Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
|---|---|---|
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... | \[ {\left( f \circ \gamma \right) }^{\prime }\left( t\right) = d{f}_{\gamma \left( t\right) }\left( {{\gamma }^{\prime }\left( t\right) }\right) . \] (11.12) 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}... | 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\rig... | 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\[ \omega = {udv} + {vdu}. \] | According to (11.16), the pullback \( {F}^{ * }\omega \) is given by\n\n\[ {F}^{ * }\omega = \left( {u \circ F}\right) d\left( {v \circ F}\right) + \left( {v \circ F}\right) d\left( {u \circ F}\right) \]\n\n\[ = \left( {{x}^{2}y}\right) d\left( {y\sin z}\right) + \left( {y\sin z}\right) d\left( {{x}^{2}y}\right) \]\n\n... | 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 |
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 restriction \( {\iota }^... | \[ {\iota }^{ * }\omega = {\iota }^{ * }{dy} = d\left( {y \circ \iota }\right) = 0. \] | 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\n. Let \( M \) be a smooth manifold with or without boundary. Suppose \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow M \) is a piecewise smooth curve segment, and \( \omega ,{\omega }_{1},{\omega }_{2} \in {\mathfrak{X}}^{ * }\left( M\right) \) . (a) For any \( {... | - Exercise 11.35. Prove Proposition 11.34. | No |
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... | Yes |
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\[ \n{\int }_{\gamma }... | Proof. Suppose first that \( \gamma \) is smooth. By Propositions 11.23 and 11.38,\n\n\[ \n{\int }_{\gamma }{df} = {\int }_{a}^{b}d{f}_{\gamma \left( t\right) }\left( {{\gamma }^{\prime }\left( t\right) }\right) {dt} = {\int }_{a}^{b}{\left( f \circ \gamma \right) }^{\prime }\left( t\right) {dt}.\n\]\n\nBy the fundamen... | 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. | To see what this condition is, suppose \( \omega \in {\mathfrak{X}}^{ * }\left( M\right) \) is exact. Let \( f \) be any potential function for \( \omega \), and let \( \left( {U,\left( {x}^{i}\right) }\right) \) be any smooth chart on \( M \) . Because \( f \) is smooth, it satisfies the following identity on \( U \) ... | Yes |
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\[ \frac{\partial \left( {y\cos {xy}}\right) }{\partial y} = \frac{\partial \left( {x\cos {xy}}\right) }{\partial x} = \cos {xy} - {xy}\sin {xy}, \]\nso \( \omega \) is closed.... | On the other hand, the covector field\n\n\[ \eta = x\cos {xydx} + y\cos {xydy} \]\n\nis not closed, because\n\n\[ \frac{\partial \left( {x\cos {xy}}\right) }{\partial y} = - {x}^{2}\sin {xy},\;\frac{\partial \left( {y\cos {xy}}\right) }{\partial x} = - {y}^{2}\sin {xy}. \]\n\nThus \( \eta \) is not exact. | No |
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\[ \omega = {e}^{{y}^{2}}{dx} + {2xy}{e}^{{y}^{2}}{dy} - {2zdz}. \]\n\nYou can check that \( \omega \) is closed. For \( f \) to be a potential for \( \omega \), it must satisfy\n\n\[ \frac{\partial f}{\partial x} = {e}^{{y}^{2}},\;\frac{\p... | Holding \( y \) and \( z \) fixed and integrating the first equation with respect to \( x \), we obtain\n\n\[ f\left( {x, y, z}\right) = \int {e}^{{y}^{2}}{dx} = x{e}^{{y}^{2}} + {C}_{1}\left( {y, z}\right) ,\]\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. | No |
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... | \[ {F}^{ * }\left( {{B}_{{i}_{1}\ldots {i}_{k}}d{y}^{{i}_{1}} \otimes \cdots \otimes d{y}^{{i}_{k}}}\right) \] \[ = \left( {{B}_{{i}_{1}\ldots {i}_{k}} \circ F}\right) d\left( {{y}^{{i}_{1}} \circ F}\right) \otimes \cdots \otimes d\left( {{y}^{{i}_{k}} \circ F}\right) . \] | Yes |
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... | \[ {F}^{ * }A = {\left( r\cos \theta \right) }^{-2}d\left( {r\sin \theta }\right) \otimes d\left( {r\sin \theta }\right) \]\n\[ = {\left( r\cos \theta \right) }^{-2}\left( {\sin {\theta dr} + r\cos {\theta d\theta }}\right) \otimes \left( {\sin {\theta dr} + r\cos {\theta d\theta }}\right) \]\n\[ = {r}^{-2}{\tan }^{2}{... | 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 |
Theorem 12.37. Let \( M \) be a smooth manifold and let \( V \in \mathfrak{X}\left( M\right) \) . A smooth covariant tensor field \( A \) is invariant under the flow of \( V \) if and only if \( {\mathcal{L}}_{V}A = 0 \) . | - Exercise 12.38. Prove Theorem 12.37. | No |
The simplest example of a Riemannian metric is the Euclidean metric \( \bar{g} \) on \( {\mathbb{R}}^{n} \), given in standard coordinates by\n\n\[ \n\bar{g} = {\delta }_{ij}d{x}^{i}d{x}^{j}\n\]\n\nwhere \( {\delta }_{ij} \) is the Kronecker delta. It is common to abbreviate the symmetric product of a tensor \( \alpha ... | Applied to vectors \( v, w \in {T}_{p}{\mathbb{R}}^{n} \), this yields\n\n\[ \n{\bar{g}}_{p}\left( {v, w}\right) = {\delta }_{ij}{v}^{i}{w}^{j} = \mathop{\sum }\limits_{{i = 1}}^{n}{v}^{i}{w}^{i} = v \cdot w.\n\]\n\nIn other words, \( \bar{g} \) is the 2-tensor field whose value at each point is the Euclidean dot produ... | Yes |
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 |
Proposition 13.6. Suppose \( \left( {{E}_{1},{E}_{2}}\right) \) is a local orthonormal frame for \( {\mathbb{R}}^{2} \smallsetminus \{ 0\} \) defined in Example 8.12, and consider the inner product \( \langle \cdot , \cdot {\rangle }_{g} \) on \( {\mathbb{R}}^{2} \smallsetminus \{ 0\} \). | The next proposition is proved in just the same way as Lemma 8.13, with the Euclidean dot product replaced by the inner product \( \langle \cdot , \cdot {\rangle }_{g} \) . | No |
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... | \[ {F}^{ * }\bar{g} = d{\left( u\cos v\right) }^{2} + d{\left( u\sin v\right) }^{2} + d{\left( v\right) }^{2} \]\n\n\[ = {\left( \cos vdu - u\sin vdv\right) }^{2} + {\left( \sin vdu + u\cos vdv\right) }^{2} + d{v}^{2} \]\n\n\[ = {\cos }^{2}{vd}{u}^{2} - {2u}\sin v\cos {vdudv} + {u}^{2}{\sin }^{2}{vd}{v}^{2} \]\n\n\[ + ... | 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 | \[ \bar{g} = d{x}^{2} + d{y}^{2} = d{\left( r\cos \theta \right) }^{2} + d{\left( r\sin \theta \right) }^{2} \] \[ = {\left( \cos \theta dr - r\sin \theta d\theta \right) }^{2} + {\left( \sin \theta dr + r\cos \theta d\theta \right) }^{2} \] \[ = \left( {{\cos }^{2}\theta + {\sin }^{2}\theta }\right) d{r}^{2} + \left( ... | 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 |
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 \( {S}_{C} \), choose any smooth local parametrization \( \gamma \left( t... | 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\[ \n= {\left( {a}^{\prime }\left( t\right) \cos \theta dt - a\left( t\right) \sin \theta d\theta \right) }^{2} \n\]... | 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 |
Example 13.26. In \( \\left( {{\\mathbb{R}}^{n},\\bar{g}}\\right) \\), Problem 13-10 shows that any straight line segment is the shortest piecewise smooth curve segment between its endpoints. Therefore, the distance function \( {d}_{\\bar{g}} \) is equal to the usual Euclidean distance: | \[ {d}_{\\bar{g}}\\left( {x, y}\\right) = \\left| {x - y}\\right| . \] | 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 |
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\[ \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\[ \left( {{a\omega } + {a}^{\prime }{\omega }^{\prime... | 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\[ \left( {{\varepsilon }^{I} \land {\varepsilon }^{J}}\right) \land {\varepsilon }^{K} = {\varepsilon }^{IJ} \land {\varepsilon }^{K} = {\varepsil... | 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 \( M \) | \[ {\left( {F}^{ * }\omega \right) }_{p}\left( {{v}_{1},\ldots ,{v}_{k}}\right) = {\omega }_{F\left( p\right) }\left( {d{F}_{p}\left( {v}_{1}\right) ,\ldots, d{F}_{p}\left( {v}_{k}\right) }\right) . \] | 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,... | Yes |
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 \land d{\widetilde{... | 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 |
Theorem 14.24 (Existence and Uniqueness of Exterior Differentiation). Suppose \( M \) is a smooth manifold with or without boundary. There are unique operators \( d : {\Omega }^{k}\left( M\right) \rightarrow {\Omega }^{k + 1}\left( M\right) \) for all \( k \), called exterior differentiation, satisfying the following f... | Proof. First, we prove existence. Suppose \( \omega \in {\Omega }^{k}\left( M\right) \) . We wish to define \( {d\omega } \) by means of the coordinate formula (14.19) in each chart; more precisely, this means that for each smooth chart \( \left( {U,\varphi }\right) \) for \( M \), we wish to set\n\n\[ {d\omega } = {\v... | Yes |
Proposition 14.26 (Naturality of the Exterior Derivative). If \( F : M \rightarrow N \) is a smooth map, then for each \( k \) the pullback map \( {F}^{ * } : {\Omega }^{k}\left( N\right) \rightarrow {\Omega }^{k}\left( M\right) \) commutes with \( d \) : for all \( \omega \in {\Omega }^{k}\left( N\right) \) , \[ {F}^{... | Proof. If \( \left( {U,\varphi }\right) \) and \( \left( {V,\psi }\right) \) are smooth charts for \( M \) and \( N \), respectively, we can apply Proposition 14.23(d) to the coordinate representation \( \psi \circ F \circ {\varphi }^{-1} \) . Using (14.22) twice, we compute as follows on \( U \cap {F}^{-1}\left( V\rig... | Yes |
Let us work out the exterior derivatives of arbitrary 1-forms and 2-forms on \( {\mathbb{R}}^{3} \) . Any smooth 1 -form can be written\n\n\[ \omega = {Pdx} + {Qdy} + {Rdz} \]\n\nfor some smooth functions \( P, Q, R \) . | Using (14.19) and the fact that the wedge product of any 1 -form with itself is zero, we compute\n\n\[ {d\omega } = {dP} \land {dx} + {dQ} \land {dy} + {dR} \land {dz} \]\n\n\[ = \left( {\frac{\partial P}{\partial x}{dx} + \frac{\partial P}{\partial y}{dy} + \frac{\partial P}{\partial z}{dz}}\right) \land {dx} + \left(... | Yes |
Proposition 14.29 (Exterior Derivative of a 1-Form). For any smooth 1-form \( \omega \) and smooth vector fields \( X \) and \( Y \) ,\n\n\[ \n{d\omega }\left( {X, Y}\right) = X\left( {\omega \left( Y\right) }\right) - Y\left( {\omega \left( X\right) }\right) - \omega \left( \left\lbrack {X, Y}\right\rbrack \right) .\n... | Proof. Since any smooth 1-form can be expressed locally as a sum of terms of the form \( {udv} \) for smooth functions \( u \) and \( v \), it suffices to consider that case. Suppose \( \omega = {udv} \), and \( X, Y \) are smooth vector fields. Then the left-hand side of (14.28) is\n\n\[ \nd\left( {udv}\right) \left( ... | Yes |
Proposition 14.33. Suppose \( M \) is a smooth manifold, \( V \in \mathfrak{X}\left( M\right) \), and \( \omega ,\eta \in \) \( {\Omega }^{ * }\left( M\right) \) . Then\n\n\[ \n{\mathcal{L}}_{V}\left( {\omega \land \eta }\right) = \left( {{\mathcal{L}}_{V}\omega }\right) \land \eta + \omega \land \left( {{\mathcal{L}}_... | - Exercise 14.34. Prove the preceding proposition. | No |
Theorem 14.35 (Cartan's Magic Formula). On a smooth manifold \( M \), for any smooth vector field \( V \) and any smooth differential form \( \omega \) , \[ {\mathcal{L}}_{V}\omega = V\lrcorner \left( {d\omega }\right) + d\left( {V\lrcorner \omega }\right) . \] | Proof. We prove that (14.32) holds for smooth \( k \) -forms by induction on \( k \) . We begin with a smooth 0 -form \( f \), in which case \[ V\lrcorner \left( {df}\right) + d\left( {V\lrcorner f}\right) = V\lrcorner {df} = {df}\left( V\right) = {Vf} = {\mathcal{L}}_{V}f, \] which is (14.32). Now let \( k \geq 1 \), ... | Yes |
Corollary 14.36 (The Lie Derivative Commutes with \( d \) ). If \( V \) is a smooth vector field and \( \omega \) is a smooth differential form, then\n\n\[ \n{\mathcal{L}}_{V}\left( {d\omega }\right) = d\left( {{\mathcal{L}}_{V}\omega }\right) \n\] | Proof. This follows from Cartan’s formula and the fact that \( d \circ d = 0 \) :\n\n\[ \n{\mathcal{L}}_{V}{d\omega } = V\lrcorner d\left( {d\omega }\right) + d\left( {V\lrcorner {d\omega }}\right) = d\left( {V\lrcorner {d\omega }}\right) \n\]\n\n\[ \nd{\mathcal{L}}_{V}\omega = d\left( {V\lrcorner {d\omega }}\right) + ... | Yes |
Proposition 15.3. Let \( V \) be a vector space of dimension \( n \) . Each nonzero element \( \omega \in {\Lambda }^{n}\left( {V}^{ * }\right) \) determines an orientation \( {\mathcal{O}}_{\omega } \) of \( V \) as follows: if \( n \geq 1 \), then \( {\mathcal{O}}_{\omega } \) is the set of ordered bases \( \left( {{... | Proof. The 0-dimensional case is immediate, since a nonzero element of \( {\Lambda }^{0}\left( {V}^{ * }\right) \) is just a nonzero real number. Thus we may assume \( n \geq 1 \) . Let \( \omega \) be a nonzero element of \( {\Lambda }^{n}\left( {V}^{ * }\right) \), and let \( {\mathcal{O}}_{\omega } \) denote the set... | Yes |
Proposition 15.5 (The Orientation Determined by an \( n \) -Form). Let \( M \) be a smooth \( n \) -manifold with or without boundary. Any nonvanishing \( n \) -form \( \omega \) on \( M \) determines a unique orientation of \( M \) for which \( \omega \) is positively oriented at each point. Conversely, if \( M \) is ... | Proof. Let \( \omega \) be a nonvanishing \( n \) -form on \( M \) . Then \( \omega \) defines a pointwise orientation by Proposition 15.3, so all we need to check is that it is continuous. This is trivially true when \( n = 0 \), so assume \( n \geq 1 \) . Given \( p \in M \), let \( \left( {E}_{i}\right) \) be any lo... | Yes |
Proposition 15.6 (The Orientation Determined by a Coordinate Atlas). Let \( M \) be a smooth positive-dimensional manifold with or without boundary. Given any consistently oriented smooth atlas for \( M \), there is a unique orientation for \( M \) with the property that each chart in the given atlas is positively orie... | Proof. First, suppose \( M \) has a consistently oriented smooth atlas. Each chart in the atlas determines a pointwise orientation at each point of its domain. Wherever two of the charts overlap, the transition matrix between their respective coordinate frames is the Jacobian matrix of the transition map, which has pos... | Yes |
Proposition 15.15 (The Pullback Orientation). Suppose \( M \) and \( N \) are smooth manifolds with or without boundary. If \( F : M \rightarrow N \) is a local diffeomorphism and \( N \) is oriented, then \( M \) has a unique orientation, called the pullback orientation induced by \( \mathbf{F} \), such that \( F \) i... | Proof. For each \( p \in M \), there is a unique orientation on \( {T}_{p}M \) that makes the isomorphism \( d{F}_{p} : {T}_{p}M \rightarrow {T}_{F\left( p\right) }N \) orientation-preserving. This defines a pointwise orientation on \( M \), and provided it is continuous, it is the unique orientation on \( M \) with re... | Yes |
Proposition 15.17. Every parallelizable smooth manifold is orientable. | Proof. Suppose \( M \) is parallelizable, and let \( \left( {{E}_{1},\ldots ,{E}_{n}}\right) \) be a global smooth frame  to be positively oriented at each \( p \in M \) . This pointwise orientation is continuous, be... | No |
Proposition 15.21. Suppose \( M \) is an oriented smooth \( n \) -manifold with or without boundary, \( S \) is an immersed hypersurface with or without boundary in \( M \), and \( N \) is a vector field along \( S \) that is nowhere tangent to \( S \) . Then \( S \) has a unique orientation such that for each \( p \in... | Proof. Let \( \omega \) be an orientation form for \( M \) . Then \( \sigma = {\iota }_{S}^{ * }\left( {N\lrcorner \omega }\right) \) is an \( \left( {n - 1}\right) \) -form on \( S \) . (Recall that the pullback \( {\iota }_{S}^{ * } \) is really just restriction to vectors tangent to \( S \) .) It will follow that \(... | Yes |
Proposition 15.23. Let \( M \) be an oriented smooth manifold, and suppose \( S \subseteq M \) is a regular level set of a smooth function \( f : M \rightarrow \mathbb{R} \) . Then \( S \) is orientable. | Proof. Choose any Riemannian metric on \( M \), and let \( N = \operatorname{grad}f{ \mid }_{S} \) . The hypotheses imply that \( N \) is a nowhere tangent vector field along \( S \), so the result follows from Proposition 15.21. | Yes |
Proposition 15.24 (The Induced Orientation on a Boundary). Let \( M \) be an oriented smooth \( n \) -manifold with boundary, \( n \geq 1 \) . Then \( \partial M \) is orientable, and all outward-pointing vector fields along \( \partial M \) determine the same orientation on \( \partial M \) . | Proof. Let \( n = \dim M \), let \( \omega \) be an orientation form for \( M \), and let \( N \) be a smooth outward-pointing vector field along \( \partial M \) . The \( \left( {n - 1}\right) \) -form \( {\iota }_{\partial M}^{ * }\left( {N\lrcorner \omega }\right) \) is an orientation form for \( \partial M \) by Pr... | Yes |
Let us determine the induced orientation on \( \partial {\mathbb{H}}^{n} \) when \( {\mathbb{H}}^{n} \) itself has the standard orientation inherited from \( {\mathbb{R}}^{n} \). | We can identify \( \partial {\mathbb{H}}^{n} \) with \( {\mathbb{R}}^{n - 1} \) under the correspondence \( \left( {{x}^{1},\ldots ,{x}^{n - 1},0}\right) \leftrightarrow \left( {{x}^{1},\ldots ,{x}^{n - 1}}\right) \). Since the vector field \( - \partial /\partial {x}^{n} \) is outward-pointing along \( \partial {\math... | Yes |
Lemma 15.27. Let \( M \) be an oriented smooth \( n \) -manifold with boundary. Suppose \( U \subseteq {\mathbb{R}}^{n - 1} \) is open, \( a, b \) are real numbers with \( a < b \), and \( F : (a, b\rbrack \times U \rightarrow M \) is a smooth embedding that restricts to an embedding of \( \{ b\} \times U \) into \( \p... | Proof. Let \( x \) be an arbitrary point of \( U \), and let \( p = f\left( x\right) = F\left( {b, x}\right) \in \partial M \) (Fig. 15.6). The hypothesis that \( F \) is an embedding means that the linear map \( d{F}_{\left( b, x\right) } : \left( {{T}_{b}\mathbb{R} \oplus {T}_{x}{\mathbb{R}}^{n - 1}}\right) \rightarr... | Yes |
Spherical coordinates (Example C.38) yield a smooth local parametrization of \( {\mathbb{S}}^{2} \) as follows. Let \( U \) be the open rectangle \( \left( {0,\pi }\right) \times \left( {0,{2\pi }}\right) \subseteq {\overline{\mathbb{R}}}^{2} \), and let \( X : U \rightarrow {\mathbb{R}}^{3} \) be the following map:\n\... | We can check whether \( X \) preserves or reverses orientation by using the fact that it is the restriction of the 3-dimensional spherical coordinate parametrization \( F : (0,1\rbrack \times U \rightarrow {\overline{\mathbb{B}}}^{3} \) defined by\n\n\[ F\left( {\rho ,\varphi ,\theta }\right) = \left( {\rho \sin \varph... | Yes |
Proposition 15.29. Suppose \( \left( {M, g}\right) \) is an oriented Riemannian n-manifold with or without boundary, and \( n \geq 1 \) . There is a unique smooth orientation form \( {\omega }_{g} \in \) \( {\Omega }^{n}\left( M\right) \), called the Riemannian volume form, that satisfies\n\n\[ \n{\omega }_{g}\left( {{... | Proof. Suppose first that such a form \( {\omega }_{g} \) exists. If \( \left( {{E}_{1},\ldots ,{E}_{n}}\right) \) is any local oriented orthonormal frame on an open subset \( U \subseteq M \) and \( \left( {{\varepsilon }^{1},\ldots ,{\varepsilon }^{n}}\right) \) is the dual coframe, we can write \( {\omega }_{g} = f{... | Yes |
Let \( \left( {M, g}\right) \) be an oriented Riemannian \( n \) -manifold with or without boundary, \( n \geq 1 \) . In any oriented smooth coordinates \( \left( {x}^{i}\right) \), the Riemannian volume form has the local coordinate expression\n\n\[ \n{\omega }_{g} = \sqrt{\det \left( {g}_{ij}\right) }d{x}^{1} \land \... | Proof. Let \( \left( {U,\left( {x}^{i}\right) }\right) \) be an oriented smooth chart, and let \( p \in M \) . In these coordinates, \( {\omega }_{g} = {fd}{x}^{1} \land \cdots \land d{x}^{n} \) for some positive coefficient function \( f \) . To compute \( f \), let \( \left( {E}_{i}\right) \) be any smooth oriented o... | Yes |
Let \( \left( {M, g}\right) \) be an oriented Riemannian manifold with or without boundary, let \( S \subseteq M \) be an immersed hypersurface with or without boundary, and let \( \widetilde{g} \) denote the induced metric on \( S \) . Suppose \( N \) is a smooth unit normal vector field along \( S \) . With respect t... | Proof. By Proposition 15.21, the \( \left( {n - 1}\right) \) -form \( {\iota }_{S}^{ * }\left( {N\lrcorner {\omega }_{g}}\right) \) is an orientation form for \( S \) . To prove that it is the volume form for the induced Riemannian metric, we need only show that it gives the value 1 whenever it is applied to an oriente... | Yes |
Proposition 15.33. Suppose \( M \) is any Riemannian manifold with boundary. There is a unique smooth outward-pointing unit normal vector field \( N \) along \( \partial M \) . | Proof. First, we prove uniqueness. At any point \( p \in \partial M \), the subspace \( {\left( {T}_{p}\partial M\right) }^{ \bot } \subseteq \) \( {T}_{p}M \) is 1 -dimensional, so there are exactly two unit vectors at \( p \) that are normal to \( \partial M \) . Since any unit normal vector \( N \) is nowhere tangen... | Yes |
Proposition 15.35. If \( \pi : E \rightarrow M \) is a smooth covering map and \( M \) is orientable, then \( E \) is also orientable. | Proof. Because a covering map is a local diffeomorphism, this follows immediately from Proposition 15.15. | No |
Theorem 15.36. Suppose \( E \) is a connected, oriented, smooth manifold with or without boundary, and \( \pi : E \rightarrow M \) is a smooth normal covering map. Then \( M \) is orientable if and only if the action of \( {\operatorname{Aut}}_{\pi }\left( E\right) \) on \( E \) is orientation-preserving. | Proof. Let \( {\mathcal{O}}_{E} \) denote the given orientation on \( E \) . First suppose \( M \) is orientable, and let \( q \) be an arbitrary point in \( E \) . Because \( M \) is connected, it has exactly two orientations, and one of them has the property that \( d{\pi }_{q} : {T}_{q}E \rightarrow {T}_{\pi \left( ... | Yes |
Let \( E \) be the total space of the Möbius bundle (Example 10.3). The quotient map \( q : {\mathbb{R}}^{2} \rightarrow E \) used to define \( E \) is a smooth normal covering map, and the covering automorphism group is isomorphic to \( \mathbb{Z} \), acting on \( {\mathbb{R}}^{2} \) by \( n \cdot \left( {x, y}\right)... | For each \( r > 0 \), the image under \( q \) of the rectangle \( \left\lbrack {0,1}\right\rbrack \times \left\lbrack {-r, r}\right\rbrack \) is a Möbius band \( {M}_{r} \) . Because \( q \) restricts to a smooth covering map from \( \mathbb{R} \times \left\lbrack {-r, r}\right\rbrack \) to \( {M}_{r} \), the same argu... | Yes |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.