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Lemma 2.26 (Extension Lemma for Smooth Functions). Suppose \( M \) is a smooth manifold with or without boundary, \( A \subseteq M \) is a closed subset, and \( f : A \rightarrow {\mathbb{R}}^{k} \) is a smooth function. For any open subset \( U \) containing \( A \), there exists a smooth function \( \widetilde{f} : M... | Proof. For each \( p \in A \), choose a neighborhood \( {W}_{p} \) of \( p \) and a smooth function \( {\widetilde{f}}_{p} : {W}_{p} \rightarrow {\mathbb{R}}^{k} \) that agrees with \( f \) on \( {W}_{p} \cap A \) . Replacing \( {W}_{p} \) by \( {W}_{p} \cap U \), we may assume that \( {W}_{p} \subseteq U \) . The fami... | Yes |
Proposition 2.28 (Existence of Smooth Exhaustion Functions). Every smooth manifold with or without boundary admits a smooth positive exhaustion function. | Proof. Let \( M \) be a smooth manifold with or without boundary, let \( {\left\{ {V}_{j}\right\} }_{j = 1}^{\infty } \) be any countable open cover of \( M \) by precompact open subsets, and let \( \left\{ {\psi }_{j}\right\} \) be a smooth partition of unity subordinate to this cover. Define \( f \in {C}^{\infty }\le... | Yes |
Lemma 3.1 (Properties of Derivations). Suppose \( a \in {\mathbb{R}}^{n}, w \in {T}_{a}{\mathbb{R}}^{n} \), and \( f, g \in \) \( {C}^{\infty }\left( {\mathbb{R}}^{n}\right) \) . (a) If \( f \) is a constant function, then \( {wf} = 0 \) . (b) If \( f\left( a\right) = g\left( a\right) = 0 \), then \( w\left( {fg}\right... | Proof. It suffices to prove (a) for the constant function \( {f}_{1}\left( x\right) \equiv 1 \), for then \( f\left( x\right) \equiv c \) implies \( {wf} = w\left( {c{f}_{1}}\right) = {cw}{f}_{1} = 0 \) by linearity. For \( {f}_{1} \), the product rule gives \[ w{f}_{1} = w\left( {{f}_{1}{f}_{1}}\right) = {f}_{1}\left(... | Yes |
Proposition 3.2. Let \( a \in {\mathbb{R}}^{n} \). (a) For each geometric tangent vector \( {v}_{a} \in {\mathbb{R}}_{a}^{n} \), the map \( {\left. {D}_{v}\right| }_{a} : {C}^{\infty }\left( {\mathbb{R}}^{n}\right) \rightarrow \mathbb{R} \) defined by (3.1) is a derivation at a. (b) The map \( {v}_{a} \mapsto {\left. {... | Proof. The fact that \( {\left. {D}_{v}\right| }_{a} \) is a derivation at \( a \) is an immediate consequence of the product rule (3.2).\n\nTo prove that the map \( {\left. {v}_{a} \mapsto {D}_{v}\right| }_{a} \) is an isomorphism, we note first that it is linear, as is easily checked. To see that it is injective, sup... | Yes |
For any \( a \in {\mathbb{R}}^{n} \), the \( n \) derivations \[ {\left. \frac{\partial }{\partial {x}^{1}}\right| }_{a},\ldots ,{\left. \frac{\partial }{\partial {x}^{n}}\right| }_{a}\text{ defined by }{\left. \frac{\partial }{\partial {x}^{i}}\right| }_{a}f = \frac{\partial f}{\partial {x}^{i}}\left( a\right) \] form... | Proof. Apply the previous proposition and note that \( {\left. \partial /\partial {x}^{i}\right| }_{a} = {\left. {D}_{{e}_{i}}\right| }_{a} \). | No |
Lemma 3.4 (Properties of Tangent Vectors on Manifolds). Suppose \( M \) is a smooth manifold with or without boundary, \( p \in M, v \in {T}_{p}M \), and \( f, g \in {C}^{\infty }\left( M\right) \) .\n\n(a) If \( f \) is a constant function, then \( {vf} = 0 \) .\n\n(b) If \( f\left( p\right) = g\left( p\right) = 0 \),... | Exercise 3.5. Prove Lemma 3.4. | No |
Proposition 3.6 (Properties of Differentials). Let \( M, N \), and \( P \) be smooth manifolds with or without boundary, let \( F : M \rightarrow N \) and \( G : N \rightarrow P \) be smooth maps, and let \( p \in M \). (a) \( d{F}_{p} : {T}_{p}M \rightarrow {T}_{F\left( p\right) }N \) is linear. (b) \( d{\left( \dot{G... | ## - Exercise 3.7. Prove Proposition 3.6. | No |
Proposition 3.8. Let \( M \) be a smooth manifold with or without boundary, \( p \in M \) , and \( v \in {T}_{p}M \) . If \( f, g \in {C}^{\infty }\left( M\right) \) agree on some neighborhood of \( p \), then \( {vf} = {vg} \) . | Proof. Let \( h = f - g \), so that \( h \) is a smooth function that vanishes in a neighborhood of \( p \) . Let \( \psi \in {C}^{\infty }\left( M\right) \) be a smooth bump function that is identically equal to 1 on the support of \( h \) and is supported in \( M \smallsetminus \{ p\} \) . Because \( \psi \equiv 1 \)... | Yes |
Proposition 3.9 (The Tangent Space to an Open Submanifold). Let \( M \) be a smooth manifold with or without boundary, let \( U \subseteq M \) be an open subset, and let \( \iota : U \hookrightarrow M \) be the inclusion map. For every \( p \in U \), the differential \( d{\iota }_{p} : {T}_{p}U \rightarrow \) \( {T}_{p... | Proof. To prove injectivity, suppose \( v \in {T}_{p}U \) and \( d{\iota }_{p}\left( v\right) = 0 \in {T}_{p}M \) . Let \( B \) be a neighborhood of \( p \) such that \( \bar{B} \subseteq U \) . If \( f \in {C}^{\infty }\left( U\right) \) is arbitrary, the extension lemma for smooth functions guarantees that there exis... | Yes |
Proposition 3.10 (Dimension of the Tangent Space). If \( M \) is an \( n \) -dimensional smooth manifold, then for each \( p \in M \), the tangent space \( {T}_{p}M \) is an \( n \) -dimensional vector space. | Proof. Given \( p \in M \), let \( \left( {U,\varphi }\right) \) be a smooth coordinate chart containing \( p \) . Because \( \varphi \) is a diffeomorphism from \( U \) onto an open subset \( \widehat{U} \subseteq {\mathbb{R}}^{n} \), it follows from Proposition 3.6(d) that \( d{\varphi }_{p} \) is an isomorphism from... | Yes |
Lemma 3.11. Let \( \iota : {\mathbb{H}}^{n} \hookrightarrow {\mathbb{R}}^{n} \) denote the inclusion map. For any \( a \in \partial {\mathbb{H}}^{n} \), the differential \( d{\iota }_{a} : {T}_{a}{\mathbb{H}}^{n} \rightarrow {T}_{a}{\mathbb{R}}^{n} \) is an isomorphism. | Proof. Suppose \( a \in \partial {\mathbb{H}}^{n} \) . To show that \( d{\iota }_{a} \) is injective, assume \( d{\iota }_{a}\left( v\right) = 0 \) . Suppose \( f : {\mathbb{H}}^{n} \rightarrow \mathbb{R} \) is smooth, and let \( \widetilde{f} \) be any extension of \( f \) to a smooth function defined on all of \( {\m... | Yes |
Proposition 3.12 (Dimension of Tangent Spaces on a Manifold with Boundary). Suppose \( M \) is an \( n \) -dimensional smooth manifold with boundary. For each \( p \in M \) , \( {T}_{p}M \) is an \( n \) -dimensional vector space. | Proof. Let \( p \in M \) be arbitrary. If \( p \) is an interior point, then because Int \( M \) is an open submanifold of \( M \), Proposition 3.9 implies that \( {T}_{p}\left( {\operatorname{Int}M}\right) \cong {T}_{p}M \) . Since Int \( M \) is a smooth \( n \) -manifold without boundary, its tangent spaces all have... | Yes |
Proposition 3.13 (The Tangent Space to a Vector Space). Suppose \( V \) is a finite-dimensional vector space with its standard smooth manifold structure. For each point \( a \in V \), the map \( v \mapsto {\left. {D}_{v}\right| }_{a} \) defined by (3.5) is a canonical isomorphism from \( V \) to \( {T}_{a}V \), such th... | Proof. Once we choose a basis for \( V \), we can use the same argument as in the proof of Proposition 3.2 to show that \( {\left. {D}_{v}\right| }_{a} \) is indeed a derivation at \( a \), and that the map \( v \mapsto {\left. {D}_{v}\right| }_{a} \) is an isomorphism.\n\nNow suppose \( L : V \rightarrow W \) is a lin... | Yes |
Proposition 3.14 (The Tangent Space to a Product Manifold). Let \( {M}_{1},\ldots ,{M}_{k} \) be smooth manifolds, and for each \( j \), let \( {\pi }_{j} : {M}_{1} \times \cdots \times {M}_{k} \rightarrow {M}_{j} \) be the projection onto the \( {M}_{j} \) factor. For any point \( p = \left( {{p}_{1},\ldots ,{p}_{k}}\... | Proof. See Problem 3-2. | No |
Proposition 3.15. Let \( M \) be a smooth \( n \) -manifold with or without boundary, and let \( p \in M \) . Then \( {T}_{p}M \) is an \( n \) -dimensional vector space, and for any smooth chart \( \left( {U,\left( {x}^{i}\right) }\right) \) containing \( p \), the coordinate vectors \( \partial /{\left. \partial {x}^... | Thus, a tangent vector \( v \in {T}_{p}M \) can be written uniquely as a linear combination\n\n\[ v = {\left. {v}^{i}\frac{\partial }{\partial {x}^{i}}\right| }_{p} \]\n\nwhere we use the summation convention as usual, with an upper index in the denominator being considered as a lower index, as explained on p. 52. The ... | Yes |
The transition map between polar coordinates and standard coordinates in suitable open subsets of the plane is given by \( \left( {x, y}\right) = \left( {r\cos \theta, r\sin \theta }\right) \). Let \( p \) be the point in \( {\mathbb{R}}^{2} \) whose polar coordinate representation is \( \left( {r,\theta }\right) = \le... | Applying (3.11) to the coordinate vectors, we find\n\n\[ {\left. \frac{\partial }{\partial r}\right| }_{p} = {\left. \cos \left( \frac{\pi }{2}\right) \frac{\partial }{\partial x}\right| }_{p} + {\left. \sin \left( \frac{\pi }{2}\right) \frac{\partial }{\partial y}\right| }_{p} = {\left. \frac{\partial }{\partial y}\ri... | Yes |
Proposition 3.20. If \( M \) is a smooth \( n \) -manifold with or without boundary, and \( M \) can be covered by a single smooth chart, then \( {TM} \) is diffeomorphic to \( M \times {\mathbb{R}}^{n} \) . | Proof. If \( \left( {U,\varphi }\right) \) is a global smooth chart for \( M \), then \( \varphi \) is, in particular, a diffeomorphism from \( U = M \) to an open subset \( \widehat{U} \subseteq {\mathbb{R}}^{n} \) or \( {\mathbb{H}}^{n} \) . The proof of the previous proposition showed that the natural coordinate cha... | Yes |
Proposition 3.21. If \( F : M \rightarrow N \) is a smooth map, then its global differential \( {dF} : {TM} \rightarrow {TN} \) is a smooth map. | Proof. From the local expression (3.9) for \( d{F}_{p} \) in coordinates, it follows that \( {dF} \) has the following coordinate representation in terms of natural coordinates for \( {TM} \) and \( {TN} \) :\n\n\[ \n{dF}\left( {{x}^{1},\ldots ,{x}^{n},{v}^{1},\ldots ,{v}^{n}}\right) = \left( {{F}^{1}\left( x\right) ,\... | Yes |
Proposition 3.23. Suppose \( M \) is a smooth manifold with or without boundary and \( p \in M \) . Every \( v \in {T}_{p}M \) is the velocity of some smooth curve in \( M \) . | Proof. First suppose that \( p \in \operatorname{Int}M \) (which includes the case \( \partial M = \varnothing \) ). Let \( \left( {U,\varphi }\right) \) be a smooth coordinate chart centered at \( p \), and write \( v = {\left. {v}^{i}\partial /\partial {x}^{i}\right| }_{p} \) in terms of the coordinate basis. For suf... | Yes |
Proposition 3.24 (The Velocity of a Composite Curve). Let \( F : M \rightarrow N \) be a smooth map, and let \( \gamma : J \rightarrow M \) be a smooth curve. For any \( {t}_{0} \in J \), the velocity at \( t = {t}_{0} \) of the composite curve \( F \circ \gamma : J \rightarrow N \) is given by\n\n\[ \n{\left( F \circ ... | Proof. Just go back to the definition of the velocity of a curve:\n\n\[ \n{\left( F \circ \gamma \right) }^{\prime }\left( {t}_{0}\right) = d\left( {F \circ \gamma }\right) \left( {\left. \frac{d}{dt}\right| }_{{t}_{0}}\right) = {dF} \circ {d\gamma }\left( {\left. \frac{d}{dt}\right| }_{{t}_{0}}\right) = {dF}\left( {{\... | Yes |
Proposition 4.1. Suppose \( F : M \rightarrow N \) is a smooth map and \( p \in M \) . If \( d{F}_{p} \) is surjective, then \( p \) has a neighborhood \( U \) such that \( {\left. F\right| }_{U} \) is a submersion. If \( d{F}_{p} \) is injective, then \( p \) has a neighborhood \( U \) such that \( {\left. F\right| }_... | Proof. If we choose any smooth coordinates for \( M \) near \( p \) and for \( N \) near \( F\left( p\right) \) , either hypothesis means that Jacobian matrix of \( F \) in coordinates has full rank at \( p \) . Example 1.28 shows that the set of \( m \times n \) matrices of full rank is an open subset of \( \mathrm{M}... | Yes |
Theorem 4.5 (Inverse Function Theorem for Manifolds). Suppose \( M \) and \( N \) are smooth manifolds, and \( F : M \rightarrow N \) is a smooth map. If \( p \in M \) is a point such that \( d{F}_{p} \) is invertible, then there are connected neighborhoods \( {U}_{0} \) of \( p \) and \( {V}_{0} \) of \( F\left( p\rig... | Proof. The fact that \( d{F}_{p} \) is bijective implies that \( M \) and \( N \) have the same dimension, say \( n \) . Choose smooth charts \( \left( {U,\varphi }\right) \) centered at \( p \) and \( \left( {V,\psi }\right) \) centered at \( F\left( p\right) \) , with \( F\left( U\right) \subseteq V \) . Then \( \wid... | Yes |
Proposition 4.8. Suppose \( M \) and \( N \) are smooth manifolds (without boundary), and \( F : M \rightarrow N \) is a map.\n\n(a) \( F \) is a local diffeomorphism if and only if it is both a smooth immersion and a smooth submersion.\n\n(b) If \( \dim M = \dim N \) and \( F \) is either a smooth immersion or a smoot... | Proof. Suppose first that \( F \) is a local diffeomorphism. Given \( p \in M \), there is a neighborhood \( U \) of \( p \) such that \( F \) is a diffeomorphism from \( U \) to \( F\left( U\right) \) . It then follows from Proposition 3.6(d) that \( d{F}_{p} : {T}_{p}M \rightarrow {T}_{F\left( p\right) }N \) is an is... | Yes |
Theorem 4.12 (Rank Theorem). Suppose \( M \) and \( N \) are smooth manifolds of dimensions \( m \) and \( n \), respectively, and \( F : M \rightarrow N \) is a smooth map with constant rank \( r \) . For each \( p \in M \) there exist smooth charts \( \left( {U,\varphi }\right) \) for \( M \) centered at \( p \) and ... | Proof. Because the theorem is local, after choosing smooth coordinates we can replace \( M \) and \( N \) by open subsets \( U \subseteq {\mathbb{R}}^{m} \) and \( V \subseteq {\mathbb{R}}^{n} \) . The fact that \( {DF}\left( p\right) \) has rank \( r \) implies that its matrix has some \( r \times r \) submatrix with ... | Yes |
Corollary 4.13. Let \( M \) and \( N \) be smooth manifolds, let \( F : M \rightarrow N \) be a smooth map, and suppose \( M \) is connected. Then the following are equivalent:\n\n(a) For each \( p \in M \) there exist smooth charts containing \( p \) and \( F\left( p\right) \) in which the coordinate representation of... | Proof. First suppose \( F \) has a linear coordinate representation in a neighborhood of each point. Since every linear map has constant rank, it follows that the rank of \( F \) is constant in a neighborhood of each point, and thus by connectedness it is constant on all of \( M \) . Conversely, if \( F \) has constant... | Yes |
Theorem 4.14 (Global Rank Theorem). Let \( M \) and \( N \) be smooth manifolds, and suppose \( F : M \rightarrow N \) is a smooth map of constant rank.\n\n(a) If \( F \) is surjective, then it is a smooth submersion.\n\n(b) If \( F \) is injective, then it is a smooth immersion.\n\n(c) If \( F \) is bijective, then it... | Proof. Let \( m = \dim M, n = \dim N \), and suppose \( F \) has constant rank \( r \) . To prove (a), assume that \( F \) is not a smooth submersion, which means that \( r < n \) . By the rank theorem, for each \( p \in M \) there are smooth charts \( \left( {U,\varphi }\right) \) for \( M \) centered at \( p \) and \... | Yes |
Theorem 4.15 (Local Immersion Theorem for Manifolds with Boundary). Suppose \( M \) is a smooth \( m \) -manifold with boundary, \( N \) is a smooth \( n \) -manifold, and \( F : M \rightarrow N \) is a smooth immersion. For any \( p \in \partial M \), there exist a smooth boundary chart \( \left( {U,\varphi }\right) \... | Proof. By choosing initial smooth charts for \( M \) and \( N \), we may assume that \( M \) and \( N \) are open subsets of \( {\mathbb{H}}^{m} \) and \( {\mathbb{R}}^{n} \), respectively, and also that \( p = 0 \in {\mathbb{H}}^{m} \) , and \( F\left( p\right) = 0 \in {\mathbb{R}}^{n} \) . By definition of smoothness... | Yes |
Consider the curve \( \beta : \left( {-\pi ,\pi }\right) \rightarrow {\mathbb{R}}^{2} \) defined by\n\n\[ \beta \left( t\right) = \left( {\sin {2t},\sin t}\right) \text{.} \]\n\nIts image is a set that looks like a figure-eight in the plane (Fig. 4.3), sometimes called a lemniscate. (It is the locus of points \( \left(... | It is easy to see that \( \beta \) is an injective smooth immersion because \( {\beta }^{\prime }\left( t\right) \) never vanishes; but it is not a topological embedding, because its image is compact in the subspace topology, while its domain is not. | Yes |
Example 4.20 (A Dense Curve on the Torus). Let \( {\mathbb{T}}^{2} = {\mathbb{S}}^{1} \times {\mathbb{S}}^{1} \subseteq {\mathbb{C}}^{2} \) denote the torus, and let \( \alpha \) be any irrational number. The map \( \gamma : \mathbb{R} \rightarrow {\mathbb{T}}^{2} \) given by\n\n\[ \gamma \left( t\right) = \left( {{e}^... | Consider the set \( \gamma \left( \mathbb{Z}\right) = \{ \gamma \left( n\right) : n \in \mathbb{Z}\} \) . It follows from Dirichlet’s approximation theorem (see below) that for every \( \varepsilon > 0 \), there are integers \( n, m \) such that \( \left| {{\alpha n} - m}\right| < \varepsilon \) . Using the fact that \... | Yes |
Lemma 4.21 (Dirichlet’s Approximation Theorem). Given \( \alpha \in \mathbb{R} \) and any positive integer \( N \), there exist integers \( n, m \) with \( 1 \leq n \leq N \) such that \( \left| {{n\alpha } - m}\right| < 1/N \) . | Proof. For any real number \( x \), let \( f\left( x\right) = x - \lfloor x\rfloor \), where \( \lfloor x\rfloor \) is the greatest integer less than or equal to \( x \) . Since the \( N + 1 \) numbers \( \{ f\left( {i\alpha }\right) : i = 0,\ldots, N\} \) all lie in the interval \( \lbrack 0,1) \), by the pigeonhole p... | Yes |
Proposition 4.22. Suppose \( M \) and \( N \) are smooth manifolds with or without boundary, and \( F : M \rightarrow N \) is an injective smooth immersion. If any of the following holds, then \( F \) is a smooth embedding. | Proof. If \( F \) is open or closed, then it is a topological embedding by Theorem A.38, so it is a smooth embedding. Either (b) or (c) implies that \( F \) is closed: if \( F \) is proper, then it is closed by Theorem A.57, and if \( M \) is compact, then \( F \) is closed by the closed map lemma. Finally, assume \( M... | Yes |
Theorem 4.25 (Local Embedding Theorem). Suppose \( M \) and \( N \) are smooth manifolds with or without boundary, and \( F : M \rightarrow N \) is a smooth map. Then \( F \) is a smooth immersion if and only if every point in \( M \) has a neighborhood \( U \subseteq M \) such that \( {\left. F\right| }_{U} : U \right... | Proof. One direction is immediate: if every point has a neighborhood on which \( F \) is a smooth embedding, then \( F \) has full rank everywhere, so it is a smooth immersion.\n\nConversely, suppose \( F \) is a smooth immersion, and let \( p \in M \) . We show first that \( p \) has a neighborhood on which \( F \) is... | Yes |
Theorem 4.26 (Local Section Theorem). Suppose \( M \) and \( N \) are smooth manifolds and \( \pi : M \rightarrow N \) is a smooth map. Then \( \pi \) is a smooth submersion if and only if every point of \( M \) is in the image of a smooth local section of \( \pi \) . | Proof. First suppose that \( \pi \) is a smooth submersion. Given \( p \in M \), let \( q = \) \( \pi \left( p\right) \in N \) . By the rank theorem, we can choose smooth coordinates \( \left( {{x}^{1},\ldots ,{x}^{m}}\right) \) centered at \( p \) and \( \left( {{y}^{1},\ldots ,{y}^{n}}\right) \) centered at \( q \) i... | Yes |
Proposition 4.28 (Properties of Smooth Submersions). Let \( M \) and \( N \) be smooth manifolds, and suppose \( \pi : M \rightarrow N \) is a smooth submersion. Then \( \pi \) is an open map, and if it is surjective it is a quotient map. | Proof. Suppose \( W \) is an open subset of \( M \) and \( q \) is a point of \( \pi \left( W\right) \) . For any \( p \in W \) such that \( \pi \left( p\right) = q \), there is a neighborhood \( U \) of \( q \) on which there exists a smooth local section \( \sigma : U \rightarrow M \) of \( \pi \) satisfying \( \sigm... | Yes |
Theorem 4.29 (Characteristic Property of Surjective Smooth Submersions). Suppose \( M \) and \( N \) are smooth manifolds, and \( \pi : M \rightarrow N \) is a surjective smooth submersion. For any smooth manifold \( P \) with or without boundary, a map \( F : N \rightarrow P \) is smooth if and only if \( F \circ \pi ... | Proof. If \( F \) is smooth, then \( F \circ \pi \) is smooth by composition. Conversely, suppose that \( F \circ \pi \) is smooth, and let \( q \in N \) be arbitrary. Since \( \pi \) is surjective, there is a point \( p \in {\pi }^{-1}\left( q\right) \), and then the local section theorem guarantees the existence of a... | Yes |
Theorem 4.30 (Passing Smoothly to the Quotient). Suppose \( M \) and \( N \) are smooth manifolds and \( \pi : M \rightarrow N \) is a surjective smooth submersion. If \( P \) is a smooth manifold with or without boundary and \( F : M \rightarrow P \) is a smooth map that is constant on the fibers of \( \pi \), then th... | Proof. Because a surjective smooth submersion is a quotient map, Theorem A. 30 shows that there exists a unique continuous map \( \widetilde{F} : N \rightarrow P \) satisfying \( \widetilde{F} \circ \pi = F \). It is smooth by Theorem 4.29. | Yes |
Proposition 4.36 (Local Section Theorem for Smooth Covering Maps). Suppose \( E \) and \( M \) are smooth manifolds with or without boundary, and \( \pi : E \rightarrow M \) is a smooth covering map. Given any evenly covered open subset \( U \subseteq M \), any \( q \in U \) , and any \( p \) in the fiber of \( \pi \) ... | Proof. Suppose \( U \subseteq M \) is evenly covered, \( q \in U \), and \( p \in {\pi }^{-1}\left( q\right) \) . Let \( {\widetilde{U}}_{0} \) be the component of \( {\pi }^{-1}\left( U\right) \) containing \( p \) . Since the restriction of \( \pi \) to \( {\widetilde{U}}_{0} \) is a diffeomorphism onto \( U \), the ... | Yes |
Proposition 4.46. Suppose \( E \) and \( M \) are nonempty connected smooth manifolds with or without boundary. If \( \pi : E \rightarrow M \) is a proper local diffeomorphism, then \( \pi \) is a smooth covering map. | Proof. Because \( \pi \) is a local diffeomorphism, it is an open map, and because it is proper, it is a closed map (Theorem A.57). Thus \( \pi \left( E\right) \) is both open and closed in \( M \) . Since it is obviously nonempty, it is all of \( M \), so \( \pi \) is surjective.\n\n![eac462f6-ce8a-4286-98b4-9fe2c2939... | Yes |
Proposition 5.1 (Open Submanifolds). Suppose \( M \) is a smooth manifold. The embedded submanifolds of codimension 0 in \( M \) are exactly the open submanifolds. | Proof. Suppose \( U \subseteq M \) is an open submanifold, and let \( \iota : U \hookrightarrow M \) be the inclusion map. Example 1.26 showed that \( U \) is a smooth manifold of the same dimension as \( M \), so it has codimension 0 . In terms of the smooth charts for \( U \) constructed in Example 1.26, \( \iota \) ... | Yes |
Proposition 5.2 (Images of Embeddings as Submanifolds). Suppose \( M \) is a smooth manifold with or without boundary, \( N \) is a smooth manifold, and \( F : N \rightarrow M \) is a smooth embedding. Let \( S = F\left( N\right) \) . With the subspace topology, \( S \) is a topological manifold, and it has a unique sm... | Proof. If we give \( S \) the subspace topology that it inherits from \( M \), then the assumption that \( F \) is an embedding means that \( F \) can be considered as a homeomorphism from \( N \) onto \( S \), and thus \( S \) is a topological manifold. We give \( S \) a smooth structure by taking the smooth charts to... | Yes |
Proposition 5.3 (Slices of Product Manifolds). Suppose \( M \) and \( N \) are smooth manifolds. For each \( p \in N \), the subset \( M \times \{ p\} \) (called a slice of the product manifold) is an embedded submanifold of \( M \times N \) diffeomorphic to \( M \) . | Proof. The set \( M \times \{ p\} \) is the image of the smooth embedding \( x \mapsto \left( {x, p}\right) \) . | No |
Proposition 5.4 (Graphs as Submanifolds). Suppose \( M \) is a smooth m-manifold (without boundary), \( N \) is a smooth \( n \) -manifold with or without boundary, \( U \subseteq M \) is open, and \( f : U \rightarrow N \) is a smooth map. Let \( \Gamma \left( f\right) \subseteq M \times N \) denote the graph of \( f ... | Proof. Define a map \( {\gamma }_{f} : U \rightarrow M \times N \) by\n\n\[ {\gamma }_{f}\left( x\right) = \left( {x, f\left( x\right) }\right) . \]\n\n(5.2)\n\nIt is a smooth map whose image is \( \Gamma \left( f\right) \) . Because the projection \( {\pi }_{M} : M \times N \rightarrow M \) satisfies \( {\pi }_{M} \ci... | Yes |
Proposition 5.5. Suppose \( M \) is a smooth manifold with or without boundary and \( S \subseteq M \) is an embedded submanifold. Then \( S \) is properly embedded if and only if it is a closed subset of \( M \) . | Proof. If \( S \) is properly embedded, then it is closed by Theorem A.57. Conversely, if \( S \) is closed in \( M \), then Proposition A.53(c) shows that the inclusion map \( S \hookrightarrow M \) is proper. | Yes |
Corollary 5.6. Every compact embedded submanifold is properly embedded. | Proof. Compact subsets of Hausdorff spaces are closed. | No |
Proposition 5.7 (Global Graphs Are Properly Embedded). Suppose \( M \) is a smooth manifold, \( N \) is a smooth manifold with or without boundary, and \( f : M \rightarrow N \) is a smooth map. With the smooth manifold structure of Proposition 5.4, \( \Gamma \left( f\right) \) is properly embedded in \( M \times N \) ... | Proof. In this case, the projection \( {\pi }_{M} : M \times N \rightarrow M \) is a smooth left inverse for the embedding \( {\gamma }_{f} : M \rightarrow M \times N \) defined by (5.2). Thus \( {\gamma }_{f} \) is proper by Proposition A.53. | Yes |
Example 5.9 (Spheres as Submanifolds). For any \( n \geq 0,{\mathbb{S}}^{n} \) is an embedded sub-manifold of \( {\mathbb{R}}^{n + 1} \) | because it is locally the graph of a smooth function: as we showed in Example 1.4, the intersection of \( {\mathbb{S}}^{n} \) with the open subset \( \left\{ {x : {x}^{i} > 0}\right\} \) is the graph of the smooth function\n\n\[ \n{x}^{i} = f\left( {{x}^{1},\ldots ,{x}^{i - 1},{x}^{i + 1},\ldots ,{x}^{n + 1}}\right) , ... | Yes |
Theorem 5.11. If \( M \) is a smooth \( n \) -manifold with boundary, then with the subspace topology, \( \partial M \) is a topological \( \left( {n - 1}\right) \) -dimensional manifold (without boundary), and has a smooth structure such that it is a properly embedded submanifold of \( M \) . | Proof. See Problem 5-2. | No |
Theorem 5.12 (Constant-Rank Level Set Theorem). Let \( M \) and \( N \) be smooth manifolds, and let \( \Phi : M \rightarrow N \) be a smooth map with constant rank \( r \) . Each level set of \( \Phi \) is a properly embedded submanifold of codimension \( r \) in \( M \). | Proof. Write \( m = \dim M, n = \dim N \), and \( k = m - r \) . Let \( c \in N \) be arbitrary, and let \( S \) denote the level set \( {\Phi }^{-1}\left( c\right) \subseteq M \) . From the rank theorem, for each \( p \in S \) there are smooth charts \( \left( {U,\varphi }\right) \) centered at \( p \) and \( \left( {... | Yes |
Corollary 5.13 (Submersion Level Set Theorem). If \( M \) and \( N \) are smooth manifolds and \( \Phi : M \rightarrow N \) is a smooth submersion, then each level set of \( \Phi \) is a properly embedded submanifold whose codimension is equal to the dimension of \( N \) . | Proof. Every smooth submersion has constant rank equal to the dimension of its codomain. | No |
Corollary 5.14 (Regular Level Set Theorem). Every regular level set of a smooth map between smooth manifolds is a properly embedded submanifold whose codimension is equal to the dimension of the codomain. | Proof. Let \( \Phi : M \rightarrow N \) be a smooth map and let \( c \in N \) be a regular value. The set \( U \) of points \( p \in M \) where \( \operatorname{rank}d{\Phi }_{p} = \dim N \) is open in \( M \) by Proposition 4.1, and contains \( {\Phi }^{-1}\left( c\right) \) because of the assumption that \( c \) is a... | Yes |
Now we can give a much easier proof that \( {\mathbb{S}}^{n} \) is an embedded submanifold of \( {\mathbb{R}}^{n + 1} \) . The sphere is a regular level set of the smooth function \( f : {\mathbb{R}}^{n + 1} \rightarrow \mathbb{R} \) given by \( f\left( x\right) = {\left| x\right| }^{2} \), since \( d{f}_{x}\left( v\ri... | The sphere is a regular level set of the smooth function \( f : {\mathbb{R}}^{n + 1} \rightarrow \mathbb{R} \) given by \( f\left( x\right) = {\left| x\right| }^{2} \), since \( d{f}_{x}\left( v\right) = 2\mathop{\sum }\limits_{i}{x}^{i}{v}^{i} \), which is surjective except at the origin. | No |
Proposition 5.16. Let \( S \) be a subset of a smooth \( m \) -manifold \( M \). Then \( S \) is an embedded \( k \) -submanifold of \( M \) if and only if every point of \( S \) has a neighborhood \( U \) in \( M \) such that \( U \cap S \) is a level set of a smooth submersion \( \Phi : U \rightarrow {\mathbb{R}}^{m ... | Proof. First suppose \( S \) is an embedded \( k \) -submanifold. If \( \left( {{x}^{1},\ldots ,{x}^{m}}\right) \) are slice coordinates for \( S \) on an open subset \( U \subseteq M \), the map \( \Phi : U \rightarrow {\mathbb{R}}^{m - k} \) given in coordinates by \( \Phi \left( x\right) = \left( {{x}^{\widehat{k} +... | Yes |
Let \( H \) be the half-plane \( \{ \left( {r, z}\right) : r > 0\} \) , and suppose \( C \subseteq H \) is an embedded 1-dimensional submanifold. The surface of revolution determined by \( C \) is the subset \( {S}_{C} \subseteq {\mathbb{R}}^{3} \) given by\n\n\[ \n{S}_{C} = \left\{ {\left( {x, y, z}\right) : \left( {\... | A computation shows that the Jacobian matrix of \( \Phi \) is\n\n\[ \n{D\Phi }\left( {x, y, z}\right) = \left( {\frac{x}{r}\frac{\partial \varphi }{\partial r}\left( {r, z}\right) \frac{y}{r}\frac{\partial \varphi }{\partial r}\left( {r, z}\right) \frac{\partial \varphi }{\partial z}\left( {r, z}\right) }\right) ,\n\]\... | Yes |
Proposition 5.18 (Images of Immersions as Submanifolds). Suppose \( M \) is a smooth manifold with or without boundary, \( N \) is a smooth manifold, and \( F : N \rightarrow \) \( M \) is an injective smooth immersion. Let \( S = F\left( N\right) \) . Then \( S \) has a unique topology and smooth structure such that i... | Proof. The proof is very similar to that of Proposition 5.2, except that now we also have to define the topology on \( S \) . We give \( S \) a topology by declaring a set \( U \subseteq S \) to be open if and only if \( {F}^{-1}\left( U\right) \subseteq N \) is open, and then give it a smooth structure by taking the s... | Yes |
Proposition 5.21. Suppose \( M \) is a smooth manifold with or without boundary, and \( S \subseteq M \) is an immersed submanifold. If any of the following holds, then \( S \) is embedded.\n\n(a) \( S \) has codimension 0 in \( M \) .\n\n(b) The inclusion map \( S \subseteq M \) is proper.\n\n(c) \( S \) is compact. | Proof. Problem 5-3. | No |
Proposition 5.22 (Immersed Submanifolds Are Locally Embedded). If \( M \) is a smooth manifold with or without boundary, and \( S \subseteq M \) is an immersed submanifold, then for each \( p \in S \) there exists a neighborhood \( U \) of \( p \) in \( S \) that is an embedded submanifold of \( M \) . | Proof. Theorem 4.25 shows that each \( p \in S \) has a neighborhood \( U \) in \( S \) such that the inclusion \( {\left. \iota \right| }_{U} : U \hookrightarrow M \) is an embedding. | Yes |
Example 5.25 (Graph Parametrizations). Suppose \( U \subseteq {\mathbb{R}}^{n} \) is an open subset and \( f : U \rightarrow {\mathbb{R}}^{k} \) is a smooth function. The map \( {\gamma }_{f} : U \rightarrow {\mathbb{R}}^{n} \times {\mathbb{R}}^{k} \) given by \( {\gamma }_{f}\left( u\right) = \left( {u, f\left( u\righ... | For example, the map \( F : {\mathbb{B}}^{2} \rightarrow {\mathbb{R}}^{3} \) given by\n\n\[ F\left( {u, v}\right) = \left( {u, v,\sqrt{1 - {u}^{2} - {v}^{2}}}\right) \]\n\nis a smooth local parametrization of \( {\mathbb{S}}^{2} \) whose image is the open upper hemisphere, and whose inverse is one of the graph coordina... | No |
Theorem 5.27 (Restricting the Domain of a Smooth Map). If \( M \) and \( N \) are smooth manifolds with or without boundary, \( F : M \rightarrow N \) is a smooth map, and \( S \subseteq M \) is an immersed or embedded submanifold (Fig. 5.9), then \( {\left. F\right| }_{S} : S \rightarrow N \) is smooth. | Proof. The inclusion map \( \iota : S \hookrightarrow M \) is smooth by definition of an immersed sub-manifold. Since \( {\left. F\right| }_{S} = F \circ \iota \), the result follows. | Yes |
Theorem 5.29 (Restricting the Codomain of a Smooth Map). Suppose \( M \) is a smooth manifold (without boundary), \( S \subseteq M \) is an immersed submanifold, and \( F : N \rightarrow M \) is a smooth map whose image is contained in \( S \) (Fig. 5.10). If \( F \) is continuous as a map from \( N \) to \( S \), then... | Proof. Let \( p \in N \) be arbitrary and let \( q = F\left( p\right) \in S \) . Proposition 5.22 guarantees that there is a neighborhood \( V \) of \( q \) in \( S \) such that \( {\left. \iota \right| }_{V} : V \hookrightarrow M \) is a smooth embedding. Thus there exists a smooth chart \( \left( {W,\psi }\right) \) ... | Yes |
Corollary 5.30 (Embedded Case). Let \( M \) be a smooth manifold and \( S \subseteq M \) be an embedded submanifold. Then every smooth map \( F : N \rightarrow M \) whose image is contained in \( S \) is also smooth as a map from \( N \) to \( S \) . | Proof. Since \( S \subseteq M \) has the subspace topology, a continuous map \( F : N \rightarrow M \) whose image is contained in \( S \) is automatically continuous into \( S \), by the characteristic property of the subspace topology (Proposition A.17(a)). | No |
Theorem 5.31. Suppose \( M \) is a smooth manifold and \( S \subseteq M \) is an embedded submanifold. The subspace topology on \( S \) and the smooth structure described in Theorem 5.8 are the only topology and smooth structure with respect to which \( S \) is an embedded or immersed submanifold. | Proof. Suppose \( S \subseteq M \) is an embedded \( k \) -dimensional submanifold. Theorem 5.8 shows that it satisfies the local \( k \) -slice condition, so it is an embedded submani-fold with the subspace topology and the smooth structure of Theorem 5.8. Suppose there were some other topology and smooth structure on... | Yes |
Theorem 5.32. Suppose \( M \) is a smooth manifold and \( S \subseteq M \) is an immersed sub-manifold. For the given topology on \( S \), there is only one smooth structure making \( S \) into an immersed submanifold. | Proof. See Problem 5-14. | No |
Theorem 5.33. If \( M \) is a smooth manifold and \( S \subseteq M \) is a weakly embedded sub-manifold, then \( S \) has only one topology and smooth structure with respect to which it is an immersed submanifold. | Proof. See Problem 5-16. | No |
Lemma 5.34 (Extension Lemma for Functions on Submanifolds). Suppose \( M \) is a smooth manifold, \( S \subseteq M \) is a smooth submanifold, and \( f \in {C}^{\infty }\left( S\right) \) . (a) If \( S \) is embedded, then there exist a neighborhood \( U \) of \( S \) in \( M \) and a smooth function \( \widetilde{f} \... | Proof. Problem 5-17. | No |
Proposition 5.37. Suppose \( M \) is a smooth manifold, \( S \subseteq M \) is an embedded sub-manifold, and \( p \in S \). As a subspace of \( {T}_{p}M \), the tangent space \( {T}_{p}S \) is characterized by\n\n\[ \n{T}_{p}S = \left\{ {v \in {T}_{p}M : {vf} = 0\text{ whenever }f \in {C}^{\infty }\left( M\right) \text... | Proof. First suppose \( v \in {T}_{p}S \subseteq {T}_{p}M \). This means, more precisely, that \( v = d{\iota }_{p}\left( w\right) \) for some \( w \in {T}_{p}S \), where \( \iota : S \rightarrow M \) is inclusion. If \( f \) is any smooth real-valued function on \( M \) that vanishes on \( S \), then \( f \circ \iota ... | Yes |
Proposition 5.41. Suppose \( M \) is a smooth \( n \) -dimensional manifold with boundary, \( p \in \partial M \), and \( \left( {x}^{i}\right) \) are any smooth boundary coordinates defined on a neighborhood of \( p \) . The inward-pointing vectors in \( {T}_{p}M \) are precisely those with positive \( {x}^{n} \) -com... | ## - Exercise 5.42. Prove Proposition 5.41. | No |
Proposition 5.43. Every smooth manifold with boundary admits a boundary defining function. | Proof. Let \( \left\{ \left( {{U}_{\alpha },{\varphi }_{\alpha }}\right) \right\} \) be a collection of smooth charts whose domains cover \( M \) . For each \( \alpha \), define a smooth function \( {f}_{\alpha } : {U}_{\alpha } \rightarrow \lbrack 0,\infty ) \) as follows: if \( {U}_{\alpha } \) is an interior chart, ... | Yes |
Consider the subset \( S = \{ \left( {x, y}\right) : y = \left| x\right| \} \subseteq {\mathbb{R}}^{2} \). It is easy to check that \( S \smallsetminus \{ \left( {0,0}\right) \} \) is an embedded 1-dimensional submanifold of \( {\mathbb{R}}^{2} \), so if \( S \) itself is a smooth submanifold at all, it must be 1-dimen... | Writing \( \gamma \left( t\right) = \left( {x\left( t\right), y\left( t\right) }\right) \), we see that \( y\left( t\right) \) takes a global minimum at \( t = 0 \), so \( {y}^{\prime }\left( 0\right) = 0 \). On the other hand, because every point \( \left( {x, y}\right) \in S \) satisfies \( {x}^{2} = {y}^{2} \), we h... | Yes |
Proposition 5.46. Suppose \( M \) is a smooth manifold without boundary and \( D \subseteq M \) is a regular domain. The topological interior and boundary of \( D \) are equal to its manifold interior and boundary, respectively. | Proof. Suppose \( p \in D \) is arbitrary. If \( p \) is in the manifold boundary of \( D \), Theorem 4.15 shows that there exist a smooth boundary chart \( \left( {U,\varphi }\right) \) for \( D \) centered at \( p \) and a smooth chart \( \left( {V,\psi }\right) \) for \( M \) centered at \( p \) in which \( F \) has... | Yes |
Proposition 5.47. Suppose \( M \) is a smooth manifold and \( f \in {C}^{\infty }\left( M\right) \). (a) For each regular value \( b \) of \( f \), the sublevel set \( {f}^{-1}(\left( {-\infty, b\rbrack }\right) \) is a regular domain in \( M \). (b) If \( a \) and \( b \) are two regular values of \( f \) with \( a < ... | Proof. Problem 5-21. | No |
Theorem 5.48. If \( M \) is a smooth manifold and \( D \subseteq M \) is a regular domain, then there exists a defining function for \( D \) . If \( D \) is compact, then \( f \) can be taken to be a smooth exhaustion function for \( M \) . | Proof. Problem 5-22. | No |
Lemma 6.2. Suppose \( A \subseteq {\mathbb{R}}^{n} \) is a compact subset whose intersection with \( \{ c\} \times {\mathbb{R}}^{n - 1} \) has \( \left( {n - 1}\right) \) -dimensional measure zero for every \( c \in \mathbb{R} \) . Then \( A \) has \( n \) -dimensional measure zero. | Proof. Choose an interval \( \left\lbrack {a, b}\right\rbrack \subseteq \mathbb{R} \) such that \( A \subseteq \left\lbrack {a, b}\right\rbrack \times {\mathbb{R}}^{n - 1} \) . For each \( c \in \) \( \left\lbrack {a, b}\right\rbrack \), let \( {A}_{c} \subseteq {\mathbb{R}}^{n - 1} \) denote the compact subset \( \lef... | Yes |
Proposition 6.3. Suppose \( A \) is an open or closed subset of \( {\mathbb{R}}^{n - 1} \) or \( {\mathbb{H}}^{n - 1} \), and \( f : A \rightarrow \mathbb{R} \) is a continuous function. Then the graph of \( f \) has measure zero in \( {\mathbb{R}}^{n} \) . | Proof. First assume \( A \) is compact. We prove the theorem in this case by induction on \( n \) . When \( n = 1 \), it is trivial because the graph of \( f \) is at most a single point. To prove the inductive step, we appeal to Lemma 6.2. For each \( c \in \mathbb{R} \), the intersection of the graph of \( f \) with ... | Yes |
Corollary 6.4. Every proper affine subspace of \( {\mathbb{R}}^{n} \) has measure zero in \( {\mathbb{R}}^{n} \) . | Proof. Let \( S \subseteq {\mathbb{R}}^{n} \) be a proper affine subspace. Suppose first that \( \dim S = n - 1 \) . Then there is at least one coordinate axis, say the \( {x}^{i} \) -axis, that is not parallel to \( S \), and in that case \( S \) is the graph of an affine function of the form \( {x}^{i} = F\left( {{x}... | Yes |
Proposition 6.5. Suppose \( A \subseteq {\mathbb{R}}^{n} \) has measure zero and \( F : A \rightarrow {\mathbb{R}}^{n} \) is a smooth map. Then \( F\left( A\right) \) has measure zero. | Proof. By definition, for each \( p \in A, F \) has an extension to a smooth map, which we still denote by \( F \), on a neighborhood of \( p \) in \( {\mathbb{R}}^{n} \) . Shrinking this neighborhood if necessary, we may assume that there is an open ball \( U \) containing \( p \) such that \( F \) extends smoothly to... | Yes |
Lemma 6.6. Let \( M \) be a smooth \( n \) -manifold with or without boundary and \( A \subseteq M \) . Suppose that for some collection \( \left\{ \left( {{U}_{\alpha },{\varphi }_{\alpha }}\right) \right\} \) of smooth charts whose domains cover \( A,{\varphi }_{\alpha }\left( {A \cap {U}_{\alpha }}\right) \) has mea... | Proof. Let \( \left( {V,\psi }\right) \) be an arbitrary smooth chart. We need to show that \( \psi \left( {A \cap V}\right) \) has measure zero. Some countable collection of the \( {U}_{\alpha } \) ’s covers \( A \cap V \) . For each such \( {U}_{\alpha } \), we have\n\n\[ \psi \left( {A \cap V \cap {U}_{\alpha }}\rig... | Yes |
Proposition 6.8. Suppose \( M \) is a smooth manifold with or without boundary and \( A \subseteq M \) has measure zero in \( M \) . Then \( M \smallsetminus A \) is dense in \( M \) . | Proof. If \( M \smallsetminus A \) is not dense, then \( A \) contains a nonempty open subset of \( M \), which implies that there is a smooth chart \( \left( {V,\psi }\right) \) such that \( \psi \left( {A \cap V}\right) \) contains a nonempty open subset of \( {\mathbb{R}}^{n} \) (where \( n = \dim M \) ). Because \(... | Yes |
Theorem 6.9. Suppose \( M \) and \( N \) are smooth \( n \) -manifolds with or without boundary, \( F : M \rightarrow N \) is a smooth map, and \( A \subseteq M \) is a subset of measure zero. Then \( F\left( A\right) \) has measure zero in \( N \) . | Proof. Let \( \left\{ \left( {{U}_{i},{\varphi }_{i}}\right) \right\} \) be a countable cover of \( M \) by smooth charts. We need to show that for each smooth chart \( \left( {V,\psi }\right) \) for \( N \), the set \( \psi \left( {F\left( A\right) \cap V}\right) \) has measure zero in \( {\mathbb{R}}^{n} \) . Note th... | Yes |
Corollary 6.11. Suppose \( M \) and \( N \) are smooth manifolds with or without boundary, and \( F : M \rightarrow N \) is a smooth map. If \( \dim M < \dim N \), then \( F\left( M\right) \) has measure zero in \( N \) . | Proof. In this case, each point of \( M \) is a critical point for \( F \) . | No |
Corollary 6.12. Suppose \( M \) is a smooth manifold with or without boundary, and \( S \subseteq M \) is an immersed submanifold with or without boundary. If \( \dim S < \dim M \) , then \( S \) has measure zero in \( M \) . | Proof. Apply Corollary 6.11 to the inclusion map \( S \hookrightarrow M \) . | No |
Lemma 6.13. Suppose \( M \subseteq {\mathbb{R}}^{N} \) is a smooth \( n \) -dimensional submanifold with or without boundary. For any \( v \in {\mathbb{R}}^{N} \smallsetminus {\mathbb{R}}^{N - 1} \), let \( {\pi }_{v} : {\mathbb{R}}^{N} \rightarrow {\mathbb{R}}^{N - 1} \) be the projection with kernel \( \mathbb{R}v \)... | Proof. In order for \( {\left. {\pi }_{v}\right| }_{M} \) to be injective, it is necessary and sufficient that \( p - q \) never be parallel to \( v \) when \( p \) and \( q \) are distinct points in \( M \) . Similarly, in order for \( {\left. {\pi }_{v}\right| }_{M} \) to be a smooth immersion, it is necessary and su... | Yes |
Corollary 6.17. Suppose \( M \) is a compact smooth \( n \) -manifold with or without boundary. If \( N \geq {2n} + 1 \), then every smooth map from \( M \) to \( {\mathbb{R}}^{N} \) can be uniformly approximated by embeddings. | Proof. Assume \( N \geq {2n} + 1 \), and let \( f : M \rightarrow {\mathbb{R}}^{N} \) be a smooth map. By the Whitney embedding theorem, there is a smooth embedding \( F : M \rightarrow {\mathbb{R}}^{{2n} + 1} \) . The map \( G = f \times F : M \rightarrow {\mathbb{R}}^{N} \times {\mathbb{R}}^{{2n} + 1} \) is also a sm... | Yes |
Theorem 6.18 (Whitney Immersion Theorem). Every smooth n-manifold with or without boundary admits a smooth immersion into \( {\mathbb{R}}^{2n} \) . | Proof. See Problem 6-2 for the case \( \partial M = \varnothing \), and Problem 9-14 for the general case. | No |
Corollary 6.22. If \( M \) is a smooth manifold with or without boundary and \( \delta : M \rightarrow \) \( \mathbb{R} \) is a positive continuous function, there is a smooth function \( e : M \rightarrow \mathbb{R} \) such that \( 0 < e\left( x\right) < \delta \left( x\right) \) for all \( x \in M \) . | Proof. Use the Whitney approximation theorem to construct a smooth function \( e : M \rightarrow \mathbb{R} \) that satisfies \( \left| {e\left( x\right) - \frac{1}{2}\delta \left( x\right) }\right| < \frac{1}{2}\delta \left( x\right) \) for all \( x \in M \) . | Yes |
Theorem 6.23. If \( M \subseteq {\mathbb{R}}^{n} \) is an embedded \( m \) -dimensional submanifold, then \( {NM} \) is an embedded \( n \) -dimensional submanifold of \( T{\mathbb{R}}^{n} \approx {\mathbb{R}}^{n} \times {\mathbb{R}}^{n} \) . | Proof. Let \( {x}_{0} \) be any point of \( M \), and let \( \left( {U,\varphi }\right) \) be a slice chart for \( M \) in \( {\mathbb{R}}^{n} \) centered at \( {x}_{0} \) . Write \( \widehat{U} = \varphi \left( U\right) \subseteq {\mathbb{R}}^{n} \), and write the coordinate functions of \( \varphi \) as \( \left( {{u... | Yes |
Proposition 6.25. Let \( M \subseteq {\mathbb{R}}^{n} \) be an embedded submanifold. If \( U \) is any tubular neighborhood of \( M \), there exists a smooth map \( r : U \rightarrow M \) that is both a retraction and a smooth submersion. | Proof. Let \( {NM} \subseteq T{\mathbb{R}}^{n} \) be the normal bundle of \( M \), and let \( {M}_{0} \subseteq {NM} \) be the set \( {M}_{0} = \{ \left( {x,0}\right) : x \in M\} \) . By definition of a tubular neighborhood, there is an open subset \( V \subseteq {NM} \) containing \( {M}_{0} \) such that \( E : V \rig... | Yes |
Theorem 6.26 (Whitney Approximation Theorem). Suppose \( N \) is a smooth manifold with or without boundary, \( M \) is a smooth manifold (without boundary), and \( F : N \rightarrow M \) is a continuous map. Then \( F \) is homotopic to a smooth map. If \( F \) is already smooth on a closed subset \( A \subseteq N \),... | Proof. By the Whitney embedding theorem, we may as well assume that \( M \) is a properly embedded submanifold of \( {\mathbb{R}}^{n} \). Let \( U \) be a tubular neighborhood of \( M \) in \( {\mathbb{R}}^{n} \), and let \( r : U \rightarrow M \) be the smooth retraction given by Proposition 6.25. For any \( x \in M \... | Yes |
Corollary 6.27 (Extension Lemma for Smooth Maps). Suppose \( N \) is a smooth manifold with or without boundary, \( M \) is a smooth manifold, \( A \subseteq N \) is a closed subset, and \( f : A \rightarrow M \) is a smooth map. Then \( f \) has a smooth extension to \( N \) if and only if it has a continuous extensio... | Proof. If \( F : N \rightarrow M \) is a continuous extension of \( f \) to all of \( N \), the Whitney approximation theorem guarantees the existence of a smooth map \( \widetilde{F} \) (homotopic to \( F \), in fact, though we do not need that here) that agrees with \( f \) on \( A \) ; in other words, \( \widetilde{... | No |
Lemma 6.28. If \( N \) and \( M \) are smooth manifolds with or without boundary, smooth homotopy is an equivalence relation on the set of all smooth maps from \( N \) to \( M \) . | Proof. Reflexivity and symmetry are proved just as for ordinary homotopy. To prove transitivity, suppose \( F, G, K : N \rightarrow M \) are smooth maps, and \( {H}_{1},{H}_{2} : N \times I \rightarrow M \) are smooth homotopies from \( F \) to \( G \) and \( G \) to \( K \), respectively. Let \( \varphi : \left\lbrack... | Yes |
Theorem 6.29. Suppose \( N \) is a smooth manifold with or without boundary, \( M \) is a smooth manifold, and \( F, G : N \rightarrow M \) are smooth maps. If \( F \) and \( G \) are homotopic, then they are smoothly homotopic. If \( F \) and \( G \) are homotopic relative to some closed subset \( A \subseteq N \), th... | Proof. Suppose \( F, G : N \rightarrow M \) are smooth, and let \( H : N \times I \rightarrow M \) be a homotopy from \( F \) to \( G \) (relative to \( A \), which may be empty). We wish to show that \( H \) can be replaced by a smooth homotopy.\n\nDefine \( \bar{H} : N \times \mathbb{R} \rightarrow M \) by\n\n\[ \bar... | Yes |
Theorem 6.30. Suppose \( N \) and \( M \) are smooth manifolds and \( S \subseteq M \) is an embedded submanifold.\n\n(a) If \( F : N \rightarrow M \) is a smooth map that is transverse to \( S \), then \( {F}^{-1}\left( S\right) \) is an embedded submanifold of \( N \) whose codimension is equal to the codimension of ... | Proof. The second statement follows easily from the first, simply by taking \( F \) to be the inclusion map \( {S}^{\prime } \hookrightarrow M \), and noting that a composition of smooth embeddings \( S \cap {S}^{\prime } \hookrightarrow S \hookrightarrow M \) is again a smooth embedding.\n\nTo prove (a), let \( m \) d... | Yes |
Theorem 6.32 (Global Characterization of Graphs). Suppose \( M \) and \( N \) are smooth manifolds and \( S \subseteq M \times N \) is an immersed submanifold. Let \( {\pi }_{M} \) and \( {\pi }_{N} \) denote the projections from \( M \times N \) onto \( M \) and \( N \), respectively. The following are equivalent.\n\n... | Proof. Problem 6-15. | No |
Corollary 6.33 (Local Characterization of Graphs). Suppose \( M \) and \( N \) are smooth manifolds, \( S \subseteq M \times N \) is an immersed submanifold, and \( \left( {p, q}\right) \in S \) . If \( S \) intersects the submanifold \( \{ p\} \times N \) transversely at \( \left( {p, q}\right) \), then there exist a ... | Proof. The hypothesis guarantees that \( d{\left( {\pi }_{M}\right) }_{\left( p, q\right) } : {T}_{\left( p, q\right) }S \rightarrow {T}_{p}M \) is an isomorphism, so \( {\left. {\pi }_{M}\right| }_{S} \) restricts to a diffeomorphism from a neighborhood \( V \) of \( \left( {p, q}\right) \) in \( S \) to a neighborhoo... | Yes |
Proposition 6.34. If \( \left\{ {{F}_{s} : s \in S}\right\} \) is a smooth family of maps from \( N \) to \( M \) and \( S \) is connected, then for any \( {s}_{1},{s}_{2} \in S \), the maps \( {F}_{{s}_{1}},{F}_{{s}_{2}} : N \rightarrow M \) are homotopic. | Proof. Because \( S \) is connected, it is path-connected. If \( \gamma : \left\lbrack {0,1}\right\rbrack \rightarrow S \) is any path from \( {s}_{1} \) to \( {s}_{2} \), then \( H\left( {x, s}\right) = F\left( {x,\gamma \left( s\right) }\right) \) is a homotopy from \( {F}_{{s}_{1}} \) to \( {F}_{{s}_{2}} \) . | Yes |
Theorem 6.36 (Transversality Homotopy Theorem). Suppose \( M \) and \( N \) are smooth manifolds and \( X \subseteq M \) is an embedded submanifold. Every smooth map \( f : N \rightarrow M \) is homotopic to a smooth map \( g : N \rightarrow M \) that is transverse to \( X \) . | Proof. The crux of the proof is constructing a smooth map \( F : N \times S \rightarrow M \) that is transverse to \( X \), where \( S = {\mathbb{B}}^{k} \) for some \( k \) and \( {F}_{0} = f \) . It then follows from the parametric transversality theorem that there is some \( s \in S \) such that \( {F}_{s} : N \righ... | Yes |
Theorem 7.5. Every Lie group homomorphism has constant rank. | Proof. Let \( F : G \rightarrow H \) be a Lie group homomorphism, and let \( e \) and \( \widetilde{e} \) denote the identity elements of \( G \) and \( H \), respectively. Suppose \( {g}_{0} \) is an arbitrary element of \( G \) . We will show that \( d{F}_{{g}_{0}} \) has the same rank as \( d{F}_{e} \) . The fact th... | Yes |
Corollary 7.6. A Lie group homomorphism is a Lie group isomorphism if and only if it is bijective. | Proof. The global rank theorem shows that a bijective Lie group homomorphism is a diffeomorphism. | Yes |
Theorem 7.9 (Uniqueness of the Universal Covering Group). For any connected Lie group \( G \), the universal covering group is unique in the following sense: if \( \widetilde{G} \) and \( {\widetilde{G}}^{\prime } \) are simply connected Lie groups that admit smooth covering maps \( \pi : \widetilde{G} \rightarrow G \)... | Proof. See Problem 7-5. | No |
For each \( n \), the map \( {\varepsilon }^{n} : {\mathbb{R}}^{n} \rightarrow {\mathbb{T}}^{n} \) given by\n\n\[ \n{\varepsilon }^{n}\left( {{x}^{1},\ldots ,{x}^{n}}\right) = \left( {{e}^{{2\pi i}{x}^{1}},\ldots ,{e}^{{2\pi i}{x}^{n}}}\right) \n\]\n\nis a Lie group homomorphism and a smooth covering map (see Example 7... | Since \( {\mathbb{R}}^{n} \) is simply connected, this shows that the universal covering group of \( {\mathbb{T}}^{n} \) is the additive Lie group \( {\mathbb{R}}^{n} \). | No |
Proposition 7.11. Let \( G \) be a Lie group, and suppose \( H \subseteq G \) is a subgroup that is also an embedded submanifold. Then \( H \) is a Lie subgroup. | Proof. We need only check that multiplication \( H \times H \rightarrow H \) and inversion \( H \rightarrow H \) are smooth maps. Because multiplication is a smooth map from \( G \times G \) into \( G \), its restriction is clearly smooth from \( H \times H \) into \( G \) (this is true even if \( H \) is merely immers... | Yes |
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