\right) \rightarrow M \) be a smooth curve satisfying \( \gamma \left( 0\right) = p \) and \( {\gamma }^{\prime }\left( 0\right) = v \) . Let \( \left( {{E}_{1},\ldots ,{E}_{n}}\right) \) be a parallel... | Yes |
Proposition 5.17. The musical isomorphisms commute with the total covariant derivative operator: if \( F \) is any smooth tensor field with a contravariant ith index position, and \( b \) represents the operation of lowering the ith index, then\n\n\[ \nabla \left( {F}^{b}\right) = {\left( \nabla F\right) }^{b}. \]\n\n(... | Proof. The discussion on page 27 shows that \( {F}^{\mathrm{b}} = \operatorname{tr}\left( {F \otimes g}\right) \), where the trace is taken on the \( i \) th and last indices of \( F \otimes g \) . Because \( g \) is parallel, for every vector field \( X \) we have \( {\nabla }_{X}\left( {F \otimes g}\right) = \left( {... | Yes |
Lemma 5.18 (Rescaling Lemma). For every \( p \in M, v \in {T}_{p}M \), and \( c, t \in \mathbb{R} \) , \[ {\gamma }_{cv}\left( t\right) = {\gamma }_{v}\left( {ct}\right) \] whenever either side is defined. | Proof. If \( c = 0 \), then both sides of (5.16) are equal to \( p \) for all \( t \in \mathbb{R} \), so we may assume that \( c \neq 0 \) . It suffices to show that \( {\gamma }_{cv}\left( t\right) \) exists and (5.16) holds whenever the right-hand side is defined. (The same argument with the substitutions \( v = {c}^... | Yes |
Proposition 5.20 (Naturality of the Exponential Map). Suppose \( \left( {M, g}\right) \) and \( \left( {\widetilde{M},\widetilde{g}}\right) \) are Riemannian or pseudo-Riemannian manifolds and \( \varphi : M \rightarrow \widetilde{M} \) is a local isometry. Then for every \( p \in M \), the following diagram commutes: ... | - Exercise 5.21. Prove Proposition 5.20. | No |
Proposition 5.22. Let \( \\left( {M, g}\\right) \) and \( \\left( {\\widetilde{M},\\widetilde{g}}\\right) \) be Riemannian or pseudo-Riemannian manifolds, with \( M \) connected. Suppose \( \\varphi ,\\psi : M \\rightarrow \\widetilde{M} \) are local isometries such that for some point \( p \\in M \), we have \( \\varp... | Proof. Problem 5-10. | No |
Proposition 5.23 (Uniqueness of Normal Coordinates). Let \( \left( {M, g}\right) \) be a Riemannian or pseudo-Riemannian n-manifold, \( p \) a point of \( M \), and \( U \) a normal neighborhood of \( p \). For every normal coordinate chart on \( U \) centered at \( p \), the coordinate basis is orthonormal at \( p \) ... | Proof. Let \( \varphi \) be a normal coordinate chart on \( U \) centered at \( p \), with coordinate functions \( \left( {x}^{i}\right) \). By definition, this means that \( \varphi = {B}^{-1} \circ {\exp }_{p}^{-1} \), where \( B : {\mathbb{R}}^{n} \rightarrow {T}_{p}M \) is the basis isomorphism determined by some o... | Yes |
Proposition 5.24 (Properties of Normal Coordinates). Let \( \left( {M, g}\right) \) be a Riemannian or pseudo-Riemannian n-manifold, and let \( \left( {U,\left( {x}^{i}\right) }\right) \) be any normal coordinate chart centered at \( p \in M \) .\n\n(a) The coordinates of \( p \) are \( \left( {0,\ldots ,0}\right) \) .... | Proof. Part (a) follows directly from the definition of normal coordinates, and parts (b) and (c) follow from Propositions 5.23 and 5.19(b), respectively.\n\nTo prove (d), let \( v = {\left. {v}^{i}{\partial }_{i}\right| }_{p} \in {T}_{p}M \) be arbitrary. The geodesic equation (4.16) for \( {\gamma }_{v}\left( t\right... | Yes |
Proposition 5.26 (Properties of Fermi Coordinates). Let \( P \) be an embedded p-dimensional submanifold of a Riemannian n-manifold \( \left( {M, g}\right) \), let \( U \) be a normal neighborhood of \( P \) in \( M \), and let \( \left( {{x}^{1},\ldots ,{x}^{p},{v}^{1},\ldots ,{v}^{n - p}}\right) \) be Fermi coordinat... | Proof. Problem 5-18. | No |
Proposition 5.27. A nonconstant curve on \( {\mathbb{S}}^{n}\left( R\right) \) is a maximal geodesic if and only if it is a periodic constant-speed curve whose image is a great circle. Thus every sphere is geodesically complete. | Proof. Let \( p \in {\mathbb{S}}^{n}\left( R\right) \) be arbitrary. Because \( f\left( x\right) = {\left| x\right| }^{2} \) is a defining function for \( {\mathbb{S}}^{n}\left( R\right) \), a vector \( v \in {T}_{p}{\mathbb{R}}^{n + 1} \) is tangent to \( {\mathbb{S}}^{n}\left( R\right) \) if and only if \( d{f}_{p}\l... | Yes |
A nonconstant curve in a hyperbolic space is a maximal geodesic if and only if it is a constant-speed embedding of \( \mathbb{R} \) whose image is one of the following:\n\n(a) HYPERBOLOID MODEL: The intersection of \( {\mathbb{H}}^{n}\left( R\right) \) with a 2-dimensional linear subspace of \( {\mathbb{R}}^{n,1} \), c... | Proof. We begin with the hyperboloid model, for which the proof is formally quite similar to what we just did for the sphere. Since the Riemannian connection on \( {\mathbb{H}}^{n}\left( R\right) \) is equal to the tangential connection by Proposition 5.12, it follows from Corollary 5.2 that a smooth curve \( \gamma : ... | Yes |
Lemma 6.1. If \( \gamma \) is an admissible curve and \( V \) is a piecewise smooth vector field along \( \gamma \), then \( V \) is the variation field of some variation of \( \gamma \) . If \( V \) is proper, the variation can be taken to be proper as well. | Proof. Suppose \( \gamma \) and \( V \) satisfy the hypotheses, and set \( \Gamma \left( {s, t}\right) = {\exp }_{\gamma \left( t\right) }\left( {{sV}\left( t\right) }\right) \) (Fig. 6.2). By compactness of \( \left\lbrack {a, b}\right\rbrack \), there is some positive \( \varepsilon \) such that \( \Gamma \) is defin... | Yes |
Lemma 6.2 (Symmetry Lemma). Let \( \Gamma : J \times \left\lbrack {a, b}\right\rbrack \rightarrow M \) be an admissible family of curves in a Riemannian manifold. On every rectangle \( J \times \left\lbrack {{a}_{i - 1},{a}_{i}}\right\rbrack \) where \( \Gamma \) is smooth,\n\n\[ \n{D}_{s}{\partial }_{t}\Gamma = {D}_{t... | Proof. This is a local question, so we may compute in local coordinates \( \left( {x}^{i}\right) \) around a point \( \Gamma \left( {{s}_{0},{t}_{0}}\right) \) . Writing the components of \( \Gamma \) as \( \Gamma \left( {s, t}\right) = \left( {{x}^{1}\left( {s, t}\right) ,\ldots ,{x}^{n}\left( {s, t}\right) }\right) \... | Yes |
Theorem 6.4. In a Riemannian manifold, every minimizing curve is a geodesic when it is given a unit-speed parametrization. | Proof. Suppose \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow M \) is minimizing and of unit speed, and \( \left( {{a}_{0},\ldots ,{a}_{k}}\right) \) is an admissible partition for \( \gamma \) . If \( \Gamma \) is any proper variation of \( \gamma \), then \( {L}_{g}\left( {\Gamma }_{s}\right) \) is a smooth... | Yes |
Corollary 6.5. A unit-speed admissible curve \( \gamma \) is a critical point for \( {L}_{g} \) if and only if it is a geodesic. | Proof. If \( \gamma \) is a critical point, the proof of Theorem 6.4 goes through without modification to show that \( \gamma \) is a geodesic. Conversely, if \( \gamma \) is a geodesic, then the first term on the right-hand side of (6.2) vanishes by the geodesic equation, and the second term vanishes because \( {\gamm... | Yes |
Lemma 6.8. In every normal neighborhood \( U \) of \( p \in M \), the radial distance function and the radial vector field are well defined, independently of the choice of normal coordinates. Both \( r \) and \( {\partial }_{r} \) are smooth on \( U \smallsetminus \{ p\} \), and \( {r}^{2} \) is smooth on all of \( U \... | Proof. Proposition 5.23 shows that any two normal coordinate charts on \( U \) are related by \( {\widetilde{x}}^{i} = {A}_{j}^{i}{x}^{j} \) for some orthogonal matrix \( \left( {A}_{j}^{i}\right) \), and a straightforward computation shows that both \( r \) and \( {\partial }_{r} \) are invariant under such coordinate... | Yes |
Corollary 6.10. Let \( U \) be a geodesic ball centered at \( p \in M \), and let \( r \) and \( {\partial }_{r} \) be the radial distance and radial vector field as defined by (6.4) and (6.5). Then \( \operatorname{grad}r = {\partial }_{r} \) on \( U \smallsetminus \{ p\} . | Proof. By the result of Problem 2-10, it suffices to show that \( {\partial }_{r} \) is orthogonal to the level sets of \( r \) and \( {\partial }_{r}\left( r\right) \equiv {\left| {\partial }_{r}\right| }_{g}^{2} \) . The first claim follows directly from the Gauss lemma, and the second from the fact that \( {\partial... | Yes |
Corollary 6.12. Let \( \left( {M, g}\right) \) be a connected Riemannian manifold and \( p \in M \) . Within every open or closed geodesic ball around \( p \), the radial distance function \( r\left( x\right) \) defined by (6.4) is equal to the Riemannian distance from \( p \) to \( x \) in \( M \). | Proof. Since every closed geodesic ball is contained in an open geodesic ball of larger radius, we need only consider the open case. If \( x \) is in the open geodesic ball \( {\exp }_{p}\left( {{B}_{c}\left( 0\right) }\right) \), the radial geodesic \( \gamma \) from \( p \) to \( x \) is minimizing by Proposition 6.1... | Yes |
Corollary 6.13. In a connected Riemannian manifold, every open or closed geodesic ball is also an open or closed metric ball of the same radius, and every geodesic sphere is a metric sphere of the same radius. | Proof. Let \( \left( {M, g}\right) \) be a Riemannian manifold, and let \( p \in M \) be arbitrary. First, let \( V = {\exp }_{p}\left( {{\bar{B}}_{c}\left( 0\right) }\right) \subseteq M \) be a closed geodesic ball of radius \( c > 0 \) around \( p \) . Suppose \( q \) is an arbitrary point of \( M \) . If \( q \in V ... | Yes |
Lemma 6.16. If \( \left( {M, g}\right) \) is a compact Riemannian manifold, then \( \operatorname{inj}\left( M\right) \) is positive. | Proof. For each \( x \in M \), there is a positive number \( \delta \left( x\right) \) such that \( x \) is contained in a uniformly \( \delta \left( x\right) \) -normal neighborhood \( {W}_{x} \), and inj \( \left( {x}^{\prime }\right) \geq \delta \left( x\right) \) for each \( {x}^{\prime } \in W \) . Since \( M \) i... | Yes |
Theorem 6.17. Let \( \left( {M, g}\right) \) be a Riemannian manifold. For each \( p \in M \), there exists \( {\varepsilon }_{0} > 0 \) such that every geodesic ball centered at \( p \) of radius less than or equal to \( {\varepsilon }_{0} \) is geodesically convex. | Proof. Problem 6-5. | No |
Theorem 6.19 (Hopf-Rinow). A connected Riemannian manifold is metrically complete if and only if it is geodesically complete. | Proof. Let \( \left( {M, g}\right) \) be a connected Riemannian manifold. Suppose first that \( M \) is geodesically complete. Then in particular, it satisfies the hypothesis of Lemma 6.18, so it is metrically complete.\n\nConversely, suppose \( M \) is metrically complete, and assume for the sake of contradiction that... | Yes |
Theorem 6.23. Suppose \( \left( {\widetilde{M},\widetilde{g}}\right) \) and \( \left( {M, g}\right) \) are connected Riemannian manifolds with \( \widetilde{M} \) complete, and \( \pi : \widetilde{M} \rightarrow M \) is a local isometry. Then \( M \) is complete and \( \pi \) is a Riemannian covering map. | Proof. A fundamental property of covering maps is the path-lifting property (Prop. A.54(b)): if \( \pi \) is a covering map, then every continuous path \( \gamma : I \rightarrow M \) lifts to a path \( \widetilde{\gamma } \) in \( \widetilde{M} \) such that \( \pi \circ \widetilde{\gamma } = \gamma \) . We begin by pro... | Yes |
Corollary 6.24. Suppose \( \widetilde{M} \) and \( M \) are connected Riemannian manifolds, and \( \pi : \widetilde{M} \rightarrow M \) is a Riemannian covering map. Then \( M \) is complete if and only if \( \widetilde{M} \) is complete. | Proof. A Riemannian covering map is, in particular, a local isometry. Thus if \( \widetilde{M} \) is complete, \( \pi \) satisfies the hypotheses of Theorem 6.23, which implies that \( M \) is also complete.\n\nConversely, suppose \( M \) is complete. Let \( \widetilde{p} \in \widetilde{M} \) and \( \widetilde{v} \in {... | Yes |
Proposition 6.25. Suppose \( \left( {M, g}\right) \) is a complete, connected Riemannian manifold, and \( p, q \in M \) . Every path-homotopy class of paths from \( p \) to \( q \) contains a geodesic segment \( \gamma \) that minimizes length among all admissible curves in the same path-homotopy class. | Proof. Let \( \pi : \widetilde{M} \rightarrow M \) be the universal covering manifold of \( M \), endowed with the pullback metric \( \widetilde{g} = {\pi }^{ * }g \) . Given \( p, q \in M \) and a path \( \sigma : \left\lbrack {0,1}\right\rbrack \rightarrow M \) from \( p \) to \( q \), choose a point \( \widetilde{p}... | Yes |
Proposition 6.28 (Existence of Closed Geodesics). Suppose \( \left( {M, g}\right) \) is a compact, connected Riemannian manifold. Every nontrivial free homotopy class in \( M \) is represented by a closed geodesic that has minimum length among all admissible loops in the given free homotopy class. | Proof. Problem 6-17. | No |
Lemma 6.29. Suppose \( \left( {M, g}\right) \) is a connected Riemannian manifold and \( S \subseteq M \) is any subset.\n\n(a) \( {d}_{g}\left( {x, S}\right) \leq {d}_{g}\left( {x, y}\right) + {d}_{g}\left( {y, S}\right) \) for all \( x, y \in M \) .\n\n(b) \( x \mapsto {d}_{g}\left( {x, S}\right) \) is a continuous f... | Exercise 6.30. Prove the preceding lemma. | No |
Theorem 6.32. Suppose \( \left( {M, g}\right) \) is a Riemannian manifold and \( f \) is a smooth local distance function on an open subset \( U \subseteq M \) . Then \( {\nabla }_{\operatorname{grad}f}\left( {\operatorname{grad}f}\right) \equiv 0 \), and each integral curve of \( \operatorname{grad}f \) is a unit-spee... | Proof. Let \( F \in \mathfrak{X}\left( U\right) \) denote the unit vector field grad \( f \) . The definition of the gradient shows that for every vector field \( W \), we have\n\n\[ \n{Wf} = {df}\left( W\right) = \langle F, W\rangle , \n\]\n\n(6.15)\n\nand therefore\n\n\[ \n{Ff} = \langle F, F\rangle = {\left| \operat... | Yes |
Lemma 6.33. Suppose \( \left( {M, g}\right) \) is a Riemannian manifold, \( K \subseteq M \), and \( f : K \rightarrow \mathbb{R} \) is a continuous function whose restriction to some open set \( W \subseteq K \) is a smooth local distance function. For every admissible curve \( \sigma : \left\lbrack {{a}_{0},{b}_{0}}\... | Proof. This is proved exactly as in (6.8), noting that the only properties of \( r \) we used in that computation were that it is continuous on the image of \( \sigma \) and continuously differentiable on \( \sigma \left( \left( {{a}_{0},{b}_{0}}\right) \right) \), and its gradient has unit length there. | Yes |
Proposition 6.37. Let \( P \) be an embedded submanifold of a Riemannian manifold \( \left( {M, g}\right) \) and let \( U \) be any normal neighborhood of \( P \) in \( M \) . There exist a unique continuous function \( r : U \rightarrow \lbrack 0,\infty ) \) and smooth vector field \( {\partial }_{r} \) on \( U \small... | Proof. The uniqueness, continuity, and smoothness claims follow immediately from the coordinate expressions (6.17) and (6.18), so we need only prove that \( r \) and \( {\partial }_{r} \) can be globally defined so as to have the indicated coordinate expressions in any Fermi coordinates.\n\nLet \( V \subseteq {NS} \) b... | Yes |
Corollary 6.39. Assume the hypotheses of Theorem 6.38.\n\n(a) \( {\partial }_{r} \) is equal to the gradient of \( r \) on \( U \smallsetminus P \) .\n\n(b) \( r \) is a local distance function.\n\n(c) Each unit-speed geodesic \( \gamma : \lbrack a, b) \rightarrow U \) with \( {\gamma }^{\prime }\left( a\right) \) norm... | Proof. By direct computation in Fermi coordinates using formulas (6.17) and (6.18), \( {\partial }_{r}\left( r\right) \equiv 1 \), which is equal to \( {\left| {\partial }_{r}\right| }_{g}^{2} \) by the previous theorem. Thus \( {\partial }_{r} = \operatorname{grad}r \) on \( U \smallsetminus P \) by Problem 2-10. Beca... | Yes |
Proposition 6.41 (Characterizations of Semigeodesic Coordinates). Let \( \left( {M, g}\right) \) be a Riemannian \( n \) -manifold, and let \( \left( {{x}^{1},\ldots ,{x}^{n}}\right) \) be smooth coordinates on an open subset of \( M \) . The following are equivalent:\n\n(a) \( \left( {x}^{i}\right) \) are semigeodesic... | Proof. We begin by showing that (b) \( \Leftrightarrow \) (c) \( \Leftrightarrow \) (d) \( \Leftrightarrow \) (e) and (c) \( \Leftrightarrow \) (f). Note that (b) is equivalent to the coordinate matrix of \( g \) having the block form \( \left( \begin{array}{ll} * & 0 \\ 0 & 1 \end{array}\right) \), where the asterisk ... | Yes |
Corollary 6.42. Let \( \left( {x}^{i}\right) \) be semigeodesic coordinates on an open subset of a Riemannian \( n \) -manifold \( \left( {M, g}\right) \) .\n\n(a) The metric has the following coordinate expression:\n\n\[ g = {\left( d{x}^{n}\right) }^{2} + {g}_{\alpha \beta }\left( {{x}^{1},\ldots ,{x}^{n}}\right) d{x... | Proof. Part (a) follows immediately from part (b) of Proposition 6.41, and (b) is proved by inserting \( {g}_{nn} = 1 \) and \( {g}_{\alpha n} = {g}_{n\alpha } = 0 \) into formula (5.8) for the Christoffel symbols. | Yes |
Suppose \( P \) is an embedded hypersurface in a Riemannian manifold \( \left( {M, g}\right) \), and let \( \left( {{x}^{1},\ldots ,{x}^{n - 1}, v}\right) \) be any Fermi coordinates for \( P \) on an open subset \( U \subseteq M \) (see (5.25)). In this case, the function \( r \) defined by (6.17) is just \( r\left( {... | It follows from Corollary 6.39 that there is a neighborhood \( {U}_{0} \) of \( P \) on which \( \left| v\right| \) is equal to the distance from \( P \) . Moreover, (6.18) reduces to \( {\partial }_{r} = \pm \partial /{\partial }_{v} \), which is equal to grad \( \left| v\right| \) by Corollary 6.39, so it follows fro... | Yes |
Example 6.45 (Polar Normal Coordinates). Polar coordinates for \( {\mathbb{R}}^{n} \) are constructed by choosing a smooth local parametrization \( \widehat{\psi } : \widehat{U} \rightarrow U \subseteq {\mathbb{S}}^{n - 1} \) for an open subset \( U \) of \( {\mathbb{S}}^{n - 1} \), and defining \( \widehat{\Psi } : \w... | Now let \( \left( {M, g}\right) \) be a Riemannian \( n \) -manifold, \( p \) a point in \( M \), and \( \varphi \) any normal coordinate chart defined on a normal neighborhood \( V \) of \( p \) . For every choice of polar coordinates \( \left( {\mathcal{U},\widehat{\Theta }}\right) \) for \( {\mathbb{R}}^{n} \smallse... | Yes |
6-1. Suppose \( M \) is a nonempty connected Riemannian 1-manifold. Show that if \( M \) is noncompact, then it is isometric to an open interval in \( \mathbb{R} \) with the Euclidean metric, while if it is compact, it is isometric to a circle \( {\mathbb{S}}^{1}\left( R\right) = \left\{ {x \in {\mathbb{R}}^{2}}\right.... | (a) Let \( \gamma : I \rightarrow M \) be any maximal unit-speed geodesic. Show that its image is open and closed, and therefore \( \gamma \) is surjective.\n\n(b) Show that if \( \gamma \) is injective, then it is an isometry between \( I \) with its Euclidean metric and \( M \) .\n\n(c) Now suppose \( \gamma \left( {... | Yes |
Proposition 7.2. If \( \left( {M, g}\right) \) is a flat Riemannian or pseudo-Riemannian manifold, then its Levi-Civita connection satisfies the flatness criterion. | Proof. We just showed that the Euclidean connection on \( {\mathbb{R}}^{n} \) satisfies (7.3). By naturality, the Levi-Civita connection on every manifold that is locally isometric to a Euclidean or pseudo-Euclidean space must also satisfy the same identity. | Yes |
Proposition 7.3. The map \( R \) defined above is multilinear over \( {C}^{\infty }\left( M\right) \), and thus defines \( a\left( {1,3}\right) \) -tensor field on \( M \) . | Proof. The map \( R \) is obviously multilinear over \( \mathbb{R} \) . For \( f \in {C}^{\infty }\left( M\right) \), \n\n\[ \nR\left( {X,{fY}}\right) Z = {\nabla }_{X}{\nabla }_{fY}Z - {\nabla }_{fY}{\nabla }_{X}Z - {\nabla }_{\left\lbrack X, fY\right\rbrack }Z \n\] \n\n\[ \n= {\nabla }_{X}\left( {f{\nabla }_{Y}Z}\rig... | No |
Proposition 7.4. Let \( \\left( {M, g}\\right) \) be a Riemannian or pseudo-Riemannian manifold. In terms of any smooth local coordinates, the components of the \( \\left( {1,3}\\right) \) -curvature tensor are given by\n\n\[ \n{R}_{ijk}{}^{l} = {\\partial }_{i}{\\Gamma }_{jk}^{l} - {\\partial }_{j}{\\Gamma }_{ik}^{l} ... | Proof. Problem 7-2. | No |
Theorem 7.10. A Riemannian or pseudo-Riemannian manifold is flat if and only if its curvature tensor vanishes identically. | Proof. One direction is immediate: Proposition 7.2 showed that the Levi-Civita connection of a flat metric satisfies the flatness criterion, so its curvature endomorphism is identically zero, which implies that the curvature tensor is also zero.\n\nNow suppose \( \left( {M, g}\right) \) has vanishing curvature tensor. ... | Yes |
Theorem 7.11. Let \( \left( {M, g}\right) \) be a Riemannian or pseudo-Riemannian manifold; let \( I \) be an open interval containing 0 ; let \( \Gamma : I \times I \rightarrow M \) be a smooth one-parameter family of curves; and let \( p = \Gamma \left( {0,0}\right), x = {\partial }_{s}\Gamma \left( {0,0}\right) \), ... | \[ R\left( {x, y}\right) z = \mathop{\lim }\limits_{{\delta ,\varepsilon \rightarrow 0}}\frac{{P}_{\delta ,0}^{0,0} \circ {P}_{\delta ,\varepsilon }^{\delta ,0} \circ {P}_{0,\varepsilon }^{\delta ,\varepsilon } \circ {P}_{0,0}^{0,\varepsilon }\left( z\right) - z}{\delta \varepsilon }. \] Proof. Define a vector field \(... | Yes |
Proposition 7.12 (Symmetries of the Curvature Tensor). Let \( \left( {M, g}\right) \) be a Riemannian or pseudo-Riemannian manifold. The \( \left( {0,4}\right) \) -curvature tensor of \( g \) has the following symmetries for all vector fields \( W, X, Y, Z \) :\n\n(a) \( \operatorname{Rm}\left( {W, X, Y, Z}\right) = - ... | Proof of Proposition 7.12. Identity (a) is immediate from the definition of the curvature tensor, because \( R\left( {W, X}\right) Y = - R\left( {X, W}\right) Y \) . To prove (b), it suffices to show that \( \operatorname{Rm}\left( {W, X, Y, Y}\right) = 0 \) for all \( Y \), for then (b) follows from the expansion of \... | Yes |
Proposition 7.13 (Differential Bianchi Identity). The total covariant derivative of the curvature tensor satisfies the following identity:\n\n\[ \n\\nabla {Rm}\\left( {X, Y, Z, V, W}\\right) + \\nabla {Rm}\\left( {X, Y, V, W, Z}\\right) + \\nabla {Rm}\\left( {X, Y, W, Z, V}\\right) = 0.\n\] | Proof. First of all, by the symmetries of \( {Rm},\left( {7.14}\right) \) is equivalent to\n\n\[ \n\\nabla {Rm}\\left( {Z, V, X, Y, W}\\right) + \\nabla {Rm}\\left( {V, W, X, Y, Z}\\right) + \\nabla {Rm}\\left( {W, Z, X, Y, V}\\right) = 0.\n\]\n\nThis can be proved by a long and tedious computation, but there is a stan... | Yes |
Theorem 7.14 (Ricci Identities). On a Riemannian or pseudo-Riemannian manifold \( M \), the second total covariant derivatives of vector and tensor fields satisfy the following identities. If \( Z \) is a smooth vector field,\n\n\[ \n{\nabla }_{X, Y}^{2}Z - {\nabla }_{Y, X}^{2}Z = R\left( {X, Y}\right) Z \n\]\n\n(7.17) | Proof. For any tensor field \( B \) and vector fields \( X, Y \), Proposition 4.21 implies\n\n\[ \n{\nabla }_{X, Y}^{2}B - {\nabla }_{Y, X}^{2}B = {\nabla }_{X}{\nabla }_{Y}B - {\nabla }_{\left( {\nabla }_{X}Y\right) }B - {\nabla }_{Y}{\nabla }_{X}B + {\nabla }_{\left( {\nabla }_{Y}X\right) }B \n\]\n\n\[ \n= {\nabla }_... | Yes |
Lemma 7.15. The Ricci curvature is a symmetric 2-tensor field. It can be expressed in any of the following ways:\n\n\[ \n{R}_{ij} = {R}_{kij}{}^{k} = {R}_{ik}{}^{k}{}_{j} = - {R}_{ki}{}^{k}{}_{j} = - {R}_{ikj}{}^{k}. \n\] | Exercise 7.16. Prove Lemma 7.15, using the symmetries of the curvature tensor. | No |
Proposition 7.17. Let \( \\left( {M, g}\\right) \) be a Riemannian or pseudo-Riemannian n-manifold. Then \( {\\operatorname{tr}}_{g}{Rc} \\equiv 0 \), and the Ricci tensor decomposes orthogonally as\n\n\[ \n{Rc} = \\overset{ \\circ }{Rc} + \\frac{1}{n}{Sg} \n\]\n\n(7.27)\n\nTherefore, in the Riemannian case,\n\n\[ \n{\... | Proof. Note that in every local frame, we have\n\n\[ \n{\\operatorname{tr}}_{g}g = {g}_{ij}{g}^{ji} = {\\delta }_{i}^{i} = n. \n\]\n\nIt then follows directly from the definition of \( \\overset{ \\circ }{Rc} \) that \( {\\operatorname{tr}}_{g}\\overset{ \\circ }{Rc} \\equiv 0 \) and (7.27) holds. The fact that the dec... | Yes |
Proposition 7.18 (Contracted Bianchi Identities). Let \( \left( {M, g}\right) \) be a Riemannian or pseudo-Riemannian manifold. The covariant derivatives of the Riemann, Ricci, and scalar curvatures of \( g \) satisfy the following identities:\n\n\[ \n{\operatorname{tr}}_{g}\left( {\nabla {Rm}}\right) = - D\left( {Rc}\... | Proof. Start with the component form (7.15) of the differential Bianchi identity, raise the index \( m \), and then contract on the indices \( i, m \) to obtain (7.32). (Note that covariant differentiation commutes with contraction by Proposition 4.15 and with the musical isomorphisms by Proposition 5.17, so it does no... | No |
Proposition 7.19 (Schur’s Lemma). Suppose \( \left( {M, g}\right) \) is a connected Riemannian or pseudo-Riemannian manifold of dimension \( n \geq 3 \) whose Ricci tensor satisfies \( {Rc} = {fg} \) for some smooth real-valued function \( f \) . Then \( f \) is constant and \( g \) is an Einstein metric. | Proof. Taking traces of both sides of \( {Rc} = {fg} \) shows that \( f = \frac{1}{n}S \), so the traceless Ricci tensor is identically zero. It follows that \( \nabla {Rc} \equiv 0 \) . Because the covariant derivative of the metric is zero, this implies the following equation in any coordinate chart:\n\n\[ 0 = {R}_{{... | Yes |
Corollary 7.20. If \( \left( {M, g}\right) \) is a connected Riemannian or pseudo-Riemannian manifold of dimension \( n \geq 3 \), then \( g \) is Einstein if and only if \( {Rc} = 0 \) . | Proof. Suppose first that \( g \) is an Einstein metric with \( {Rc} = {\lambda g} \) . Taking traces of both sides, we find that \( \lambda = \frac{1}{n}S \), and therefore \( \overset{ \circ }{Rc} = {Rc} - {\lambda g} = 0 \) . Conversely, if \( \overset{ \circ }{Rc} = 0 \) , Schur’s lemma implies that \( g \) is Eins... | Yes |
Lemma 7.22 (Properties of the Kulkarni-Nomizu Product). Let \( V \) be an \( n \) - dimensional vector space endowed with a scalar product \( g \), let \( h \) and \( k \) be symmetric 2-tensors on \( V \), let \( T \) be an algebraic curvature tensor on \( V \), and let \( {\operatorname{tr}}_{g} \) denote the trace o... | Proof. It is evident from the definition that \( h \odot k \) has three of the four symmetries of an algebraic curvature tensor: it is antisymmetric in its first two arguments and also in its last two, and its value is unchanged when the first two and last two arguments are interchanged. Thus to prove (a), only the alg... | No |
Proposition 7.23. Let \( \left( {V, g}\right) \) be an \( n \) -dimensional scalar product space with \( n \geq 3 \) , and define a linear map \( G : {\sum }^{2}\left( {V}^{ * }\right) \rightarrow \mathcal{R}\left( {V}^{ * }\right) \) by\n\n\[ \nG\left( h\right) = \frac{1}{n - 2}\left( {h - \frac{{\operatorname{tr}}_{g... | Proof. The fact that \( G \) is a right inverse is a straightforward computation based on the definition and Lemma 7.22(c, d). This implies that \( G \) is injective and \( {\operatorname{tr}}_{g} \) is surjective, so the dimension of \( \operatorname{Im}G \) is equal to the codimension of \( \operatorname{Ker}\left( {... | Yes |
Proposition 7.24. For every Riemannian or pseudo-Riemannian manifold \( \left( {M, g}\right) \) of dimension \( n \geq 3 \), the trace of the Weyl tensor is zero, and \( {Rm} = W + P \odot g \) is the orthogonal decomposition of \( {Rm} \) corresponding to \( \mathcal{R}\left( {V}^{ * }\right) = \operatorname{Ker}\left... | Proof. This follows immediately from Proposition 7.23 and the fact that \( P \otimes g = \) \( G\left( {Rc}\right) = G\left( {{\operatorname{tr}}_{g}{Rm}}\right) \) | No |
Corollary 7.25 Let \( V \) be an \( n \) -dimensional real vector space.\n\n(a) If \( n = 0 \) or \( n = 1 \), then \( \mathcal{R}\left( {V}^{ * }\right) = \{ 0\} \).\n\n(b) If \( n = 2 \), then \( \mathcal{R}\left( {V}^{ * }\right) \) is 1 -dimensional, spanned by \( g \oslash g \).\n\n(c) If \( n = 3 \), then \( \mat... | Proof. The dimensional results follow immediately from Proposition 7.21. In the case \( n = 2 \), Lemma 7.22(d) shows that \( {\operatorname{tr}}_{g}\left( {g \odot g}\right) = {2g} \neq 0 \), which implies that \( g \otimes g \) is nonzero and therefore spans the 1-dimensional space \( \mathcal{R}\left( {V}^{ * }\righ... | Yes |
Corollary 7.26 (The Curvature Tensor in Dimension 3). On every Riemannian or pseudo-Riemannian manifold \( \left( {M, g}\right) \) of dimension 3, the Weyl tensor is zero, and the Riemann curvature tensor is determined by the Ricci tensor via the formula\n\n\[ \n{Rm} = P \oslash g = {Rc} \oslash g - \frac{1}{4}{Sg} \os... | Proof. Corollary 7.25 shows that \( G : {\sum }^{2}\left( {V}^{ * }\right) \rightarrow \mathcal{R}\left( {V}^{ * }\right) \) is an isomorphism in dimension 3. Since \( {\operatorname{tr}}_{g} \circ G \) is the identity, it follows that \( {\operatorname{tr}}_{g} \) is also an isomorphism. Because \( {\operatorname{tr}}... | Yes |
Corollary 7.27 (The Curvature Tensor in Dimension 2). On every Riemannian or pseudo-Riemannian manifold \( \left( {M, g}\right) \) of dimension 2, the Riemann and Ricci tensors are determined by the scalar curvature as follows:\n\n\[ \n{Rm} = \frac{1}{4}{Sg} \otimes g,\;{Rc} = \frac{1}{2}{Sg}.\n\] | Proof. In dimension 2, it follows from Corollary 7.25(b) that there is some scalar function \( f \) such that \( {Rm} = {fg} \odot g \) . Taking traces, we find from Lemma 7.22(d) that \( {Rc} = {\operatorname{tr}}_{g}\left( {Rm}\right) = {2fg} \), and then \( S = {\operatorname{tr}}_{g}\left( {Rc}\right) = {2f}{\opera... | Yes |
Proposition 7.28 (The Ricci Decomposition of the Curvature Tensor). Let \( \left( {M, g}\right) \) be a Riemannian or pseudo-Riemannian manifold of dimension \( n \geq 3 \) . Then the \( \left( {0,4}\right) \) -curvature tensor of \( g \) has the following orthogonal decomposition:\n\n\[ \n{Rm} = W + \frac{1}{n - 2}\ov... | Proof. The decomposition (7.40) follows immediately by substituting (7.27) into the definition of the Weyl tensor and simplifying. The decomposition is orthogonal thanks to Lemma 7.22(e), and (7.41) follows from Lemma 7.22(f). | Yes |
Proposition 7.29 (Conformal Transformation of the Levi-Civita Connection). Let \( \left( {M, g}\right) \) be a Riemannian or pseudo-Riemannian \( n \) -manifold (with or without boundary), and let \( \widetilde{g} = {e}^{2f}g \) be any metric conformal to \( g \) . If \( \nabla \) and \( \widetilde{\nabla } \) denote t... | Proof. Formula (7.43) is a straightforward computation using formula (5.10) for the Christoffel symbols in coordinates, and then (7.42) follows by expanding everything in coordinates and using (7.43). | No |
Theorem 7.30 (Conformal Transformation of the Curvature). Let \( g \) be a Riemannian or pseudo-Riemannian metric on an n-manifold \( M \) with or without boundary, \( f \in {C}^{\infty }\left( M\right) \), and \( \widetilde{g} = {e}^{2f}g \). In the Riemannian case, the curvature tensors of \( \widetilde{g} \) (repres... | Proof. We begin with (7.44). The plan is to choose local coordinates and insert formula (7.43) for the Christoffel symbols into the coordinate formula (7.8) for the coefficients of the curvature tensor. As in the proof of the differential Bianchi identity, we can make the computations much more tractable by computing t... | Yes |
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