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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 |
Every smooth vector field is invariant under its own flow. | 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 |
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\\[ \n\\left( {u + 1}\\right) \\frac{\\partial u}{\\partial x} + \\frac{\\partial u}{\\partial y} = 0 \n\\]\n\n\\[ \nu\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 i... | 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... | No |
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 3.1... | 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 |
Corollary 10.36 (The Normal Bundle to a Submanifold of \( {\mathbb{R}}^{n} \) ). If \( M \subseteq {\mathbb{R}}^{n} \) is an immersed m-dimensional submanifold with or without boundary, its normal bundle \( {NM} \) is a smooth rank- \( \left( {n - m}\right) \) subbundle of \( {\left. T{\mathbb{R}}^{n}\right| }_{M} \) .... | Proof. Apply Lemma 10.35 to the smooth subbundle \( {\left. TM \subseteq T{\mathbb{R}}^{n}\right| }_{M} \) . | No |
Proposition 11.1. Let \( V \) be a finite-dimensional vector space. Given any basis \( \left( {{E}_{1},\ldots ,{E}_{n}}\right) \) for \( V \), let \( {\varepsilon }^{1},\ldots ,{\varepsilon }^{n} \in {V}^{ * } \) be the covectors defined by\n\n\[ \n{\varepsilon }^{i}\left( {E}_{j}\right) = {\delta }_{j}^{i} \n\] \n\nwh... | ## - Exercise 11.2. Prove Proposition 11.1. | No |
Proposition 11.4. The dual map satisfies the following properties:\n\n(a) \( {\left( A \circ B\right) }^{ * } = {B}^{ * } \circ {A}^{ * } \) .\n\n(b) \( {\left( {\operatorname{Id}}_{V}\right) }^{ * } : {V}^{ * } \rightarrow {V}^{ * } \) is the identity map of \( {V}^{ * } \) . | Exercise 11.5. Prove the preceding proposition. | No |
Proposition 11.8. For any finite-dimensional vector space \( V \), the map \( \xi : V \rightarrow {V}^{* * } \) is an isomorphism. | Proof. Because \( \dim V = \dim {V}^{* * } \), it suffices to verify that \( \xi \) is injective (see Exercise B.22(c)). Suppose \( v \in V \) is not zero. Extend \( v \) to a basis \( \left( {v = {E}_{1},\ldots ,{E}_{n}}\right) \) for \( V \), and let \( \left( {{\varepsilon }^{1},\ldots ,{\varepsilon }^{n}}\right) \)... | No |
Proposition 11.9 (The Cotangent Bundle as a Vector Bundle). Let \( M \) be a smooth \( n \) -manifold with or without boundary. With its standard projection map and the natural vector space structure on each fiber, the cotangent bundle \( {T}^{ * }M \) has a unique topology and smooth structure making it into a smooth ... | Proof. The proof is just like that of Theorem 10.4. Given a smooth chart \( \left( {U,\varphi }\right) \) on \( M \), with coordinate functions \( \left( {x}^{i}\right) \), define \( \Phi : {\pi }^{-1}\left( U\right) \rightarrow U \times {\mathbb{R}}^{n} \) by\n\n\[ \Phi \left( {\left. {\xi }_{i}{\lambda }^{i}\right| }... | Yes |
Proposition 11.11 (Smoothness Criteria for Covector Fields). Let \( M \) be a smooth manifold with or without boundary, and let \( \omega : M \rightarrow {T}^{ * }M \) be a rough covector field. The following are equivalent:\n\n(a) \( \omega \) is smooth.\n\n(b) In every smooth coordinate chart, the component functions... | - Exercise 11.12. Prove this proposition. [Suggestion: try proving (a) \( \Rightarrow \) (b) \( \Rightarrow \) (c) \( \Rightarrow \) (a), and (c) \( \Rightarrow \) (d) \( \Rightarrow \) (e) \( \Rightarrow \) (b). The only tricky part is (d) \( \Rightarrow \) (e); look at the proof of Proposition 8.14 for ideas.] | No |
Lemma 11.14. Let \( M \) be a smooth manifold with or without boundary. If \( \left( {E}_{i}\right) \) is a rough local frame over an open subset \( U \subseteq M \) and \( \left( {\varepsilon }^{i}\right) \) is its dual coframe, then \( \left( {E}_{i}\right) \) is smooth if and only if \( \left( {\varepsilon }^{i}\rig... | Proof. It suffices to show that for each \( p \in U \), the frame \( \left( {E}_{i}\right) \) is smooth in a neighborhood of \( p \) if and only if \( \left( {\varepsilon }^{i}\right) \) is. Given \( p \in U \), let \( \left( {V,\left( {x}^{i}\right) }\right) \) be a smooth coordinate chart such that \( p \in V \subset... | Yes |
Proposition 11.18. The differential of a smooth function is a smooth covector field. | Proof. It is straightforward to verify that at each point \( p \in M, d{f}_{p}\left( v\right) \) depends linearly on \( v \), so that \( d{f}_{p} \) is indeed a covector at \( p \) . To see that \( {df} \) is smooth, we use Proposition 11.11(d): for any smooth vector field \( X \) on \( M \), the function \( {df}\left(... | Yes |
If \( f\left( {x, y}\right) = {x}^{2}y\cos x \) on \( {\mathbb{R}}^{2} \), then \( {df} \) is given by the formula | \[ {df} = \frac{\partial \left( {{x}^{2}y\cos x}\right) }{\partial x}{dx} + \frac{\partial \left( {{x}^{2}y\cos x}\right) }{\partial y}{dy} \] \[ = \left( {{2xy}\cos x - {x}^{2}y\sin x}\right) {dx} + {x}^{2}\cos {xdy}. \] | Yes |
Proposition 11.22 (Functions with Vanishing Differentials). If \( f \) is a smooth real-valued function on a smooth manifold \( M \) with or without boundary, then \( {df} = \) 0 if and only if \( f \) is constant on each component of \( M \) . | Proof. It suffices to assume that \( M \) is connected and show that \( {df} = 0 \) if and only if \( f \) is constant. One direction is immediate: if \( f \) is constant, then \( {df} = 0 \) by Proposition 11.20(e). Conversely, suppose \( {df} = 0 \), let \( p \in M \), and let \( \mathcal{C} = \{ q \in M \) : \( f\le... | Yes |
Proposition 11.23 (Derivative of a Function Along a Curve). Suppose \( M \) is a smooth manifold with or without boundary, \( \gamma : J \rightarrow M \) is a smooth curve, and \( f : M \rightarrow \mathbb{R} \) is a smooth function. Then the derivative of the real-valued function \( f \circ \gamma : J \rightarrow \mat... | Proof. See Fig. 11.3. Directly from the definitions, for any \( {t}_{0} \in J \) , \[ d{f}_{\gamma \left( {t}_{0}\right) }\left( {{\gamma }^{\prime }\left( {t}_{0}\right) }\right) = {\gamma }^{\prime }\left( {t}_{0}\right) f\;\text{ (definition of }{df}\text{ ) } \] \[ = d{\gamma }_{{t}_{0}}\left( {\left. \frac{d}{dt}\... | Yes |
Proposition 11.25. Let \( F : M \rightarrow N \) be a smooth map between smooth manifolds with or without boundary. Suppose \( u \) is a continuous real-valued function on \( N \) , and \( \omega \) is a covector field on \( N \) . Then\n\n\[ \n{F}^{ * }\left( {u\omega }\right) = \left( {u \circ F}\right) {F}^{ * }\ome... | Proof. To prove (11.14) we compute\n\n\[ \n{\left( {F}^{ * }\left( u\omega \right) \right) }_{p} = d{F}_{p}^{ * }\left( {\left( u\omega \right) }_{F\left( p\right) }\right) \;\left( {\text{by }\left( {11.13}\right) }\right)\n\]\n\n\[ \n= d{F}_{p}^{ * }\left( {u\left( {F\left( p\right) }\right) {\omega }_{F\left( p\righ... | Yes |
Proposition 11.26. Suppose \( F : M \rightarrow N \) is a smooth map between smooth manifolds with or without boundary, and let \( \omega \) be a covector field on \( N \) . Then \( {F}^{ * }\omega \) is a (continuous) covector field on \( M \) . If \( \omega \) is smooth, then so is \( {F}^{ * }\omega \) . | Proof. Let \( p \in M \) be arbitrary, and choose smooth coordinates \( \left( {y}^{j}\right) \) for \( N \) in a neighborhood \( V \) of \( F\left( p\right) \) . Let \( U = {F}^{-1}\left( V\right) \), which is a neighborhood of \( p \) . Writing \( \omega \) in coordinates as \( \omega = {\omega }_{j}d{y}^{j} \) for c... | Yes |
Let \( F : {\mathbb{R}}^{3} \rightarrow {\mathbb{R}}^{2} \) be the map given by\n\n\[ \left( {u, v}\right) = F\left( {x, y, z}\right) = \left( {{x}^{2}y, y\sin z}\right) ,\]\n\nand let \( \omega \in {\mathfrak{X}}^{ * }\left( {\mathbb{R}}^{2}\right) \) be the covector field\n\n\[ \omega = {udv} + {vdu}. \]\n\nAccording... | \[ = \left( {{x}^{2}y}\right) d\left( {y\sin z}\right) + \left( {y\sin z}\right) d\left( {{x}^{2}y}\right) \]\n\n\[ = {x}^{2}y\left( {\sin {zdy} + y\cos {zdz}}\right) + y\sin z\left( {{2xydx} + {x}^{2}{dy}}\right) \]\n\n\[ = {2x}{y}^{2}\sin {zdx} + 2{x}^{2}y\sin {zdy} + {x}^{2}{y}^{2}\cos {zdz}. \] | Yes |
Let \( \left( {r,\theta }\right) \) be polar coordinates on, say, the right half-plane \( H = \) \( \{ \left( {x, y}\right) : x > 0\} \) . We can think of the change of coordinates \( \left( {x, y}\right) = \left( {r\cos \theta, r\sin \theta }\right) \) as the coordinate expression for the identity map of \( H \), but ... | \n\[
{xdy} - {ydx} = {\operatorname{Id}}^{ * }\left( {{xdy} - {ydx}}\right)
\]
\n\[
= \left( {r\cos \theta }\right) d\left( {r\sin \theta }\right) - \left( {r\sin \theta }\right) d\left( {r\cos \theta }\right)
\]
\n\[
= \left( {r\cos \theta }\right) \left( {\sin {\theta dr} + r\cos {\theta d\theta }}\right) - \left( {r... | Yes |
Example 11.29. Let \( \omega = {dy} \) on \( {\mathbb{R}}^{2} \), and let \( S \) be the \( x \) -axis, considered as an embedded submanifold of \( {\mathbb{R}}^{2} \). As a covector field on \( {\mathbb{R}}^{2},\omega \) is nonzero everywhere, because one of its component functions is always 1 . However, the restricti... | \[ {\iota }^{ * }\omega = {\iota }^{ * }{dy} = d\left( {y \circ \iota }\right) = 0. \] | Yes |
Proposition 11.31 (Diffeomorphism Invariance of the Integral). Let \( \omega \) be a smooth covector field on the compact interval \( \left\lbrack {a, b}\right\rbrack \subseteq \mathbb{R} \). If \( \varphi : \left\lbrack {c, d}\right\rbrack \rightarrow \left\lbrack {a, b}\right\rbrack \) is an increasing diffeomorphism... | Proof. If we let \( s \) denote the standard coordinate on \( \left\lbrack {c, d}\right\rbrack \) and \( t \) that on \( \left\lbrack {a, b}\right\rbrack \) , then (11.17) shows that the pullback \( {\varphi }^{ * }\omega \) has the coordinate expression \( {\left( {\varphi }^{ * }\omega \right) }_{s} = \) \( f\left( {... | Yes |
Proposition 11.33. If \( M \) is a connected smooth manifold with or without boundary, any two points of \( M \) can be joined by a piecewise smooth curve segment. | Proof. Let \( p \) be an arbitrary point of \( M \), and define a subset \( \mathcal{C} \subseteq M \) by\n\n\( \mathcal{C} = \{ q \in M \) : there is a piecewise smooth curve segment in \( M \) from \( p \) to \( q\} \) .\n\nClearly, \( p \in \mathcal{C} \), so \( \mathcal{C} \) is nonempty. To show that \( \mathcal{C... | Yes |
Let \( M = {\mathbb{R}}^{2} \smallsetminus \{ 0\} \), let \( \omega \) be the covector field on \( M \) given by\n\n\[ \omega = \frac{{xdy} - {ydx}}{{x}^{2} + {y}^{2}} \]\n\nand let \( \gamma : \left\lbrack {0,{2\pi }}\right\rbrack \rightarrow M \) be the curve segment defined by \( \gamma \left( t\right) = \left( {\co... | Since \( {\gamma }^{ * }\omega \) can be computed by substituting \( x = \cos t \) and \( y = \sin t \) everywhere in the formula for \( \omega \), we find that\n\n\[ {\int }_{\gamma }\omega = {\int }_{\left\lbrack 0,2\pi \right\rbrack }\frac{\cos t\left( {\cos {tdt}}\right) - \sin t\left( {-\sin {tdt}}\right) }{{\sin ... | Yes |
Proposition 11.37 (Parameter Independence of Line Integrals). Suppose \( M \) is a smooth manifold with or without boundary, \( \omega \in {\mathfrak{X}}^{ * }\left( M\right) \), and \( \gamma \) is a piecewise smooth curve segment in \( M \) . For any reparametrization \( \widetilde{\gamma } \) of \( \gamma \), we hav... | Proof. First assume that \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow M \) is smooth, and suppose \( \widetilde{\gamma } = \gamma \circ \varphi \), where \( \varphi : \left\lbrack {c, d}\right\rbrack \rightarrow \left\lbrack {a, b}\right\rbrack \) is an increasing diffeomorphism. Then Proposition 11.31 impl... | No |
Proposition 11.38. If \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow M \) is a piecewise smooth curve segment, the line integral of \( \omega \) over \( \gamma \) can also be expressed as the ordinary integral\n\n\[{\int }_{\gamma }\omega = {\int }_{a}^{b}{\omega }_{\gamma \left( t\right) }\left( {{\gamma }^{... | Proof. First suppose that \( \gamma \) is smooth and that its image is contained in the domain of a single smooth chart. Writing the coordinate representations of \( \gamma \) and \( \omega \) as \( \left( {{\gamma }^{1}\left( t\right) ,\ldots ,{\gamma }^{n}\left( t\right) }\right) \) and \( {\omega }_{i}d{x}^{i} \), r... | Yes |
Theorem 11.39 (Fundamental Theorem for Line Integrals). Let \( M \) be a smooth manifold with or without boundary. Suppose \( f \) is a smooth real-valued function on \( M \) and \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow M \) is a piecewise smooth curve segment in \( M \) . Then\n\n\[{\int }_{\gamma }{df... | Proof. Suppose first that \( \gamma \) is smooth. By Propositions 11.23 and 11.38,\n\n\[{\int }_{\gamma }{df} = {\int }_{a}^{b}d{f}_{\gamma \left( t\right) }\left( {{\gamma }^{\prime }\left( t\right) }\right) {dt} = {\int }_{a}^{b}{\left( f \circ \gamma \right) }^{\prime }\left( t\right) {dt}.\n\nBy the fundamental the... | Yes |
Proposition 11.40. A smooth covector field \( \omega \) is conservative if and only if its line integrals are path-independent, in the sense that \( {\int }_{\gamma }\omega = {\int }_{\widetilde{\gamma }}\omega \) whenever \( \gamma \) and \( \widetilde{\gamma } \) are piecewise smooth curve segments with the same star... | Exercise 11.41. Prove Proposition 11.40. [Remark: this would be harder to prove if we defined conservative fields in terms of smooth curves instead of piecewise smooth ones.] | No |
The covector field \( \omega \) of Example 11.36 cannot be exact on \( {\mathbb{R}}^{2} \smallsetminus \{ 0\} \), because it is not conservative: the computation in that example showed that \( {\int }_{\gamma }\omega = {2\pi } \neq 0 \), where \( \gamma \) is the unit circle traversed counterclockwise. | Because exactness has such important consequences for the evaluation of line integrals, we would like to have an easy way to check whether a given covector field is exact. Fortunately, there is a very simple necessary condition, which follows from the fact that partial derivatives of smooth functions can be taken in an... | Yes |
Proposition 11.44. Every exact covector field is closed. | One technical difficulty in checking directly from the definition that a covector field is closed is that it would require checking that (11.21) holds in every coordinate chart. The next proposition gives an alternative characterization of closed covector fields that is coordinate independent, and incidentally shows th... | No |
Proposition 11.45. Let \( \omega \) be a smooth covector field on a smooth manifold \( M \) with or without boundary. The following are equivalent:\n\n(a) \( \omega \) is closed.\n\n(b) \( \omega \) satisfies (11.21) in some smooth chart around every point.\n\n(c) For any open subset \( U \subseteq M \) and smooth vect... | Proof. We will prove that (a) \( \Rightarrow \) (b) \( \Rightarrow \) (c) \( \Rightarrow \) (a). The implication (a) \( \Rightarrow \) (b) is immediate from the definition of closed covector fields.\n\nTo prove (b) \( \Rightarrow \) (c), assume (b) holds, and suppose \( U \subseteq M \) and \( X, Y \in \mathfrak{X}\lef... | Yes |
Corollary 11.46. Suppose \( F : M \rightarrow N \) is a local diffeomorphism. Then the pullback \( {F}^{ * } : {\mathfrak{X}}^{ * }\left( N\right) \rightarrow {\mathfrak{X}}^{ * }\left( M\right) \) takes closed covector fields to closed covector fields, and exact ones to exact ones. | Proof. The result for exact covector fields follows immediately from (11.15). For closed covector fields, if \( \left( {U,\varphi }\right) \) is any smooth chart for \( N \), then \( \varphi \circ F \) is a smooth chart for \( M \) in a neighborhood of each point of \( {F}^{-1}\left( U\right) \) . In these coordinates,... | Yes |
Consider the following covector field on \( {\mathbb{R}}^{2} \) :\n\n\[ \omega = y\cos {xydx} + x\cos {xydy}. \]\n\nIt is easy to check that\n\n\[ \frac{\partial \left( {y\cos {xy}}\right) }{\partial y} = \frac{\partial \left( {x\cos {xy}}\right) }{\partial x} = \cos {xy} - {xy}\sin {xy}, \] | so \( \omega \) is closed. In fact, you might guess that \( \omega = d\left( {\sin {xy}}\right) \) . | Yes |
Theorem 11.49 (Poincaré Lemma for Covector Fields). If \( U \) is a star-shaped open subset of \( {\mathbb{R}}^{n} \) or \( {\mathbb{H}}^{n} \), then every closed covector field on \( U \) is exact. | Proof. Suppose \( U \) is star-shaped with respect to \( c \in U \), and let \( \omega = {\omega }_{i}d{x}^{i} \) be a closed covector field on \( U \) . As in the proof of Theorem 11.42, we will construct a potential function for \( \omega \) by integrating along smooth curve segments from \( c \) . However, in this c... | Yes |
Corollary 11.50 (Local Exactness of Closed Covector Fields). Let \( \omega \) be a closed covector field on a smooth manifold \( M \) with or without boundary. Then every point of \( M \) has a neighborhood on which \( \omega \) is exact. | Proof. Let \( p \in M \) be arbitrary. The hypothesis implies that \( \omega \) satisfies (11.21) in some smooth coordinate ball or half-ball \( \left( {U,\varphi }\right) \) containing \( p \) . Because balls and half-balls are convex, we can apply Theorem 11.49 to the coordinate representation of \( \omega \) and con... | Yes |
Let \( \\omega \) be the following covector field on \( {\\mathbb{R}}^{3} \) :\n\n\[ \n\\omega = {e}^{{y}^{2}}{dx} + {2xy}{e}^{{y}^{2}}{dy} - {2zdz}.\n\]\n\nYou can check that \( \\omega \) is closed. For \( f \) to be a potential for \( \\omega \), it must satisfy\n\n\[ \n\\frac{\\partial f}{\\partial x} = {e}^{{y}^{2... | Holding \( y \) and \( z \) fixed and integrating the first equation with respect to \( x \), we obtain\n\n\[ \nf\\left( {x, y, z}\\right) = \\int {e}^{{y}^{2}}{dx} = x{e}^{{y}^{2}} + {C}_{1}\\left( {y, z}\\right) ,\n\]\n\nwhere the \ | No |
Proposition 12.4 (A Basis for the Space of Multilinear Functions). Let \( {V}_{1},\ldots ,{V}_{k} \) be real vector spaces of dimensions \( {n}_{1},\ldots ,{n}_{k} \), respectively. For each \( j \in \{ 1,\ldots, k\} \) , let \( \left( {{E}_{1}^{\left( j\right) },\ldots ,{E}_{{n}_{j}}^{\left( j\right) }}\right) \) be a... | Proof. We need to show that \( \mathcal{B} \) is linearly independent and spans \( \mathrm{L}\left( {{V}_{1},\ldots ,{V}_{k};\mathbb{R}}\right) \) . Suppose \( F \in \mathrm{L}\left( {{V}_{1},\ldots ,{V}_{k};\mathbb{R}}\right) \) is arbitrary. For each ordered \( k \) -tuple \( \left( {{i}_{1},\ldots ,{i}_{k}}\right) \... | Yes |
Proposition 12.7 (Characteristic Property of the Tensor Product Space). Let \( {V}_{1},\ldots ,{V}_{k} \) be finite-dimensional real vector spaces. If \( A : {V}_{1} \times \cdots \times {V}_{k} \rightarrow X \) is any multilinear map into a vector space \( X \), then there is a unique linear map \( \widetilde{A} : {V}... | Proof. First note that any map \( A : {V}_{1} \times \cdots \times {V}_{k} \rightarrow X \) extends uniquely to a linear map \( \bar{A} : \mathcal{F}\left( {{V}_{1} \times \cdots \times {V}_{k}}\right) \rightarrow X \) by the characteristic property of the free vector space. This map is characterized by the fact that \... | Yes |
Proposition 12.8 (A Basis for the Tensor Product Space). Suppose \( {V}_{1},\ldots ,{V}_{k} \) are real vector spaces of dimensions \( {n}_{1},\ldots ,{n}_{k} \), respectively. For each \( j = 1,\ldots, k \) , suppose \( \left( {{E}_{1}^{\left( j\right) },\ldots ,{E}_{{n}_{j}}^{\left( j\right) }}\right) \) is a basis f... | Proof. Elements of the form \( {v}_{1} \otimes \cdots \otimes {v}_{k} \) span the tensor product space by definition; expanding each \( {v}_{i} \) in such an expression in terms of its basis representation shows that \( \mathcal{C} \) spans \( {V}_{1} \otimes \cdots \otimes {V}_{k} \) . \n\nTo prove that \( \mathcal{C}... | Yes |
Proposition 12.9 (Associativity of Tensor Product Spaces). Let \( {V}_{1},{V}_{2},{V}_{3} \) be finite-dimensional real vector spaces. There are unique isomorphisms\n\n\[ \n{V}_{1} \otimes \left( {{V}_{2} \otimes {V}_{3}}\right) \cong {V}_{1} \otimes {V}_{2} \otimes {V}_{3} \cong \left( {{V}_{1} \otimes {V}_{2}}\right)... | Proof. We construct the isomorphism \( {V}_{1} \otimes {V}_{2} \otimes {V}_{3} \cong \left( {{V}_{1} \otimes {V}_{2}}\right) \otimes {V}_{3} \) ; the other one is constructed similarly. The map \( \alpha : {V}_{1} \times {V}_{2} \times {V}_{3} \rightarrow \left( {{V}_{1} \otimes {V}_{2}}\right) \otimes {V}_{3} \) defin... | Yes |
Proposition 12.10 (Abstract vs. Concrete Tensor Products). If \( {V}_{1},\ldots ,{V}_{k} \) are finite-dimensional vector spaces, there is a canonical isomorphism\n\n\[ \n{V}_{1}^{ * } \otimes \cdots \otimes {V}_{k}^{ * } \cong \mathrm{L}\left( {{V}_{1},\ldots ,{V}_{k};\mathbb{R}}\right) \n\]\n\nunder which the abstrac... | Proof. First, define a map \( \Phi : {V}_{1}^{ * } \times \cdots \times {V}_{k}^{ * } \rightarrow \mathrm{L}\left( {{V}_{1},\ldots ,{V}_{k};\mathbb{R}}\right) \) by\n\n\[ \n\Phi \left( {{\omega }^{1},\ldots ,{\omega }^{k}}\right) \left( {{v}_{1},\ldots ,{v}_{k}}\right) = {\omega }^{1}\left( {v}_{1}\right) \cdots {\omeg... | Yes |
Corollary 12.12. Let \( V \) be an \( n \) -dimensional real vector space. Suppose \( \left( {E}_{i}\right) \) is any basis for \( V \) and \( \left( {\varepsilon }^{j}\right) \) is the dual basis for \( {V}^{ * } \) . Then the following sets constitute bases for the tensor spaces over \( V \) :\n\n\[ \left\{ {{\vareps... | In particular, once a basis is chosen for \( V \), every covariant \( k \) -tensor \( \alpha \in {T}^{k}\left( {V}^{ * }\right) \) can be written uniquely in the form\n\n\[ \alpha = {\alpha }_{{i}_{1}\ldots {i}_{k}}{\varepsilon }^{{i}_{1}} \otimes \cdots \otimes {\varepsilon }^{{i}_{k}} \]\n\nwhere the \( {n}^{k} \) co... | Yes |
Proposition 12.14 (Properties of Symmetrization). Let \( \alpha \) be a covariant tensor on a finite-dimensional vector space.\n\n(a) Sym \( \alpha \) is symmetric.\n\n(b) \( \operatorname{Sym}\alpha = \alpha \) if and only if \( \alpha \) is symmetric. | Proof. Suppose \( \alpha \in {T}^{k}\left( {V}^{ * }\right) \) . If \( \tau \in {S}_{k} \) is any permutation, then\n\n\[ \n\left( {\operatorname{Sym}\alpha }\right) \left( {{v}_{\tau \left( 1\right) },\ldots ,{v}_{\tau \left( k\right) }}\right) = \frac{1}{k!}\mathop{\sum }\limits_{{\sigma \in {S}_{k}}}{}^{\sigma }\alp... | No |
Proposition 12.19(d) shows that if \( A \) is a smooth covariant \( k \) -tensor field on \( M \) and \( {X}_{1},\ldots ,{X}_{k} \) are smooth vector fields, then \( A\left( {{X}_{1},\ldots ,{X}_{k}}\right) \) is a smooth real-valued function on \( M \) . Thus \( A \) induces a map\n\n\[ \underset{k\text{ copies }}{\un... | This property turns out to be characteristic of smooth tensor fields, as the next lemma shows. | Yes |
Corollary 12.28. Let \( F : M \rightarrow N \) be smooth, and let \( B \) be a covariant \( k \) -tensor field on \( N \) . If \( p \in M \) and \( \left( {y}^{i}\right) \) are smooth coordinates for \( N \) on a neighborhood of \( F\left( p\right) \), then \( {F}^{ * }B \) has the following expression in a neighborhoo... | \[ {F}^{ * }\left( {{B}_{{i}_{1}\ldots {i}_{k}}d{y}^{{i}_{1}} \otimes \cdots \otimes d{y}^{{i}_{k}}}\right) \] \[ = \left( {{B}_{{i}_{1}\ldots {i}_{k}} \circ F}\right) d\left( {{y}^{{i}_{1}} \circ F}\right) \otimes \cdots \otimes d\left( {{y}^{{i}_{k}} \circ F}\right) . \] | No |
Example 12.29 (Pullback of a Tensor Field). Let \( M = \{ \left( {r,\theta }\right) : r > 0,\left| \theta \right| < \pi /2\} \) and \( N = \{ \left( {x, y}\right) : x > 0\} \), and let \( F : M \rightarrow {\mathbb{R}}^{2} \) be the smooth map \( F\left( {r,\theta }\right) = \) \( \left( {r\cos \theta, r\sin \theta }\r... | \[ {F}^{ * }A = {\left( r\cos \theta \right) }^{-2}d\left( {r\sin \theta }\right) \otimes d\left( {r\sin \theta }\right) \] \[ = {\left( r\cos \theta \right) }^{-2}\left( {\sin {\theta dr} + r\cos {\theta d\theta }}\right) \otimes \left( {\sin {\theta dr} + r\cos {\theta d\theta }}\right) \] \[ = {r}^{-2}{\tan }^{2}{\t... | Yes |
Proposition 12.32. Let \( M \) be a smooth manifold and let \( V \in \mathfrak{X}\left( M\right) \) . Suppose \( f \) is a smooth real-valued function (regarded as a 0-tensor field) on \( M \), and \( A, B \) are smooth covariant tensor fields on \( M \) .\n\n(a) \( {\mathcal{L}}_{V}f = {Vf} \) . | Proof. Let \( \theta \) be the flow of \( V \) . For a real-valued function \( f \), we can write\n\n\[ {\theta }_{t}^{ * }f\left( p\right) = f\left( {{\theta }_{t}\left( p\right) }\right) = f \circ {\theta }^{\left( p\right) }\left( t\right) . \]\n\nThus the definition of \( {\mathcal{L}}_{V}f \) reduces to the ordina... | Yes |
Corollary 12.33. If \( V \) is a smooth vector field and \( A \) is a smooth covariant \( k \) -tensor field, then for any smooth vector fields \( {X}_{1},\ldots ,{X}_{k} \) , | Proof. Just solve (12.9) for \( \left( {{\mathcal{L}}_{V}A}\right) \left( {{X}_{1},\ldots ,{X}_{k}}\right) \), and replace \( {\mathcal{L}}_{V}f \) by \( {Vf} \) and \( {\mathcal{L}}_{V}{X}_{i} \) by \( \left\lbrack {V,{X}_{i}}\right\rbrack \) . | No |
Corollary 12.34. If \( f \in {C}^{\infty }\left( M\right) \), then \( {\mathcal{L}}_{V}\left( {df}\right) = d\left( {{\mathcal{L}}_{V}f}\right) \) . | Proof. Using (12.10), for any \( X \in \mathfrak{X}\left( M\right) \) we compute\n\n\[ \left( {{\mathcal{L}}_{V}{df}}\right) \left( X\right) = V\left( {{df}\left( X\right) }\right) - {df}\left( \left\lbrack {V, X}\right\rbrack \right) = {VXf} - \left\lbrack {V, X}\right\rbrack f \]\n\n\[ = {VXf} - \left( {{VXf} - {XVf}... | Yes |
Suppose \( A \) is an arbitrary smooth covariant 2-tensor field, and \( V \) is a smooth vector field. We compute the Lie derivative \( {\mathcal{L}}_{V}A \) in smooth local coordinates \( \left( {x}^{i}\right) \) . | First, we observe that \( {\mathcal{L}}_{V}d{x}^{i} = d\left( {{\mathcal{L}}_{V}{x}^{i}}\right) = d\left( {V{x}^{i}}\right) = d{V}^{i} \) . Therefore,\n\n\[ \n{\mathcal{L}}_{V}A = {\mathcal{L}}_{V}\left( {{A}_{ij}d{x}^{i} \otimes d{x}^{j}}\right) \n\]\n\n\[ \n= {\mathcal{L}}_{V}\left( {A}_{ij}\right) d{x}^{i} \otimes d... | Yes |
Proposition 12.36. Suppose \( M \) is a smooth manifold with or without boundary and \( V \in \mathfrak{X}\left( M\right) \) . If \( \partial M \neq \varnothing \), assume in addition that \( V \) is tangent to \( \partial M \) . Let \( \theta \) be the flow of \( V \) . For any smooth covariant tensor field \( A \) an... | Proof. After expanding the definitions of the pullbacks in (12.12), we see that we have to prove \[ {\left. \frac{d}{dt}\right| }_{t = {t}_{0}}d{\left( {\theta }_{t}\right) }_{p}^{ * }\left( {A}_{{\theta }_{t}\left( p\right) }\right) = d{\left( {\theta }_{{t}_{0}}\right) }_{p}^{ * }\left( {\left( {\mathcal{L}}_{V}A\rig... | Yes |
The simplest example of a Riemannian metric is the Euclidean metric \( \bar{g} \) on \( {\mathbb{R}}^{n} \), given in standard coordinates by\n\n\[ \bar{g} = {\delta }_{ij}d{x}^{i}d{x}^{j} \] | where \( {\delta }_{ij} \) is the Kronecker delta. It is common to abbreviate the symmetric product of a tensor \( \alpha \) with itself by \( {\alpha }^{2} \), so the Euclidean metric can also be written\n\n\[ \bar{g} = {\left( d{x}^{1}\right) }^{2} + \cdots + {\left( d{x}^{n}\right) }^{2}. \]\n\nApplied to vectors \(... | Yes |
Example 13.2 (Product Metrics). If \( \left( {M, g}\right) \) and \( \left( {\widetilde{M},\widetilde{g}}\right) \) are Riemannian manifolds, we can define a Riemannian metric \( \widehat{g} = g \oplus \widetilde{g} \) on the product manifold \( M \times \widetilde{M} \), called the product metric, as follows: | \[ \widehat{g}\left( {\left( {v,\widetilde{v}}\right) ,\left( {w,\widetilde{w}}\right) }\right) = g\left( {v, w}\right) + \widetilde{g}\left( {\widetilde{v},\widetilde{w}}\right) \] for any \( \left( {v,\widetilde{v}}\right) ,\left( {w,\widetilde{w}}\right) \in {T}_{p}M \oplus {T}_{q}\widetilde{M} \cong {T}_{\left( p, ... | Yes |
Proposition 13.3 (Existence of Riemannian Metrics). Every smooth manifold with or without boundary admits a Riemannian metric. | Proof. Let \( M \) be a smooth manifold with or without boundary, and choose a covering of \( M \) by smooth coordinate charts \( \left( {{U}_{\alpha },{\varphi }_{\alpha }}\right) \) . In each coordinate domain, there is a Riemannian metric \( {g}_{\alpha } = {\varphi }_{\alpha }^{ * }\bar{g} \), whose coordinate expr... | Yes |
Corollary 13.8 (Existence of Local Orthonormal Frames). Let \( \\left( {M, g}\\right) \) be a Riemannian manifold with or without boundary. For each \( p \\in M \), there is a smooth orthonormal frame on a neighborhood of \( p \) . | Proof. Start with a smooth coordinate frame and apply Proposition 13.6. | No |
Consider the smooth map \( F : {\mathbb{R}}^{2} \rightarrow {\mathbb{R}}^{3} \) given by\n\n\[ F\left( {u, v}\right) = \left( {u\cos v, u\sin v, v}\right) . \]\n\nIt is a proper injective smooth immersion, and thus it is an embedding by Proposition 4.22. Its image is a surface called a helicoid; it looks like an infini... | The pullback metric \( {F}^{ * }\bar{g} \) can be computed by substituting the coordinate functions for \( F \) in place of \( x, y, z \) in the formula for \( \bar{g} \) :\n\n\[ {F}^{ * }\bar{g} = d{\left( u\cos v\right) }^{2} + d{\left( u\sin v\right) }^{2} + d{\left( v\right) }^{2} \]\n\n\[ = {\left( \cos vdu - u\si... | Yes |
To illustrate, we compute the coordinate expression for the Euclidean metric \( \bar{g} = d{x}^{2} + d{y}^{2} \) on \( {\mathbb{R}}^{2} \) in polar coordinates. Substituting \( x = r\cos \theta \) and \( y = r\sin \theta \) and expanding, we obtain | \n\[
\bar{g} = d{x}^{2} + d{y}^{2} = d{\left( r\cos \theta \right) }^{2} + d{\left( r\sin \theta \right) }^{2}
\]
\[
= {\left( \cos \theta dr - r\sin \theta d\theta \right) }^{2} + {\left( \sin \theta dr + r\cos \theta d\theta \right) }^{2}
\]
\[
= \left( {{\cos }^{2}\theta + {\sin }^{2}\theta }\right) d{r}^{2} + \le... | Yes |
Theorem 13.14. For a Riemannian manifold \( \left( {M, g}\right) \), the following are equivalent:\n\n(a) \( g \) is flat.\n\n(b) Each point of \( M \) is contained in the domain of a smooth coordinate chart in which \( g \) has the coordinate representation \( g = {\delta }_{ij}d{x}^{i}d{x}^{j} \) .\n\n(c) Each point ... | Proof. The implications (a) \( \Rightarrow \) (b) \( \Rightarrow \) (c) \( \Rightarrow \) (d) are easy consequences of the definitions, and are left to the reader. The remaining implication,(d) \( \Rightarrow \) (a), follows from the canonical form theorem for commuting frames: if \( \left( {E}_{i}\right) \) is a commu... | No |
Example 13.17 (Induced Metrics in Graph Coordinates). Let \( U \subseteq {\mathbb{R}}^{n} \) be an open subset, and let \( S \subseteq {\mathbb{R}}^{n + 1} \) be the graph of a smooth function \( f : U \rightarrow \mathbb{R} \) . The map \( X : U \rightarrow {\mathbb{R}}^{n + 1} \) given by \( X\left( {{u}^{1},\ldots ,... | \[ {X}^{ * }\bar{g} = {X}^{ * }\left( {{\left( d{x}^{1}\right) }^{2} + \cdots + {\left( d{x}^{n + 1}\right) }^{2}}\right) = {\left( d{u}^{1}\right) }^{2} + \cdots + {\left( d{u}^{n}\right) }^{2} + d{f}^{2}. \] | Yes |
Example 13.18 (Induced Metrics on Surfaces of Revolution). Let \( C \) be an embedded 1-dimensional submanifold of the half-plane \( \{ \left( {r, z}\right) : r > 0\} \), and let \( {S}_{C} \) be the surface of revolution generated by \( C \) as described in Example 5.17. To compute the induced metric on \( {S}_{C} \),... | Thus we can compute\n\n\[ \n{X}^{ * }\bar{g} = d{\left( a\left( t\right) \cos \theta \right) }^{2} + d{\left( a\left( t\right) \sin \theta \right) }^{2} + d{\left( b\left( t\right) \right) }^{2} \]\n\n\[ \n= {\left( {a}^{\prime }\left( t\right) \cos \theta dt - a\left( t\right) \sin \theta d\theta \right) }^{2} \]\n\n\... | Yes |
Proposition 13.25 (Parameter Independence of Length). Let \( \\left( {M, g}\\right) \) be a Riemannian manifold with or without boundary, and let \( \\gamma : \\left\\lbrack {a, b}\\right\\rbrack \\rightarrow M \) be a piecewise smooth curve segment. If \( \\widetilde{\\gamma } \) is a reparametrization of \( \\gamma \... | Proof. First suppose that \( \\gamma \) is smooth, and \( \\varphi : \\left\\lbrack {c, d}\\right\\rbrack \\rightarrow \\left\\lbrack {a, b}\\right\\rbrack \) is a diffeomorphism such that \( \\widetilde{\\gamma } = \\gamma \\circ \\varphi \) . The fact that \( \\varphi \) is a diffeomorphism implies that either \( {\\... | Yes |
Lemma 13.28. Let \( g \) be a Riemannian metric on an open subset \( U \subseteq {\mathbb{R}}^{n} \). Given a compact subset \( K \subseteq U \), there exist positive constants \( c, C \) such that for all \( x \in K \) and all \( v \in {T}_{x}{\mathbb{R}}^{n} \), \[ c{\left| v\right| }_{\bar{g}} \leq {\left| v\right| ... | Proof. For any compact subset \( K \subseteq U \), let \( L \subseteq T{\mathbb{R}}^{n} \) be the set \[ L = \left\{ {\left( {x, v}\right) \in T{\mathbb{R}}^{n} : x \in K,{\left| v\right| }_{\bar{g}} = 1}\right\} . \] Under the canonical identification of \( T{\mathbb{R}}^{n} \) with \( {\mathbb{R}}^{n} \times {\mathbb... | Yes |
Corollary 13.30. Every smooth manifold with or without boundary is metrizable. | Proof. First suppose \( M \) is a smooth manifold without boundary, and choose any Riemannian metric \( g \) on \( M \) . If \( M \) is connected, Theorem 13.29 shows that \( M \) is metrizable. More generally, let \( \left\{ {M}_{i}\right\} \) be the connected components of \( M \), and choose a point \( {p}_{i} \in {... | Yes |
Let us compute the gradient of a function \( f \in {C}^{\infty }\left( {\mathbb{R}}^{2}\right) \) with respect to the Euclidean metric in polar coordinates. | From Example 13.12 we see that the matrix of \( \bar{g} \) in polar coordinates is \( \left( \begin{matrix} 1 & 0 \\ 0 & {r}^{2} \end{matrix}\right) \), so its inverse matrix is \( \left( \begin{matrix} 1 & 0 \\ 0 & 1/{r}^{2} \end{matrix}\right) \) . Inserting this into the formula for the gradient, we obtain\n\n\[\n\o... | Yes |
Lemma 14.1. Let \( \alpha \) be a covariant \( k \) -tensor on a finite-dimensional vector space V. The following are equivalent:\n\n(a) \( \alpha \) is alternating.\n\n(b) \( \alpha \left( {{v}_{1},\ldots ,{v}_{k}}\right) = 0 \) whenever the \( k \) -tuple \( \left( {{v}_{1},\ldots ,{v}_{k}}\right) \) is linearly depe... | Proof. The implications (a) \( \Rightarrow \) (c) and (b) \( \Rightarrow \) (c) are immediate. We complete the proof by showing that (c) implies both (a) and (b).\n\nAssume that \( \alpha \) satisfies (c). For any vectors \( {v}_{1},\ldots ,{v}_{k} \), the hypothesis implies\n\n\[ 0 = \alpha \left( {{v}_{1},\ldots ,{v}... | Yes |
Proposition 14.3 (Properties of Alternation). Let \( \alpha \) be a covariant tensor on a finite-dimensional vector space.\n\n(a) Alt \( \alpha \) is alternating.\n\n(b) Alt \( \alpha = \alpha \) if and only if \( \alpha \) is alternating. | Exercise 14.4. Prove Proposition 14.3. | No |
Lemma 14.7 (Properties of Elementary \( k \) -Covectors). Let \( \left( {E}_{i}\right) \) be a basis for \( V \) , let \( \left( {\varepsilon }^{i}\right) \) be the dual basis for \( {V}^{ * } \), and let \( {\varepsilon }^{I} \) be as defined above.\n\n(a) If \( I \) has a repeated index, then \( {\varepsilon }^{I} = ... | Proof. If \( I \) has a repeated index, then for any vectors \( {v}_{1},\ldots ,{v}_{k} \), the determinant in (14.1) has two identical rows and thus is equal to zero, which proves (a). On the other hand, if \( J \) is obtained from \( I \) by interchanging two indices, then the corresponding determinants have opposite... | Yes |
Proposition 14.8 (A Basis for \( {\Lambda }^{k}\left( {V}^{ * }\right) \) ). Let \( V \) be an \( n \) -dimensional vector space. If \( \left( {\varepsilon }^{i}\right) \) is any basis for \( {V}^{ * } \), then for each positive integer \( k \leq n \), the collection of \( k \) -covectors\n\n\[ \mathcal{E} = \left\{ {{... | Proof. The fact that \( {\Lambda }^{k}\left( {V}^{ * }\right) \) is the trivial vector space when \( k > n \) follows immediately from Lemma 14.1(b), since every \( k \) -tuple of vectors is linearly dependent in that case. For the case \( k \leq n \), we need to show that the set \( \mathcal{E} \) spans \( {\Lambda }^... | Yes |
Proposition 14.9. Suppose \( V \) is an \( n \) -dimensional vector space and \( \omega \in {\Lambda }^{n}\left( {V}^{ * }\right) \) . If \( T : V \rightarrow V \) is any linear map and \( {v}_{1},\ldots ,{v}_{n} \) are arbitrary vectors in \( V \), then\n\n\[ \omega \left( {T{v}_{1},\ldots, T{v}_{n}}\right) = \left( {... | Proof. Let \( \left( {E}_{i}\right) \) be any basis for \( V \), and let \( \left( {\varepsilon }^{i}\right) \) be the dual basis. Let \( \left( {T}_{i}^{j}\right) \) denote the matrix of \( T \) with respect to this basis, and let \( {T}_{i} = T{E}_{i} = {T}_{i}^{j}{E}_{j} \) . By Proposition 14.8, we can write \( \om... | Yes |
Proposition 14.11 (Properties of the Wedge Product). Suppose \( \omega ,{\omega }^{\prime },\eta ,{\eta }^{\prime } \), and \( \xi \) are multicovectors on a finite-dimensional vector space \( V \) .\n\n(a) BILINEARITY: For \( a,{a}^{\prime } \in \mathbb{R} \) ,\n\n\[ \n\left( {{a\omega } + {a}^{\prime }{\omega }^{\pri... | Proof. Bilinearity follows immediately from the definition, because the tensor product is bilinear and Alt is linear. To prove associativity, note that Lemma 14.10 gives\n\n\[ \n\left( {{\varepsilon }^{I} \land {\varepsilon }^{J}}\right) \land {\varepsilon }^{K} = {\varepsilon }^{IJ} \land {\varepsilon }^{K} = {\vareps... | Yes |
Lemma 14.13. Let \( V \) be a finite-dimensional vector space and \( v \in V \) .\n\n(a) \( {i}_{v} \circ {i}_{v} = 0 \) .\n\n(b) If \( \omega \in {\Lambda }^{k}\left( {V}^{ * }\right) \) and \( \eta \in {\Lambda }^{l}\left( {V}^{ * }\right) \), | Proof. On \( k \) -covectors for \( k \geq 2 \), part (a) is immediate from the definition, because any alternating tensor gives zero when two of its arguments are identical. On 1- covectors and 0 -covectors, it follows from the fact that \( {i}_{v} \equiv 0 \) on 0 -covectors.\n\nTo prove (b), it suffices to consider ... | Yes |
If \( F : M \rightarrow N \) is a smooth map and \( \omega \) is a differential form on \( N \), the pullback \( {F}^{ * }\omega \) is a differential form on \( M \), defined as for any covariant tensor field: | \[ {\left( {F}^{ * }\omega \right) }_{p}\left( {{v}_{1},\ldots ,{v}_{k}}\right) = {\omega }_{F\left( p\right) }\left( {d{F}_{p}\left( {v}_{1}\right) ,\ldots, d{F}_{p}\left( {v}_{k}\right) }\right) . \] | Yes |
Lemma 14.16. Suppose \( F : M \rightarrow N \) is smooth.\n\n(a) \( {F}^{ * } : {\Omega }^{k}\left( N\right) \rightarrow {\Omega }^{k}\left( M\right) \) is linear over \( \mathbb{R} \) .\n\n(b) \( {F}^{ * }\left( {\omega \land \eta }\right) = \left( {{F}^{ * }\omega }\right) \land \left( {{F}^{ * }\eta }\right) \) .\n\... | Exercise 14.17. Prove this lemma. | No |
Define \( F : {\mathbb{R}}^{2} \rightarrow {\mathbb{R}}^{3} \) by \( F\left( {u, v}\right) = \left( {u, v,{u}^{2} - {v}^{2}}\right) \), and let \( \omega \) be the 2-form \( {ydx} \land {dz} + {xdy} \land {dz} \) on \( {\mathbb{R}}^{3} \). Compute the pullback \( {F}^{ * }\omega \). | \[ {F}^{ * }\left( {{ydx} \land {dz} + {xdy} \land {dz}}\right) = {vdu} \land d\left( {{u}^{2} - {v}^{2}}\right) + {udv} \land d\left( {{u}^{2} - {v}^{2}}\right) \] \[ = {vdu} \land \left( {{2udu} - {2vdv}}\right) + {udv} \land \left( {{2udu} - {2vdv}}\right) \] \[ = - 2{v}^{2}{du} \land {dv} + 2{u}^{2}{dv} \land {du} ... | Yes |
Let \( \omega = {dx} \land {dy} \) on \( {\mathbb{R}}^{2} \) . Thinking of the transformation to polar coordinates \( x = r\cos \theta, y = r\sin \theta \) as an expression for the identity map with respect to different coordinates on the domain and codomain, we obtain | \[ {dx} \land {dy} = d\left( {r\cos \theta }\right) \land d\left( {r\sin \theta }\right) \] \[ = \left( {\cos {\theta dr} - r\sin {\theta d\theta }}\right) \land \left( {\sin {\theta dr} + r\cos {\theta d\theta }}\right) \] \[ = {rdr} \land {d\theta }\text{.} \] | Yes |
Proposition 14.20 (Pullback Formula for Top-Degree Forms). Let \( F : M \rightarrow N \) be 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, a... | Proof. Because the fiber of \( {\Lambda }^{n}{T}^{ * }M \) is spanned by \( d{x}^{1} \land \cdots \land d{x}^{n} \) at each point, it suffices to show that both sides of (14.15) give the same result when evaluated on \( \left( {\partial /\partial {x}^{1},\ldots ,\partial /\partial {x}^{n}}\right) \) . From Lemma 14.16,... | No |
Proposition 14.11(e) shows that\n\n\[ d{F}^{1} \land \cdots \land d{F}^{n}\left( {\frac{\partial }{\partial {x}^{1}},\ldots ,\frac{\partial }{\partial {x}^{n}}}\right) = \det \left( {d{F}^{j}\left( \frac{\partial }{\partial {x}^{i}}\right) }\right) = \det \left( \frac{\partial {F}^{j}}{\partial {x}^{i}}\right) . \] | Therefore, the left-hand side of (14.15) gives \( \left( {u \circ F}\right) \det {DF} \) when applied to \( \left( {\partial /\partial {x}^{1},\ldots ,\partial /\partial {x}^{n}}\right) \) . On the other hand, the right-hand side gives the same thing, because \( d{x}^{1} \land \cdots \land d{x}^{n}\left( {\partial /\pa... | Yes |
Corollary 14.21. If \( \\left( {U,\\left( {x}^{i}\\right) }\\right) \) and \( \\left( {\\widetilde{U},\\left( {\\widetilde{x}}^{j}\\right) }\\right) \) are overlapping smooth coordinate charts on \( M \), then the following identity holds on \( U \\cap \\widetilde{U} \) :\n\n\[ d{\\widetilde{x}}^{1} \\land \\cdots \\la... | Proof. Apply the previous proposition with \( F \) equal to the identity map of \( U \\cap \\widetilde{U} \) , but using coordinates \( \\left( {x}^{i}\\right) \) in the domain and \( \\left( {\\widetilde{x}}^{j}\\right) \) in the codomain. | Yes |
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