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Lemma 3.15 For all integers \( 0 < a < b \) we have\n\n(i) \( \mathop{\sum }\limits_{{n = a}}^{b}\frac{\chi \left( n\right) }{{n}^{1/2}} = O\left( {a}^{-1/2}\right) \), \n\n(ii) \( \mathop{\sum }\limits_{{n = a}}^{b}\frac{\chi \left( n\right) }{n} = O\left( {a}^{-1}\right) \). | Proof. This argument is similar to the proof of Proposition 3.4; we use summation by parts. Let \( {s}_{n} = \mathop{\sum }\limits_{{1 \leq k \leq n}}\chi \left( k\right) \), and remember that \( \left| {s}_{n}\right| \leq q \) for all \( n \) . Then\n\n\[ \mathop{\sum }\limits_{{n = a}}^{b}\frac{\chi \left( n\right) }... | Yes |
Lemma 1.2 If \( f \) is real-valued integrable on \( \left\lbrack {a, b}\right\rbrack \) and \( \varphi \) is a real-valued continuous function on \( \mathbb{R} \), then \( \varphi \circ f \) is also integrable on \( \left\lbrack {a, b}\right\rbrack \) . | Proof. Let \( \epsilon > 0 \) and remember that \( f \) is bounded, say \( \left| f\right| \leq M \) . Since \( \varphi \) is uniformly continuous on \( \left\lbrack {-M, M}\right\rbrack \) we may choose \( \delta > 0 \) so that if \( s, t \in \left\lbrack {-M, M}\right\rbrack \) and \( \left| {s - t}\right| < \delta \... | Yes |
Proposition 1.3 A bounded monotonic function \( f \) on an interval \( \left\lbrack {a, b}\right\rbrack \) is integrable. | Proof. We may assume without loss of generality that \( a = 0, b = 1 \) , and \( f \) is monotonically increasing. Then, for each \( N \), we choose the uniform partition \( {P}_{N} \) given by \( {x}_{j} = j/N \) for all \( j = 0,\ldots, N \) . If \( {\alpha }_{j} = \) \( f\left( {x}_{j}\right) \), then we have\n\n\[ ... | Yes |
Proposition 1.4 Let \( f \) be a bounded function on the compact interval \( \left\lbrack {a, b}\right\rbrack \) . If \( c \in \left( {a, b}\right) \), and if for all small \( \delta > 0 \) the function \( f \) is integrable on the intervals \( \left\lbrack {a, c - \delta }\right\rbrack \) and \( \left\lbrack {c + \del... | Proof. Suppose \( \left| f\right| \leq M \) and let \( \epsilon > 0 \) . Choose \( \delta > 0 \) (small) so that \( {4\delta M} \leq \epsilon /3 \) . Now let \( {P}_{1} \) and \( {P}_{2} \) be partitions of \( \left\lbrack {a, c - \delta }\right\rbrack \) and \( \lbrack c + \) \( \delta, b\rbrack \) so that for each \(... | Yes |
Lemma 1.6 The union of countably many sets of measure 0 has measure 0. | Proof. Say \( {E}_{1},{E}_{2},\ldots \) are sets of measure 0, and let \( E = { \cup }_{i = 1}^{\infty }{E}_{i} \) . Let \( \epsilon > 0 \), and for each \( i \) choose open interval \( {I}_{i,1},{I}_{i,2},\ldots \) so that\n\n\[ \n{E}_{i} \subset \mathop{\bigcup }\limits_{{k = 1}}^{\infty }{I}_{i, k}\;\text{ and }\;\m... | Yes |
Lemma 1.8 If \( \epsilon > 0 \), then the set \( {A}_{\epsilon } \) is closed and therefore compact. | Proof. The argument is simple. Suppose \( {c}_{n} \in {A}_{\epsilon } \) converges to \( c \) and assume that \( c \notin {A}_{\epsilon } \) . Write \( \operatorname{osc}\left( {f, c}\right) = \epsilon - \delta \) where \( \delta > 0 \) . Select \( r \) so that \( \operatorname{osc}\left( {f, c, r}\right) < \epsilon - ... | No |
Theorem 2.1 Let \( f \) be a continuous function defined on a closed rectangle \( R \subset {\mathbb{R}}^{d} \). Suppose \( R = {R}_{1} \times {R}_{2} \) where \( {R}_{1} \subset {\mathbb{R}}^{{d}_{1}} \) and \( {R}_{2} \subset {\mathbb{R}}^{{d}_{2}} \) with \( d = {d}_{1} + {d}_{2} \). If we write \( x = \left( {{x}_{... | Proof. The continuity of \( F \) follows from the uniform continuity of \( f \) on \( R \) and the fact that \[ \left| {F\left( {x}_{1}\right) - F\left( {x}_{1}^{\prime }\right) }\right| \leq {\int }_{{R}_{2}}\left| {f\left( {{x}_{1},{x}_{2}}\right) - f\left( {{x}_{1}^{\prime },{x}_{2}}\right) }\right| d{x}_{2}. \] To ... | Yes |
Theorem 2.2 Suppose \( A \) and \( B \) are compact subsets of \( {\mathbb{R}}^{d} \) and \( g : A \rightarrow B \) is a diffeomorphism of class \( {C}^{1} \) . If \( f \) is continuous on \( B \) , then\n\n\[ \n{\int }_{g\left( A\right) }f\left( x\right) {dx} = {\int }_{A}f\left( {g\left( y\right) }\right) \left| {\de... | The proof of this theorem consists first of an analysis of the special situation when \( g \) is a linear transformation \( L \) . In this case, if \( R \) is a rectangle, then\n\n\[ \n\left| {g\left( R\right) }\right| = \left| {\det \left( L\right) }\right| \left| R\right|\n\]\nwhich explains the term \( \left| {\det ... | Yes |
Theorem 1.6. (Total Curvature Theorem) If \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow {\mathbf{R}}^{2} \) is a unit speed simple closed curve such that \( \dot{\gamma }\left( a\right) = \dot{\gamma }\left( b\right) \), and \( N \) is the inward-pointing normal, then\n\n\[{\int }_{a}^{b}{\kappa }_{N}\left( ... | The second will be derived as a consequence of a more general result in Chapter 9; the proof of the first is left to Problem 9-6. | No |
Lemma 2.1. Let \( V \) be a finite-dimensional vector space. There is a natural (basis-independent) isomorphism between \( {T}_{l + 1}^{k}\left( V\right) \) and the space of multilinear maps\n\n\[ \underset{l}{\underbrace{{V}^{ * } \times \cdots \times {V}^{ * }}} \times \underset{k}{\underbrace{V \times \cdots \times ... | Exercise 2.1. Prove Lemma 2.1. [Hint: In the special case \( k = 1, l = 0 \) , consider the map \( \Phi : \operatorname{End}\left( V\right) \rightarrow {T}_{1}^{1}\left( V\right) \) by letting \( {\Phi A} \) be the \( \left( \begin{array}{l} 1 \\ 1 \end{array}\right) \) -tensor defined by \( {\Phi A}\left( {\omega, X}\... | No |
Lemma 2.2. Let \( M \) be a smooth manifold, \( E \) a set, and \( \pi : E \rightarrow M \) a surjective map. Suppose we are given an open covering \( \left\{ {U}_{\alpha }\right\} \) of \( M \) together with bijective maps \( {\varphi }_{\alpha } : {\pi }^{-1}\left( {U}_{\alpha }\right) \rightarrow {U}_{\alpha } \time... | Proof. For each \( p \in M \), let \( {E}_{p} = {\pi }^{-1}\left( p\right) \). If \( p \in {U}_{\alpha } \), observe that the map \( {\left( {\varphi }_{\alpha }\right) }_{p} : {E}_{p} \rightarrow \{ p\} \times {\mathbf{R}}^{k} \) obtained by restricting \( {\varphi }_{\alpha } \) is a bijection. We can define a vector... | Yes |
Lemma 3.1. Let \( g \) be a Riemannian metric on a manifold \( M \) . There is a unique fiber metric on each tensor bundle \( {T}_{l}^{k}M \) with the property that if \( \left( {{E}_{1},\ldots ,{E}_{n}}\right) \) is an orthonormal basis for \( {T}_{p}M \) and \( \left( {{\varphi }^{1},\ldots ,{\varphi }^{n}}\right) \)... | Exercise 3.8. Prove Lemma 3.1 by showing that in any local coordinate system, the required inner product is given by\n\n\[ \langle F, G\rangle = {g}^{{i}_{1}{r}_{1}}\cdots {g}^{{i}_{k}{r}_{k}}{g}_{{j}_{1}{s}_{1}}\cdots {g}_{{j}_{l}{s}_{l}}{F}_{{i}_{1}\ldots {i}_{k}}^{{j}_{1}\ldots {j}_{l}}{G}_{{r}_{1}\ldots {r}_{k}}^{{... | No |
Lemma 3.2. On any oriented Riemannian n-manifold \( \left( {M, g}\right) \), there is a unique \( n \) -form \( {dV} \) satisfying the property that \( {dV}\left( {{E}_{1},\ldots ,{E}_{n}}\right) = 1 \) whenever \( \left( {{E}_{1},\ldots ,{E}_{n}}\right) \) is an oriented orthonormal basis for some tangent space \( {T}... | Exercise 3.9. Prove Lemma 3.2, and show that the expression for \( {dV} \) with respect to any oriented local frame \( \left\{ {E}_{i}\right\} \) is\n\n\[ \n{dV} = \sqrt{\det \left( {g}_{ij}\right) }{\varphi }^{1} \land \cdots \land {\varphi }^{n}, \n\] \n\nwhere \( {g}_{ij} = \left\langle {{E}_{i},{E}_{j}}\right\rangl... | No |
Proposition 3.3. \( O\left( {n + 1}\right) \) acts transitively on orthonormal bases on \( {\mathbf{S}}_{R}^{n} \) . More precisely, given any two points \( p,\widetilde{p} \in {\mathbf{S}}_{R}^{n} \), and orthonormal bases \( \left\{ {E}_{i}\right\} \) for \( {T}_{p}{\mathbf{S}}_{R}^{n} \) and \( \left\{ {\widetilde{E... | Proof. It suffices to show that given any \( p \in {\mathbf{S}}_{R}^{n} \) and any orthonormal basis \( \left\{ {E}_{i}\right\} \) for \( {T}_{p}{\mathbf{S}}_{R}^{n} \), there is an orthogonal map that takes the \ | No |
Lemma 3.4. Stereographic projection is a conformal equivalence between \( {\mathbf{S}}_{R}^{n} - \{ N\} \) and \( {\mathbf{R}}^{n} \) . | Proof. The inverse map \( {\sigma }^{-1} \) is a local parametrization, so we will use it to compute the pullback metric. Consider an arbitrary point \( q \in {\mathbf{R}}^{n} \) and a vector \( V \in {T}_{q}{\mathbf{R}}^{n} \), and compute\n\n\[ \n{\left( {\sigma }^{-1}\right) }^{ * }{\overset{ \circ }{g}}_{R}\left( {... | Yes |
Proposition 3.6. \( {O}_{ + }\left( {n,1}\right) \) acts transitively on the set of orthonormal bases on \( {\mathbf{H}}_{R}^{n} \), and therefore \( {\mathbf{H}}_{R}^{n} \) is homogeneous and isotropic. | Proof. The argument is entirely analogous to the proof of Proposition 3.3, so we give only a sketch. If \( p \in {\mathbf{H}}_{R}^{n} \) and \( \left\{ {E}_{i}\right\} \) is an orthonormal basis for \( {T}_{p}{\mathbf{H}}_{R}^{n} \), an easy computation shows that \( \left\{ {{E}_{1},\ldots ,{E}_{n},{E}_{n + 1} = p/R}\... | Yes |
Lemma 4.1. If \( \nabla \) is a connection in a bundle \( E, X \in \mathcal{T}\left( M\right), Y \in \mathcal{E}\left( M\right) \) , and \( p \in M \), then \( {\left. {\nabla }_{X}Y\right| }_{p} \) depends only on the values of \( X \) and \( Y \) in an arbitrarily small neighborhood of \( p \) . More precisely, if \(... | Proof. First consider \( Y \) . Replacing \( Y \) by \( Y - \widetilde{Y} \), it clearly suffices to show that \( {\left. {\nabla }_{X}Y\right| }_{p} = 0 \) if \( Y \) vanishes on a neighborhood \( U \) of \( p \) .\n\nChoose a bump function \( \varphi \in {C}^{\infty }\left( M\right) \) with support in \( U \) such th... | Yes |
Lemma 4.2. With notation as in 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. By linearity, it suffices to show that \( {\left. {\nabla }_{X}Y\right| }_{p} = 0 \) whenever \( {X}_{p} = \) 0 . Choose a coordinate neighborhood \( U \) of \( p \), and write \( X = {X}^{i}{\partial }_{i} \) in coordinates on \( U \), with \( {X}^{i}\left( p\right) = 0 \) . Then, for any \( Y \in \mathcal{E}\l... | Yes |
Lemma 4.3. Let \( \nabla \) be a linear connection, and let \( X, Y \in \mathcal{T}\left( U\right) \) be expressed in terms of a local frame by \( X = {X}^{i}{E}_{i}, Y = {Y}^{j}{E}_{j} \) . Then\n\n\[ \n{\nabla }_{X}Y = \left( {X{Y}^{k} + {X}^{i}{Y}^{j}{\Gamma }_{ij}^{k}}\right) {E}_{k} \n\]\n\n(4.3) | Proof. Just use the defining rules for a connection and compute:\n\n\[ \n{\nabla }_{X}Y = {\nabla }_{X}\left( {{Y}^{j}{E}_{j}}\right) \n\]\n\n\[ \n= \left( {X{Y}^{j}}\right) {E}_{j} + {Y}^{j}{\nabla }_{{X}^{i}{E}_{i}}{E}_{j} \n\]\n\n\[ \n= \left( {X{Y}^{j}}\right) {E}_{j} + {X}^{i}{Y}^{j}{\nabla }_{{E}_{i}}{E}_{j} \n\]... | Yes |
Lemma 4.4. Suppose \( M \) is a manifold covered by a single coordinate chart. There is a one-to-one correspondence between linear connections on \( M \) and choices of \( {n}^{3} \) smooth functions \( \left\{ {\Gamma }_{ij}^{k}\right\} \) on \( M \), by the rule\n\n\[{\nabla }_{X}Y = \left( {{X}^{i}{\partial }_{i}{Y}... | Proof. Observe that (4.5) is equivalent to (4.3) when \( {E}_{i} = {\partial }_{i} \) is a coordinate frame, so for every connection the functions \( \left\{ {\Gamma }_{ij}^{k}\right\} \) defined by (4.2) satisfy (4.5). On the other hand, given \( \left\{ {\Gamma }_{ij}^{k}\right\} \), it is easy to see by inspection t... | No |
Proposition 4.5. Every manifold admits a linear connection. | Proof. Cover \( M \) with coordinate charts \( \left\{ {U}_{\alpha }\right\} \) ; the preceding lemma guarantees the existence of a connection \( {\nabla }^{\alpha } \) on each \( {U}_{\alpha } \) . Choosing a partition of unity \( \left\{ {\varphi }_{\alpha }\right\} \) subordinate to \( \left\{ {U}_{\alpha }\right\} ... | Yes |
Lemma 4.7. If \( \nabla \) is a linear connection on \( M \), and \( F \in {\mathcal{T}}_{l}^{k}\left( M\right) \), the map \( \nabla F : {\mathcal{T}}^{1}\left( M\right) \times \cdots \times {\mathcal{T}}^{1}\left( M\right) \times \mathcal{T}\left( M\right) \times \cdots \times \mathcal{T}\left( M\right) \rightarrow {... | Proof. This follows immediately from the tensor characterization lemma: \( {\nabla }_{X}F \) 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 |
Lemma 4.8. Let \( \nabla \) be a linear connection. The components of the total covariant derivative of a \( \left( \begin{array}{l} k \\ l \end{array}\right) \) -tensor field \( F \) with respect to a coordinate system are given by \[ {F}_{{i}_{1}\ldots {i}_{k};m}^{{j}_{1}\ldots {j}_{l}} = {\partial }_{m}{F}_{{i}_{1}\... | Exercise 4.6. Prove Lemma 4.8. | No |
Lemma 4.9. Let \( \nabla \) be a linear connection on \( M \) . For each curve \( \gamma : I \rightarrow \) \( M,\nabla \) determines a unique operator\n\n\[ \n{D}_{t} : \mathcal{T}\left( \gamma \right) \rightarrow \mathcal{T}\left( \gamma \right)\n\]\n\nsatisfying the following properties:\n\n(a) Linearity over \( \ma... | Proof. First we show uniqueness. Suppose \( {D}_{t} \) is such an operator, and let \( {t}_{0} \in I \) be arbitrary. An argument similar to that of Lemma 4.1 shows that the value of \( {D}_{t}V \) at \( {t}_{0} \) depends only on the values of \( V \) in any interval \( \left( {{t}_{0} - \varepsilon ,{t}_{0} + \vareps... | Yes |
Theorem 4.10. (Existence and Uniqueness of Geodesics) Let \( M \) be a manifold with a linear connection. For any \( p \in M \), any \( V \in {T}_{p}M \), and any \( {t}_{0} \in \mathbf{R} \), there exist an open interval \( I \subset \mathbf{R} \) containing \( {t}_{0} \) and a geodesic \( \gamma : I \rightarrow M \) ... | Proof. Choose coordinates \( \left( {x}^{i}\right) \) on some neighborhood \( U \) of \( p \). From (4.10), a curve \( \gamma : I \rightarrow U \) is a geodesic if and only if its component functions \( \gamma \left( t\right) = \left( {{x}^{1}\left( t\right) ,\ldots ,{x}^{n}\left( t\right) }\right) \) satisfy the geode... | Yes |
Theorem 4.11. (Parallel Translation) Given a curve \( \gamma : I \rightarrow M,{t}_{0} \in \) \( I \), and a vector \( {V}_{0} \in {T}_{\gamma \left( {t}_{0}\right) }M \), there exists a unique parallel vector field \( V \) along \( \gamma \) such that \( V\left( {t}_{0}\right) = {V}_{0} \) . | Proof of Theorem 4.11. First suppose \( \gamma \left( I\right) \) is contained in a single coordinate chart. Then, using formula (4.10), \( V \) is parallel along \( \gamma \) if and only if\n\n\[{\dot{V}}^{k}\left( t\right) = - {V}^{j}\left( t\right) {\dot{\gamma }}^{i}\left( t\right) {\Gamma }_{ij}^{k}\left( {\gamma ... | Yes |
Theorem 4.12. (Existence and Uniqueness for Linear ODEs) Let \( I \subset \mathbf{R} \) be an interval, and for \( 1 \leq j, k \leq n \) let \( {A}_{j}^{k} : I \rightarrow \mathbf{R} \) be arbitrary smooth functions. The linear initial-value problem\n\n\[ \n{\dot{V}}^{k}\left( t\right) = {A}_{j}^{k}\left( t\right) {V}^... | Exercise 4.11. Prove Theorem 4.12, as follows. Consider the vector field \( Y \) on \( I \times {\mathbf{R}}^{n} \) given by\n\n\[ \n{Y}^{0}\left( {{x}^{0},\ldots ,{x}^{n}}\right) = 1 \n\]\n\n\[ \n{Y}^{k}\left( {{x}^{0},\ldots ,{x}^{n}}\right) = {A}_{j}^{k}\left( {x}^{0}\right) {x}^{j},\;k = 1,\ldots, n. \n\]\n\n(a) Sh... | No |
Lemma 5.1. The operator \( {\nabla }^{\top } \) is well defined, and is a connection on \( M \) . | Proof. Since the value of \( {\bar{\nabla }}_{X}Y \) at a point \( p \in M \) depends only on \( {X}_{p} \) , \( {\nabla }_{X}^{\top }Y \) is clearly independent of the choice of vector field extending \( X \) . On the other hand, because of the result of Exercise 4.7, the value of \( {\bar{\nabla }}_{X}Y \) at \( p \)... | Yes |
Lemma 5.3. The tangential connection on an embedded submanifold \( M \subset \) \( {\mathbf{R}}^{n} \) is symmetric. | Exercise 5.3. Prove Lemma 5.3. [Hint: If \( X \) and \( Y \) are vector fields on \( {\mathbf{R}}^{n} \) that are tangent to \( M \) at points of \( M \), so is \( \left\lbrack {X, Y}\right\rbrack \) by Exercise 2.3.] | No |
Theorem 5.4. (Fundamental Lemma of Riemannian Geometry) Let \( \left( {M, g}\right) \) be a Riemannian (or pseudo-Riemannian) manifold. There exists a unique linear connection \( \nabla \) on \( M \) that is compatible with \( g \) and symmetric. | Proof. We prove uniqueness first, by deriving a formula for \( \nabla \) . Suppose, therefore, that \( \nabla \) is such a connection, and let \( X, Y, Z \in \mathcal{T}\left( M\right) \) be arbitrary vector fields. Writing the compatibility equation three times with \( X, Y, Z \) cyclically permuted, we obtain\n\n\[ X... | Yes |
Lemma 5.5. All Riemannian geodesics are constant speed curves. | Proof. Let \( \gamma \) be a Riemannian geodesic. Since \( \dot{\gamma } \) is parallel along \( \gamma \), its length \( \left| \dot{\gamma }\right| = \langle \dot{\gamma },\dot{\gamma }{\rangle }^{1/2} \) is constant by Lemma 5.2(d). | Yes |
Proposition 5.6. (Naturality of the Riemannian Connection) Suppose \( \varphi : \left( {M, g}\right) \rightarrow \left( {\widetilde{M},\widetilde{g}}\right) \) is an isometry.\n\n(a) \( \varphi \) takes the Riemannian connection \( \nabla \) of \( g \) to the Riemannian connection \( \widetilde{\nabla } \) of \( \widet... | Exercise 5.4. Prove Proposition 5.6 as follows. For part (a), define a map\n\n\[{\varphi }^{ * }\widetilde{\nabla } : \mathfrak{T}\left( M\right) \times \mathfrak{T}\left( M\right) \rightarrow \mathfrak{T}\left( M\right)\]\n\nby\n\n\[{\left( {\varphi }^{ * }\widetilde{\nabla }\right) }_{X}Y = {\varphi }_{ * }^{-1}\left... | No |
Proposition 5.7. (Properties of the Exponential Map)\n\n(a) \( \\mathcal{E} \) is an open subset of TM containing the zero section, and each set \( {\\mathcal{E}}_{p} \) is star-shaped with respect to 0 .\n\n(b) For each \( V \\in {TM} \), the geodesic \( {\\gamma }_{V} \) is given by\n\n\[ \n{\\gamma }_{V}\\left( t\\r... | Proof of Proposition 5.7. The rescaling lemma with \( t = 1 \) says precisely that \( \\exp \\left( {cV}\\right) = {\\gamma }_{cV}\\left( 1\\right) = {\\gamma }_{V}\\left( c\\right) \) whenever either side is defined; this is (b). Moreover, if \( V \\in {\\mathcal{E}}_{p} \), by definition \( {\\gamma }_{V} \) is defin... | No |
Lemma 5.8. (Rescaling Lemma) For any \( V \in {TM} \) and \( c, t \in R \) , \[ {\gamma }_{cV}\left( t\right) = {\gamma }_{V}\left( {ct}\right) \] whenever either side is defined. | Proof. It suffices to show that \( {\gamma }_{cV}\left( t\right) \) exists and (5.5) holds whenever the right-hand side is defined, for then the converse statement follows by replacing \( V \) by \( {cV}, t \) by \( {ct} \), and \( c \) by \( 1/c \) . Suppose the domain of \( {\gamma }_{V} \) is the open interval \( I ... | Yes |
Lemma 5.10. (Normal Neighborhood Lemma) For any \( p \in M \), there is a neighborhood \( \mathcal{V} \) of the origin in \( {T}_{p}M \) and a neighborhood \( \mathcal{U} \) of \( p \) in \( M \) such that \( {\exp }_{p} : \mathcal{V} \rightarrow \mathcal{U} \) is a diffeomorphism. | Proof. This follows immediately from the inverse function theorem, once we show that \( {\left( {\exp }_{p}\right) }_{ * } \) is invertible at 0 . Since \( {T}_{p}M \) is a vector space, there is a natural identification \( {T}_{0}\left( {{T}_{p}M}\right) = {T}_{p}M \) . Under this identification, we will show that \( ... | Yes |
Lemma 6.1. For any curve segment \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow M \), and any reparametrization \( \widetilde{\gamma } \) of \( \gamma, L\left( \gamma \right) = L\left( \widetilde{\gamma }\right) \) . | Exercise 6.1. Prove Lemma 6.1. | No |
Lemma 6.3. (Symmetry Lemma) Let \( \Gamma : \left( {-\varepsilon ,\varepsilon }\right) \times \left\lbrack {a, b}\right\rbrack \rightarrow M \) be an admissible family of curves in a Riemannian (or pseudo-Riemannian) manifold. On any rectangle \( \left( {-\varepsilon ,\varepsilon }\right) \times \left\lbrack {{a}_{i - ... | Proof. This is a local question, so we may compute in coordinates \( \left( {x}^{i}\right) \) around any 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 |
Lemma 6.4. If \( \gamma \) is an admissible curve and \( V \) is a 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. Set \( \Gamma \left( {s, t}\right) = \exp \left( {{sV}\left( t\right) }\right) \) (Figure 6.5). By compactness of \( \left\lbrack {a, b}\right\rbrack \), there is some positive \( \varepsilon \) such that \( \Gamma \) is defined on \( \left( {-\varepsilon ,\varepsilon }\right) \times \left\lbrack {a, b}\right\rb... | Yes |
Theorem 6.6. 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 unit speed, and let \( a = \) \( {a}_{0} < \cdots < {a}_{k} = b \) be a subdivision such that \( \gamma \) is smooth on \( \left\lbrack {{a}_{i - 1},{a}_{i}}\right\rbrack \) . If \( \Gamma \) is any proper variation of \( \ga... | Yes |
Corollary 6.7. A unit speed admissible curve \( \gamma \) is a critical point for \( L \) if and only if it is a geodesic. | Proof. If \( \gamma \) is a critical point, the proof of Theorem 6.6 goes through without modification to show that \( \gamma \) is a geodesic. Conversely, if \( \gamma \) is a geodesic, then the first term in the second variation formula vanishes by the geodesic equation, and the second term vanishes because \( \dot{\... | Yes |
Corollary 6.9. Let \( \left( {x}^{i}\right) \) be normal coordinates on a geodesic ball \( \mathcal{U} \) centered at \( p \in M \), and let \( r \) be the radial distance function as defined in (5.9). Then \( \operatorname{grad}r = \partial /\partial r \) on \( \mathcal{U} - \{ p\} \) . | Proof. For any \( q \in \mathcal{U} - \{ p\} \) and \( Y \in {T}_{q}M \), we need to show that\n\n\[ \n{dr}\left( Y\right) = \left\langle {\frac{\partial }{\partial r}, Y}\right\rangle \n\]\n\n(6.4)\n\nThe geodesic sphere \( {\exp }_{p}\left( {\partial {B}_{R}\left( 0\right) }\right) \) through \( q \) is characterized... | Yes |
Corollary 6.11. Within any geodesic ball around \( p \in M \), the radial distance function \( r\left( x\right) \) defined by (5.9) is equal to the Riemannian distance from \( p \) to \( x \) . | Proof. The radial geodesic \( \gamma \) from \( p \) to \( x \) is minimizing by Proposition 6.10. Since its velocity is equal to \( \partial /\partial r \), which is a unit vector in both the \( g \) norm and the Euclidean norm in normal coordinates, the \( g \) -length of \( \gamma \) is equal to its Euclidean length... | Yes |
Theorem 6.12. Every Riemannian geodesic is locally minimizing. | Proof. Let \( \gamma : I \rightarrow M \) be a geodesic, which we may assume to be defined on an open interval, and let \( {t}_{0} \in I \) . Let \( \mathcal{W} \) be a uniformly normal neighborhood of \( \gamma \left( {t}_{0}\right) \), and let \( \mathcal{U} \subset I \) be the connected component of \( {\gamma }^{-1... | Yes |
Theorem 6.13. (Hopf-Rinow) A connected Riemannian manifold is geodesically complete if and only if it is complete as a metric space. | Proof. Suppose first that \( M \) is complete as a metric space but not geodesically complete. Then there is some unit speed geodesic \( \gamma : \lbrack 0, b) \rightarrow M \) that extends to no interval \( \lbrack 0, b + \varepsilon ) \) for \( \varepsilon > 0 \) . Let \( \left\{ {t}_{i}\right\} \) be any increasing ... | Yes |
Lemma 7.2. The Riemann curvature endomorphism and curvature tensor are local isometry invariants. More precisely, if \( \varphi : \left( {M, g}\right) \rightarrow \left( {\widetilde{M},\widetilde{g}}\right) \) is a local isometry, then\n\n\[{\varphi }^{ * }\widetilde{Rm} = {Rm}\]\n\n\[ \widetilde{R}\left( {{\varphi }_{... | Exercise 7.2. Prove Lemma 7.2. | No |
Proposition 7.4. (Symmetries of the Curvature Tensor) The curvature tensor has the following symmetries for any vector fields \( W, X, Y \) , \( Z \) :\n\n(a) \( \operatorname{Rm}\left( {W, X, Y, Z}\right) = - \operatorname{Rm}\left( {X, W, Y, Z}\right) \) .\n\n(b) \( \operatorname{Rm}\left( {W, X, Y, Z}\right) = - \op... | Proof of Proposition 7.4. Identity (a) is immediate from the obvious fact that \( R\left( {W, X}\right) Y = - R\left( {X, W}\right) Y \) . To prove (b), it suffices to show that \( {Rm}\left( {W, X, Y, Y}\right) = 0 \) for all \( Y \), for then (b) follows from the expansion of \( \operatorname{Rm}\left( {W, X, Y + Z, ... | Yes |
Proposition 7.5. (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 \( {Rm} \) ,(7.6) 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 standard shortcut for such ... | Yes |
Lemma 7.7. (Contracted Bianchi Identity) The covariant derivatives of the Ricci and scalar curvatures satisfy the following identity:\n\n\[ \operatorname{div}{Rc} = \frac{1}{2}\nabla S \]\n\nwhere div is the divergence operator (Problem 3-3). In components, this is\n\n\[ {R}_{{ij};}{}^{j} = \frac{1}{2}{S}_{;i} \] | Proof. Formula (7.9) follows immediately by contracting the component form (7.7) of the differential Bianchi identity on the indices \( i, l \) and then again on \( j, k \), after raising one index of each pair. | No |
Proposition 7.8. If \( g \) is an Einstein metric on a connected manifold of dimension \( n \geq 3 \), its scalar curvature is constant. | Proof. Taking the covariant derivative of each side of (7.10) and noting that the covariant derivative of the metric is zero, we see that the Einstein condition implies\n\n\[ {R}_{{ij};k} = \frac{1}{n}{S}_{;k}{g}_{ij} \]\n\nTracing this equation on \( j \) and \( k \), and comparing with the contracted Bianchi identity... | Yes |
Lemma 8.1. The second fundamental form is\n\n(a) independent of the extensions of \( X \) and \( Y \) ;\n\n(b) bilinear over \( {C}^{\infty }\left( M\right) \) ; and\n\n(c) symmetric in \( X \) and \( Y \) . | Proof. First we show that the symmetry of \( \Pi \) follows from the symmetry of the connection \( \widetilde{\nabla } \) . Let \( X \) and \( Y \) be extended arbitrarily to \( M \) . Then\n\n\[ \Pi \left( {X, Y}\right) - \Pi \left( {Y, X}\right) = {\left( {\widetilde{\nabla }}_{X}Y - {\widetilde{\nabla }}_{Y}X\right)... | No |
Theorem 8.2. (The Gauss Formula) If \( X, Y \in \mathcal{T}\left( M\right) \) are extended arbitrarily to vector fields on \( \widetilde{M} \), the following formula holds along \( M \) : | 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{T}\left( M\right) \times \mathcal{T}\left( M\right) ... | Yes |
Lemma 8.3. (The Weingarten Equation) Suppose \( X, Y \in \mathfrak{T}\left( M\right) \) and \( N \in \mathcal{N}\left( M\right) \) . When \( X, Y, N \) are extended arbitrarily to \( \widetilde{M} \), the following equation holds at points of \( M \) : | Proof. Since \( \langle N, Y\rangle \) vanishes identically along \( M \) and \( X \) is tangent to \( M \) , the following holds along \( M \) :\n\n\[ 0 = X\langle N, Y\rangle \]\n\n\[ = \left\langle {{\widetilde{\nabla }}_{X}N, Y}\right\rangle + \left\langle {N,{\widetilde{\nabla }}_{X}Y}\right\rangle \]\n\n\[ = \lef... | Yes |
Theorem 8.4. (The Gauss Equation) For any \( X, Y, Z, W \in {T}_{p}M \), the following equation holds:\n\n\[ \widetilde{\operatorname{Rm}}\left( {X, Y, Z, W}\right) = \operatorname{Rm}\left( {X, Y, Z, W}\right) \]\n\n\[ - \langle \Pi \left( {X, W}\right) ,\Pi \left( {Y, Z}\right) \rangle + \langle \Pi \left( {X, Z}\rig... | Proof. Let \( X, Y, Z, W \) be extended arbitrarily to vector fields on \( M \), and then to vector fields on \( \widetilde{M} \) that are tangent to \( M \) at points of \( M \) . Along \( M \), the Gauss formula gives\n\n\[ \widetilde{Rm}\left( {X, Y, Z, W}\right) = \left\langle {{\widetilde{\nabla }}_{X}{\widetilde{... | Yes |
Lemma 8.5. (The Gauss Formula Along a Curve) Let \( M \) be a Riemannian submanifold of \( \widetilde{M} \), and \( \gamma \) a curve in \( M \) . For any vector field \( V \) tangent to \( M \) along \( \gamma \) ,\n\n\[{\widetilde{D}}_{t}V = {D}_{t}V + \Pi \left( {\dot{\gamma }, V}\right)\] | Proof. In terms of an adapted orthonormal frame, \( V \) can be written \( V\left( t\right) = \) \( {V}^{i}\left( t\right) {E}_{i} \), where the sum is only over \( i = 1,\ldots, n \) . Applying the product rule and the Gauss formula, we get\n\n\[{\widetilde{D}}_{t}V = {\dot{V}}^{i}{E}_{i} + {V}^{i}{\widetilde{\nabla }... | Yes |
Theorem 8.6. (Gauss’s Theorema Egregium) Let \( M \subset {\mathbf{R}}^{3} \) be a 2-dimensional submanifold and \( g \) the induced metric on \( M \). For any \( p \in M \) and any basis \( \left( {X, Y}\right) \) for \( {T}_{p}M \), the Gaussian curvature of \( M \) at \( p \) is given \( {by} \)\n\n\[ K = \frac{\ope... | Proof. We begin with the special case in which \( \left( {X, Y}\right) = \left( {{E}_{1},{E}_{2}}\right) \) is an orthonormal basis for \( {T}_{p}M \). In this case the denominator in (8.5) is equal to 1. If we write \( {h}_{ij} = \dot{h}\left( {{E}_{i},{E}_{j}}\right) \), then in this basis \( K = \det s = \det \left(... | Yes |
Lemma 8.7. The Gaussian curvature of a Riemannian 2-manifold is related to the curvature tensor, Ricci tensor, and scalar curvature by the formulas\n\n\[ \n{Rm}\left( {X, Y, Z, W}\right) = K\left( {\langle X, W\rangle \langle Y, Z\rangle -\langle X, Z\rangle \langle Y, W\rangle }\right) \]\n\n\[ \n{Rc}\left( {X, Y}\rig... | Proof. Since both sides of the first equation are tensors, we can compute them in terms of any basis. Let \( \left( {{E}_{1},{E}_{2}}\right) \) be any orthonormal basis for \( {T}_{p}M \) , and consider the components \( {R}_{ijkl} = {Rm}\left( {{E}_{i},{E}_{j},{E}_{k},{E}_{l}}\right) \) of the curvature tensor. In ter... | Yes |
Proposition 8.8. If \( \\left( {X, Y}\\right) \) is any basis for a 2-plane \( \\Pi \\subset {T}_{p}M \), then\n\n\[ K\\left( {X, Y}\\right) = \\frac{\\operatorname{Rm}\\left( {X, Y, Y, X}\\right) }{{\\left| X\\right| }^{2}{\\left| Y\\right| }^{2}-\\langle X, Y{\\rangle }^{2}}.\] | Proof. For this proof, we denote the induced metric on \( {S}_{\\Pi } \) by \( \\widetilde{g} \), and continue to denote the metric on \( M \) by \( g \) . As in the first part of this chapter, we use tildes to denote geometric quantities associated with \( \\widetilde{g} \), but note that now the roles of \( g \) and ... | Yes |
Lemma 8.9. Suppose \( {\mathcal{R}}_{1} \) and \( {\mathcal{R}}_{2} \) are covariant 4-tensors on a vector space \( V \) with an inner product, and both have the symmetries of the curvature tensor (as described in Proposition 7.4). If for every pair of independent vectors \( X, Y \in V \), \[ \frac{{\mathcal{R}}_{1}\le... | Proof. Setting \( \mathcal{R} = {\mathcal{R}}_{1} - {\mathcal{R}}_{2} \), it suffices to show \( \mathcal{R} = 0 \) under the assumption that \( \mathcal{R}\left( {X, Y, Y, X}\right) = 0 \) for all \( X, Y \). For any vectors \( X, Y, Z \), since \( \mathcal{R} \) also has the symmetries of the curvature tensor, \[ 0 =... | Yes |
Lemma 8.10. Suppose \( \left( {M, g}\right) \) is any Riemannian \( n \) -manifold with constant sectional curvature \( C \) . The curvature endomorphism, curvature tensor, Ricci tensor, and scalar curvature of \( g \) are given by the formulas\n\n\[ R\left( {X, Y}\right) Z = C\left( {\\langle Y, Z\\rangle X-\\langle X... | Exercise 8.8. Prove Lemma 8.10. | No |
Lemma 9.2. If \( \gamma \) is a positively oriented curved polygon in \( M \), the rotation angle of \( \gamma \) is \( {2\pi } \) . | Proof. If we use the given coordinate chart to consider \( \gamma \) as a curved polygon in the plane, we can compute its tangent angle function either with respect to \( g \) or with respect to the Euclidean metric \( \bar{g} \) . In either case, \( \operatorname{Rot}\left( \gamma \right) \) is an integral multiple of... | Yes |
Theorem 9.7. (The Gauss-Bonnet Theorem) If \( M \) is a triangulated, compact, oriented, Riemannian 2-manifold, then\n\n\[{\int }_{M}{KdA} = {2\pi \chi }\left( M\right)\] | Proof. Let \( \left\{ {{\Omega }_{i} : i = 1,\ldots ,{N}_{f}}\right\} \) denote the faces of the triangulation, and for each \( i \) let \( \left\{ {{\gamma }_{ij} : j = 1,2,3}\right\} \) be the edges of \( {\Omega }_{i} \) and \( \left\{ {{\theta }_{ij} : j = 1,2,3}\right\} \) its interior angles. Since each exterior ... | Yes |
Corollary 9.8. Let \( M \) be a compact Riemannian 2-manifold and \( K \) 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 \( K \equiv 0 \) or \( K \) takes on both... | Proof. If \( M \) is orientable, the result follows immediately from the Gauss-Bonnet theorem, because a function whose integral is positive, negative, or zero must satisfy the claimed sign condition. If \( M \) is nonorientable, the result follows by applying the Gauss-Bonnet theorem to the orientable double cover \( ... | Yes |
Lemma 10.1. If \( \\Gamma \) is any smooth admissible family of curves, and \( V \) is a smooth vector field along \( \\Gamma \), then\n\n\[ \n{D}_{s}{D}_{t}V - {D}_{t}{D}_{s}V = R\\left( {S, T}\\right) V.\n\] | Proof. This is a local issue, so we can compute in any local coordinates.\n\nWriting \( V\\left( {s, t}\\right) = {V}^{i}\\left( {s, t}\\right) {\\partial }_{i} \), we compute\n\n\[ \n{D}_{t}V = \\frac{\\partial {V}^{i}}{\\partial t}{\\partial }_{i} + {V}^{i}{D}_{t}{\\partial }_{i}\n\]\n\nTherefore,\n\n\[ \n{D}_{s}{D}_... | Yes |
Theorem 10.2. (The Jacobi Equation) Let \( \gamma \) be a geodesic and \( V \) a vector field along \( \gamma \) . If \( V \) is the variation field of a variation through geodesics, then \( V \) satisfies\n\n\[ \n{D}_{t}^{2}V + R\left( {V,\dot{\gamma }}\right) \dot{\gamma } = 0 \n\] | Proof. With \( S \) and \( T \) as before, the preceding lemma implies\n\n\[ \n0 = {D}_{s}{D}_{t}T \n\]\n\n\[ \n= {D}_{t}{D}_{s}T + R\left( {S, T}\right) T \n\]\n\n\[ \n= {D}_{t}{D}_{t}S + R\left( {S, T}\right) T \n\]\n\nwhere the last step follows from the symmetry lemma. Evaluating at \( s = 0 \) , where \( S\left( {... | Yes |
Lemma 10.3. Every Jacobi field along a geodesic \( \gamma \) is the variation field of some variation of \( \gamma \) through geodesics. | Exercise 10.1. Prove Lemma 10.3. [Hint: Let \( \Gamma \left( {s, t}\right) = {\exp }_{\sigma \left( s\right) }{tW}\left( s\right) \) for a suitable curve \( \sigma \) and vector field \( W \) along \( \sigma \) .] | No |
Proposition 10.4. (Existence and Uniqueness of Jacobi Fields) Let \( \gamma : I \rightarrow M \) be a geodesic, \( a \in I \), and \( p = \gamma \left( a\right) \) . For any pair of vectors \( X, Y \in {T}_{p}M \), there is a unique Jacobi field \( J \) along \( \gamma \) satisfying the initial conditions\n\n\[ J\left(... | Proof. Choose an orthonormal basis \( \left\{ {E}_{i}\right\} \) for \( {T}_{p}M \), and extend it to a parallel orthonormal frame along all of \( \gamma \) . Writing \( J\left( t\right) = {J}^{i}\left( t\right) {E}_{i} \), we can express the Jacobi equation as\n\n\[ {\ddot{J}}^{i} + {R}_{jkl}{}^{i}{J}^{j}{\dot{\gamma ... | Yes |
Corollary 10.5. Along any geodesic \( \gamma \), the set of Jacobi fields is a \( {2n} \) - dimensional linear subspace of \( \mathcal{T}\left( \gamma \right) \) . | Proof. Let \( p = \gamma \left( a\right) \) be any point on \( \gamma \), and consider the map from the set of Jacobi fields along \( \gamma \) to \( {T}_{p}M \oplus {T}_{p}M \) by sending \( J \) to \( \left( {J\left( a\right) ,{D}_{t}J\left( a\right) }\right) \) . The preceding proposition says precisely that this ma... | Yes |
Lemma 10.6. Let \( \gamma : I \rightarrow M \) be a geodesic, and \( a \in I \) .\n\n(a) A Jacobi field J along \( \gamma \) is normal if and only if\n\n\[ J\left( a\right) \bot \dot{\gamma }\left( a\right) \text{ and }{D}_{t}J\left( a\right) \bot \dot{\gamma }\left( a\right) . \] | Proof. Using compatibility with the metric and the fact that \( {D}_{t}\dot{\gamma } \equiv 0 \), we compute\n\n\[ \frac{{d}^{2}}{d{t}^{2}}\langle J,\dot{\gamma }\rangle = \left\langle {{D}_{t}^{2}J,\dot{\gamma }}\right\rangle \]\n\n\[ = - \langle R\left( {J,\dot{\gamma }}\right) \dot{\gamma },\dot{\gamma }\rangle \]\n... | Yes |
Lemma 10.7. Let \( p \in M \), let \( \left( {x}^{i}\right) \) be normal coordinates on a neighborhood \( \mathcal{U} \) of \( p \), and let \( \gamma \) be a radial geodesic starting at \( p \) . For any \( W = {W}^{i}{\partial }_{i} \in {T}_{p}M \), the Jacobi field \( J \) along \( \gamma \) such that \( J\left( 0\r... | Proof. An easy computation using formula (4.10) for covariant derivatives in coordinates shows that \( J \) satisfies the specified initial conditions, so it suffices to show that \( J \) is a Jacobi field. If we set \( V = \dot{\gamma }\left( 0\right) \in {T}_{p}M \), then we know from Lemma 5.11 that \( \gamma \) is ... | Yes |
Lemma 10.8. 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 precisely the vector fields\n\n\[ J\\left( t\\right) = u\\left( t\\right) ... | Proof. Since \( g \) has constant curvature, its curvature endomorphism is given by the formula of Lemma 8.10:\n\n\[ R\\left( {X, Y}\\right) Z = C\\left( {\\langle Y, Z\\rangle X-\\langle X, Z\\rangle Y}\\right) \]\n\nSubstituting this into the Jacobi equation, we find that a normal Jacobi field \( J \) satisfies\n\n\[... | Yes |
Proposition 10.9. Suppose \( \left( {M, g}\right) \) is a Riemannian manifold with constant sectional curvature \( C \) . Let \( \left( {x}^{i}\right) \) be Riemannian normal coordinates on a normal neighborhood \( \mathcal{U} \) of \( p \in M \), let \( {\left| \cdot \right| }_{\bar{q}} \) be the Euclidean norm in the... | Proof. By the Gauss lemma, the decomposition \( V = {V}^{\top } + {V}^{ \bot } \) is orthogonal, so \( {\left| V\right| }_{g}^{2} = {\left| {V}^{ \bot }\right| }_{g}^{2} + {\left| {V}^{\top }\right| }_{g}^{2} \) . Since \( \partial /\partial r \) is a unit vector in both the \( g \) and \( \bar{g} \) norms, it is immed... | Yes |
Proposition 10.10. (Local Uniqueness of Constant Curvature Metrics) Let \( \left( {M, g}\right) \) and \( \left( {\widetilde{M},\widetilde{g}}\right) \) be Riemannian manifolds with constant sectional curvature \( C \) . For any points \( p \in M,\widetilde{p} \in \widetilde{M} \), there exist neighborhoods \( \mathcal... | Proof. Choose \( p \in M \) and \( \widetilde{p} \in \widetilde{M} \), and let \( \mathcal{U} \) and \( \widetilde{\mathcal{U}} \) be geodesic balls of small radius \( \varepsilon \) around \( p \) and \( \widetilde{p} \), respectively. Riemannian normal coordinates give maps \( \varphi : \mathcal{U} \rightarrow {B}_{\... | Yes |
Proposition 10.11. Suppose \( p \in M, V \in {T}_{p}M \), and \( q = {\exp }_{p}V \) . Then \( {\exp }_{p} \) is a local diffeomorphism in a neighborhood of \( V \) if and only if \( q \) is not conjugate to \( p \) along the geodesic \( \gamma \left( t\right) = {\exp }_{p}{tV},\;t \in \left\lbrack {0,1}\right\rbrack \... | Proof. By the inverse function theorem, \( {\exp }_{p} \) is a local diffeomorphism near \( V \) if and only if \( {\left( {\exp }_{p}\right) }_{ * } \) is an isomorphism at \( V \), and by dimensional considerations, this occurs if and only if \( {\left( {\exp }_{p}\right) }_{ * } \) is injective at \( V \) .\n\nIdent... | Yes |
Proposition 10.14. For any pair of proper normal vector fields \( V, W \) along a geodesic segment \( \gamma \) , | Proof. On any 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 theorem of calcul... | Yes |
If \( \gamma \) is a geodesic segment from \( p \) to \( q \) that has an interior conjugate point to \( p \), then there exists a proper normal vector field \( X \) along \( \gamma \) such that \( I\left( {X, X}\right) < 0 \) . In particular, \( \gamma \) is not minimizing. | Proof. Suppose \( \gamma : \left\lbrack {0, b}\right\rbrack \rightarrow M \) is a unit speed parametrization of \( \gamma \), and \( \gamma \left( a\right) \) is conjugate to \( \gamma \left( 0\right) \) for some \( 0 < a < b \) . This means there is a nontrivial normal Jacobi field \( J \) along \( {\left. \gamma \rig... | Yes |
Theorem 11.1. (Sturm Comparison Theorem) Suppose \( u \) and \( v \) are differentiable real-valued functions on \( \left\lbrack {0, T}\right\rbrack \), twice differentiable on \( \left( {0, T}\right) \) , and \( u > 0 \) on \( \left( {0, T}\right) \) . Suppose further that \( u \) and \( v \) satisfy\n\n\[ \ddot{u}\le... | Proof. Consider the function \( f\left( t\right) = v\left( t\right) /u\left( t\right) \) defined on \( \left( {0, T}\right) \) . It follows from l’Hôpital’s rule that \( \mathop{\lim }\limits_{{t \rightarrow 0}}f\left( t\right) = \dot{v}\left( 0\right) /\dot{u}\left( 0\right) = 1 \) . Since \( f \) is differentiable on... | Yes |
Corollary 11.3. (Conjugate Point Comparison Theorem) Suppose all sectional curvatures of \( \left( {M, g}\right) \) are bounded above by a constant \( C \) . If\n\n\( C \leq 0 \), then no point of \( M \) has conjugate points along any geodesic. If \( C = 1/{R}^{2} > 0 \), then the first conjugate point along any geode... | Proof. If \( C \leq 0 \), the Jacobi field comparison theorem implies that any nontrivial normal Jacobi field vanishing at \( t = 0 \) satisfies \( \left| {J\left( t\right) }\right| > 0 \) for all \( t > 0 \) . Similarly, if \( C > 0 \), then \( \left| {J\left( t\right) }\right| \geq \left( \text{constant}\right) \sin ... | Yes |
Corollary 11.4. (Metric Comparison Theorem) Suppose all sectional curvatures of \( \left( {M, g}\right) \) are bounded above by a constant \( C \) . In any normal coordinate chart, \( g\left( {V, V}\right) \geq {g}_{C}\left( {V, V}\right) \), where \( {g}_{C} \) is the constant curvature metric given by formula (10.8). | Proof. Decomposing a vector \( V \) into components \( {V}^{\top } \) tangent to the geodesic sphere and \( {V}^{ \bot } \) tangent to the radial geodesics as in the proof of Proposition 10.9 gives\n\n\[ g\left( {V, V}\right) = g\left( {{V}^{ \bot },{V}^{ \bot }}\right) + g\left( {{V}^{\top },{V}^{\top }}\right) .\n\]\... | Yes |
Theorem 11.5. (The Cartan-Hadamard Theorem) If \( M \) is a complete, connected manifold all of whose sectional curvatures are nonpositive, then for any point \( p \in M,{\exp }_{p} : {T}_{p}M \rightarrow M \) is a covering map. In particular, the universal covering space of \( M \) is diffeomorphic to \( {\mathbf{R}}^... | Proof. The assumption of nonpositive curvature guarantees that \( p \) has no conjugate points along any geodesic, which can be shown by using either the conjugate point comparison theorem above or Problem 10-2. Therefore, by Proposition 10.11, \( {\exp }_{p} \) is a local diffeomorphism on all of \( {T}_{p}M \) . Let ... | Yes |
Lemma 11.6. Suppose \( \widetilde{M} \) and \( M \) 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 covering map. | Proof. A fundamental property of covering maps is the path-lifting property: any continuous path \( \gamma \) in \( M \) lifts to a path \( \widetilde{\gamma } \) in \( \widetilde{M} \) such that \( \pi \circ \widetilde{\gamma } = \gamma \) . We begin by proving that \( \pi \) possesses the path-lifting property for ge... | Yes |
Theorem 11.8. (Myers’s Theorem) Suppose \( M \) is a complete, connected Riemannian n-manifold whose Ricci tensor satisfies the following inequality for all \( V \in {TM} \) :\n\n\[ \operatorname{Rc}\left( {V, V}\right) \geq \frac{n - 1}{{R}^{2}}{\left| V\right| }^{2} \]\n\nThen \( M \) is compact, with a finite fundam... | Proof. As in the proof of Bonnet's theorem, it suffices to prove the diameter estimate. As before, let \( \gamma \) be a minimizing unit speed geodesic segment of\n\nlength \( L > {\pi R} \) . Let \( \left( {{E}_{1},\ldots ,{E}_{n}}\right) \) be a parallel orthonormal frame along \( \gamma \) such that \( {E}_{n} = \do... | Yes |
Theorem 11.9. (Rauch Comparison Theorem) Let \( M \) and \( \widetilde{M} \) be Riemannian manifolds, let \( \gamma : \left\lbrack {0, T}\right\rbrack \rightarrow M \) and \( \widetilde{\gamma } : \left\lbrack {0, T}\right\rbrack \rightarrow \widetilde{M} \) be unit speed geodesic segments such that \( \widetilde{\gamm... | You can find proofs in [dC92], [CE75], and [Spi79, volume 4]. Letting \( \widetilde{M} \) be one of our constant curvature model spaces, we recover the Jacobi field comparison theorem above. On the other hand, if instead we take \( M \) to have constant curvature, we get the same result with the inequalities reversed. | No |
Theorem 11.10. (The Sphere Theorem) Suppose \( M \) is a complete, simply-connected, Riemannian n-manifold that is strictly \( \frac{1}{4} \) -pinched. Then \( M \) is homeomorphic to \( {\mathbf{S}}^{n} \) . | The proof, which can be found in [CE75] or [dC92], is an elaborate application of the Rauch comparison theorem together with the Morse index theorem mentioned in Chapter 10. This result is sharp, at least in even dimensions, because the Fubini-Study metrics on complex projective spaces are \( \frac{1}{4} \) -pinched (P... | No |
Corollary 11.13. (Classification of Constant Curvature Metrics) Suppose \( M \) is a complete, connected Riemannian manifold with constant sectional curvature. Then \( M \) is isometric to \( \widetilde{M}/\Gamma \), where \( \widetilde{M} \) is one of the constant curvature model spaces \( {\mathbf{R}}^{n},{\mathbf{S}... | Proof. If \( \pi : \widetilde{M} \rightarrow M \) is the universal covering space of \( M \) with the lifted metric \( \widetilde{g} = {\pi }^{ * }g \), the preceding theorem shows that \( \left( {\widetilde{M},\widetilde{g}}\right) \) is isometric to one of the model spaces. From covering space theory [Sie92, Mas67] i... | Yes |
Lemma 1. Let \( S \) have density \( \alpha \) and \( 0 \in S \) . Then \( S \oplus S \) has density at least \( {2\alpha } - {\alpha }^{2} \) . | Proof. All the gaps in the set \( S \) are covered in part by the translation of \( S \) by the term of \( S \) just before this gap. Hence, at least the fraction \( \alpha \) of this gap gets covered. So from this covering we have density \( \alpha \) from \( S \) itself and \( \alpha \) times the gaps. Altogether, th... | Yes |
Lemma 2. If \( S \) has density \( \alpha > \frac{1}{2} \), then \( S \oplus S \) contains all the positive integers. | Proof. Fix an integer \( n \) which is arbitrary, let \( A \) be the subset of \( S \) which lies \( \leq n \), and let \( B \) be the set of all \( n \) minus elements of \( S \) . Since \( A \) contains more than \( n/2 \) elements and \( B \) contains at least \( n/2 \) elements, the Pigeonhole principle guarantees ... | Yes |
Lemma 4. There exists \( \epsilon > 0 \) and \( {C}_{2} \) such that, throughout any interval \( {I}_{a, b, N} \) , \[ \left| {\mathop{\sum }\limits_{{n = 1}}^{N}e\left( {x{n}^{k}}\right) }\right| \leq \frac{{C}_{2}N}{{\left( b + j\right) }^{\epsilon }} \] | Proof. This is almost trivial if \( b > {N}^{2/3} \), for, since the derivative of \( \left| {\mathop{\sum }\limits_{{n = 1}}^{N}e\left( {x{n}^{k}}\right) }\right| \) is bounded by \( {2\pi }{N}^{k + 1} \) , \[ \left| {\mathop{\sum }\limits_{{n = 1}}^{N}e\left( {x{n}^{k}}\right) }\right| \leq \left| {\mathop{\sum }\lim... | No |
Theorem 1.1.4 Let \( {A}_{0},{A}_{1},{A}_{2},\ldots \) be countable sets. Then their union \( A = \mathop{\bigcup }\limits_{0}^{\infty }{A}_{n} \) is countable. | Proof. For each \( n \), choose an enumeration \( {a}_{n0},{a}_{n1},{a}_{n2},\ldots \) of \( {A}_{n} \) . We enumerate \( A = \mathop{\bigcup }\limits_{n}{A}_{n} \) following the above diagonal method. | No |
Theorem 1.1.4 Let \( {A}_{0},{A}_{1},{A}_{2},\ldots \) be countable sets. Then their union \( A = \mathop{\bigcup }\limits_{0}^{\infty }{A}_{n} \) is countable. | Proof. For each \( n \), choose an enumeration \( {a}_{n0},{a}_{n1},{a}_{n2},\ldots \) of \( {A}_{n} \) . We enumerate \( A = \mathop{\bigcup }\limits_{n}{A}_{n} \) following the above diagonal method. | Yes |
Theorem 1.1.8 (Cantor) For any two real numbers \( a, b \) with \( a < b \), the interval \( \left\lbrack {a, b}\right\rbrack \) is uncountable. | Proof. (Cantor) Let \( \left( {a}_{n}\right) \) be a sequence in \( \left\lbrack {a, b}\right\rbrack \) . Define an increasing sequence \( \left( {b}_{n}\right) \) and a decreasing sequence \( \left( {c}_{n}\right) \) in \( \left\lbrack {a, b}\right\rbrack \) inductively as follows: Put \( {b}_{0} = a \) and \( {c}_{0}... | Yes |
Theorem 1.1.8 (Cantor) For any two real numbers \( a, b \) with \( a < b \), the interval \( \left\lbrack {a, b}\right\rbrack \) is uncountable. | Proof. (Cantor) Let \( \left( {a}_{n}\right) \) be a sequence in \( \left\lbrack {a, b}\right\rbrack \) . Define an increasing sequence \( \left( {b}_{n}\right) \) and a decreasing sequence \( \left( {c}_{n}\right) \) in \( \left\lbrack {a, b}\right\rbrack \) inductively as follows: Put \( {b}_{0} = a \) and \( {c}_{0}... | Yes |
Theorem 1.1.9 The set \( \{ 0,1{\} }^{\mathbb{N}} \), consisting of all sequences of 0’s and 1’s, is uncountable. | Proof. Let \( \left( {\alpha }_{n}\right) \) be a sequence in \( \{ 0,1{\} }^{\mathbb{N}} \) . Define \( \alpha \in \{ 0,1{\} }^{\mathbb{N}} \) by\n\n\[ \alpha \left( n\right) = 1 - {\alpha }_{n}\left( n\right), n \in \mathbb{N}. \]\n\nThen \( \alpha \neq {\alpha }_{i} \) for all \( i \) . Since \( \left( {\alpha }_{n}... | Yes |
Theorem 1.2.1 (Cantor) For any set \( X, X{ < }_{c}\mathcal{P}\left( X\right) \) . | Proof. First assume that \( X = \varnothing \) . Then \( \mathcal{P}\left( X\right) = \{ \varnothing \} \) . The only function on \( X \) is the empty function \( \varnothing \), which is not onto \( \{ \varnothing \} \) . This observation proves the result when \( X = \varnothing \) .\n\nNow assume that \( X \) is non... | Yes |
Theorem 1.2.3 (Schröder - Bernstein Theorem) For any two sets \( X \) and \( Y \) , \[ \left( {X{ \leq }_{c}Y\& Y{ \leq }_{c}X}\right) \Rightarrow X \equiv Y. \] | Proof. (Dedekind) Let \( X{ \leq }_{c}Y \) and \( Y{ \leq }_{c}X \) . Fix one-to-one maps \( f : X \rightarrow Y \) and \( g : Y \rightarrow X \) . We have to show that \( X \) and \( Y \) have the same cardinality; i.e., that there is a bijection \( h \) from \( X \) onto \( Y \) .\n\nWe first show that there is a set... | Yes |
Corollary 1.2.4 For sets \( A \) and \( B \) , | \[ A{ < }_{c}B \Leftrightarrow A{ \leq }_{c}B\& B{ \nleq }_{c}A. \] | Yes |
Example 1.2.5 Define \( f : \mathcal{P}\left( \mathbb{N}\right) \rightarrow \mathbb{R} \), the set of all real numbers, by\n\n\[ f\left( A\right) = \mathop{\sum }\limits_{{n \in A}}\frac{2}{{3}^{n + 1}}, A \subseteq \mathbb{N}. \]\n\nThen \( f \) is one-to-one. Therefore, \( \mathcal{P}\left( \mathbb{N}\right) { \leq }... | Now consider the map \( g \) :\n\n\( \mathbb{R} \rightarrow \mathcal{P}\left( \mathbb{Q}\right) \) by\n\n\[ g\left( x\right) = \{ r \in \mathbb{Q} \mid r < x\}, x \in \mathbb{R}. \]\n\nClearly, \( g \) is one-to-one and so \( \mathbb{R}{ \leq }_{c}\mathcal{P}\left( \mathbb{Q}\right) \) . As \( \mathbb{Q} \equiv \mathbb... | Yes |
Theorem 1.3.1 If \( X \) is infinite and \( A \subseteq X \) finite, then \( X \smallsetminus A \) and \( X \) have the same cardinality. | Proof. Let \( A = \left\{ {{a}_{0},{a}_{1},\ldots ,{a}_{n}}\right\} \) with the \( {a}_{i} \) ’s distinct. By \( \mathbf{{AC}} \), there exist distinct elements \( {a}_{n + 1},{a}_{n + 2},\ldots \) in \( X \smallsetminus A \) . To see this, fix a choice function \( f : \mathcal{P}\left( X\right) \smallsetminus \{ \varn... | Yes |
Example 1.3.5 Let \( V \) be a vector space over any field \( F \) and \( P \) the set of all independent subsets of \( V \) ordered by the inclusion \( \subseteq \) . Then \( P \) is a poset that is not a linearly ordered set. | In 1.3.5, Let \( C \) be a chain in \( P \) . Then for any two elements \( E \) and \( F \) of \( P \), either \( E \subseteq F \) or \( F \subseteq E \) . It follows that \( \bigcup C \) itself is an independent set and so is an upper bound of \( C \) . | "No" |
Proposition 1.3.7 Every vector space \( V \) has a basis. | Proof. Let \( P \) be the poset defined in 1.3.5; i.e., \( P \) is the set of all independent subsets of \( V \) . Since every singleton set \( \{ v\}, v \neq 0 \), is an independent set, \( P \neq \varnothing \) . As shown earlier, every chain in \( P \) has an upper bound. Therefore, by Zorn’s lemma, \( P \) has a ma... | Yes |
Theorem 1.4.1 For any two sets \( X \) and \( Y \), at least one of\n\n\[ X{ \leq }_{c}Y\text{or}Y{ \leq }_{c}X \]\n\nholds. | Proof. Without loss of generality we can assume that both \( X \) and \( Y \) are nonempty. We need to show that either there exists a one-to-one map \( f : X \rightarrow Y \) or there exists a one-to-one map \( g : Y \rightarrow X \) . To show this, consider the poset \( {Fn}\left( {X, Y}\right) \) of all one-to-one p... | Yes |
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