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Lemma 7.12. Suppose \( G \) is a Lie group and \( H \subseteq G \) is an open subgroup. Then \( H \) is an embedded Lie subgroup. In addition, \( H \) is closed, so it is a union of connected components of \( G \) . | Proof. If \( H \) is open in \( G \), it is embedded by Proposition 5.1. In addition, every left coset \( {gH} = \{ {gh} : h \in H\} \) is open in \( G \) because it is the image of the open subset \( H \) under the diffeomorphism \( {L}_{g} \) . Because \( G \smallsetminus H \) is the union of the cosets of \( H \) ot... | Yes |
Proposition 7.14. Suppose \( G \) is a Lie group, and \( W \subseteq G \) is any neighborhood of the identity.\n\n(a) \( W \) generates an open subgroup of \( G \) .\n\n(b) If \( W \) is connected, it generates a connected open subgroup of \( G \) .\n\n(c) If \( G \) is connected, then \( W \) generates \( G \) . | Proof. Let \( W \subseteq G \) be any neighborhood of the identity, and let \( H \) be the subgroup generated by \( W \) . As a matter of notation, if \( A \) and \( B \) are subsets of \( G \), let us write\n\n\[{AB} = \{ {ab} : a \in A, b \in B\} ,\;{A}^{-1} = \left\{ {{a}^{-1} : a \in A}\right\} .\n\]\n\n(7.5)\n\nFo... | Yes |
Proposition 7.15. Let \( G \) be a Lie group and let \( {G}_{0} \) be its identity component. Then \( {G}_{0} \) is a normal subgroup of \( G \), and is the only connected open subgroup. Every connected component of \( G \) is diffeomorphic to \( {G}_{0} \) . | Proof. Problem 7-7. | No |
Proposition 7.16. Let \( F : G \rightarrow H \) be a Lie group homomorphism. The kernel of \( F \) is a properly embedded Lie subgroup of \( G \), whose codimension is equal to the rank of \( F \) . | Proof. Because \( F \) has constant rank, its kernel \( {F}^{-1}\left( e\right) \) is a properly embedded submanifold of codimension equal to rank \( F \) . It is thus a Lie subgroup by Proposition 7.11. | Yes |
Proposition 7.17. If \( F : G \rightarrow H \) is an injective Lie group homomorphism, the image of \( F \) has a unique smooth manifold structure such that \( F\left( G\right) \) is a Lie subgroup of \( H \) and \( F : G \rightarrow F\left( G\right) \) is a Lie group isomorphism. | Proof. Since a Lie group homomorphism has constant rank, it follows from the global rank theorem that \( F \) is a smooth immersion. Proposition 5.18 shows that \( F\left( G\right) \) has a unique smooth manifold structure such that it is an immersed subman-ifold of \( H \) and \( F \) is a diffeomorphism onto its imag... | Yes |
Theorem 7.21. Suppose \( G \) is a Lie group and \( H \subseteq G \) is a Lie subgroup. Then \( H \) is closed in \( G \) if and only if it is embedded. | Proof. Assume first that \( H \) is embedded in \( G \) . To prove that \( H \) is closed, let \( g \) be an arbitrary point of \( \bar{H} \) . Then there is a sequence \( \left( {h}_{i}\right) \) of points in \( H \) converging to \( g \) (Fig. 7.1). Let \( U \) be the domain of a slice chart for \( H \) containing th... | No |
Proposition 7.23. Suppose \( E \) and \( M \) are smooth manifolds with or without boundary, and \( \pi : E \rightarrow M \) is a smooth covering map. With the discrete topology, the automorphism group \( {\operatorname{Aut}}_{\pi }\left( E\right) \) is a zero-dimensional Lie group acting smoothly and freely on \( E \)... | Proof. Suppose \( \varphi \in {\operatorname{Aut}}_{\pi }\left( E\right) \) is an automorphism that fixes a point \( p \in E \) . Simply by rotating diagram (7.8), we can consider \( \varphi \) as a lift of \( \pi \) :\n\n. Let \( M \) and \( N \) be smooth manifolds and let \( G \) be a Lie group. Suppose \( F : M \rightarrow N \) is a smooth map that is equivariant with respect to a transitive smooth \( G \) -action on \( M \) and any smooth \( G \) -action on \( N \) . Then \( F \) has constant ... | Proof. Let \( \theta \) and \( \varphi \) denote the \( G \) -actions on \( M \) and \( N \), respectively, and let \( p \) and \( q \) be arbitrary points in \( M \) . Choose \( g \in G \) such that \( {\theta }_{g}\left( p\right) = q \) . (Such a \( g \) exists because we are assuming that \( G \) acts transitively o... | Yes |
Proposition 7.26 (Properties of the Orbit Map). Suppose \( \theta \) is a smooth left action of a Lie group \( G \) on a smooth manifold \( M \). For each \( p \in M \), the orbit map \( {\theta }^{\left( p\right) } : G \rightarrow M \) is smooth and has constant rank, so the isotropy group \( {G}_{p} = {\left( {\theta... | Proof. The orbit map is smooth because it is equal to the composition\n\n\[ G \approx G \times \{ p\} \hookrightarrow G \times M\overset{\theta }{ \rightarrow }M.\]\n\nIt follows from the definition of a group action that \( {\theta }^{\left( p\right) } \) is equivariant with respect to the action of \( G \) on itself ... | Yes |
A real \( n \times n \) matrix \( A \) is said to be orthogonal if as a linear map \( A : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \) it preserves the Euclidean dot product:\n\n\[ \left( {Ax}\right) \cdot \left( {Ay}\right) = x \cdot y\;\text{ for all }x, y \in {\mathbb{R}}^{n}. \] | The set \( \mathrm{O}\left( n\right) \) of all orthogonal \( n \times n \) matrices is a subgroup of \( \mathrm{{GL}}\left( {n,\mathbb{R}}\right) \), called the orthogonal group of degree \( \mathbf{n} \). It is easy to check that a matrix \( A \) is orthogonal if and only if it takes the standard basis of \( {\mathbb{... | Yes |
The special orthogonal group of degree \( n \) is defined as \( \mathrm{{SO}}\left( n\right) = \mathrm{O}\left( n\right) \cap \mathrm{{SL}}\left( {n,\mathbb{R}}\right) \subseteq \mathrm{{GL}}\left( {n,\mathbb{R}}\right) \). | Because every matrix \( A \in \mathrm{O}\left( n\right) \) satisfies\n\n\[ 1 = \det {I}_{n} = \det \left( {{A}^{T}A}\right) = \left( {\det A}\right) \left( {\det {A}^{T}}\right) = {\left( \det A\right) }^{2}, \]\n\n it follows that \( \det A = \pm 1 \) for all \( A \in \mathrm{O}\left( n\right) \). Therefore, \( \mathr... | Yes |
For any positive integer \( n \), the unitary group of degree \( \mathbf{n} \) is the subgroup \( \mathrm{U}\left( n\right) \subseteq \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \) consisting of complex \( n \times n \) matrices whose columns form an orthonormal basis for \( {\mathbb{C}}^{n} \) with respect to the Hermit... | It is straightforward to check that \( \mathrm{U}\left( n\right) \) consists of those matrices \( A \) such that \( {A}^{ * }A = {I}_{n} \). | No |
Example 7.32 (The Euclidean Group). If we consider \( {\mathbb{R}}^{n} \) as a Lie group under addition, then the natural action of \( \mathrm{O}\left( n\right) \) on \( {\mathbb{R}}^{n} \) is an action by automorphisms. The resulting semidirect product \( \mathrm{E}\left( n\right) = {\mathbb{R}}^{n} \rtimes \mathrm{O}... | This action preserves lines, distances, and angle measures, and thus all of the relationships of Euclidean geometry. | Yes |
Theorem 7.35 (Characterization of Semidirect Products). Suppose \( G \) is a Lie group, and \( N, H \subseteq G \) are closed Lie subgroups such that \( N \) is normal, \( N \cap H = \) \( \{ e\} \), and \( {NH} = G \) . Then the map \( \left( {n, h}\right) \mapsto {nh} \) is a Lie group isomorphism between \( N{ \rtim... | Proof. Problem 7-18. | No |
Proposition 7.37. Let \( G \) be a Lie group and \( V \) be a finite-dimensional vector space. A smooth left action of \( G \) on \( V \) is linear if and only if it is of the form \( g \cdot x = \rho \left( g\right) x \) for some representation \( \rho \) of \( G \) . | Proof. Every action induced by a representation is evidently linear. To prove the converse, assume that we are given a linear action of \( G \) on \( V \) . The hypothesis implies that for each \( g \in G \) there is a linear map \( \rho \left( g\right) \in \mathrm{{GL}}\left( V\right) \) such that \( g \cdot x = \rho ... | Yes |
Proposition 8.1 (Smoothness Criterion for Vector Fields). Let \( M \) be a smooth manifold with or without boundary, and let \( X : M \rightarrow {TM} \) be a rough vector field. \( {If}\left( {U,\left( {x}^{i}\right) }\right) \) is any smooth coordinate chart on \( M \), then the restriction of \( X \) to \( U \) is s... | Proof. Let \( \left( {{x}^{i},{v}^{i}}\right) \) be the natural coordinates on \( {\pi }^{-1}\left( U\right) \subseteq {TM} \) associated with the chart \( \left( {U,\left( {x}^{i}\right) }\right) \) . By definition of natural coordinates, the coordinate representation of \( X : M \rightarrow {TM} \) on \( U \) is\n\n\... | Yes |
Example 8.2 (Coordinate Vector Fields). If \( \left( {U,\left( {x}^{i}\right) }\right) \) is any smooth chart on \( M \) , the assignment \[ p \mapsto {\left. \frac{\partial }{\partial {x}^{i}}\right| }_{p} \] determines a vector field on \( U \), called the \( i \) th coordinate vector field and denoted by \( \partial... | It is smooth because its component functions are constants. | No |
Example 8.4 (The Angle Coordinate Vector Field on the Circle). Let \( \theta \) be any angle coordinate on a proper open subset \( U \subseteq {\mathbb{S}}^{1} \) (see Problem 1-8), and let \( d/{d\theta } \) denote the corresponding coordinate vector field. Because any other angle coordinate \( \widetilde{\theta } \) ... | It is a smooth vector field because its component function is constant in any such chart. We denote this global vector field by \( d/{d\theta } \), even though, strictly speaking, it cannot be considered as a coordinate vector field on the entire circle at once. | Yes |
Lemma 8.6 (Extension Lemma for Vector Fields). Let \( M \) be a smooth manifold with or without boundary, and let \( A \subseteq M \) be a closed subset. Suppose \( X \) is a smooth vector field along \( A \). Given any open subset \( U \) containing \( A \), there exists a smooth global vector field \( \widetilde{X} \... | Proof. See Problem 8-1. | No |
Proposition 8.7. Let \( M \) be a smooth manifold with or without boundary. Given \( p \in M \) and \( v \in {T}_{p}M \), there is a smooth global vector field \( X \) on \( M \) such that \( {X}_{p} = v \) . | Proof. The assignment \( p \mapsto v \) is an example of a vector field along the set \( \{ p\} \) as defined above. It is smooth because it can be extended, say, to a constant-coefficient vector field in a coordinate neighborhood of \( p \) . Thus, the proposition follows from the extension lemma with \( A = \{ p\} \)... | No |
Proposition 8.11 (Completion of Local Frames). Let \( M \) be a smooth \( n \) -manifold with or without boundary.\n\n(a) If \( \left( {{X}_{1},\ldots ,{X}_{k}}\right) \) is a linearly independent \( k \) -tuple of smooth vector fields on an open subset \( U \subseteq M \), with \( 1 \leq k < n \), then for each \( p \... | Proof. See Problem 8-5. | No |
The standard coordinate frame is a global orthonormal frame on \( {\mathbb{R}}^{n} \) . For a less obvious example, consider the smooth vector fields defined on \( {\mathbb{R}}^{2} \smallsetminus \{ 0\} \) by\n\n\[ \n{E}_{1} = \frac{x}{r}\frac{\partial }{\partial x} + \frac{y}{r}\frac{\partial }{\partial y},\;{E}_{2} =... | A straightforward computation shows that \( \left( {{E}_{1},{E}_{2}}\right) \) is an orthonormal frame for \( {\mathbb{R}}^{2} \) over the open subset \( {\mathbb{R}}^{2} \smallsetminus \{ 0\} \) . Geometrically, \( {E}_{1} \) and \( {E}_{2} \) are unit vector fields tangent to radial lines and circles centered at the ... | Yes |
Lemma 8.13 (Gram-Schmidt Algorithm for Frames). Suppose \( \left( {X}_{j}\right) \) is a smooth local frame for \( T{\mathbb{R}}^{n} \) over an open subset \( U \subseteq {\mathbb{R}}^{n} \) . Then there is a smooth orthonormal frame \( \left( {E}_{j}\right) \) over \( U \) such that \( \operatorname{span}\left( {{\lef... | Proof. Applying the Gram-Schmidt algorithm to the vectors \( \left( {\left. {X}_{j}\right| }_{p}\right) \) at each \( p \in U \) , we obtain an \( n \) -tuple of rough vector fields \( \left( {{E}_{1},\ldots ,{E}_{n}}\right) \) given inductively by\n\n\[ \n{E}_{j} = \frac{{X}_{j} - \mathop{\sum }\limits_{{i = 1}}^{{j -... | Yes |
Proposition 8.14. Let \( M \) be a smooth manifold with or without boundary, and let\n\n\( X : M \rightarrow {TM} \) be a rough vector field. The following are equivalent:\n\n(a) \( X \) is smooth.\n\n(b) For every \( f \in {C}^{\infty }\left( M\right) \), the function \( {Xf} \) is smooth on \( M \) .\n\n(c) For every... | Proof. We will prove that (a) \( \Rightarrow \) (b) \( \Rightarrow \) (c) \( \Rightarrow \) (a).\n\nTo prove (a) \( \Rightarrow \) (b), assume \( X \) is smooth, and let \( f \in {C}^{\infty }\left( M\right) \) . For any \( p \in M \) , we can choose smooth coordinates \( \left( {x}^{i}\right) \) on a neighborhood \( U... | Yes |
Proposition 8.15. Let \( M \) be a smooth manifold with or without boundary. A map \( D : {C}^{\infty }\left( M\right) \rightarrow {C}^{\infty }\left( M\right) \) is a derivation if and only if it is of the form \( {Df} = {Xf} \) for some smooth vector field \( X \in \mathfrak{X}\left( M\right) \) . | Proof. We just showed that every smooth vector field induces a derivation. Conversely, suppose \( D : {C}^{\infty }\left( M\right) \rightarrow {C}^{\infty }\left( M\right) \) is a derivation. We need to concoct a vector field \( X \) such that \( {Df} = {Xf} \) for all \( f \) . From the discussion above, it is clear t... | Yes |
Proposition 8.16. Suppose \( F : M \rightarrow N \) is a smooth map between manifolds with or without boundary, \( X \in \mathfrak{X}\left( M\right) \), and \( Y \in \mathfrak{X}\left( N\right) \) . Then \( X \) and \( Y \) are \( F \) -related if and only if for every smooth real-valued function \( f \) defined on an ... | Proof. For any \( p \in M \) and any smooth real-valued \( f \) defined in a neighborhood of \( F\left( p\right) \) , \[ X\left( {f \circ F}\right) \left( p\right) = {X}_{p}\left( {f \circ F}\right) = d{F}_{p}\left( {X}_{p}\right) f, \] while \[ \left( {Yf}\right) \circ F\left( p\right) = \left( {Yf}\right) \left( {F\l... | Yes |
Proposition 8.19. Suppose \( M \) and \( N \) are smooth manifolds with or without boundary, and \( F : M \rightarrow N \) is a diffeomorphism. For every \( X \in \mathfrak{X}\left( M\right) \), there is a unique smooth vector field on \( N \) that is \( F \) -related to \( X \) . | Proof. For \( Y \in \mathfrak{X}\left( N\right) \) to be \( F \) -related to \( X \) means that \( d{F}_{p}\left( {X}_{p}\right) = {Y}_{F\left( p\right) } \) for every \( p \in M \) . If \( F \) is a diffeomorphism, therefore, we define \( Y \) by\n\n\[ \n{Y}_{q} = d{F}_{{F}^{-1}\left( q\right) }\left( {X}_{{F}^{-1}\le... | Yes |
Example 8.20 (Computing the Pushforward of a Vector Field). Let \( M \) and \( N \) be the following open submanifolds of \( {\mathbb{R}}^{2} \) :\n\n\[ M = \{ \left( {x, y}\right) : y > 0\text{ and }x + y > 0\} ,\]\n\n\[ N = \{ \left( {u, v}\right) : u > 0\text{ and }v > 0\} \]\n\nand define \( F : M \rightarrow N \) ... | The differential of \( F \) at a point \( \left( {x, y}\right) \in M \) is represented by its Jacobian matrix,\n\n\[ {DF}\left( {x, y}\right) = \left( \begin{matrix} 1 & 1 \\ \frac{1}{y} & - \frac{x}{{y}^{2}} \end{matrix}\right) \]\n\nand thus \( d{F}_{{F}^{-1}\left( {u, v}\right) } \) is represented by the matrix\n\n\... | Yes |
Proposition 8.22. Let \( M \) be a smooth manifold, \( S \subseteq M \) be an embedded subman-ifold with or without boundary, and \( X \) be a smooth vector field on \( M \) . Then \( X \) is tangent to \( S \) if and only if \( {\left. \left( Xf\right) \right| }_{S} = 0 \) for every \( f \in {C}^{\infty }\left( M\righ... | Proof. This is an immediate consequence of Proposition 5.37. | No |
Proposition 8.23 (Restricting Vector Fields to Submanifolds). Let \( M \) be a smooth manifold, let \( S \subseteq M \) be an immersed submanifold with or without boundary, and let \( \iota : S \hookrightarrow M \) denote the inclusion map. If \( Y \in \mathfrak{X}\left( M\right) \) is tangent to \( S \), then there is... | Proof. The fact that \( Y \) is tangent to \( S \) means by definition that \( {Y}_{p} \) is in the image of \( d{\iota }_{p} \) for each \( p \) . Thus, for each \( p \) there is a vector \( {X}_{p} \in {T}_{p}S \) such that \( {Y}_{p} = d{\iota }_{p}\left( {X}_{p}\right) \) . Since \( d{\iota }_{p} \) is injective, \... | Yes |
Lemma 8.25. The Lie bracket of any pair of smooth vector fields is a smooth vector field. | Proof. By Proposition 8.15, it suffices to show that \( \left\lbrack {X, Y}\right\rbrack \) is a derivation of \( {C}^{\infty }\left( M\right) \) . For arbitrary \( f, g \in {C}^{\infty }\left( M\right) \), we compute\n\n\[ \left\lbrack {X, Y}\right\rbrack \left( {fg}\right) = X\left( {Y\left( {fg}\right) }\right) - Y\... | Yes |
Proposition 8.26 (Coordinate Formula for the Lie Bracket). Let \( X, Y \) be smooth vector fields on a smooth manifold \( M \) with or without boundary, and let \( X = \) \( {X}^{i}\partial /\partial {x}^{i} \) and \( Y = {Y}^{j}\partial /\partial {x}^{j} \) be the coordinate expressions for \( X \) and \( Y \) in term... | Proof. Because we know already that \( \left\lbrack {X, Y}\right\rbrack \) is a smooth vector field, its action on a function is determined locally: \( {\left. \left( \left\lbrack X, Y\right\rbrack f\right) \right| }_{U} = \left\lbrack {X, Y}\right\rbrack \left( {\left. f\right| }_{U}\right) \) . Thus it suffices to co... | Yes |
Define smooth vector fields \( X, Y \in \mathfrak{X}\left( {\mathbb{R}}^{3}\right) \) by\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\nThen (8.9) yields\n\n\[ \l... | \n\[ = 0\frac{\partial }{\partial x} + 1\frac{\partial }{\partial z} - 1\frac{\partial }{\partial x} - 0\frac{\partial }{\partial y} - \left( {y + 1}\right) \frac{\partial }{\partial z} \]\n\n\[ = - \frac{\partial }{\partial x} - y\frac{\partial }{\partial z} \] | Yes |
Proposition 8.28 (Properties of the Lie Bracket). The Lie bracket satisfies the following identities for all \( X, Y, Z \in \mathfrak{X}\left( M\right) \):\n\n(a) BILINEARITY: For \( a, b \in \mathbb{R} \),\n\n\[ \left\lbrack {{aX} + {bY}, Z}\right\rbrack = a\left\lbrack {X, Z}\right\rbrack + b\left\lbrack {Y, Z}\right... | Proof. Bilinearity and antisymmetry are obvious consequences of the definition. The proof of the Jacobi identity is just a computation:\n\n\[ \left\lbrack {X,\left\lbrack {Y, Z}\right\rbrack }\right\rbrack f + \left\lbrack {Y,\left\lbrack {Z, X}\right\rbrack }\right\rbrack f + \left\lbrack {Z,\left\lbrack {X, Y}\right\... | No |
Proposition 8.30 (Naturality of the Lie Bracket). Let \( F : M \rightarrow N \) be a smooth map between manifolds with or without boundary, and let \( {X}_{1},{X}_{2} \in \mathfrak{X}\left( M\right) \) and \( {Y}_{1},{Y}_{2} \in \mathfrak{X}\left( N\right) \) be vector fields such that \( {X}_{i} \) is \( F \) -related... | Proof. Using Proposition 8.16 and the fact that \( {X}_{i} \) and \( {Y}_{i} \) are \( F \) -related,\n\n\[ \n{X}_{1}{X}_{2}\left( {f \circ F}\right) = {X}_{1}\left( {\left( {{Y}_{2}f}\right) \circ F}\right) = \left( {{Y}_{1}{Y}_{2}f}\right) \circ F.\n\]\n\nSimilarly,\n\n\[ \n{X}_{2}{X}_{1}\left( {f \circ F}\right) = \... | Yes |
Corollary 8.31 (Pushforwards of Lie Brackets). Suppose \( F : M \rightarrow N \) is a diffeomorphism and \( {X}_{1},{X}_{2} \in \mathfrak{X}\left( M\right) \) . Then \( {F}_{ * }\left\lbrack {{X}_{1},{X}_{2}}\right\rbrack = \left\lbrack {{F}_{ * }{X}_{1},{F}_{ * }{X}_{2}}\right\rbrack \) . | Proof. This is just the special case of Proposition 8.30 in which \( F \) is a diffeomorphism and \( {Y}_{i} = {F}_{ * }{X}_{i} \) . | No |
Corollary 8.32 (Brackets of Vector Fields Tangent to Submanifolds). Let \( M \) \( {be} \) a smooth manifold and let \( S \) be an immersed submanifold with or without boundary in \( M \) . If \( {Y}_{1} \) and \( {Y}_{2} \) are smooth vector fields on \( M \) that are tangent to \( S \), then \( \left\lbrack {{Y}_{1},... | Proof. By Proposition 8.23, there exist smooth vector fields \( {X}_{1} \) and \( {X}_{2} \) on \( S \) such that \( {X}_{i} \) is \( \iota \) -related to \( {Y}_{i} \) for \( i = 1,2 \) (where \( \iota : S \rightarrow M \) is the inclusion). By Proposition \( {8.30},\left\lbrack {{X}_{1},{X}_{2}}\right\rbrack \) is \(... | Yes |
Proposition 8.33. Let \( G \) be a Lie group, and suppose \( X \) and \( Y \) are smooth left-invariant vector fields on \( G \) . Then \( \left\lbrack {X, Y}\right\rbrack \) is also left-invariant. | Proof. Let \( g \in G \) be arbitrary. Since \( {\left( {L}_{g}\right) }_{ * }X = X \) and \( {\left( {L}_{g}\right) }_{ * }Y = Y \) by definition of left-invariance, it follows from Corollary 8.31 that\n\n\[ \n{\left( {L}_{g}\right) }_{ * }\left\lbrack {X, Y}\right\rbrack = \left\lbrack {{\left( {L}_{g}\right) }_{ * }... | Yes |
Corollary 8.38. Every left-invariant rough vector field on a Lie group is smooth. | Proof. Let \( X \) be a left-invariant rough vector field on a Lie group \( G \), and let \( v = {X}_{e} \) . The fact that \( X \) is left-invariant implies that \( X = {v}^{\mathrm{L}} \), which is smooth. | Yes |
Every Lie group admits a left-invariant smooth global frame, and therefore every Lie group is parallelizable. | If \( G \) is a Lie group, every basis for \( \operatorname{Lie}\left( G\right) \) is a left-invariant smooth global frame for \( G \) . | Yes |
Let us determine the Lie algebras of some familiar Lie groups. | (a) Euclidean space \( {\mathbb{R}}^{n} \) : If we consider \( {\mathbb{R}}^{n} \) as a Lie group under addition, left translation by an element \( b \in {\mathbb{R}}^{n} \) is given by the affine map \( {L}_{b}\left( x\right) = b + x \) , whose differential \( d\left( {L}_{b}\right) \) is represented by the identity m... | Yes |
Proposition 8.41 (Lie Algebra of the General Linear Group). The composition of the natural maps\n\n\[ \operatorname{Lie}\left( {\mathrm{{GL}}\left( {n,\mathbb{R}}\right) }\right) \rightarrow {T}_{{I}_{n}}\mathrm{{GL}}\left( {n,\mathbb{R}}\right) \rightarrow \mathrm{g}\mathrm{I}\left( {n,\mathbb{R}}\right) \]\n\n(8.14)\... | Proof. Using the matrix entries \( {X}_{j}^{i} \) as global coordinates on \( \mathrm{{GL}}\left( {n,\mathbb{R}}\right) \subseteq \mathrm{{gI}}\left( {n,\mathbb{R}}\right) \) , the natural isomorphism \( {T}_{{I}_{n}}\mathrm{{GL}}\left( {n,\mathbb{R}}\right) \leftrightarrow \mathrm{g}\mathrm{I}\left( {n,\mathbb{R}}\rig... | No |
Corollary 8.42. If \( V \) is any finite-dimensional real vector space, the composition of the canonical isomorphisms in (8.16) yields a Lie algebra isomorphism between \( \operatorname{Lie}\left( {\mathrm{{GL}}\left( V\right) }\right) \) and \( \mathfrak{{gl}}\left( V\right) \) . | - Exercise 8.43. Prove the preceding corollary by choosing a basis for \( V \) and applying Proposition 8.41. | No |
Theorem 8.44 (Induced Lie Algebra Homomorphisms). Let \( G \) and \( H \) be Lie groups, and let \( \mathfrak{g} \) and \( \mathfrak{h} \) be their Lie algebras. Suppose \( F : G \rightarrow H \) is a Lie group homomorphism. For every \( X \in \mathfrak{g} \), there is a unique vector field in \( \mathfrak{h} \) that i... | Proof. If there is any vector field \( Y \in \mathfrak{h} \) that is \( F \) -related to \( X \), it must satisfy \( {Y}_{e} = \) \( d{F}_{e}\left( {X}_{e}\right) \), and thus it must be uniquely determined by\n\n\[ Y = {\left( d{F}_{e}\left( {X}_{e}\right) \right) }^{\mathrm{L}}. \]\n\nTo show that this \( Y \) is \( ... | Yes |
Theorem 8.46 (The Lie Algebra of a Lie Subgroup). Suppose \( H \subseteq G \) is a Lie subgroup, and \( \iota : H \hookrightarrow G \) is the inclusion map. There is a Lie subalgebra \( \mathfrak{h} \subseteq \) \( \operatorname{Lie}\left( G\right) \) that is canonically isomorphic to \( \operatorname{Lie}\left( H\righ... | Proof. Because the inclusion map \( \iota : H \hookrightarrow G \) is a Lie group homomorphism, \( {\iota }_{ * }\left( {\operatorname{Lie}\left( H\right) }\right) \) is a Lie subalgebra of \( \operatorname{Lie}\left( G\right) \) . By the way we defined the induced Lie algebra homomorphism, this subalgebra is precisely... | Yes |
Example 8.47 (The Lie Algebra of \( \mathbf{O}\left( n\right) \) ). The orthogonal group \( \mathrm{O}\left( n\right) \) is a Lie subgroup of \( \mathrm{{GL}}\left( {n,\mathbb{R}}\right) \) . By Example 7.27, it is equal to the level set \( {\Phi }^{-1}\left( {I}_{n}\right) \), where \( \Phi : \mathrm{{GL}}\left( {n,\m... | By the computation in Example 7.27, this differential is \( d{\Phi }_{{I}_{n}}\left( B\right) = {B}^{\widetilde{T}} + B \), so\n\n\[ \n{T}_{{I}_{n}}\mathrm{O}\left( n\right) = \left\{ {B \in \mathfrak{{gl}}\left( {n,\mathbb{R}}\right) : {B}^{T} + B = 0}\right\} \n\]\n\n\[ \n= \{ \text{skew-symmetric}n \times n\text{mat... | Yes |
Proposition 8.48 (The Lie Algebra of \( \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \) ). The composition of the maps in (8.18) yields a Lie algebra isomorphism between \( \operatorname{Lie}\left( {\mathrm{{GL}}\left( {n,\mathbb{C}}\right) }\right) \) and the matrix algebra \( \mathfrak{{gl}}\left( {n,\mathbb{C}}\right) ... | Proof. The Lie group homomorphism \( \beta : \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \rightarrow \mathrm{{GL}}\left( {{2n},\mathbb{R}}\right) \) that we constructed in Example 7.18(d) induces a Lie algebra homomorphism\n\n\[ \n{\beta }_{ * } : \operatorname{Lie}\left( {\mathrm{{GL}}\left( {n,\mathbb{C}}\right) }\righ... | Yes |
Every finite-dimensional real Lie algebra is isomorphic to a Lie subalgebra of some matrix algebra \( \mathfrak{{gl}}\left( {n,\mathbb{R}}\right) \) with the commutator bracket. | Let \( \mathfrak{g} \) be a finite-dimensional real Lie algebra. By Ado’s theorem, \( \mathfrak{g} \) has a faithful representation \( \rho : \mathfrak{g} \rightarrow \mathfrak{{gl}}\left( V\right) \) for some finite-dimensional real vector space \( V \) . Choosing a basis for \( V \) yields an isomorphism of \( \mathf... | Yes |
Proposition 9.2. Let \( V \) be a smooth vector field on a smooth manifold \( M \). For each point \( p \in M \), there exist \( \varepsilon > 0 \) and a smooth curve \( \gamma : \left( {-\varepsilon ,\varepsilon }\right) \rightarrow M \) that is an integral curve of \( V \) starting at \( p \). | Proof. This is just the existence statement of Theorem D. 1 applied to the coordinate representation of \( V \). | No |
Lemma 9.3 (Rescaling Lemma). Let \( V \) be a smooth vector field on a smooth manifold \( M \), let \( J \subseteq \mathbb{R} \) be an interval, and let \( \gamma : J \rightarrow M \) be an integral curve of \( V \). For any \( a \in \mathbb{R} \), the curve \( \widetilde{\gamma } : \widetilde{J} \rightarrow M \) defin... | Proof. One way to see this is as a straightforward application of the chain rule in local coordinates. Somewhat more invariantly, we can examine the action of \( {\widetilde{\gamma }}^{\prime }\left( t\right) \) on a smooth real-valued function \( f \) defined in a neighborhood of a point \( \widetilde{\gamma }\left( {... | Yes |
Proposition 9.6 (Naturality of Integral Curves). Suppose \( M \) and \( N \) are smooth manifolds and \( F : M \rightarrow N \) is a smooth map. Then \( X \in \mathfrak{X}\left( M\right) \) and \( Y \in \mathfrak{X}\left( N\right) \) are \( F \) -related if and only if \( F \) takes integral curves of \( X \) to integr... | Proof. Suppose first that \( X \) and \( Y \) are \( F \) -related, and \( \gamma : J \rightarrow M \) is an integral curve of \( X \) . If we define \( \sigma : J \rightarrow N \) by \( \sigma = F \circ \gamma \) (see Fig. 9.3), then\n\n\[ \n{\sigma }^{\prime }\left( t\right) = {\left( F \circ \gamma \right) }^{\prime... | Yes |
Proposition 9.7. Let \( \theta : \mathbb{R} \times M \rightarrow M \) be a smooth global flow on a smooth manifold \( M \) . The infinitesimal generator \( V \) of \( \theta \) is a smooth vector field on \( M \), and each curve \( {\theta }^{\left( p\right) } \) is an integral curve of \( V \) . | Proof. To show that \( V \) is smooth, it suffices by Proposition 8.14 to show that \( {Vf} \) is smooth for every smooth real-valued function \( f \) defined on an open subset \( U \subseteq M \) . For any such \( f \) and any \( p \in U \), just note that\n\n\[ \n{Vf}\left( p\right) = {V}_{p}f = {\theta }^{\left( p\r... | Yes |
The flow of \( V = \partial /\partial x \) in \( {\mathbb{R}}^{2} \) is the map \( \tau : \mathbb{R} \times {\mathbb{R}}^{2} \rightarrow {\mathbb{R}}^{2} \) given by | \[ {\tau }_{t}\left( {x, y}\right) = \left( {x + t, y}\right) . \] For each nonzero \( t \in \mathbb{R},{\tau }_{t} \) translates the plane to the right \( \left( {t > 0}\right) \) or left \( \left( {t < 0}\right) \) by a distance \( \left| t\right| \) . | Yes |
Let \( M = {\mathbb{R}}^{2} \smallsetminus \{ 0\} \) with standard coordinates \( \left( {x, y}\right) \), and let \( V \) be the vector field \( \partial /\partial x \) on \( M \) . The unique integral curve of \( V \) starting at \( \left( {-1,0}\right) \in M \) is \( \gamma \left( t\right) = \left( {t - 1,0}\right) ... | This is intuitively evident because of the \ | No |
For a more subtle example, let \( M \) be all of \( {\mathbb{R}}^{2} \) and let \( W = {x}^{2}\partial /\partial x \) . You can check easily that the unique integral curve of \( W \) starting at \( \left( {1,0}\right) \) is | \[ \gamma \left( t\right) = \left( {\frac{1}{1 - t},0}\right) \] This curve also cannot be extended past \( t = 1 \), because its \( x \) -coordinate is unbounded as \( t \nearrow 1 \). | Yes |
Proposition 9.11. If \( \theta : \mathcal{D} \rightarrow M \) is a smooth flow, then the infinitesimal generator \( V \) of \( \theta \) is a smooth vector field, and each curve \( {\theta }^{\left( p\right) } \) is an integral curve of \( V \) . | Proof. The proof is essentially identical to the analogous proof for global flows, Proposition 9.7. In the proof that \( V \) is smooth, we need only note that for any \( {p}_{0} \in M,\theta \left( {t, p}\right) \) is defined and smooth for all \( \left( {t, p}\right) \) sufficiently close to \( \left( {0,{p}_{0}}\rig... | Yes |
Proposition 9.13 (Naturality of Flows). Suppose \( M \) and \( N \) are smooth manifolds, \( F : M \rightarrow N \) is a smooth map, \( X \in \mathfrak{X}\left( M\right) \), and \( Y \in \mathfrak{X}\left( N\right) \) . Let \( \theta \) be the flow of \( X \) and \( \eta \) the flow of \( Y \) . If \( X \) and \( Y \) ... | Proof. By Proposition 9.6, for any \( p \in M \), the curve \( F \circ {\theta }^{\left( p\right) } \) is an integral curve of \( Y \) starting at \( F \circ {\theta }^{\left( p\right) }\left( 0\right) = F\left( p\right) \) . By uniqueness of integral curves, therefore, the maximal integral curve \( {\eta }^{\left( F\l... | Yes |
Lemma 9.15 (Uniform Time Lemma). Let \( V \) be a smooth vector field on a smooth manifold \( M \), and let \( \theta \) be its flow. Suppose there is a positive number \( \varepsilon \) such that for every \( p \in M \), the domain of \( {\theta }^{\left( p\right) } \) contains \( \left( {-\varepsilon ,\varepsilon }\r... | Proof. Suppose for the sake of contradiction that for some \( p \in M \), the domain \( {\mathcal{D}}^{\left( p\right) } \) of \( {\theta }^{\left( p\right) } \) is bounded above. (A similar proof works if it is bounded below.) Let \( b = \) \( \sup {\mathcal{D}}^{\left( p\right) } \), let \( {t}_{0} \) be a positive n... | Yes |
Theorem 9.16. Every compactly supported smooth vector field on a smooth manifold is complete. | Proof. Suppose \( V \) is a compactly supported vector field on a smooth manifold \( M \) , and let \( K = \operatorname{supp}V \) . For each \( p \in K \), there is a neighborhood \( {U}_{p} \) of \( p \) and a positive number \( {\varepsilon }_{p} \) such that the flow of \( V \) is defined at least on \( \left( {-{\... | Yes |
Theorem 9.18. Every left-invariant vector field on a Lie group is complete. | Proof. Let \( G \) be a Lie group, let \( X \in \operatorname{Lie}\left( G\right) \), and let \( \theta : \mathcal{D} \rightarrow G \) denote the flow of \( X \) . There is some \( \varepsilon > 0 \) such that \( {\theta }^{\left( e\right) } \) is defined on \( \left( {-\varepsilon ,\varepsilon }\right) \) . Let \( g \... | Yes |
Lemma 9.19 (Escape Lemma). Suppose \( M \) is a smooth manifold and \( V \in \mathfrak{X}\left( M\right) \) . If \( \gamma : J \rightarrow M \) is a maximal integral curve of \( V \) whose domain \( J \) has a finite least upper bound \( b \), then for any \( {t}_{0} \in J,\gamma \left( \left\lbrack {{t}_{0}, b}\right)... | Proof. Problem 9-6. | No |
Theorem 9.20 (Flowout Theorem). Suppose \( M \) is a smooth manifold, \( S \subseteq M \) is an embedded \( k \) -dimensional submanifold, and \( V \in \mathfrak{X}\left( M\right) \) is a smooth vector field that is nowhere tangent to \( S \) . Let \( \theta : \mathcal{D} \rightarrow M \) be the flow of \( V \), let \(... | Proof. First we prove (b). Fix \( p \in S \), and let \( \sigma : {\mathcal{D}}^{\left( p\right) } \rightarrow \mathbb{R} \times S \) be the curve \( \sigma \left( t\right) = \) \( \left( {t, p}\right) \) . Then \( \Phi \circ \sigma \left( t\right) = \theta \left( {t, p}\right) \) is an integral curve of \( V \), so fo... | Yes |
Proposition 9.21. Let \( V \) be a smooth vector field on a smooth manifold \( M \), and let \( \theta : \mathcal{D} \rightarrow M \) be the flow generated by \( V \). If \( p \in M \) is a singular point of \( V \), then \( {\mathcal{D}}^{\left( p\right) } = \mathbb{R} \) and \( {\theta }^{\left( p\right) } \) is the ... | Proof. If \( {V}_{p} = 0 \), then the constant curve \( \gamma : \mathbb{R} \rightarrow M \) given by \( \gamma \left( t\right) \equiv p \) is clearly an integral curve of \( V \), so by uniqueness and maximality it must be equal to \( {\theta }^{\left( p\right) } \).\n\nTo verify the second statement, we prove its con... | Yes |
Theorem 9.22 (Canonical Form Near a Regular Point). Let \( V \) be a smooth vector field on a smooth manifold \( M \), and let \( p \in M \) be a regular point of \( V \) . There exist smooth coordinates \( \left( {s}^{i}\right) \) on some neighborhood of \( p \) in which \( V \) has the coordinate representation \( \p... | Proof. If no hypersurface \( S \) is given, choose any smooth coordinates \( \left( {U,\left( {x}^{i}\right) }\right) \) centered at \( p \), and let \( S \subseteq U \) be the hypersurface defined by \( {x}^{j} = 0 \), where \( j \) is chosen so that \( {V}^{j}\left( p\right) \neq 0 \) . (Recall that \( p \) is a regu... | Yes |
Let \( W = x\partial /\partial y - y\partial /\partial x \) on \( {\mathbb{R}}^{2} \) . We computed the flow of \( W \) in Example 9.8(b). The point \( \left( {1,0}\right) \in {\mathbb{R}}^{2} \) is a regular point of \( W \), because \( {W}_{\left( 1,0\right) } = \) \( \partial /{\left. \partial y\right| }_{\left( 1,0... | \[ \Psi \left( {t, s}\right) = {\theta }_{t}\left( {s,0}\right) = \left( {s\cos t, s\sin t}\right) ,\] and then solve locally for \( \left( {t, s}\right) \) in terms of \( \left( {x, y}\right) \) to obtain the following coordinate map in a neighborhood of \( \left( {1,0}\right) \) : \[ \left( {t, s}\right) = {\Psi }^{-... | Yes |
Theorem 9.24 (Boundary Flowout Theorem). Let \( M \) be a smooth manifold with nonempty boundary, and let \( N \) be a smooth vector field on \( M \) that is inward-pointing at each point of \( \partial M \) . There exist a smooth function \( \delta : \partial M \rightarrow {\mathbb{R}}^{ + } \) and a smooth embedding ... | Proof. Problem 9-11. | No |
Theorem 9.25 (Collar Neighborhood Theorem). If \( M \) is a smooth manifold with nonempty boundary, then \( \partial M \) has a collar neighborhood. | Proof. By the result of Problem 8-4, there exists a smooth vector field \( N \in \mathfrak{X}\left( M\right) \) whose restriction to \( \partial M \) is everywhere inward-pointing. Let \( \delta : M \rightarrow {\mathbb{R}}^{ + } \) and \( \Phi : {\mathcal{P}}_{\delta } \rightarrow M \) be as in Theorem 9.24, and defin... | Yes |
Theorem 9.26. Let \( M \) be a smooth manifold with nonempty boundary, and let \( \iota : \) Int \( M \hookrightarrow M \) denote inclusion. There exists a proper smooth embedding \( R : M \rightarrow \) Int \( M \) such that both \( \iota \circ R : M \rightarrow M \) and \( R \circ \iota : \) Int \( M \rightarrow \) I... | Proof. Theorem 9.25 shows that \( \partial M \) has a collar neighborhood \( C \) in \( M \), which is the image of a smooth embedding \( E : \lbrack 0,1) \times \partial M \rightarrow M \) satisfying \( E\left( {0, x}\right) = x \) for all \( x \in \partial M \) . To simplify notation, we will use this embedding to id... | Yes |
Theorem 9.27 (Whitney Approximation for Manifolds with Boundary). If \( M \) and \( N \) are smooth manifolds with boundary, then every continuous map from \( M \) to \( N \) is homotopic to a smooth map. | Proof. Theorem 6.26 takes care of the case in which \( \partial N = \varnothing \), so we may assume that \( \partial N \neq \varnothing \) . Let \( F : M \rightarrow N \) be a continuous map, let \( \iota : \) Int \( N \hookrightarrow N \) be inclusion, and let \( R : N \rightarrow \operatorname{Int}N \) be the map co... | Yes |
Theorem 9.28. Suppose \( M \) and \( N \) are smooth manifolds with or without boundary. If \( F, G : M \rightarrow N \) are homotopic smooth maps, then they are smoothly homotopic. | Proof. Theorem 6.29 takes care of the case \( \partial N = \varnothing \), so we may assume that \( N \) has nonempty boundary. Let \( \iota : \operatorname{Int}N \hookrightarrow N \) and \( R : N \rightarrow \operatorname{Int}N \) be as in Theorem 9.26. Then \( R \circ F \) and \( R \circ G \) are homotopic smooth map... | Yes |
Lemma 9.33. Suppose \( M \) is a smooth manifold and \( D \subseteq M \) is a regular domain. If \( V \) is a smooth vector field on \( M \) that is tangent to \( \partial D \), then every integral curve of \( V \) that starts in \( D \) remains in \( D \) as long as it is defined. | Proof. Suppose \( \gamma : J \rightarrow M \) is an integral curve of \( V \) with \( \gamma \left( 0\right) \in D \) . Define \( \mathcal{T} \subseteq J \) by \( \mathcal{T} = \{ t \in J : \gamma \left( t\right) \in D\} \) . We will show that \( \mathcal{T} \) is both open and closed in \( J \) ; since \( J \) is an i... | Yes |
Theorem 9.34 (Flows on Manifolds with Boundary). The conclusions of Theorem 9.12 remain true if \( M \) is a smooth manifold with boundary and \( V \) is a smooth vector field on \( M \) that is tangent to \( \partial M \) . | Proof. Example 9.32 shows that we can consider \( M \) as a regular domain in its double \( D\left( M\right) \) . By the extension lemma for vector fields, we can extend \( V \) to a smooth vector field \( \widetilde{V} \) on \( D\left( M\right) \) . Let \( \widetilde{\theta } : \widetilde{\mathcal{D}} \rightarrow D\le... | Yes |
Theorem 9.35 (Canonical Form Near a Regular Point on the Boundary). Let \( M \) be a smooth manifold with boundary and let \( V \) be a smooth vector field on \( M \) that is tangent to \( \partial M \) . If \( p \in \partial M \) is a regular point of \( V \), there exist smooth boundary coordinates \( \left( {s}^{i}\... | Proof. Problem 9-15. | No |
Lemma 9.36. Suppose \( M \) is a smooth manifold with or without boundary, and \( V, W \in \mathfrak{X}\left( M\right) \) . If \( \partial M \neq \varnothing \), assume in addition that \( V \) is tangent to \( \partial M \) . Then \( {\left( {\mathcal{L}}_{V}W\right) }_{p} \) exists for every \( p \in M \), and \( {\m... | Proof. Let \( \theta \) be the flow of \( V \) . For arbitrary \( p \in M \), let \( \left( {U,\left( {x}^{i}\right) }\right) \) be a smooth chart containing \( p \) . Choose an open interval \( {J}_{0} \) containing 0 and an open subset \( {U}_{0} \subseteq U \) containing \( p \) such that \( \theta \) maps \( {J}_{0... | Yes |
Proposition 9.41. Suppose \( M \) is a smooth manifold with or without boundary and \( V, W \in \mathfrak{X}\left( M\right) \) . If \( \partial M \neq \varnothing \), assume also that \( V \) is tangent to \( \partial M \) . Let \( \theta \) be the flow of \( V \) . For any \( \left( {{t}_{0}, p}\right) \) in the domai... | Proof. Let \( p \in M \) be arbitrary, let \( {\mathcal{D}}^{\left( p\right) } \subseteq \mathbb{R} \) denote the domain of the integral curve \( {\theta }^{\left( p\right) } \), and consider the map \( X : {\mathcal{D}}^{\left( p\right) } \rightarrow {T}_{p}M \) given by \( X\left( t\right) = \) \( d{\left( {\theta }_... | Yes |
Theorem 9.42. For smooth vector fields \( V \) and \( W \) on a smooth manifold \( M \), the following are equivalent:\n\n(a) \( V \) and \( W \) commute.\n\n(b) \( W \) is invariant under the flow of \( V \).\n\n(c) \( V \) is invariant under the flow of \( W \). | Proof. Suppose \( V, W \in \mathfrak{X}\left( M\right) \), and let \( \theta \) denote the flow of \( V \) . If (b) holds, then \( {W}_{{\theta }_{t}\left( p\right) } = d{\left( {\theta }_{t}\right) }_{p}\left( {W}_{p}\right) \) whenever \( \left( {t, p}\right) \) is in the domain of \( \theta \) . Applying \( d{\left(... | Yes |
Corollary 9.43. Every smooth vector field is invariant under its own flow. | Proof. Use the preceding proposition together with the fact that \( \left\lbrack {V, V}\right\rbrack \equiv 0 \) . | No |
Consider the following two vector fields on \( {\mathbb{R}}^{2} \) :\n\n\[ V = x\frac{\partial }{\partial y} - y\frac{\partial }{\partial x},\;W = x\frac{\partial }{\partial x} + y\frac{\partial }{\partial y}. \]\n\nA computation shows that \( \left\lbrack {V, W}\right\rbrack = 0 \) . | Example 9.8 showed that the flow of \( V \) is\n\n\[ {\theta }_{t}\left( {x, y}\right) = \left( {x\cos t - y\sin t, x\sin t + y\cos t}\right) ,\]\n\nand an easy verification shows that the flow of \( W \) is\n\n\[ {\eta }_{t}\left( {x, y}\right) = \left( {{e}^{t}x,{e}^{t}y}\right) \]\n\nAt \( p = \left( {1,0}\right) ,{... | Yes |
Theorem 9.48 (Fundamental Theorem on Time-Dependent Flows). Let \( M \) \( {be} \) a smooth manifold, let \( J \subseteq \mathbb{R} \) be an open interval, and let \( V : J \times M \rightarrow {TM} \) be a smooth time-dependent vector field on \( M \). There exist an open subset \( \mathcal{E} \subseteq J \times \) \(... | Proof. This can be proved by following the outline of the proof of Theorem 9.12, using Theorem D. 6 in place of Theorem D.1. However, it is much quicker to use the following trick to reduce it to the time-independent case.\n\nConsider the smooth vector field \( \widetilde{V} \) on \( J \times M \) defined by\n\n\[{\wid... | Yes |
Define a time-dependent vector field \( V \) on \( {\mathbb{R}}^{n} \) by\n\n\[ V\left( {t, x}\right) = {\left. \frac{1}{t}{x}^{i}\frac{\partial }{\partial {x}^{i}}\right| }_{x},\;\left( {t, x}\right) \in \left( {0,\infty }\right) \times {\mathbb{R}}^{n}. \]\n\nSuppose \( {t}_{0} \in \left( {0,\infty }\right) \) and \(... | The maximal solution to this system, as you can easily check, is \( {x}^{i}\left( t\right) = t{x}_{0}^{i}/{t}_{0} \), defined for all \( t > 0 \) . Therefore, the time-dependent flow of \( V \) is given by \( \psi \left( {t,{t}_{0}, x}\right) = \) \( {tx}/{t}_{0} \) for \( \left( {t,{t}_{0}, x}\right) \in \left( {0,\in... | Yes |
Theorem 9.51 (The Linear First-Order Cauchy Problem). Let \( M \) be a smooth manifold. Suppose we are given an embedded hypersurface \( S \subseteq M \), a smooth vector field \( A \in \mathfrak{X}\left( M\right) \) that is nowhere tangent to \( S \), and functions \( b, f \in {C}^{\infty }\left( M\right) \) and \( \v... | Proof. The flowout theorem gives us a neighborhood \( {\mathcal{O}}_{\delta } \) of \( \{ 0\} \times S \) in \( \mathbb{R} \times S \) , a neighborhood \( U \) of \( S \) in \( M \), and a diffeomorphism \( \Phi : {\mathcal{O}}_{\delta } \rightarrow U \) that satisfies \( \Phi \left( {0, p}\right) = p \) for \( p \in S... | Yes |
Suppose we wish to solve the following Cauchy problem for a smooth function \( u\left( {x, y}\right) \) in the plane:\n\n\[ x\frac{\partial u}{\partial y} - y\frac{\partial u}{\partial x} = x \]\n\n(9.27)\n\n\[ u\left( {x,0}\right) = x\;\text{ when }x > 0. \]\n\n(9.28) | The vector field acting on \( u \) on the left-hand side of (9.27) is the vector field \( W \) of Example 9.23. The initial hypersurface \( S \) is the positive \( x \) -axis, and this problem is noncharacteristic because \( W \) is nowhere tangent to \( S \) . (Notice that this would not be the case if we took \( S \)... | Yes |
Suppose we wish to solve the following quasilinear Cauchy problem in the plane:\n\n\[ \left( {u + 1}\right) \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 0 \]\n\n\[ u\left( {x,0}\right) = x. \] | The initial hypersurface \( S \) is the \( x \) -axis, and the initial value is \( \varphi \left( {x,0}\right) = x \) . The vector field \( {A}^{\varphi } \) is \( \left( {x + 1}\right) \partial /\partial x + \partial /\partial y \), which is nowhere tangent to the \( x \) -axis, so this problem is noncharacteristic.\n... | Yes |
Example 10.2 (Product Bundles). One particularly simple example of a rank- \( k \) vector bundle over any space \( M \) is the product space \( E = M \times {\mathbb{R}}^{k} \) with \( \pi = {\pi }_{1} : M \times {\mathbb{R}}^{k} \rightarrow M \) as its projection. Any such bundle, called a product bundle, is trivial (... | If \( M \) is a smooth manifold with or without boundary, then \( M \times {\mathbb{R}}^{k} \) is smoothly trivial. | Yes |
Define an equivalence relation on \( {\mathbb{R}}^{2} \) by declaring that \( \left( {x, y}\right) \sim \left( {{x}^{\prime },{y}^{\prime }}\right) \) if and only if \( \left( {{x}^{\prime },{y}^{\prime }}\right) = \left( {x + n,{\left( -1\right) }^{n}y}\right) \) for some \( n \in \mathbb{Z} \) . Let \( E = {\mathbb{R... | To visualize \( E \), let \( S \) denote the strip \( \left\lbrack {0,1}\right\rbrack \times \mathbb{R} \subseteq {\mathbb{R}}^{2} \) . The restriction of \( q \) to \( S \) is surjective and closed, so it is a quotient map. The only nontrivial identifications made by \( {\left. q\right| }_{S} \) are on the two boundar... | Yes |
Proposition 10.4 (The Tangent Bundle as a Vector Bundle). Let \( M \) be a smooth \( n \) -manifold with or without boundary, and let TM be its tangent bundle. With its standard projection map, its natural vector space structure on each fiber, and the topology and smooth structure constructed in Proposition 3.18, TM is... | Proof. Given any smooth chart \( \left( {U,\varphi }\right) \) for \( M \) with coordinate functions \( \left( {x}^{i}\right) \), define a map \( \Phi : {\pi }^{-1}\left( U\right) \rightarrow U \times {\mathbb{R}}^{n} \) by\n\n\[ \Phi \left( {\left. {v}^{i}\frac{\partial }{\partial {x}^{i}}\right| }_{p}\right) = \left(... | Yes |
Lemma 10.5. Let \( \pi : E \rightarrow M \) be a smooth vector bundle of rank \( k \) over \( M \) . Suppose \( \Phi : {\pi }^{-1}\left( U\right) \rightarrow U \times {\mathbb{R}}^{k} \) and \( \Psi : {\pi }^{-1}\left( V\right) \rightarrow V \times {\mathbb{R}}^{k} \) are two smooth local trivi-alizations of \( E \) wi... | Proof. The following diagram commutes:\n\n\n\n(10.2)\n\nwhere the maps on top are to be interpreted as the restrictions of \( \Psi \) and \( \Phi \) to \( {\pi }^{-1}\left( {U \cap V}\right) \) . It follows that \( {... | No |
Lemma 10.6 (Vector Bundle Chart Lemma). Let \( M \) be a smooth manifold with or without boundary, and suppose that for each \( p \in M \) we are given a real vector space \( {E}_{p} \) of some fixed dimension \( k \) . Let \( E = \mathop{\coprod }\limits_{{p \in M}}{E}_{p} \), and let \( \pi : E \rightarrow M \) be th... | Proof. For each point \( p \in M \), choose some \( {U}_{\alpha } \) containing \( p \) ; choose a smooth chart \( \left( {{V}_{p},{\varphi }_{p}}\right) \) for \( M \) such that \( p \in {V}_{p} \subseteq {U}_{\alpha } \) ; and let \( {\widehat{V}}_{p} = {\varphi }_{p}\left( {V}_{p}\right) \subseteq {\mathbb{R}}^{n} \... | Yes |
Given a smooth manifold \( M \) and smooth vector bundles \( {E}^{\prime } \rightarrow M \) and \( {E}^{\prime \prime } \rightarrow M \) of ranks \( {k}^{\prime } \) and \( {k}^{\prime \prime } \), respectively, we will construct a new vector bundle over \( M \) called the Whitney sum of \( {E}^{\prime } \) and \( {E}^... | For each \( p \in M \), choose a neighborhood \( U \) of \( p \) small enough that there exist local trivializations \( \left( {U,{\Phi }^{\prime }}\right) \) of \( {E}^{\prime } \) and \( \left( {U,{\Phi }^{\prime \prime }}\right) \) of \( {E}^{\prime \prime } \), and define \( \Phi : {\pi }^{-1}\left( U\right) \right... | Yes |
Example 10.8 (Restriction of a Vector Bundle). Suppose \( \pi : E \rightarrow M \) is a rank- \( k \) vector bundle and \( S \subseteq M \) is any subset. We define the restriction of \( E \) to \( S \) to be the set \( {\left. E\right| }_{S} = \mathop{\bigcup }\limits_{{p \in S}}{E}_{p} \), with the projection \( {\le... | If \( E \) is a smooth vector bundle and \( S \subseteq M \) is an immersed or embedded submanifold, it follows easily from the chart lemma that \( {\left. E\right| }_{S} \) is a smooth vector bundle. In particular, if \( S \subseteq M \) is a smooth (embedded or immersed) submanifold, then the restricted bundle \( {\l... | Yes |
Example 10.17 (A Global Frame for a Product Bundle). If \( E = M \times {\mathbb{R}}^{k} \rightarrow M \) is a product bundle, the standard basis \( \left( {{e}_{1},\ldots ,{e}_{k}}\right) \) for \( {\mathbb{R}}^{k} \) yields a global frame \( \left( {\widetilde{e}}_{i}\right) \) for \( E \), defined by \( {\widetilde{... | If \( M \) is a smooth manifold with or without boundary, then this global frame is smooth. | No |
Example 10.18 (Local Frames Associated with Local Trivializations). Suppose \( \pi : E \rightarrow M \) is a smooth vector bundle. If \( \Phi : {\pi }^{-1}\left( U\right) \rightarrow U \times {\mathbb{R}}^{k} \) is a smooth local trivialization of \( E \), we can use the same idea as in the preceding example to constru... | Then \( {\sigma }_{i} \) is smooth because \( \Phi \) is a diffeomorphism, and the fact that \( {\pi }_{1} \circ \Phi = \pi \) implies that \[ \pi \circ {\sigma }_{i}\left( p\right) = \pi \circ {\Phi }^{-1}\left( {p,{e}_{i}}\right) = {\pi }_{1}\left( {p,{e}_{i}}\right) = p, \] so \( {\sigma }_{i} \) is a section. To se... | Yes |
Corollary 10.20. A smooth vector bundle is smoothly trivial if and only if it admits a smooth global frame. | Proof. Example 10.18 and Proposition 10.19 show that there is a smooth local trivialization over an open subset \( U \subseteq M \) if and only if there is a smooth local frame over \( U \) . The corollary is just the special case of this statement when \( U = M \) . | No |
Corollary 10.21. Let \( \pi : E \rightarrow M \) be a smooth vector bundle of rank \( k \), let \( \left( {V,\varphi }\right) \) be a smooth chart on \( M \) with coordinate functions \( \left( {x}^{i}\right) \), and suppose there exists a smooth local frame \( \left( {\sigma }_{i}\right) \) for \( E \) over \( V \) . ... | Proof. Just check that \( \widetilde{\varphi } \) is equal to the composition \( \left( {\varphi \times {\operatorname{Id}}_{{\mathbb{R}}^{k}}}\right) \circ \Phi \), where \( \Phi \) is the local trivialization associated with \( \left( {\sigma }_{i}\right) \) . As a composition of diffeomorphisms, it is a diffeomorphi... | Yes |
Proposition 10.22 (Local Frame Criterion for Smoothness). Let \( \pi : E \rightarrow M \) be a smooth vector bundle, and let \( \tau : M \rightarrow E \) be a rough section. If \( \left( {\sigma }_{i}\right) \) is a smooth local frame for \( E \) over an open subset \( U \subseteq M \), then \( \tau \) is smooth on \( ... | Proof. Let \( \Phi : {\pi }^{-1}\left( U\right) \rightarrow U \times {\mathbb{R}}^{k} \) be the local trivialization associated with the local frame \( \left( {\sigma }_{i}\right) \) . Because \( \Phi \) is a diffeomorphism, \( \tau \) is smooth on \( U \) if and only if the composite map \( \Phi \circ \tau \) is smoot... | Yes |
Proposition 10.24 (Uniqueness of the Smooth Structure on \( {TM} \) ). Let \( M \) be a smooth \( n \) -manifold with or without boundary. The topology and smooth structure on TM constructed in Proposition 3.18 are the unique ones with respect to which \( \pi : {TM} \rightarrow M \) is a smooth vector bundle with the g... | Proof. Suppose \( {TM} \) is endowed with some topology and smooth structure making it into a smooth vector bundle with the given properties. If \( \left( {U,\varphi }\right) \) is any smooth chart for \( M \), the corresponding coordinate frame \( \left( {\partial /\widehat{\partial }{x}^{i}}\right) \) is a smooth loc... | Yes |
Proposition 10.25. Suppose \( \pi : E \rightarrow M \) and \( {\pi }^{\prime } : E \rightarrow {M}^{\prime } \) are vector bundles and \( F : E \rightarrow {E}^{\prime } \) is a bundle homomorphism covering \( f : M \rightarrow {M}^{\prime } \) . Then \( f \) is continuous and is uniquely determined by \( F \) . If the... | Proof. All of the conclusions follow from the easily verified fact that \( f = {\pi }^{\prime } \circ F \circ \zeta \) , where \( \zeta : M \rightarrow E \) is the zero section. | Yes |
Proposition 10.26. Suppose \( E \) and \( {E}^{\prime } \) are smooth vector bundles over a smooth manifold \( M \) with or without boundary, and \( F : E \rightarrow {E}^{\prime } \) is a bijective smooth bundle homomorphism over \( M \) . Then \( F \) is a smooth bundle isomorphism. | Proof. Problem 10-11. | No |
Lemma 10.32 (Local Frame Criterion for Subbundles). Let \( \pi : E \rightarrow M \) be a smooth vector bundle, and suppose that for each \( p \in M \) we are given an \( m \) - dimensional linear subspace \( {D}_{p} \subseteq {E}_{p} \) . Then \( D = \mathop{\bigcup }\limits_{{p \in M}}{D}_{p} \subseteq E \) is a smoot... | Proof. If \( D \) is a smooth subbundle, then by definition each \( p \in M \) has a neighborhood \( U \) over which there exists a smooth local trivialization of \( D \), and Example 10.18 shows that there exists a smooth local frame for \( D \) over each such set \( U \) . Such a local frame is by definition a collec... | Yes |
Lemma 10.35 (Orthogonal Complement Bundles). Let \( M \) be an immersed sub-manifold with or without boundary in \( {\mathbb{R}}^{n} \), and \( D \) be a smooth rank-k subbundle of \( {\left. T{\mathbb{R}}^{n}\right| }_{M} \) . For each \( p \in M \), let \( {D}_{p}^{ \bot } \) denote the orthogonal complement of \( {D... | Proof. Let \( p \in M \) be arbitrary, and let \( \left( {{X}_{1},\ldots ,{X}_{k}}\right) \) be a smooth local frame for \( D \) over some neighborhood \( V \) of \( p \) in \( M \) . Because immersed submanifolds are locally embedded, by shrinking \( V \) if necessary, we may assume that it is a single slice in some c... | Yes |
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