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Lemma 15.39. Suppose \( N \) and \( M \) are topological spaces and \( \pi : N \rightarrow M \) is a generalized covering map. If \( M \) is connected, then the restriction of \( \pi \) to each component of \( N \) is a covering map. | Proof. Suppose \( W \) is a component of \( N \) . If \( U \) is any open subset of \( M \) that is evenly covered by \( \pi \), then each component of \( {\pi }^{-1}\left( U\right) \) is connected and therefore contained in a single component of \( N \) . It follows that \( {\left( {\left. \pi \right| }_{W}\right) }^{... | Yes |
Theorem 15.41 (Orientation Covering Theorem). Suppose \( M \) is a connected smooth manifold with or without boundary, and let \( \widehat{\pi } : \widehat{M} \rightarrow M \) be its orientation covering.\n\n(a) If \( M \) is orientable, then \( \widehat{M} \) has exactly two components, and the restriction of \( \wide... | Proof. If \( M \) is orientable, then Proposition 15.40(b) shows that \( M \) is evenly covered by \( \widehat{\pi } \), which means that \( \widehat{M} \) has two components, each mapped diffeomorphically onto \( M \) .\n\nNow assume \( M \) is nonorientable. We show first that \( \widehat{M} \) is connected. Let \( W... | Yes |
Theorem 15.42 (Uniqueness of the Orientation Covering). Let \( M \) be a nonorientable connected smooth manifold with or without boundary, and let \( \widehat{\pi } : \widehat{M} \rightarrow M \) be its orientation covering. If \( \widetilde{M} \) is an oriented smooth manifold with or without boundary that admits a tw... | ## Proof. See Problem 15-11. | No |
Theorem 15.43. Let \( M \) be a connected smooth manifold with or without boundary, and suppose the fundamental group of \( M \) has no subgroup of index 2 . Then \( M \) is orientable. In particular, if \( M \) is simply connected then it is orientable. | Proof. Suppose \( M \) is not orientable, and let \( \widehat{\pi } : \widehat{M} \rightarrow M \) be its orientation covering, which is an honest covering map in this case. Choose any point \( q \in \widehat{M} \), and let \( p = \widehat{\pi }\left( q\right) \in M \) . Let \( \alpha : \widehat{M} \rightarrow \widehat... | Yes |
Proposition 16.1. Suppose \( D \) and \( E \) are open domains of integration in \( {\mathbb{R}}^{n} \) or \( {\mathbb{H}}^{n} \) , and \( G : \bar{D} \rightarrow \bar{E} \) is a smooth map that restricts to an orientation-preserving or orientation-reversing diffeomorphism from \( D \) to \( E \) . If \( \omega \) is a... | Proof. Let us use \( \left( {{y}^{1},\ldots ,{y}^{n}}\right) \) to denote standard coordinates on \( E \), and \( \left( {{x}^{1},\ldots ,{x}^{n}}\right) \) to denote those on \( D \) . Suppose first that \( G \) is orientation-preserving. With \( \omega = \) \( {fd}{y}^{1} \land \cdots \land d{y}^{n} \), the change of... | Yes |
Lemma 16.2. Suppose \( U \) is an open subset of \( {\mathbb{R}}^{n} \) or \( {\mathbb{H}}^{n} \), and \( K \) is a compact subset of \( U \) . Then there is an open domain of integration \( D \) such that \( K \subseteq D \subseteq \) \( \bar{D} \subseteq U \) . | Proof. For each \( p \in K \), there is an open ball or half-ball containing \( p \) whose closure is contained in \( U \) . By compactness, finitely many such sets \( {B}_{1},\ldots ,{B}_{m} \) cover \( K \) (Fig. 16.4). Since the boundary of an open ball is a codimension-1 submani-fold, and the boundary of an open ha... | Yes |
Proposition 16.3. Suppose \( U, V \) are open subsets of \( {\mathbb{R}}^{n} \) or \( {\mathbb{H}}^{n} \), and \( G : U \rightarrow \) \( V \) is an orientation-preserving or orientation-reversing diffeomorphism. If \( \omega \) is a compactly supported \( n \) -form on \( V \), then\n\n\[ \n{\int }_{V}\omega = \pm {\i... | Proof. Let \( E \) be an open domain of integration such that supp \( \omega \subseteq E \subseteq \bar{E} \subseteq V \) (Fig. 16.5). Since diffeomorphisms take interiors to interiors, boundaries to boundaries, and sets of measure zero to sets of measure zero, \( D = {G}^{-1}\left( E\right) \subseteq U \) is an open d... | Yes |
Proposition 16.4. With \( \omega \) as above, \( {\int }_{M}\omega \) does not depend on the choice of smooth chart whose domain contains supp \( \omega \) . | Proof. Suppose \( \left( {U,\varphi }\right) \) and \( \left( {\widetilde{U},\widetilde{\varphi }}\right) \) are two smooth charts such that supp \( \omega \subseteq U \cap \widetilde{U} \) (Fig. 16.7). If both charts are positively oriented or both are negatively oriented, then \( \widetilde{\varphi } \circ {\varphi }... | Yes |
Proposition 16.5. The definition of \( {\int }_{M}\omega \) given above does not depend on the choice of open cover or partition of unity. | Proof. Suppose \( \left\{ {\widetilde{U}}_{j}\right\} \) is another finite open cover of supp \( \omega \) by domains of positively or negatively oriented smooth charts, and \( \left\{ {\widetilde{\psi }}_{j}\right\} \) is a subordinate smooth parti-\n\n. Suppose \( M \) and \( N \) are nonempty oriented smooth \( n \) -manifolds with or without boundary, and \( \omega ,\eta \) are compactly supported \( n \) -forms on \( M \) .\n\n(a) LINEARITY: If \( a, b \in \mathbb{R} \), then\n\n\[{\int }_{M}{a\omega } + {b\eta }... | Proof. Parts (a) and (b) are left as an exercise. Suppose \( \omega \) is a positively oriented orientation form for \( M \) . This means that if \( \left( {U,\varphi }\right) \) is a positively oriented smooth chart, then \( {\left( {\varphi }^{-1}\right) }^{ * }\omega \) is a positive function times \( d{x}^{1} \land... | No |
Proposition 16.8 (Integration Over Parametrizations). Let \( M \) be an oriented smooth \( n \) -manifold with or without boundary, and let \( \omega \) be a compactly supported \( n \) -form on \( M \) . Suppose \( {D}_{1},\ldots ,{D}_{k} \) are open domains of integration in \( {\mathbb{R}}^{n} \), and for \( i = 1,\... | Proof. As in the preceding proof, it suffices to assume that \( \omega \) is supported in the domain of a single oriented smooth chart \( \left( {U,\varphi }\right) \) . In fact, by restricting to sufficiently nice charts, we may assume that \( U \) is precompact, \( Y = \varphi \left( U\right) \) is a domain of integr... | Yes |
Let us use this technique to compute the integral of a 2-form over \( {\mathbb{S}}^{2} \), oriented as the boundary of \( {\overline{\mathbb{B}}}^{3} \). Let \( \omega \) be the following 2-form on \( {\mathbb{R}}^{3} \): \[ \omega = {xdy} \land {dz} + {ydz} \land {dx} + {zdx} \land {dy}. \] | Let \( D \) be the open rectangle \( \left( {0,\pi }\right) \times \left( {0,{2\pi }}\right) \), and let \( F : \bar{D} \rightarrow {\mathbb{S}}^{2} \) be the spherical coordinate parametrization \( F\left( {\varphi ,\theta }\right) = \left( {\sin \varphi \cos \theta ,\sin \varphi \sin \theta ,\cos \varphi }\right) \).... | Yes |
Let \( M \) be a smooth manifold and suppose \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow M \) is a smooth embedding, so that \( S = \gamma \left( \left\lbrack {a, b}\right\rbrack \right) \) is an embedded 1-submanifold with boundary in \( M \). If we give \( S \) the orientation such that \( \gamma \) is o... | \[ {\int }_{\gamma }{df} = {\int }_{\left\lbrack a, b\right\rbrack }{\gamma }^{ * }{df} = {\int }_{S}{df} = {\int }_{\partial S}f = f\left( {\gamma \left( b\right) }\right) - f\left( {\gamma \left( a\right) }\right) . \] | Yes |
Corollary 16.13 (Integrals of Exact Forms). If \( M \) is a compact oriented smooth manifold without boundary, then the integral of every exact form over \( M \) is zero: | \[ {\int }_{M}{d\omega } = 0\;\text{ if }\partial M = \varnothing \] | No |
Corollary 16.14 (Integrals of Closed Forms over Boundaries). Suppose \( M \) is a compact oriented smooth manifold with boundary. If \( \omega \) is a closed form on \( M \), then the integral of \( \omega \) over \( \partial M \) is zero: | \[ {\int }_{\partial M}\omega = 0\;\text{ if }{d\omega } = 0\text{ on }M \] | Yes |
Theorem 16.17 (Green's Theorem). Suppose \( D \) is a compact regular domain in \( {\mathbb{R}}^{2} \), and \( P, Q \) are smooth real-valued functions on \( D \) . Then\n\n\[ \n{\int }_{D}\left( {\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}}\right) {dxdy} = {\int }_{\partial D}{Pdx} + {Qdy}.\n\] | Proof. This is just Stokes’s theorem applied to the 1-form \( {Pdx} + {Qdy} \) . | No |
Proposition 16.20 (Invariance of Corner Points). Let \( M \) be a smooth \( n \) -manifold with corners, \( n \geq 2 \), and let \( p \in M \) . If \( \varphi \left( p\right) \) is a corner point for some smooth chart with corners \( \left( {U,\varphi }\right) \), then the same is true for every such chart whose domain... | Proof. Suppose \( \left( {U,\varphi }\right) \) and \( \left( {V,\psi }\right) \) are two smooth charts with corners such that \( \varphi \left( p\right) \) is a corner point but \( \psi \left( p\right) \) is not (Fig. 16.11). To simplify notation, let us assume without loss of generality that \( \varphi \left( p\right... | Yes |
Let \( I \times I = \left\lbrack {0,1}\right\rbrack \times \left\lbrack {0,1}\right\rbrack \) be the unit square in \( {\mathbb{R}}^{2} \), and suppose \( \omega \) is a 1 -form on \( \partial \left( {I \times I}\right) \) . Then it is not hard to check that the maps \( {F}_{i} : I \rightarrow I \times I \) given by\n\... | \[ \n{\int }_{\partial \left( {I \times I}\right) }\omega = {\int }_{{F}_{1}}\omega + {\int }_{{F}_{2}}\omega + {\int }_{{F}_{3}}\omega + {\int }_{{F}_{4}}\omega . \n\] | Yes |
Theorem 16.25 (Stokes’s Theorem on Manifolds with Corners). Let \( M \) be an oriented smooth \( n \) -manifold with corners, and let \( \omega \) be a compactly supported smooth \( \left( {n - 1}\right) \) -form on \( M \) . Then\n\n\[{\int }_{M}{d\omega } = {\int }_{\partial M}\omega\] | Proof. The proof is nearly identical to the proof of Stokes's theorem proper, so we just indicate where changes need to be made. By means of smooth charts and a partition of unity, we may reduce the theorem to the case in which either \( M = {\mathbb{R}}^{n} \) or \( M = {\overline{\mathbb{R}}}_{ + }^{n} \) . The \( {\... | Yes |
Theorem 16.26. Suppose \( M \) is a smooth manifold and \( {\gamma }_{0},{\gamma }_{1} : \left\lbrack {a, b}\right\rbrack \rightarrow M \) are path-homotopic piecewise smooth curve segments. For every closed 1-form \( \omega \) on \( M \) , \[ {\int }_{{\gamma }_{0}}\omega = {\int }_{{\gamma }_{1}}\omega \] | Proof. By means of an affine reparametrization, we may as well assume for simplicity that \( \left\lbrack {a, b}\right\rbrack = \left\lbrack {0,1}\right\rbrack \) . Assume first that \( {\gamma }_{0} \) and \( {\gamma }_{1} \) are smooth. By Theorem 6.29, \( {\gamma }_{0} \) and \( {\gamma }_{1} \) are smoothly homotop... | Yes |
Corollary 16.27. On a simply connected smooth manifold, every closed 1-form is exact. | Proof. Suppose \( M \) is simply connected and \( \omega \) is a closed 1 -form on \( M \) . Since every piecewise smooth closed curve segment in \( M \) is path-homotopic to a constant curve, the preceding theorem shows that the integral of \( \omega \) over every such curve is equal to 0 . Thus, \( \omega \) is conse... | Yes |
Proposition 16.28. Let \( \\left( {M, g}\\right) \) be a nonempty oriented Riemannian manifold with or without boundary, and suppose \( f \) is a compactly supported continuous real-valued function on \( M \) satisfying \( f \\geq 0 \) . Then \( {\\int }_{M}{fd}{V}_{g} \\geq 0 \), with equality if and only if \( f \\eq... | Proof. If \( f \) is supported in the domain of a single oriented smooth chart \( \\left( {U,\\varphi }\\right) \) , then Proposition 15.31 shows that\n\n\\[ \n{\\int }_{M}{fd}{V}_{g} = {\\int }_{\\varphi \\left( U\\right) }f\\left( x\\right) \\sqrt{\\det \\left( {g}_{ij}\\right) }d{x}^{1}\\cdots d{x}^{n} \\geq 0.\n\\]... | Yes |
Lemma 16.30. Let \( \\left( {M, g}\\right) \) be an oriented Riemannian manifold with or without boundary. Suppose \( S \\subseteq M \) is an immersed hypersurface with the orientation determined by a unit normal vector field \( N \), and \( \\widetilde{g} \) is the induced metric on \( S \) . If \( X \) is any vector ... | Proof. Define two vector fields \( {X}^{\\top } \) and \( {X}^{ \\bot } \) along \( S \) by\n\n\[ \n{X}^{ \\bot } = \\langle X, N{\\rangle }_{g}N\n\]\n\n\[ \n{X}^{\\top } = X - {X}^{ \\bot }\n\]\n\nThen \( X = {X}^{ \\bot } + {X}^{\\top } \), where \( {X}^{ \\bot } \) is normal to \( S \) and \( {X}^{\\top } \) is tang... | Yes |
Theorem 16.32 (The Divergence Theorem). Let \( \\left( {M, g}\\right) \) be an oriented Riemannian manifold with boundary. For any compactly supported smooth vector field \( X \) on \( M \) , | Proof. By Stokes's theorem,\n\n\[ {\\int }_{M}\\left( {\\operatorname{div}X}\\right) d{V}_{g} = {\\int }_{M}d\\left( {\\beta \\left( X\\right) }\\right) = {\\int }_{\\partial M}{\\iota }_{S}^{ * }\\beta \\left( X\\right) .\n\]\n\nThe divergence theorem then follows from Lemma 16.30. | No |
Proposition 16.33 (Geometric Interpretation of the Divergence). Let \( M \) be an oriented Riemannian manifold, let \( X \in \mathfrak{X}\left( M\right) \), and let \( \theta \) be the flow of \( X \). Then \( \theta \) is\n\n(a) volume-preserving if and only if \( \operatorname{div}X = 0 \) everywhere on \( M \).\n\n(... | Proof. First we establish some preliminary results. For each \( t \in \mathbb{R} \), let \( {M}_{t} \) be the domain of \( {\theta }_{t} \). If \( D \) is a compact regular domain contained in \( {M}_{t} \), then \( {\theta }_{t} \) is an orientation-preserving diffeomorphism from \( D \) to \( {\theta }_{t}\left( D\ri... | No |
Theorem 16.34 (Stokes’s Theorem for Surface Integrals). Suppose \( M \) is an oriented Riemannian 3-manifold with or without boundary, and \( S \) is a compact oriented 2-dimensional smooth submanifold with boundary in \( M \) . For any smooth vector field \( X \) on \( M \) , \[ {\int }_{S}\langle \operatorname{curl}X... | Proof. The general version of Stokes’s theorem applied to the 1-form \( {X}^{\mathrm{b}} \) yields \[ {\int }_{S}d\left( {X}^{b}\right) = {\int }_{\partial S}{X}^{b} \] Thus the theorem follows from the following two identities: \[ {\iota }_{S}^{ * }d\left( {X}^{b}\right) = \langle \operatorname{curl}X, N{\rangle }_{g}... | Yes |
Proposition 16.35 (Properties of Densities). Let \( V \) be a vector space of dimension \( n \geq 1 \). (a) \( \mathcal{D}\left( V\right) \) is a vector space under the obvious vector operations:\n\n\[ \left( {{c}_{1}{\mu }_{1} + {c}_{2}{\mu }_{2}}\right) \left( {{v}_{1},\ldots ,{v}_{n}}\right) = {c}_{1}{\mu }_{1}\left... | Proof. Part (a) is immediate from the definition. | No |
Proposition 16.36. If \( M \) is a smooth manifold with or without boundary, its density bundle is a smooth line bundle over \( M \) . | Proof. We will construct local trivializations and use the vector bundle chart lemma (Lemma 10.6). Let \( \left( {U,\left( {x}^{i}\right) }\right) \) be any smooth coordinate chart on \( M \), and let \( \omega = \) \( d{x}^{1} \land \cdots \land d{x}^{n} \) . Proposition 16.35 shows that \( \left| {\omega }_{p}\right|... | Yes |
Proposition 16.37. If \( M \) is a smooth manifold with or without boundary, there exists a smooth positive density on \( M \) . | Proof. Because the set of positive elements of \( \mathcal{D}M \) is an open subset whose intersection with each fiber is convex, the usual partition of unity argument (Problem 13-2) allows us to piece together local positive densities to obtain a global smooth positive density. | No |
Proposition 16.40. Suppose \( F : M \rightarrow N \) is a smooth map between n-manifolds with or without boundary. If \( \left( {x}^{i}\right) \) and \( \left( {y}^{j}\right) \) are smooth coordinates on open subsets \( U \subseteq M \) and \( V \subseteq N \), respectively, and \( u \) is a continuous real-valued func... | Proof. Using Propositions 14.20 and 16.38, we obtain\n\n\[ \n{F}^{ * }\left( {u\left| {d{y}^{1} \land \cdots \land d{y}^{n}}\right| }\right) = \left( {u \circ F}\right) {F}^{ * }\left| {d{y}^{1} \land \cdots \land d{y}^{n}}\right| \n\]\n\n\[ \n= \left( {u \circ F}\right) \left| {{F}^{ * }\left( {d{y}^{1} \land \cdots \... | Yes |
Proposition 16.41. Suppose \( U \) and \( V \) are open subsets of \( {\mathbb{R}}^{n} \) or \( {\mathbb{H}}^{n} \), and \( G : U \rightarrow \) \( V \) is a diffeomorphism. If \( \mu \) is a compactly supported density on \( V \), then\n\n\[{\int }_{V}\mu = {\int }_{U}{G}^{ * }\mu\] | Proof. The proof is essentially identical to that of Proposition 16.3, using (16.19) instead of (14.15). | No |
Proposition 16.45 (The Riemannian Density). Let \( \left( {M, g}\right) \) be a Riemannian manifold with or without boundary. There is a unique smooth positive density \( {\mu }_{g} \) on \( M \) , called the Riemannian density, with the property that\n\n\[{\mu }_{g}\left( {{E}_{1},\ldots ,{E}_{n}}\right) = 1\]\n\n(16.... | Proof. Uniqueness is immediate, because any two densities that agree on a basis must be equal. Given any point \( p \in M \), let \( U \) be a connected smooth coordinate neighborhood of \( p \) . Since \( U \) is diffeomorphic to an open subset of Euclidean space, it is orientable. Any choice of orientation of \( U \)... | Yes |
Theorem 16.48 (The Divergence Theorem in the Nonorientable Case). Suppose \( \\left( {M, g}\\right) \) is a nonorientable Riemannian manifold with boundary. For any compactly supported smooth vector field \( X \) on \( M \) ,\n\n\[ \n{\\int }_{M}\\left( {\\operatorname{div}X}\\right) {\\mu }_{g} = {\\int }_{\\partial M... | Proof. Let \( \\widehat{\\pi } : \\widehat{M} \\rightarrow M \) be the orientation covering of \( M \) . Problem 5-12 shows that \( \\widehat{\\pi } \) restricts to a smooth covering map from each component of \( \\partial \\widehat{M} \) to a component of \( \\partial M \), so in the terminology of Chapter \( {15},\\w... | Yes |
The fact that there is a closed 1-form on \( {\mathbb{R}}^{2} \smallsetminus \{ 0\} \) that is not exact means that \( {H}_{\mathrm{{dR}}}^{1}\left( {{\mathbb{R}}^{2}\smallsetminus \{ 0\} }\right) \neq 0 \) (see Example 11.48). | On the other hand, the Poincaré lemma for 1-forms (Theorem 11.49) implies that \( {H}_{\mathrm{{dR}}}^{1}\left( U\right) = 0 \) for any star-shaped open subset \( U \subseteq {\mathbb{R}}^{n} \). | No |
Proposition 17.2 (Induced Cohomology Maps). For any smooth map \( F : M \rightarrow \) \( N \) between smooth manifolds with or without boundary, the pullback \( {F}^{ * } : {\Omega }^{p}\left( N\right) \rightarrow \) \( {\Omega }^{p}\left( M\right) \) carries \( {\mathcal{Z}}^{p}\left( N\right) \) into \( {\mathcal{Z}... | Proof. If \( \omega \) is closed, then \( d\left( {{F}^{ * }\omega }\right) = {F}^{ * }\left( {d\omega }\right) = 0 \), so \( {F}^{ * }\omega \) is also closed. If \( \omega = \) \( {d\eta } \) is exact, then \( {F}^{ * }\omega = {F}^{ * }\left( {d\eta }\right) = d\left( {{F}^{ * }\eta }\right) \), which is also exact.... | Yes |
Proposition 17.5 (Cohomology of Disjoint Unions). Let \( \left\{ {M}_{j}\right\} \) be a countable collection of smooth \( n \) -manifolds with or without boundary, and let \( M = \mathop{\coprod }\limits_{j}{M}_{j} \) . For each \( p \), the inclusion maps \( {\iota }_{j} : {M}_{j} \hookrightarrow M \) induce an isomo... | Proof. The pullback maps \( {\iota }_{j}^{ * } : {\Omega }^{p}\left( M\right) \rightarrow {\Omega }^{p}\left( {M}_{j}\right) \) already induce an isomorphism from \( {\Omega }^{p}\left( M\right) \) to \( \mathop{\prod }\limits_{j}{\Omega }^{p}\left( {M}_{j}\right) \), namely\n\n\[ \omega \mapsto \left( {{\iota }_{1}^{ ... | Yes |
Proposition 17.6 (Cohomology in Degree Zero). If \( M \) is a connected smooth manifold with or without boundary, then \( {H}_{\mathrm{{dR}}}^{0}\left( M\right) \) is equal to the space of constant functions and is therefore 1-dimensional. | Proof. Because there are no \( \left( {-1}\right) \) -forms, \( {\mathcal{B}}^{0}\left( M\right) = 0 \) . A closed 0 -form is a smooth real-valued function \( f \) such that \( {df} = 0 \), and since \( M \) is connected, this is true if and only if \( f \) is constant. Therefore, \( {H}_{\mathrm{{dR}}}^{0}\left( M\rig... | Yes |
Corollary 17.7 (Cohomology of Zero-Manifolds). Suppose \( M \) is a manifold of dimension 0 . Then \( {H}_{\mathrm{{dR}}}^{0}\left( M\right) \) is a direct product of 1-dimensional vector spaces, one for each point of \( M \), and all other de Rham cohomology groups of \( M \) are zero. | Proof. The statement about \( {H}_{\mathrm{{dR}}}^{0}\left( M\right) \) follows from Propositions 17.5 and 17.6, and the cohomology groups in nonzero degrees vanish for dimensional reasons. | Yes |
Proposition 17.8. Suppose \( M \) and \( N \) are smooth manifolds with or without boundary. If \( F, G : M \rightarrow N \) are smooth maps and there exists a homotopy operator between the pullback maps \( {F}^{ * } \) and \( {G}^{ * } \), then the induced cohomology maps \( {F}^{ * },{G}^{ * } : {H}_{\mathrm{{dR}}}^{... | The key to our proof of homotopy invariance is to construct a homotopy operator first in the following special case. Let \( M \) be a smooth manifold with or without boundary, and for each \( t \in I \), let \( {i}_{t} : M \rightarrow M \times I \) be the map\n\n\[ \n{i}_{t}\left( x\right) = \left( {x, t}\right) \text{... | No |
Proposition 17.10. Suppose \( M \) and \( N \) are smooth manifolds with or without boundary, and \( F, G : M \rightarrow N \) are homotopic smooth maps. For every \( p \), the induced cohomology maps \( {F}^{ * },{G}^{ * } : {H}_{\mathrm{{dR}}}^{p}\left( N\right) \rightarrow {H}_{\mathrm{{dR}}}^{p}\left( M\right) \) a... | Proof. The preceding lemma implies that the two cohomology maps \( {i}_{0}^{ * } \) and \( {i}_{1}^{ * } \) from \( {H}_{\mathrm{{dR}}}^{p}\left( {M \times I}\right) \) to \( {H}_{\mathrm{{dR}}}^{p}\left( M\right) \) are equal. By Theorem 9.28, there is a smooth homotopy \( H : M \times I \rightarrow N \) from \( F \) ... | Yes |
Theorem 17.11 (Homotopy Invariance of de Rham Cohomology). If \( M \) and \( N \) are homotopy equivalent smooth manifolds with or without boundary, then \( {H}_{\mathrm{{dR}}}^{p}\left( M\right) \cong {H}_{\mathrm{{dR}}}^{p}\left( N\right) \) for each \( p \) . The isomorphisms are induced by any smooth homotopy equiv... | Proof. Suppose \( F : M \rightarrow N \) is a homotopy equivalence, with homotopy inverse \( G : N \rightarrow M \) . By the Whitney approximation theorem (Theorem 6.26 or 9.27), there are smooth maps \( \widetilde{F} : M \rightarrow N \) homotopic to \( F \) and \( \widetilde{G} : N \rightarrow M \) homotopic to \( G ... | Yes |
Theorem 17.13 (Cohomology of Contractible Manifolds). If \( M \) is a contractible smooth manifold with or without boundary, then \( {H}_{\mathrm{{dR}}}^{p}\left( M\right) = 0 \) for \( p \geq 1 \) . | Proof. The assumption means there is some point \( q \in M \) such that the identity map of \( M \) is homotopic to the constant map \( {c}_{q} : M \rightarrow M \) sending all of \( M \) to \( q \) . If \( {\iota }_{q} : \{ q\} \hookrightarrow M \) denotes the inclusion map, it follows that \( {c}_{q} \circ {\iota }_{... | Yes |
Theorem 17.14 (The Poincaré Lemma). If \( U \) is a star-shaped open subset of \( {\mathbb{R}}^{n} \) or \( {\mathbb{H}}^{n} \), then \( {H}_{\mathrm{{dR}}}^{p}\left( U\right) = 0 \) for \( p \geq 1 \) . | Proof. If \( U \) is star-shaped with respect to \( c \), then it is contractible by the following straight-line homotopy:\n\n\[ H\left( {x, t}\right) = c + t\left( {x - c}\right) . \] | Yes |
Corollary 17.15 (Local Exactness of Closed Forms). Let \( M \) be a smooth manifold with or without boundary. Each point of \( M \) has a neighborhood on which every closed form is exact. | Proof. Every point of \( M \) has a neighborhood diffeomorphic to an open ball in \( {\mathbb{R}}^{n} \) or an open half-ball in \( {\mathbb{H}}^{n} \), each of which is star-shaped. The result follows from the Poincaré lemma and the diffeomorphism invariance of de Rham cohomology. | Yes |
Corollary 17.16 (Cohomology of Euclidean Spaces and Half-Spaces). For any integers \( n \geq 0 \) and \( p \geq 1,{H}_{\mathrm{{dR}}}^{p}\left( {\mathbb{R}}^{n}\right) = 0 \) and \( {H}_{\mathrm{{dR}}}^{p}\left( {\mathbb{H}}^{n}\right) = 0 \) . | Proof. Both \( {\mathbb{R}}^{n} \) and \( {\mathbb{H}}^{n} \) are star-shaped. | No |
Theorem 17.17 (First Cohomology and the Fundamental Group). Suppose \( M \) is a connected smooth manifold. For each \( q \in M \), the linear map \( \Phi : {H}_{\mathrm{{dR}}}^{1}\left( M\right) \rightarrow \) \( \operatorname{Hom}\left( {{\pi }_{1}\left( {M, q}\right) ,\mathbb{R}}\right) \) is well defined and inject... | Proof. Given \( \left\lbrack \gamma \right\rbrack \in {\pi }_{1}\left( {M, q}\right) \), it follows from the Whitney approximation theorem that there is some smooth closed curve segment \( \widetilde{\gamma } \) in the same path class as \( \gamma \), and from Theorem 16.26 that \( {\int }_{\widetilde{\gamma }}\omega \... | Yes |
Corollary 17.18. If \( M \) is a connected smooth manifold with finite fundamental group, then \( {H}_{\mathrm{{dR}}}^{1}\left( M\right) = 0 \) . | Proof. There are no nontrivial homomorphisms from a finite group to \( \mathbb{R} \) . | No |
Theorem 17.20 (Mayer-Vietoris). Let \( M \) be a smooth manifold with or without boundary, and let \( U, V \) be open subsets of \( M \) whose union is \( M \) . For each \( p \), there is a linear map \( \delta : {H}_{\mathrm{{dR}}}^{p}\left( {U \cap V}\right) \rightarrow {H}_{\mathrm{{dR}}}^{p + 1}\left( M\right) \) ... | \[ \cdots \overset{\delta }{ \rightarrow }{H}_{\mathrm{{dR}}}^{p}\left( M\right) \xrightarrow[]{{k}^{ * } \oplus {l}^{ * }}{H}_{\mathrm{{dR}}}^{p}\left( U\right) \oplus {H}_{\mathrm{{dR}}}^{p}\left( V\right) \xrightarrow[]{{i}^{ * } - {j}^{ * }}{H}_{\mathrm{{dR}}}^{p}\left( {U \cap V}\right) \] \[ \overset{\delta }{ \r... | Yes |
Theorem 17.21 (Cohomology of Spheres). For \( n \geq 1 \), the de Rham cohomology groups of \( {\mathbb{S}}^{n} \) are\n\n\[ \n{H}_{\mathrm{{dR}}}^{p}\left( {\mathbb{S}}^{n}\right) \cong \left\{ \begin{array}{ll} \mathbb{R} & \text{ if }p = 0\text{ or }p = n, \\ 0 & \text{ if }0 < p < n. \end{array}\right.\n\]\n\n(17.1... | Proof. Proposition 17.6 shows that \( {H}_{\mathrm{{dR}}}^{0}\left( {\mathbb{S}}^{n}\right) \cong \mathbb{R} \), so we need only prove (17.11) for \( p \geq 1 \) . We do so by induction on \( n \) . For \( n = 1 \), note first that any orientation form on \( {\mathbb{S}}^{1} \) has nonzero integral, so it is not exact ... | Yes |
Corollary 17.23 (Cohomology of Punctured Euclidean Space). Suppose \( n \geq 2 \) and \( x \in {\mathbb{R}}^{n} \), and let \( M = {\mathbb{R}}^{n} \smallsetminus \{ x\} \) . The only nontrivial de Rham groups of \( M \) are \( {H}_{\mathrm{{dR}}}^{0}\left( M\right) \) and \( {H}_{\mathrm{{dR}}}^{n - 1}\left( M\right) ... | Proof. Let \( S \subseteq M \) be any \( \left( {n - 1}\right) \) -dimensional sphere centered at \( x \) . Because inclusion \( \iota : S \hookrightarrow M \) is a homotopy equivalence, \( {\iota }^{ * } : {H}_{\mathrm{{dR}}}^{p}\left( M\right) \rightarrow {H}_{\mathrm{{dR}}}^{p}\left( S\right) \) is an isomorphism fo... | Yes |
Corollary 17.25. Suppose \( n \geq 2, U \subseteq {\mathbb{R}}^{n} \) is any open subset, and \( x \in U \) . Then \( {H}_{\mathrm{{dR}}}^{n - 1}\left( {U\smallsetminus \{ x\} }\right) \neq 0 \) . | Proof. Because \( U \) is open, there is an \( \left( {n - 1}\right) \) -dimensional sphere \( S \) centered at \( x \) such that \( S \subseteq U \smallsetminus \{ x\} \) . Let \( \iota : S \hookrightarrow U \smallsetminus \{ x\} \) be inclusion and \( r : U \smallsetminus \{ x\} \rightarrow S \) be the radial project... | Yes |
Theorem 17.26 (Topological Invariance of Dimension). A nonempty n-dimensional topological manifold cannot be homeomorphic to an m-dimensional manifold unless \( m = n \) . | Proof. If \( M \) is a topological \( n \) -manifold that is homeomorphic to an \( m \) -manifold, then \( M \) is itself both an \( n \) -manifold and an \( m \) -manifold. The case in which \( m \) or \( n \) is zero was already taken care of in Chapter 1, so assume that \( m > n \geq 1 \) . Because \( M \) is an \( ... | Yes |
Theorem 17.28 (Compactly Supported Cohomology of \( {\mathbb{R}}^{n} \) ). For \( n \geq 1 \), the compactly supported de Rham cohomology groups of \( {\mathbb{R}}^{n} \) are\n\n\[ \n{H}_{c}^{p}\left( {\mathbb{R}}^{n}\right) \cong \left\{ \begin{array}{ll} 0 & \text{ if }0 \leq p < n \\ \mathbb{R} & \text{ if }p = n \e... | Exercise 17.29. Prove this theorem. | No |
Theorem 17.30 (Top Cohomology, Orientable Compact Support Case). If \( M \) is a connected oriented smooth \( n \) -manifold, then the integration map \( I : {H}_{c}^{n}\left( M\right) \rightarrow \mathbb{R} \) is an isomorphism, so \( {H}_{c}^{n}\left( M\right) \) is 1-dimensional. | Proof. Because a connected 0-manifold is a single point, the 0-dimensional case is an immediate consequence of Corollary 17.7, so we may assume \( n \geq 1 \) . Let \( \left( {U,\left( {x}^{i}\right) }\right) \) be an oriented smooth coordinate chart on \( M \), and let \( f \) be a smooth bump function with compact su... | Yes |
Theorem 17.31 (Top Cohomology, Orientable Compact Case). If \( M \) is a compact connected orientable smooth \( n \) -manifold, then \( {H}_{\mathrm{{dR}}}^{n}\left( M\right) \) is 1 -dimensional, and is spanned by the cohomology class of any smooth orientation form. | Proof. This follows from the preceding theorem, because \( {H}_{\mathrm{{dR}}}^{p}\left( M\right) = {H}_{c}^{p}\left( M\right) \) in that case, and the integral of any orientation form is nonzero. | Yes |
Theorem 17.32 (Top Cohomology, Orientable Noncompact Case). If \( M \) is a noncompact connected orientable smooth \( n \) -manifold, then \( {H}_{\mathrm{{dR}}}^{n}\left( M\right) = 0 \) . | Proof. Choose an orientation on \( M \) . Let \( f \in {C}^{\infty }\left( M\right) \) be a smooth exhaustion function. By adding a constant, we can arrange that \( \mathop{\inf }\limits_{M}f = 0 \), and then connectedness and noncompactness of \( M \) imply that \( f\left( M\right) = \lbrack 0,\infty ) \) . For each p... | Yes |
Lemma 17.33. Suppose \( M \) is a connected nonorientable smooth manifold and \( \widehat{\pi } : \widehat{M} \rightarrow M \) is its orientation covering. For each \( p \), the induced cohomology maps \( {\widehat{\pi }}^{ * } : {H}_{\mathrm{{dR}}}^{p}\left( M\right) \rightarrow {H}_{\mathrm{{dR}}}^{p}\left( \widehat{... | Proof. First, we prove the lemma for compactly supported cohomology. Suppose \( \omega \) is a closed, compactly supported \( p \) -form on \( M \) such that \( {\widehat{\pi }}^{ * }\left\lbrack \omega \right\rbrack = 0 \in {H}_{c}^{p}\left( \widehat{M}\right) \) . Then there exists \( \eta \in {\Omega }_{c}^{p}\left(... | Yes |
Theorem 17.34 (Top Cohomology, Nonorientable Case). If \( M \) is a connected nonorientable smooth \( n \) -manifold, then \( {H}_{c}^{n}\left( M\right) = 0 \) and \( {H}_{\mathrm{{dR}}}^{n}\left( M\right) = 0 \) . | Proof. First consider the case of compactly supported cohomology. By the preceding lemma, it suffices to show that \( {\widehat{\pi }}^{ * } : {H}_{c}^{n}\left( M\right) \rightarrow {H}_{c}^{n}\left( \widehat{M}\right) \) is the zero map, where \( \widehat{\pi } : \widehat{M} \rightarrow M \) is the orientation coverin... | Yes |
Theorem 17.35 (Degree of a Smooth Map). Suppose \( M \) and \( N \) are compact, connected, oriented, smooth manifolds of dimension \( n \), and \( F : M \rightarrow N \) is a smooth map. There exists a unique integer \( k \), called the degree of \( \mathbf{F} \), that satisfies both of the following conditions.\n\n(a... | Proof. By Theorem 17.31, two smooth \( n \) -forms on either \( M \) or \( N \) are cohomolo-gous if and only if they have the same integral. Let \( \theta \) be any smooth \( n \) -form on \( N \) such that \( {\int }_{N}\theta = 1 \), and let \( k = {\int }_{M}{F}^{ * }\theta \) . If \( \omega \in {\Omega }^{n}\left(... | Yes |
Theorem 17.38. Suppose \( N \) is a compact, connected, oriented, smooth \( n \) -manifold, and \( X \) is a compact, oriented, smooth \( \left( {n + 1}\right) \) -manifold with connected boundary. If \( f : \partial X \rightarrow N \) is a continuous map that has a continuous extension to \( X \), then \( \deg f = 0 \... | Proof. Suppose \( f \) has an extension to a continuous map \( F : X \rightarrow N \) . By the Whitney approximation theorem, there is a smooth map \( \widetilde{F} : X \rightarrow N \) that is homotopic to \( F \) . Replacing \( F \) by \( \widetilde{F} \) and \( f \) by \( {\left. \widetilde{F}\right| }_{\partial X} ... | Yes |
Theorem 17.39 (Brouwer Fixed-Point Theorem). Every continuous map from \( {\overline{\mathbb{B}}}^{n} \) to itself has a fixed point. | Proof. Suppose for the sake of contradiction that \( F : {\overline{\mathbb{B}}}^{n} \rightarrow {\overline{\mathbb{B}}}^{n} \) is continuous and has no fixed points. We can define a continuous map \( G : {\overline{\mathbb{B}}}^{n} \rightarrow {\mathbb{S}}^{n - 1} \) by\n\n\[ G\left( x\right) = \frac{x - F\left( x\rig... | Yes |
Lemma 17.40 (The Zigzag Lemma). Given a short exact sequence of complexes as above, for each \( p \) there is a linear map\n\n\[ \delta : {H}^{p}\left( {C}^{ * }\right) \rightarrow {H}^{p + 1}\left( {A}^{ * }\right) \]\n\ncalled the connecting homomorphism, such that the following sequence is exact:\n\n\[ \cdots \overs... | Proof. We sketch only the main idea; you can either carry out the details yourself or look them up.\n\nThe hypothesis means that the following diagram commutes and has exact horizontal rows:\n\n\n\nSuppose \( {c}^{p}... | No |
The connecting homomorphism in the Mayer-Vietoris sequence, \( \delta : {H}_{\mathrm{{dR}}}^{p}\left( {U \cap V}\right) \rightarrow {H}_{\mathrm{{dR}}}^{p + 1}\left( M\right) \), is defined as follows. For each \( \omega \in {\mathcal{Z}}^{p}\left( {U \cap V}\right) \) , there are \( p \) -forms \( \eta \in {\Omega }^{... | Proof. A characterization of the connecting homomorphism was given in the proof of the zigzag lemma. Specializing this characterization to the situation of the short exact sequence (17.7), we find that \( \delta \left\lbrack \omega \right\rbrack = \left\lbrack \sigma \right\rbrack \), provided there exists \( \left( {\... | Yes |
Lemma 18.2. If \( c \) is any singular chain, then \( \partial \left( {\partial c}\right) = 0 \) . | Sketch of Proof. The starting point is the fact that\n\n\[ \n{F}_{i, p} \circ {F}_{j, p - 1} = {F}_{j, p} \circ {F}_{i - 1, p - 1} \n\]\n\n(18.1)\n\nwhen \( i > j \), which can be verified by following what both compositions do to each of the vertices of \( {\Delta }_{p - 2} \) . Using this, the proof of the lemma is j... | No |
Theorem 18.4 (Mayer-Vietoris for Singular Homology). Let \( M \) be a topological space and let \( U, V \) be open subsets of \( M \) whose union is \( M \) . For each \( p \) there is a connecting homomorphism \( {\partial }_{ * } : {H}_{p}\left( M\right) \rightarrow {H}_{p - 1}\left( {U \cap V}\right) \) such that th... | Sketch of Proof. The basic idea, of course, is to construct a short exact sequence of complexes and use the zigzag lemma. The hardest part of the proof is showing that every homology class \( \left\lbrack e\right\rbrack \in {H}_{p}\left( M\right) \) can be represented in the form \( \beta \left( {\left\lbrack c\right\r... | No |
Theorem 18.6 (Mayer-Vietoris for Singular Cohomology). Suppose \( M, U \), and \( V \) satisfy the hypotheses of Theorem 18.4. The following sequence is exact:\n\n\[ \cdots \xrightarrow[]{{\partial }^{ * }}{H}^{p}\left( {M;\mathbb{R}}\right) \xrightarrow[]{{k}^{ * } \oplus {l}^{ * }}{H}^{p}\left( {U;\mathbb{R}}\right) ... | Sketch of Proof. For any homomorphism \( F : A \rightarrow B \) between abelian groups, there is a dual homomorphism \( {F}^{ * } : \operatorname{Hom}\left( {B,\mathbb{R}}\right) \rightarrow \operatorname{Hom}\left( {A,\mathbb{R}}\right) \) given by \( {F}^{ * }\left( \gamma \right) = \gamma \circ F \) . Applying this ... | Yes |
Lemma 18.8. Let \( M \) be a smooth manifold. For each integer \( p \geq 0 \) and each singular p-simplex \( \sigma : {\Delta }_{p} \rightarrow M \), there exists a continuous map \( {H}_{\sigma } : {\Delta }_{p} \times I \rightarrow M \) such that the following properties hold:\n\n(i) \( {H}_{\sigma } \) is a homotopy... | Proof. We will construct the homotopies \( {H}_{\sigma } \) (see Fig. 18.4) by induction on the dimension of \( \sigma \) . To get started, for each 0 -simplex \( \sigma : {\Delta }_{0} \rightarrow M \), we just define \( {H}_{\sigma }\left( {x, t}\right) = \sigma \left( x\right) \) . Since each 0 -simplex is smooth an... | Yes |
Lemma 18.10 (The Five Lemma). Consider the following commutative diagram of modules and linear maps: \n\nIf the horizontal rows are exact and \( {f}_{1},{f}_{2},{f}_{4} \), and \( {f}_{5} \) are isomorphisms, then \(... | - Exercise 18.11. Prove (or look up) the five lemma. | No |
Theorem 18.12 (Stokes’s Theorem for Chains). If \( c \) is a smooth p-chain in a smooth manifold \( M \), and \( \omega \) is a smooth \( \left( {p - 1}\right) \) -form on \( M \), then\n\n\[ \n{\int }_{\partial c}\omega = {\int }_{c}{d\omega } \n\] | Proof. It suffices to prove the theorem when \( c \) is just a smooth simplex \( \sigma \) . Since \( {\Delta }_{p} \) is a manifold with corners, Stokes's theorem says that\n\n\[ \n{\int }_{\sigma }{d\omega } = {\int }_{{\Delta }_{p}}{\sigma }^{ * }{d\omega } = {\int }_{{\Delta }_{p}}d{\sigma }^{ * }\omega = {\int }_{... | Yes |
Proposition 18.13 (Naturality of the de Rham Homomorphism). For a smooth manifold \( M \) and nonnegative integer \( p \), let \( \vartheta : {H}_{\mathrm{{dR}}}^{p}\left( M\right) \rightarrow {H}^{p}\left( {M;\mathbb{R}}\right) \) denote the de Rham homomorphism.\n\n(a) If \( F : M \rightarrow N \) is a smooth map, th... | Proof. Directly from the definitions, if \( \sigma \) is a smooth \( p \) -simplex in \( M \) and \( \omega \) is a smooth \( p \) -form on \( N \),\n\n\[{\int }_{\sigma }{F}^{ * }\omega = {\int }_{{\Delta }_{p}}{\sigma }^{ * }{F}^{ * }\omega = {\int }_{{\Delta }_{p}}{\left( F \circ \sigma \right) }^{ * }\omega = {\int... | Yes |
Proposition 19.2. Let \( D \subseteq {TM} \) be a smooth distribution, and let \( \Gamma \left( D\right) \subseteq \mathfrak{X}\left( M\right) \) denote the space of smooth global sections of \( D \) . Then \( D \) is involutive if and only if \( \Gamma \left( D\right) \) is a Lie subalgebra of \( \mathfrak{X}\left( M\... | Proof. If \( D \) is involutive, the definition implies that \( \Gamma \left( D\right) \) is closed under Lie brackets. Because it is also a linear subspace of \( \mathfrak{X}\left( M\right) \), it is a Lie subalgebra.\n\nConversely, suppose \( \Gamma \left( D\right) \) is a Lie subalgebra of \( \mathcal{X}\left( M\rig... | Yes |
Proposition 19.3. Every integrable distribution is involutive. | Proof. Let \( D \subseteq {TM} \) be an integrable distribution. Suppose \( X \) and \( Y \) are smooth local sections of \( D \) defined on some open subset \( U \subseteq M \) . Let \( p \) be any point in \( U \) , and let \( N \) be an integral manifold of \( D \) containing \( p \) . The fact that \( X \) and \( Y... | Yes |
Lemma 19.4 (Local Frame Criterion for Involutivity). Let \( D \subseteq {TM} \) be a distribution. If in a neighborhood of every point of \( M \) there exists a smooth local frame \( \left( {{V}_{1},\ldots ,{V}_{k}}\right) \) for \( D \) such that \( \left\lbrack {{V}_{i},{V}_{j}}\right\rbrack \) is a section of \( D \... | Proof. Suppose the hypothesis holds, and suppose \( X \) and \( Y \) are smooth local sections of \( D \) over some open subset \( U \subseteq M \) . Given \( p \in U \), choose a smooth local frame \( \left( {{V}_{1},\ldots ,{V}_{k}}\right) \) satisfying the hypothesis in a neighborhood of \( p \), and write \( X = {X... | Yes |
Lemma 19.5 (1-Form Criterion for Smooth Distributions). Suppose \( M \) is a smooth \( n \) -manifold and \( D \subseteq {TM} \) is a distribution of rank \( k \) . Then \( D \) is smooth if and only if each point \( p \in M \) has a neighborhood \( U \) on which there are smooth 1 -forms \( {\omega }^{1},\ldots ,{\ome... | Proof. First suppose that there exist such forms \( {\omega }^{1},\ldots ,{\omega }^{n - k} \) in a neighborhood of each point. The assumption (19.1) together with the fact that \( D \) has rank \( k \) implies that the forms \( {\omega }^{1},\ldots ,{\omega }^{n - k} \) are independent on \( U \) for dimensional reaso... | Yes |
Lemma 19.6. Suppose \( M \) is a smooth \( n \) -manifold and \( D \) is a smooth rank- \( k \) distribution on \( M \) . Let \( {\omega }^{1},\ldots ,{\omega }^{n - k} \) be smooth local defining forms for \( D \) over an open subset \( U \subseteq M \) . A smooth p-form \( \eta \) defined on \( U \) annihilates \( D ... | Proof. It is easy to check that any form \( \eta \) that satisfies (19.2) in a neighborhood of each point annihilates \( D \) . Conversely, suppose \( \eta \) annihilates \( D \) on \( U \) . In a neighborhood of each point we can complete the \( \left( {n - k}\right) \) -tuple \( \left( {{\omega }^{1},\ldots ,{\omega ... | Yes |
Theorem 19.7 (1-Form Criterion for Involutivity). Suppose \( D \subseteq {TM} \) is a smooth distribution. Then \( D \) is involutive if and only if the following condition is satisfied: If \( \eta \) is any smooth 1 -form that annihilates \( D \) on an open subset \( U \subseteq M \), then \( {d\eta } \) also annihila... | Proof. First, assume that \( D \) is involutive, and suppose \( \eta \) is a smooth 1 -form that annihilates \( D \) on \( U \subseteq M \) . Then for any smooth local sections \( X, Y \) of \( D \), formula (14.28) for \( {d\eta } \) gives \[ {d\eta }\left( {X, Y}\right) = X\left( {\eta \left( Y\right) }\right) - Y\le... | Yes |
Proposition 19.11 (Differential Ideal Criterion for Involutivity). Let \( M \) be a smooth manifold. A smooth distribution \( D \subseteq {TM} \) is involutive if and only if \( J\left( D\right) \) is a differential ideal in \( {\Omega }^{ * }\left( M\right) \) . | Proof. Problem 19-1. | No |
Corollary 19.13. Suppose \( M \) is a smooth manifold, \( D \) is an involutive rank- \( k \) distribution on \( M \), and \( S \subseteq M \) is a codimension-k embedded submanifold. If \( p \in S \) is a point such that \( {T}_{p}S \) is complementary to \( {D}_{p} \), then there is a flat chart \( \left( {U,\left( {... | Proof. The proof of the theorem showed that locally \( D \) is spanned by \( k \) commuting vector fields, and then the corollary follows from Theorem 9.46. | No |
Let \( D \subseteq T{\mathbb{R}}^{3} \) be the distribution spanned by the vector fields\n\n\[ X = x\frac{\partial }{\partial x} + \frac{\partial }{\partial y} + x\left( {y + 1}\right) \frac{\partial }{\partial z} \]\n\n\[ Y = \frac{\partial }{\partial x} + y\frac{\partial }{\partial z} \]\n\nThe computation of Example... | The proof of the Frobenius theorem shows that if we can find smooth local sections \( V, W \) of \( D \) that are \( \pi \) -related to \( \partial /\partial x \) and \( \partial /\partial y \) , respectively, they will be commuting vector fields spanning \( D \) . It is easy to check that \( V, W \) have this property... | Yes |
Proposition 19.16 (Local Structure of Integral Manifolds). Let \( D \) be an involutive distribution of rank \( k \) on a smooth manifold \( M \), and let \( \left( {U,\left( {x}^{i}\right) }\right) \) be a flat chart for \( D \) . If \( H \) is any integral manifold of \( D \), then \( H \cap U \) is a union of counta... | Proof. Let \( H \) be an integral manifold of \( D \) . Because the inclusion map \( \iota : H \hookrightarrow M \) is continuous, \( H \cap U = {\iota }^{-1}\left( U\right) \) is open in \( H \), and thus consists of a countable disjoint union of connected components, each of which is open in \( H \) . Let \( V \) be ... | Yes |
Theorem 19.17. Every integral manifold of an involutive distribution is weakly embedded. | Proof. Let \( M \) be a smooth \( n \) -manifold, let \( H \subseteq M \) be an integral manifold of an involutive rank- \( k \) distribution \( D \) on \( M \), and suppose \( F : N \rightarrow M \) is a smooth map such that \( F\left( N\right) \subseteq H \) . Let \( p \in N \) be arbitrary, and set \( q = F\left( p\... | Yes |
Proposition 19.23. Let \( M \) be a smooth manifold and \( \Phi : M \rightarrow M \) be a diffeomorphism. Suppose \( D \) is an involutive distribution on \( M \) and \( \mathcal{F} \) is the foliation it determines. Then \( D \) is \( \Phi \) -invariant if and only if \( \mathcal{F} \) is \( \Phi \) -invariant. | Proof. Problem 19-9. | No |
Lemma 19.24. Let \( G \) be a Lie group. If \( \mathfrak{h} \) is a Lie subalgebra of \( \operatorname{Lie}\left( G\right) \), then the subset \( D = \mathop{\bigcup }\limits_{{g \in G}}{D}_{g} \subseteq {TG} \), where\n\n\[ \n{D}_{g} = \left\{ {{X}_{g} : X \in \mathfrak{h}}\right\} \subseteq {T}_{g}G \n\]\n\n(19.7)\n\... | Proof. Each \( X \in \mathfrak{h} \) is a left-invariant vector field on \( G \) . Thus, for any \( g,{g}^{\prime } \in G \) , the differential \( d\left( {L}_{{g}^{\prime }{g}^{-1}}\right) \) restricts to an isomorphism from \( {D}_{g} \) to \( {D}_{{g}^{\prime }} \) . It follows that \( {D}_{g} \) has the same dimens... | Yes |
Theorem 19.25 (Lie Subgroups Are Weakly Embedded). Every Lie subgroup is an integral manifold of an involutive distribution, and therefore is a weakly embedded submanifold. | Proof. Suppose \( G \) is a Lie group and \( H \subseteq G \) is a Lie subgroup. Theorem 8.46 shows that the Lie algebra of \( H \) is canonically isomorphic to the Lie subalgebra \( \mathfrak{h} = {\iota }_{ * }\left( {\operatorname{Lie}\left( H\right) }\right) \subseteq \operatorname{Lie}\left( G\right) \), where \( ... | Yes |
Theorem 19.26 (The Lie Subgroup Associated with a Lie Subalgebra). Suppose \( G \) is a Lie group and \( \mathfrak{g} \) is its Lie algebra. If \( \mathfrak{h} \) is any Lie subalgebra of \( \mathfrak{g} \), then there is a unique connected Lie subgroup of \( G \) whose Lie algebra is \( \mathfrak{h} \) . | Proof. Suppose \( \mathfrak{h} \) is a Lie subalgebra of \( \mathfrak{g} \) . Let \( D \subseteq {TG} \) be the involutive distribution defined by (19.7). Let \( \mathcal{H} \) denote the foliation determined by \( D \), and for any \( g \in G \), let \( {\mathcal{H}}_{g} \) denote the leaf of \( \mathcal{H} \) contain... | Yes |
Theorem 19.27. Let \( W \subseteq {\mathbb{R}}^{n} \) be an open subset and let \( m \) be an integer such that \( 1 \leq m \leq n \) . Suppose we are given an embedded codimension- \( m \) submanifold \( S \subseteq W \), a linearly independent \( m \) -tuple of smooth vector fields \( \left( {{A}_{1},\ldots ,{A}_{m}}... | Proof. Let \( D \) be the distribution on \( W \) spanned by \( {A}_{1},\ldots ,{A}_{m} \), and let \( p \in S \) be arbitrary. It follows from (19.9) that \( D \) is involutive, so by Corollary 19.13, on some neighborhood \( U \) of \( p \) there is a flat chart for \( D \) centered at \( p \) that is also a slice cha... | Yes |
Theorem 20.1 (Characterization of One-Parameter Subgroups). Let \( G \) be a Lie group. The one-parameter subgroups of \( G \) are precisely the maximal integral curves of left-invariant vector fields starting at the identity. | Proof. First suppose \( \gamma \) is the maximal integral curve of some left-invariant vector field \( X \in \operatorname{Lie}\left( G\right) \) starting at the identity. Because left-invariant vector fields are complete (Theorem 9.18), \( \gamma \) is defined on all of \( \mathbb{R} \) . Left-invariance means that \(... | Yes |
Proposition 20.3. Suppose \( G \) is a Lie group and \( H \subseteq G \) is a Lie subgroup. The one-parameter subgroups of \( H \) are precisely those one-parameter subgroups of \( G \) whose initial velocities lie in \( {T}_{e}H \) . | Proof. Let \( \gamma : \mathbb{R} \rightarrow H \) be a one-parameter subgroup. Then the composite map\n\n\[ \mathbb{R}\overset{\gamma }{ \rightarrow }H \hookrightarrow G \]\n\n is a Lie group homomorphism and thus a one-parameter subgroup of \( G \), which clearly satisfies \( {\gamma }^{\prime }\left( 0\right) \in {T... | Yes |
Proposition 20.5. Let \( G \) be a Lie group. For any \( X \in \operatorname{Lie}\left( G\right) ,\gamma \left( s\right) = \exp {sX} \) is the one-parameter subgroup of \( G \) generated by \( X \). | Proof. Let \( \gamma : \mathbb{R} \rightarrow G \) be the one-parameter subgroup generated by \( X \), which is the integral curve of \( X \) starting at \( e \) . For any fixed \( s \in \mathbb{R} \), it follows from the rescaling lemma (Lemma 9.3) that \( \widetilde{\gamma }\left( t\right) = \gamma \left( {st}\right)... | Yes |
If \( V \) is a finite-dimensional real vector space, a choice of basis for \( V \) yields isomorphisms \( \mathrm{{GL}}\left( V\right) \cong \mathrm{{GL}}\left( {n,\mathbb{R}}\right) \) and \( \mathrm{{gI}}\left( V\right) \cong \mathrm{{gI}}\left( {n,\mathbb{R}}\right) \) . The analysis of the \( \mathrm{{GL}}\left( {... | \[ \exp A = \mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{1}{k!}{A}^{k} \] where we consider \( A \in \mathfrak{{gl}}\left( V\right) \) as a linear map from \( V \) to itself, and \( {A}^{k} = A \circ \cdots \circ A \) is the \( k \) -fold composition of \( A \) with itself. | Yes |
Proposition 20.8 (Properties of the Exponential Map). Let \( G \) be a Lie group and let \( \mathfrak{g} \) be its Lie algebra.\n\n(a) The exponential map is a smooth map from \( \mathfrak{g} \) to \( G \) . | Proof. In this proof, for any \( X \in \mathfrak{g} \) we let \( {\theta }_{\left( X\right) } \) denote the flow of \( X \) . To prove (a), we need to show that the expression \( {\theta }_{\left( X\right) }^{\left( e\right) }\left( 1\right) \) depends smoothly on \( X \), which amounts to showing that the flow varies ... | Yes |
Proposition 20.9. Let \( G \) be a Lie group, and let \( H \subseteq G \) be a Lie subgroup. With \( \operatorname{Lie}\left( H\right) \) considered as a subalgebra of \( \operatorname{Lie}\left( G\right) \) in the usual way, the exponential map of \( H \) is the restriction to \( \operatorname{Lie}\left( H\right) \) o... | Proof. The fact that the exponential map of \( H \) is the restriction of that of \( G \) is an immediate consequence of Proposition 20.3. To prove the second assertion, by the way we have identified \( \operatorname{Lie}\left( H\right) \) as a subalgebra of \( \operatorname{Lie}\left( G\right) \), we need to establish... | Yes |
Proposition 20.10. Let \( G \) be a Lie group and let \( \mathfrak{g} \) be its Lie algebra. For any \( X, Y \in \mathfrak{g} \), there is a smooth function \( Z : \left( {-\varepsilon ,\varepsilon }\right) \rightarrow \mathfrak{g} \) for some \( \varepsilon > 0 \) such that the following identity holds for all \( t \i... | Proof. Since the exponential map is a diffeomorphism on some neighborhood of the origin in \( \mathfrak{g} \), there is some \( \varepsilon > 0 \) such that the map \( \varphi : \left( {-\varepsilon ,\varepsilon }\right) \rightarrow \mathfrak{g} \) defined by\n\n\[ \varphi \left( t\right) = {\exp }^{-1}\left( {\exp {tX... | Yes |
Corollary 20.11. Under the hypotheses of the preceding proposition, \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}{\left( \left( \exp \frac{t}{n}X\right) \left( \exp \frac{t}{n}Y\right) \right) }^{n} = \exp t\left( {X + Y}\right) . \] | Proof. Formula (20.4) implies that for any \( t \in \mathbb{R} \) and any sufficiently large \( n \in \mathbb{Z} \) , \[ \left( {\exp \frac{t}{n}X}\right) \left( {\exp \frac{t}{n}Y}\right) = \exp \left( {\frac{t}{n}\left( {X + Y}\right) + \frac{{t}^{2}}{{n}^{2}}Z\left( \frac{t}{n}\right) }\right) , \] and then Proposit... | Yes |
Lemma 20.14. Suppose \( G \) is a Lie group and \( \theta \) is a smooth right action of \( G \) on a smooth manifold \( M \) . For any \( X \in \operatorname{Lie}\left( G\right) \) and \( p \in M \), the vector fields \( X \) and \( \widehat{\theta }\left( X\right) \) are \( {\theta }^{\left( p\right) } \) -related. | Proof. Let \( X \in \operatorname{Lie}\left( G\right) \) and \( p \in M \) be arbitrary, and write \( \widehat{X} = \widehat{\theta }\left( X\right) \) . Note that the group law \( p \cdot g{g}^{\prime } = \left( {p \cdot g}\right) \cdot {g}^{\prime } \) translates to\n\n\[{\theta }^{\left( p\right) } \circ {L}_{g}\lef... | Yes |
Theorem 20.15. Suppose \( G \) is a Lie group and \( \theta \) is a smooth right action of \( G \) on a smooth manifold \( M \) . Then the map \( \widehat{\theta } : \operatorname{Lie}\left( G\right) \rightarrow \mathfrak{X}\left( M\right) \) defined above is a Lie algebra homomorphism. | Proof. For each \( p \in M \), it follows from (20.8) that \( {\widehat{X}}_{p} \) depends linearly on \( X \), so \( \widehat{\theta } \) is a linear map. Given \( p \in M \), Lemma 20.14 together with the naturality of Lie brackets implies that \( \left\lbrack {X, Y}\right\rbrack \) is \( {\theta }^{\left( p\right) }... | Yes |
Suppose \( G \) is a Lie group and \( M \) is a smooth manifold. (a) If \( \theta \) is a smooth left action of \( G \) on \( M \), the map \( \widehat{\theta } : \operatorname{Lie}\left( G\right) \rightarrow \mathfrak{X}\left( M\right) \) defined by (20.11) is an antihomomorphism (a linear map satisfying \( \widehat{\... | Proof. Problem 20-15. | No |
Theorem 20.19. Suppose \( G \) and \( H \) are Lie groups with \( G \) simply connected, and let \( \mathfrak{g} \) and \( \mathfrak{h} \) be their Lie algebras. For any Lie algebra homomorphism \( \varphi : \mathfrak{g} \rightarrow \mathfrak{h} \) , there is a unique Lie group homomorphism \( \Phi : G \rightarrow H \)... | Proof. The Lie algebra homomorphism \( \varphi : \mathfrak{g} \rightarrow \mathfrak{h} \subseteq \mathfrak{X}\left( H\right) \) is, in particular, a complete g-action on \( H \) (since every left-invariant vector field is complete). Thus, by Theorem 20.16, there is a unique smooth right \( G \) -action \( \theta : H \t... | Yes |
Corollary 20.20. If \( G \) and \( H \) are simply connected Lie groups with isomorphic Lie algebras, then \( G \) and \( H \) are isomorphic. | Proof. Let \( \mathfrak{g},\mathfrak{h} \) be the Lie algebras of \( G \) and \( H \), respectively, and let \( \varphi : \mathfrak{g} \rightarrow \mathfrak{h} \) be a Lie algebra isomorphism between them. By the preceding theorem, there are Lie group homomorphisms \( \Phi : G \rightarrow H \) and \( \Psi : H \rightarr... | Yes |
Theorem 20.21 (The Lie Correspondence). There is a one-to-one correspondence between isomorphism classes of finite-dimensional Lie algebras and isomorphism classes of simply connected Lie groups, given by associating each simply connected Lie group with its Lie algebra. | Proof. We need to show that the functor that sends a simply connected Lie group to its Lie algebra is both surjective and injective up to isomorphism. Injectivity is precisely the content of Corollary 20.20.\n\nTo prove surjectivity, suppose \( g \) is any finite-dimensional Lie algebra. By Corollary 8.50 to Ado’s theo... | Yes |
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