Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
|---|---|---|
Theorem 20.6. Let \( h : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \) . Then \( h \) is an isometry if and only if it equals an orthogonal transformation followed by a translation, that is, if and only if h has the form\n\n\[ h\left( \mathrm{x}\right) = A \cdot \mathrm{x} + \mathbf{p}, \]\n\nwhere \( A \) is an ort... | Proof. Given \( h \), let \( \mathbf{p} = h\left( 0\right) \), and define \( k\left( \mathbf{x}\right) = h\left( \mathbf{x}\right) - \mathbf{p} \) . Then\n\n\[ \parallel k\left( \mathrm{x}\right) - k\left( \mathrm{y}\right) \parallel = \parallel h\left( \mathrm{x}\right) - h\left( \mathrm{y}\right) \parallel \text{,} \... | Yes |
Theorem 20.7. Let \( h : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \) be an isometry. If \( S \) is a rectifiable set in \( {\mathbb{R}}^{n} \), then the set \( T = h\left( S\right) \) is rectifiable, and \( v\left( T\right) = v\left( S\right) \) . | Proof. The map \( h \) is of the form \( h\left( \mathbf{x}\right) = A \cdot \mathbf{x} + \mathbf{p} \), where \( A \) is orthogonal. Then \( {Dh}\left( \mathbf{x}\right) = A \), and it follows from the change of variables theorem that\n\n\[ v\left( T\right) = \left| {\det A}\right| \cdot v\left( S\right) = v\left( S\r... | Yes |
Lemma 21.1. Let \( W \) be a linear subspace of \( {\mathbb{R}}^{n} \) of dimension \( k \) . Then there is an orthonormal basis for \( {\mathbb{R}}^{n} \) whose first \( k \) elements form a basis for W. | Proof. By Theorem 1.2, there is a basis \( {\mathbf{a}}_{1},\ldots ,{\mathbf{a}}_{n} \) for \( {\mathbb{R}}^{n} \) whose first \( k \) elements form a basis for \( W \) . There is a standard procedure for forming from these vectors an orthogonal set of vectors \( {\mathbf{b}}_{1},\ldots ,{\mathbf{b}}_{n} \) such that f... | Yes |
Theorem 21.2. Let \( W \) be a \( k \) -dimensional linear subspace of \( {\mathbb{R}}^{n} \) . There is an orthogonal transformation \( h : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \) that carries \( W \) onto the subspace \( {\mathbb{R}}^{k} \times 0 \) of \( {\mathbb{R}}^{n} \) . | Proof. Choose an orthonormal basis \( {\mathbf{b}}_{1},\ldots ,{\mathbf{b}}_{n} \) for \( {\mathbb{R}}^{n} \) such that the first \( k \) basis elements \( {\mathbf{b}}_{1},\ldots ,{\mathbf{b}}_{k} \) form a basis for \( W \) . Let \( g : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \) be the linear transformation \( ... | Yes |
Theorem 21.3. There is a unique function \( V \) that assigns, to each \( k \) -tuple \( \left( {{\mathbf{x}}_{1},\ldots ,{\mathbf{x}}_{k}}\right) \) of elements of \( {\mathbb{R}}^{n} \), a non-negative number such that:\n\n(1) If \( h : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \) is an orthogonal transformation,... | Proof. Given \( X = \left\lbrack {{\mathbf{x}}_{1}\ldots {\mathbf{x}}_{k}}\right\rbrack \), define\n\n\[ F\left( X\right) = \det \left( {{X}^{\mathrm{{tr}}} \cdot X}\right) .\n\nStep 1. If \( h : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \) is an orthogonal transformation, given by the equation \( h\left( \mathbf{x... | Yes |
Theorem 22.1. Let \( g : A \rightarrow B \) be a diffeomorphism of open sets in \( {\mathbb{R}}^{k} \) . Let \( \beta : B \rightarrow {\mathbb{R}}^{n} \) be a map of class \( {C}^{r} \) ; let \( Y = \beta \left( B\right) \) . Let \( \alpha = \beta \circ g \) ; then \( \alpha : A \rightarrow {\mathbb{R}}^{n} \) and \( Y... | Proof. We must show that\n\n\[{\int }_{B}\left( {f \circ \beta }\right) V\left( {D\beta }\right) = {\int }_{A}\left( {f \circ \alpha }\right) V\left( {D\alpha }\right) ,\]\n\nwhere one integral exists if the other does. See Figure 22.2.\n\nThe change of variables theorem tells us that\n\n\[{\int }_{B}\left( {f \cdot \b... | Yes |
Lemma 23.1. Let \( S \) be a subset of \( {\mathbb{R}}^{k} \) ; let \( f : S \rightarrow {\mathbb{R}}^{n} \) . If for each \( \mathbf{x} \in S \) , there is a neighborhood \( {U}_{\mathbf{x}} \) of \( \mathbf{x} \) and a function \( {g}_{\mathbf{x}} : {U}_{\mathbf{x}} \rightarrow {\mathbb{R}}^{n} \) of class \( {C}^{r}... | Proof. The lemma was given as an exercise in \( §{16} \) ; we provide a proof here. Cover \( S \) by the neighborhoods \( {U}_{\mathbf{x}} \) ; let \( A \) be the union of these neighborhoods; let \( \left\{ {\phi }_{i}\right\} \) be a partition of unity on \( A \) of class \( {C}^{r} \) dominated by the collection \( ... | Yes |
Lemma 23.2. Let \( U \) be open in \( {\mathbb{H}}^{k} \) but not in \( {\mathbb{R}}^{k} \); let \( \alpha : U \rightarrow {\mathbb{R}}^{n} \) be of class \( {C}^{r} \). Let \( \beta : {U}^{\prime } \rightarrow {\mathbb{R}}^{n} \) be a \( {C}^{r} \) extension of \( \alpha \) defined on an open set \( {U}^{\prime } \) o... | Proof. Note that to calculate the partial derivative \( \partial \beta /\partial {x}_{j} \) at \( \mathbf{x} \), we form the difference quotient\n\n\[ \left\lbrack {\beta \left( {\mathbf{x} + h{\mathbf{e}}_{j}}\right) - \beta \left( \mathbf{x}\right) }\right\rbrack /h \]\n\nand take the limit as \( h \) approaches 0. F... | Yes |
Lemma 23.3. Let \( M \) be a manifold in \( {\mathbb{R}}^{n} \), and let \( \alpha : U \rightarrow V \) be a coordinate patch on \( M \). If \( {U}_{0} \) is a subset of \( U \) that is open in \( U \), then the restriction of \( \alpha \) to \( {U}_{0} \) is also a coordinate patch on \( M \). | Proof. The fact that \( {U}_{0} \) is open in \( U \) and \( {\alpha }^{-1} \) is continuous implies that the set \( {V}_{0} = \alpha \left( {U}_{0}\right) \) is open in \( V \). Then \( {U}_{0} \) is open in \( {\mathbb{R}}^{k} \) or \( {\mathbb{H}}^{k} \) (according as \( U \) is open in \( {\mathbb{R}}^{k} \) or \( ... | Yes |
Lemma 24.2. Let \( M \) be a \( k \) -manifold in \( {\mathbb{R}}^{n} \) ; let \( \alpha : U \rightarrow V \) be a coordinate patch about the point \( \mathbf{p} \) of \( M \) .\n\n(a) If \( U \) is open in \( {\mathbb{R}}^{k} \), then \( \mathbf{p} \) is an interior point of \( M \) .\n\n(b) If \( U \) is open in \( {... | Proof. (a) is immediate from the definition. (b) is almost as easy. Given \( \alpha \) : \( U \rightarrow V \) as in (b), let \( {U}_{0} : U \cap {H}_{ + }^{k} \) . and let \( {V}_{0} = \alpha \left( {U}_{0}\right) \) . Then \( \alpha \mid {U}_{0} \) , mapping \( {U}_{0} \) onto \( {V}_{0} \), is a coordinate patch abo... | Yes |
Theorem 24.3. Let \( M \) be a \( k \) -manifold in \( {\mathbb{R}}^{n} \), of class \( {C}^{r} \). If \( \partial M \) is nonempty, then \( \partial M \) is a \( k - 1 \) manifold without boundary in \( {\mathbb{R}}^{n} \) of class \( {C}^{r} \). | Proof. Let \( \mathbf{p} \in \partial M \). Let \( \alpha : U \rightarrow V \) be a coordinate patch on \( M \) about \( \mathbf{p} \). Then \( U \) is open in \( {\mathbb{H}}^{k} \) and \( \mathbf{p} = \alpha \left( {\mathbf{x}}_{0}\right) \) for some \( {\mathbf{x}}_{0} \in \partial {\mathbb{H}}^{k} \). By the preced... | Yes |
Theorem 24.4. Let \( \mathbf{O} \) be open in \( {\mathbb{R}}^{n} \) ; let \( f : \mathbf{O} \rightarrow \mathbb{R} \) be of class \( {C}^{r} \) . Let \( M \) be the set of points \( \mathbf{x} \) for which \( f\left( \mathbf{x}\right) = 0 \) ; let \( N \) be the set of points for which \( f\left( \mathbf{x}\right) \ge... | Proof. Suppose first that \( \mathbf{p} \) is a point of \( N \) such that \( f\left( \mathbf{p}\right) > 0 \) . Let \( U \) be the open set in \( {\mathbb{R}}^{n} \) consisting of all points \( \mathbf{x} \) for which \( f\left( \mathbf{x}\right) > 0 \) ; let \( \alpha : U \rightarrow U \) be the identity map. Then \(... | Yes |
Corollary 24.5. The n-ball \( {B}^{n}\left( a\right) \) is an n-manifold in \( {\mathbb{R}}^{n} \) of class \( {C}^{\infty } \) , and \( {S}^{n} \) \( {}^{-1}\left( a\right) = \partial {B}^{n}\left( a\right) \) . | Proof. We apply the preceding theorem to the function \( f\left( \mathbf{x}\right) = {a}^{2} - \parallel \mathbf{x}{\parallel }^{2} \) . Then\n\n\[ \n{Df}\left( x\right) = \left\lbrack \begin{array}{lll} \left( {-2{x}_{1}}\right) & \cdots & \left( {-2{x}_{n}}\right) \end{array}\right\rbrack ,\n\]\n\nwhich is non-zero a... | Yes |
Lemma 25.1. If the support of \( f \) can be covered by a single coordinate patch, the integral \( {\int }_{M}{fdV} \) is well-defined, independent of the choice of coordinate patch. | Proof. We prove a preliminary result. Let \( \alpha : U \rightarrow V \) be a coordinate patch containing the support of \( f \) . Let \( W \) be an open set in \( U \) such that \( \alpha \left( W\right) \) also contains the support of \( f \) . Then\n\n\[ \n{\int }_{\operatorname{Int}W}\left( {f \circ \alpha }\right)... | Yes |
Lemma 25.2. Let \( M \) be a compact \( k \) -manifold in \( {\mathbb{R}}^{n} \), of class \( {C}^{r} \) . Given a covering of \( M \) by coordinate patches, there exists a finite collection of \( {C}^{\infty } \) functions \( {\phi }_{1},\ldots ,{\phi }_{\ell } \) mapping \( {\mathbb{R}}^{n} \) into \( \mathbb{R} \) s... | Proof. For each coordinate patch \( \alpha : U \rightarrow V \) belonging to the given collection, choose an open set \( {A}_{V} \) of \( {\mathbb{R}}^{n} \) such that \( {A}_{V} \cap M = V \) . Let \( A \) be the union of the sets \( {A}_{V} \) . Choose a partition of unity on \( A \) that is dominated by this open co... | Yes |
Theorem 26.1. The set of all \( k \) -tensors on \( V \) constitutes a vector space if we define\n\n\[ \left( {f + g}\right) \left( {{\mathbf{v}}_{1},\ldots ,{\mathbf{v}}_{k}}\right) = f\left( {{\mathbf{v}}_{1},\ldots {\mathbf{v}}_{k}}\right) + g\left( {{\mathbf{v}}_{1},\ldots ,{\mathbf{v}}_{k}}\right) ,\] \n\n\[ \left... | Proof. The proof is left as an exercise. The zero tensor is the function whose value is zero on every \( k \) -tuple of vectors. | No |
Lemma 26.2. Let \( {a}_{1},\ldots ,{a}_{n} \) be a basis for \( V \) . If \( f, g : {V}^{k} \rightarrow \mathbb{R} \) are \( k \) -tensors on \( V \), and if\n\n\[ f\left( {{\mathrm{a}}_{{i}_{1}},\ldots ,{\mathrm{a}}_{{i}_{k}}}\right) = g\left( {{\mathrm{a}}_{{i}_{1}},\ldots ,{\mathrm{a}}_{{i}_{k}}}\right) \]\n\nfor ev... | Proof. Given an arbitrary \( k \) -tuple \( \left( {{\mathbf{v}}_{1},\ldots ,{\mathbf{v}}_{k}}\right) \) of vectors of \( V \), let us express each \( {\mathbf{v}}_{i} \) in terms of the given basis, writing\n\n\[ {\mathbf{v}}_{i} = \mathop{\sum }\limits_{{j = 1}}^{n}{c}_{ij}{\mathrm{a}}_{j} \]\n\nThen we compute\n\n\[... | Yes |
Theorem 26.3. Let \( V \) be a vector space with basis \( {\mathbf{a}}_{1},\ldots ,{\mathbf{a}}_{n} \) . Let \( I = \left( {{i}_{1},\ldots }\right. \) , \( {i}_{k} \) ) be a \( k \) -tuple of integers from the set \( \{ 1,\ldots, n) \) . There is a unique \( k \) -tensor \( {\phi }_{I} \) on \( V \) such that, for ever... | Proof. Uniqueness follows from the preceding lemma. We prove existence as follows: First, consider the case \( k = 1 \) . We know that we can determine a linear transformation \( {\phi }_{i} : V \rightarrow \mathbb{R} \) by specifying its values arbitrarily on basis elements. So we can define \( {\phi }_{i} \) by the e... | Yes |
Theorem 26.4. Let \( f, g, h \) be tensors on \( V \) . Then the following properties hold:\n\n(1) (Associativity). \( f \otimes \left( {g \otimes h}\right) = \left( {f \otimes g}\right) \otimes h \) . | Proof. The proofs are straightforward. Associativity is proved, for instance, by noting that (if \( f, g, h \) have orders \( k,\ell, m \), respectively)\n\n\[ \left( {f \otimes \left( {g \otimes h}\right) }\right) \left( {{\mathbf{v}}_{1},\ldots ,{\mathbf{v}}_{k + 1 + m}}\right) \]\n\n\[ = f\left( {{\mathbf{v}}_{1},\l... | No |
Theorem 26.5. Let \( T : V \rightarrow W \) be a linear transformation; let\n\n\[ \n{T}^{ * } : {\mathrm{L}}^{k}\left( W\right) \rightarrow {\mathrm{L}}^{k}\left( V\right)\n\]\n\nbe the dual transformation. Then:\n\n(1) \( {T}^{ * } \) is linear.\n\n(2) \( {T}^{ * }\left( {f \otimes g}\right) = {T}^{ * }f \otimes {T}^{... | Proof. The proofs are straightforward. One verifies (1), for instance, as follows:\n\n\[ \n\left( {{T}^{ * }\left( {{af} + {bg}}\right) }\right) \left( {{v}_{1},\ldots ,{v}_{k}}\right) = \left( {{af} + {bg}}\right) \left( {T\left( {v}_{1}\right) ,\ldots, T\left( {v}_{k}\right) }\right)\n\]\n\n\[ \n= {af}\left( {T\left(... | No |
Lemma 27.1. If \( \sigma \in {S}_{k} \), then \( \sigma \) equals a composite of elementary permutations. | Proof. Given \( 0 \leq i \leq k \), we say that \( \sigma \) fixes the first \( i \) integers if \( \sigma \left( j\right) = j \) for 1 \( \leq j \leq i \) . If \( i = 0 \), then \( \sigma \) need not fix any integers at all. If \( i = k \), then \( \sigma \) fixes all the integers \( 1,\ldots, k \), so that \( \sigma ... | Yes |
Lemma 27.2. Let \( \sigma ,\tau \in {S}_{k} \) .\n\n(a) If \( \sigma \) equals a composite of \( m \) elementary permutations, then \( \operatorname{sgn}\sigma = \) \( {\left( -1\right) }^{m} \) . | Proof. Step 1. We show that for any \( \sigma \) ,\n\n\[ \operatorname{sgn}\left( {\sigma \circ {e}_{1}}\right) \text{-}\operatorname{sgn}\sigma \text{.} \]\n\nGiven \( \sigma \), let us write down the values of \( \sigma \) in order as follows:\n\n\[ \left( {\sigma \left( 1\right) ,\sigma \left( 2\right) ,\ldots ,\sig... | Yes |
Lemma 27.3. Let \( f \) be a \( k \) -tensor on \( V \) ; let \( \sigma ,\tau \in {S}_{k} \) .\n\n(a) The transformation \( f \rightarrow {f}^{\sigma } \) is a linear transformation of \( {\mathbf{L}}^{k}\left( V\right) \) to \( {\mathbf{L}}^{k}\left( V\right) \) . It has the property that for all \( \sigma ,\tau \) ,\... | Proof. (a) The linearity property is straightforward; it states simply that ( \( {af} \) \( + {bg}{)}^{\sigma } = a{f}^{\sigma } + b{g}^{\sigma } \) . To complete the proof of (a), we compute\n\n\[{\left( {f}^{\sigma }\right) }^{\tau }\left( {{\mathbf{v}}_{1},\ldots ,{\mathbf{v}}_{k}}\right) = {f}^{\sigma }\left( {{\ma... | Yes |
Lemma 27.4. Let \( {\mathbf{a}}_{1},\ldots ,{\mathbf{a}}_{n} \) be a basis for \( V \). If \( f, g \) are alternating \( k \)-tensors on \( V \), and if\n\n\[ f\left( {{\mathrm{a}}_{{i}_{1}},\ldots ,{\mathrm{a}}_{{i}_{k}}}\right) = g\left( {{\mathrm{a}}_{{i}_{1}},\ldots ,{\mathrm{a}}_{{i}_{k}}}\right) \]\n\nfor every a... | Proof. In view of Lemma 26.2, it suffices to prove that \( f \) and \( g \) have the same values on an arbitrary \( k \)-tuple \( \left( {{\mathbf{a}}_{{j}_{1}}\ldots ,{\mathbf{a}}_{{j}_{k}}}\right) \) of basis elements. Let \( J = \left( {j}_{1}\right. \), \( \ldots ,{j}_{k} \). \n\nIf two of the indices, say, \( {j}_... | Yes |
Theorem 27.5. Let \( V \) be a vector space with basis \( {\mathbf{a}}_{1},\ldots ,{\mathbf{a}}_{n} \) . Let \( I = \left( {{i}_{1},\ldots }\right. \) , \( \left. {i}_{k}\right) \) be an ascending \( k \) -tuple from the set \( \{ 1,\ldots, n\} \) . There is a unique alternating \( k \) -tensor \( {\psi }_{I} \) on \( ... | Proof. Uniqueness follows from the preceding lemma. To prove existence, we define \( {\psi }_{I} \) by the formula given in the theorem, and show that \( {\psi }_{I} \) satisfies the requirements of the theorem.\n\nFirst, we show \( {\psi }_{I} \) is alternating. If \( \tau \in {S}_{k} \), we compute\n\n\[ \n{\left( {\... | Yes |
Theorem 27.7. Let \( {\psi }_{I} \) be an elementary alternating tensor on \( {\mathbb{R}}^{n} \) corresponding to the usual basis for \( {\mathbb{R}}^{n} \), where \( I = \left( {{i}_{1}.,\ldots ,{i}_{k}}\right) \) . Given vectors \( {\mathbf{x}}_{1},\ldots ,{\mathbf{x}}_{k} \) of \( {\mathbb{R}}^{n} \), let \( X \) b... | Proof. We compute\n\n\[ \n{\psi }_{I}\left( {{\mathbf{x}}_{1},\ldots ,{\mathbf{x}}_{k}}\right) = \mathop{\sum }\limits_{\sigma }\left( {\operatorname{sgn}\sigma }\right) {\varphi }_{I}\left( {{\mathbf{x}}_{\sigma \left( 1\right) },\ldots ,{\mathbf{x}}_{\sigma \left( k\right) }}\right)\n\]\n\n\[ \n= \mathop{\sum }\limit... | Yes |
Theorem 28.1. Let \( V \) be a vector space. There is an operation that assigns, to each \( f \in {\mathrm{A}}^{k}\left( V\right) \) and each \( g \in {\mathrm{A}}^{\ell }\left( V\right) \), an element \( f \land g \in {\mathrm{A}}^{k + \ell }\left( V\right) \), such that the following properties hold:\n\n(1) (Associat... | Proof. Step 1, Let \( F \) be a \( k \) -tensor on \( V \) (not necessarily alternating). For purposes of this proof, it is convenient to define a transformation \( A : {\mathbf{L}}^{k}\left( V\right) \rightarrow \) \( {\mathrm{L}}^{k}\left( V\right) \) by the formula\n\n\[ {AF} = \mathop{\sum }\limits_{\sigma }\left( ... | Yes |
Lemma 29.1. Let \( A \) be open in \( {\mathbb{R}}^{k} \) or \( {\mathbb{H}}^{k} \) ; let \( \alpha : A \rightarrow {\mathbb{R}}^{m} \) be of class \( {C}^{r} \) . Let \( B \) be an open set of \( {\mathbb{R}}^{m} \) or \( {\mathbb{H}}^{m} \) containing \( \alpha \left( A\right) \) ; let \( \beta : B \rightarrow {\math... | Proof. This formula is just the chain rule. Let \( \mathbf{y} = \alpha \left( \mathbf{x}\right) \) and let \( \mathbf{z} = \beta \left( \mathbf{y}\right) \) . We compute\n\n\[{\left( \beta \circ \alpha \right) }_{ * }\left( {\mathbf{x},\mathbf{v}}\right) = \left( {\beta \left( {\alpha \left( \mathbf{x}\right) }\right) ... | Yes |
Lemma 29.2. Let \( \omega \) be a \( k \) -form on the open set \( A \) of \( {\mathbb{R}}^{n} \) . Then \( \omega \) is of class \( {C}^{r} \) if and only if its component functions \( {b}_{I} \) are of class \( {C}^{r} \) on \( A \) . | Proof. Given \( \omega \), let us express it in terms of elementary forms by the equation\n\n\[ \omega = \mathop{\sum }\limits_{\left\lbrack I\right\rbrack }{b}_{I}{\widetilde{\psi }}_{I} \]\n\nThe functions \( {\widetilde{\psi }}_{I} \) are of class \( {C}^{\infty } \) . Therefore, if the functions \( {b}_{I} \) are o... | Yes |
Lemma 29.3. Let \( \omega \) and \( \eta \) be \( k \) -forms, and let \( \theta \) be an \( \ell \) -form, on the open set \( A \) of \( {\mathbb{R}}^{n} \) . If \( \omega \) and \( \eta \) and \( \theta \) are of class \( {C}^{r} \), so are \( {a\omega } + {b\eta } \) and \( \omega \land \theta \) . | Proof. It is immediate that \( {a\omega } + {b\eta } \) is of class \( {\mathrm{C}}^{\mathrm{r}} \), since it is a linear combination of \( {C}^{r} \) functions. To show that \( \omega \land \theta \) is of class \( {C}^{r} \), one could use the formula for the wedge product given in the proof of Theorem 28.1. Alternat... | Yes |
Theorem 30.1. The operator \( d \) is linear on \( 0 \) -forms. | Proof. Let \( f, g : A \rightarrow \mathbb{R} \) be of class \( {C}^{r} \) . Let \( h = {af} + {bg} \) . Then\n\n\[ \n{Dh}\left( \mathbf{x}\right) = {aDf}\left( \mathbf{x}\right) + {bDg}\left( \mathbf{x}\right) , \n\]\n\nso that\n\n\[ \n{dh}\left( \mathbf{x}\right) \left( {\mathbf{x};\mathbf{v}}\right) = {adf}\left( \m... | Yes |
Lemma 30.2. Let \( {\widetilde{\varphi }}_{1},\ldots ,{\widetilde{\varphi }}_{n} \) be the elementary 1 -forms in \( {\mathbb{R}}^{n} \) . Let \( {\pi }_{i} : {\mathbb{R}}^{n} \rightarrow \) R be the \( {i}^{\text{th }} \) projection function, defined by the equation\n\n\[ \n{\pi }_{i}\left( {{x}_{1},\ldots ,{x}_{n}}\r... | Proof. Since \( {\pi }_{i} \) is a \( {C}^{\infty } \) function, \( d{\pi }_{i} \) is a 1 -form of class \( {C}^{\infty } \) . We compute\n\n\[ \nd{\pi }_{i}\left( \mathbf{x}\right) \left( {\mathbf{x};\mathbf{v}}\right) = D{\pi }_{i}\left( \mathbf{x}\right) \cdot \mathbf{v}\n\]\n\n\[ \n= \left\lbrack \begin{array}{lll}... | Yes |
Theorem 30.3. Let \( A \) be open in \( {\mathbb{R}}^{n} \) ; let \( f : A \rightarrow \mathbb{R} \) be of class \( {C}^{r} \) . Then\n\n\[ \n{df} = \left( {{D}_{1}f}\right) d{x}_{1} + \cdots + \left( {{D}_{n}f}\right) d{x}_{n}.\n\]\n\nIn particular, \( {df} = 0 \) if \( f \) is a constant function.\n\nIn Leibnitz's no... | Proof. We evaluate both sides of the equation on the tangent vector \( \left( {\mathbf{x};\mathbf{v}}\right) \) . We have\n\n\[ \n{df}\left( \mathbf{x}\right) \left( {\mathbf{x};\mathbf{v}}\right) = {Df}\left( \mathbf{x}\right) \cdot \mathbf{v}\n\]\n\nby definition, whereas\n\n\[ \n\mathop{\sum }\limits_{{i = 1}}^{n}{D... | Yes |
Theorem 30.4. Let \( A \) be an open set in \( {\mathbb{R}}^{n} \) . There exists a unique linear transformation\n\n\[ d : {\Omega }^{k}\left( A\right) \rightarrow {\Omega }^{k + 1}\left( A\right) ,\]\n\ndefined for \( k \geq 0 \), such that:\n\n(1) If \( f \) is a 0 -form, then \( {df} \) is the 1 -form\n\n\[ {df}\lef... | Proof. Step 1. We verify uniqueness. First, we show that conditions (2) and\n\n(3) imply that for any forms \( {\omega }_{1},\ldots ,{\omega }_{k} \), we have\n\n\[ d\left( {d{\omega }_{1} \land \cdots \land d{\omega }_{k}}\right) = 0. \]\n\nIf \( k = 1 \), this equation is a consequence of (3). Supposing it true for \... | Yes |
Theorem 31.1. Let \( A \) be an open set in \( {\mathbb{R}}^{n} \). There exist vector space isomorphisms \( {\alpha }_{i} \) and \( {\beta }_{j} \) as in the following diagram:\n\n\[ \text{Scalar fields in}A\overset{{\alpha }_{0}}{ \rightarrow }\;{\Omega }^{0}\left( A\right) \]\n\n\[ \text{Vector fields in}A\overset{{... | Proof. Let \( f \) and \( h \) be scalar fields in \( A \) ; let\n\n\[ F\left( x\right) = \left( {x;\sum {f}_{i}\left( x\right) {\mathbf{e}}_{i}}\right) \;\text{ and }\;G\left( x\right) = \left( {x;\sum {g}_{i}\left( x\right) {\mathbf{e}}_{i}}\right) \]\n\nbe vector fields in \( A \), We define the transformations \( {... | No |
Theorem 31.2. Let \( A \) be an open set in \( {\mathbb{R}}^{3} \) . There exist vector space isomorphisms \( {\alpha }_{i} \) and \( {\beta }_{j} \) as in the following diagram:\n\n\n\nsuch that\n\n\[ d \cdot {\alph... | Proof. The maps \( {\alpha }_{i} \) and \( {\beta }_{j} \) are those defined in the proof of the preceding theorem. Only the second equation needs checking; we leave it to you. | No |
Theorem 32.1. Let \( A \) be open in \( {\mathbb{R}}^{k} \) ; let \( \alpha : A \rightarrow {\mathbb{R}}^{m} \) be a \( {C}^{\infty } \) map. Let \( B \) be open in \( {\mathbb{R}}^{m} \) and contain \( \alpha \left( A\right) \) ; let \( \beta : B \rightarrow {\mathbb{R}}^{n} \) be a \( {C}^{\infty } \) map. Let \( \om... | Proof. See Figure 32.1. In the case of forms of positive order, properties (1) and (3) are merely restatements, in the language of forms, of Theorem 26.5, and (2) is a restatement of (6) of Theorem 28.1.\n\nChecking the properties when some or all of the forms have order zero is a computation we leave to you. | No |
Theorem 32.2. Let \( A \) be open in \( {\mathbb{R}}^{k} \) ; let \( \alpha : A \rightarrow {\mathbb{R}}^{n} \) be a \( {C}^{\infty } \) map. Let \( \mathbf{x} \) denote the general point of \( {\mathbb{R}}^{k} \) ; let \( \mathbf{y} \) denote the general point of \( {\mathbb{R}}^{n} \) . Then \( d{x}_{i} \) and \( d{y... | Proof. (a) Set \( \mathbf{y} = \alpha \left( \mathbf{x}\right) \) . We compute the value of \( {\alpha }^{ * }\left( {d{y}_{i}}\right) \) on a typical\ntangent vector as follows:\n\n\[ \left( {{\alpha }^{ * }\left( {d{y}_{i}}\right) }\right) \left( \mathbf{x}\right) \left( {\mathbf{x};\mathbf{v}}\right) = d{y}_{i}\left... | Yes |
Theorem 33.1. Let \( g : A \rightarrow B \) be a diffeomorphism of open sets in \( {\mathbb{R}}^{k} \) . Assume \( \det {Dg} \) does not change sign on \( A \) . Let \( \beta : B \rightarrow {\mathbb{R}}^{n} \) be a map of class \( {C}^{\infty } \) ; let \( Y = \beta \left( B\right) \) . Let \( \alpha = \beta \circ g \... | Proof. Let \( \mathbf{x} \) denote the general point of \( A \) ; let \( \mathbf{y} \) denote the general point of B. See Figure 33.1. We wish to show that\n\n\[{\int }_{A}{\alpha }^{ * }\omega = \in {\int }_{B}{\beta }^{ * }\omega\]\n\nwhere \( \epsilon = \pm 1 \) and agrees with the sign of det \( {Dg} \) . If we set... | Yes |
Theorem 33.2. Let \( A \) be open in \( {\mathbb{R}}^{k} \) ; let \( \alpha : A \rightarrow {\mathbb{R}}^{n} \) be of class \( {C}^{\infty } \) ; let \( Y = \) \( \alpha \left( A\right) \) . Let \( \mathbf{x} \) denote the general point of \( A \) ; and let \( \mathbf{z} \) denote the general point of \( {\mathbb{R}}^{... | Proof. Applying Theorem 32.2, we have\n\n\[ {\alpha }^{ * }\omega = \left( {f \circ \alpha }\right) \det \left( {\partial {\alpha }_{I}/\partial \mathrm{x}}\right) d{x}_{1} \land \cdots \land d{x}_{k}. \]\n\nThe theorem follows. | Yes |
Theorem 34.1. Let \( k > 1 \) . If \( M \) is an orientable \( k \) -manifold with non-empty boundary, then \( \partial M \) is orientable. | Proof. Let \( \mathbf{p} \in \partial M \) ; let \( \alpha : U \rightarrow V \) be a coordinate patch about \( \mathbf{p} \) . There is a corresponding coordinate patch \( {\alpha }_{0} \) on \( \partial M \) that is said to be obtained by restricting \( \alpha \) . (See §24.) Formally, if we define \( b : {\mathbb{R}}... | Yes |
Let \( M \) be a compact oriented \( k \) -manifold in \( {\mathbb{R}}^{n} \) ; let \( \omega \) be a \( k \) -form defined in an open set of \( {\mathbb{R}}^{n} \) containing \( M \) . Let \( \lambda \) be the scalar function on \( M \) defined by the equation\n\n\[ \lambda \left( \mathbf{p}\right) = \omega \left( \ma... | Proof. By linearity, it suffices to consider the case where the support of \( \omega \) is covered by a single coordinate patch \( \alpha : U \rightarrow V \) belonging to the orientation of \( M \) . We have\n\n\[ {\alpha }^{ * }\omega = {hd}{x}_{1} \land \cdots \land d{x}_{k} \]\n\nfor some scalar function \( h \) . ... | Yes |
Theorem 38.2 (The gradient theorem). Let \( M \) be a compact 1-manifold in \( {\mathbb{R}}^{n} \) ; let \( T \) be a unit tangent vector field to \( M \) . Let \( f \) be a \( {C}^{\infty } \) function defined in an open set about \( M \) . If \( \partial M \) is empty, then\n\n\[ \n{\int }_{M}\langle \operatorname{gr... | Proof. The 1-form \( {df} \) corresponds to the vector field grad \( f \), by Theorem 31.1. Therefore\n\n\[ \n{\int }_{M}{df} = {\int }_{M}\langle \operatorname{grad}f, T\rangle \mathrm{d}s\n\]\n\nby the preceding lemma. Our theorem then follows from the 1-dimensional version of Stokes' theorem. | Yes |
Lemma 38.3. Given independent vectors \( {\mathbf{x}}_{1},\ldots ,{\mathbf{x}}_{n - 1} \) in \( {\mathbb{R}}^{n} \), let \( X \) be the \( n \) by \( n - 1 \) matrix \( X = \left\lbrack {{\mathbf{x}}_{1}\ldots {\mathbf{x}}_{n - 1}}\right\rbrack \), and let \( \mathrm{c} \) be the vector \( \mathrm{c} = \sum {c}_{i}{e}_... | Proof. We begin with a preliminary calculation. Let \( {\mathbf{x}}_{1},\ldots ,{\mathbf{x}}_{n - 1} \) be fixed. Given \( \mathbf{a} \in {\mathbb{R}}^{n} \), we compute the following determinant; expanding by cofactors of the first column, we have:\n\n\[ \n\det \left\lbrack {\mathbf{a}{\mathbf{x}}_{1}\ldots {\mathbf{x... | Yes |
Corollary 38.4. If \( M \) is an oriented \( n - 1 \) manifold in \( {\mathbb{R}}^{n} \), then the unit normal vector \( N\left( \mathbf{p}\right) \) corresponding to the orientation of \( M \) is a \( {C}^{\infty } \) function of p. | Proof. If \( \alpha : U \rightarrow V \) is a coordinate patch on \( M \) about \( \mathbf{p} \), let\n\n\[ \n{c}_{i}\left( \mathbf{x}\right) = {\left( -1\right) }^{i - 1}\det {D\alpha }\left( {1,\ldots ,{\iota }^{ \land },\ldots, n}\right) \left( \mathbf{x}\right) \n\]\n\nfor \( \mathbf{x} \in U \), and let \( c\left(... | Yes |
Lemma 38.5. Let \( M \) be a compact oriented \( n - 1 \) manifold in \( {\mathbb{R}}^{n} \) ; let \( N \) be the corresponding unit normal vector field. Let \( G \) be a vector field defined in an open set \( U \) of \( {\mathbb{R}}^{n} \) containing \( M \) . If we denote the general point of \( {\mathbb{R}}^{n} \) b... | Proof. We give two proofs of this theorem. The first relies on the results of §36 and the second does not.\n\nFirst proof. By Theorem 36.2, we have\n\n\[ {\int }_{M}\omega = {\int }_{M}{\lambda dV} \]\nwhere \( \lambda \left( \mathbf{p}\right) \) is the value of \( \omega \left( \mathbf{p}\right) \) on an orthonormal b... | Yes |
Lemma 38.6. Let \( M \) be a compact \( n \) -manifold in \( {\mathbb{R}}^{n} \), oriented naturally. Let \( \omega = {hd}{x}_{1} \land \ldots \land d{x}_{n} \) be an \( n \) -form defined in an open set of \( {\mathbb{R}}^{n} \) containing M. Then \( h \) is the corresponding scalar field, and\n\n\[ \n{\int }_{M}\omeg... | Proof. First proof. We use the results of \( §{36} \) . We have\n\n\[ \n{\int }_{M}\omega = {\int }_{M}\lambda \mathrm{d}V \n\]\n\nwhere \( \lambda \) is obtained by evaluating \( \omega \) on an orthonormal basis for \( {\mathbf{T}}_{\mathbf{p}}\left( M\right) \) that belongs to its natural orientation. Now \( \alpha ... | Yes |
Lemma 38.7. Let \( M \) be an n-manifold in \( {\mathbb{R}}^{n} \). If \( M \) is oriented naturally, then the induced orientation of \( \partial M \) corresponds to the unit normal field \( N \) to \( \partial M \) that points outwards from \( M \) at each point of \( \partial M \). | Proof. Let \( \alpha : U \rightarrow V \) be a coordinate patch on \( M \) about \( \mathbf{p} \) belonging to the orientation of \( M \). Then det \( {D\alpha } > 0 \). Let \( b : {\mathbb{R}}^{n - 1} \rightarrow {\mathbb{R}}^{n} \) be the map\n\n\[ b\left( {{x}_{1},\ldots ,{x}_{n - 1}}\right) = \left( {{x}_{1},\ldots... | Yes |
Theorem 38.8 (The divergence theorem). Let \( M \) be a compact \( n \) - manifold in \( {\mathbb{R}}^{n} \) . Let \( N \) be the unit normal vector field to \( \partial M \) that points outwards from \( M \) . If \( G \) is a vector field defined in an open set of \( {\mathbb{R}}^{n} \) containing \( M \), then\n\n\[{... | Proof. Given \( G \), let \( \omega = {\beta }_{n - 1}G \) be the corresponding \( n - 1 \) form. Orient \( M \)\nnaturally and give \( \partial M \) the induced orientation. Then the normal field \( N \) corresponds to the orientation of \( \partial M \), by Lemma 38.7, so that\n\n\[{\int }_{\partial M}\omega = {\int ... | Yes |
Theorem 38.9 (Stokes’ theorem—classical version). Let \( M \) be a compact orientable 2-manifold in \( {\mathbb{R}}^{3} \) . Let \( N \) be a unit normal field to \( M \) . Let \( F \) be a \( {C}^{\infty } \) vector field defined in an open set about \( M \) . If \( \partial M \) is empty, then\n\n\[ \n{\int }_{M}\lan... | Proof. Given \( F \), let \( \omega = {\alpha }_{1}F \) be the corresponding 1 -form. Then according to Theorem 31.2, the vector field curl \( F \) corresponds to the 2-form \( {d\omega } \) . Orient \( M \) so that \( N \) is the corresponding unit normal field. Then by Lemma 38.5,\n\n\[ \n{\int }_{M}{d\omega } = {\in... | Yes |
Theorem 39.1 (Leibnitz’s rule). Let \( Q \) be a rectangle in \( {\mathbb{R}}^{n} \) ; let \( f : Q \times \lbrack a \) , \( b\rbrack \rightarrow \mathbb{R} \) be a continuous function. Denote \( f \) by \( f\left( {\mathbf{x}, t}\right) \) for \( \mathbf{x} \in Q \) and \( t \in \lbrack a \) , b], Then the function\n\... | Proof. Step 1. We show that \( F \) is continuous. The rectangle \( Q \times \left\lbrack {a, b}\right\rbrack \) is compact; therefore \( f \) is uniformly continuous on \( Q \times \left\lbrack {a, b}\right\rbrack \) . That is, given \( \in \) \( > 0 \), there is \( \delta > 0 \) such that\n\n\[ \left| {f\left( {{x}_{... | Yes |
Theorem 39.2. Let \( A \) and \( B \) be open sets in \( {\mathbb{R}}^{n} \) and \( {\mathbb{R}}^{m} \), respectively. Let \( g, h : A \rightarrow B \) be \( {C}^{\infty } \) maps that are differentiably homotopic. Then there is a linear transformation\n\n\[ \text{ P } : {\Omega }^{k + 1}\left( B\right) \rightarrow {\O... | Proof. Step 1. We consider first a very special case. Given an open set \( A \) in \( {\mathbb{R}}^{n} \), let \( U \) be a neighborhood of \( A \times I \) in \( {\mathbb{R}}^{n + 1} \) and let \( \alpha ,\beta : A \rightarrow U \) be the maps given by the equations\n\n\[ \alpha \left( \mathbf{x}\right) = \left( {\mat... | Yes |
Theorem 39.3 (The Poincaré lemma). Let A be a star-convex open set in \( {\mathbb{R}}^{n} \). If \( \omega \) is a closed \( k \)-form on \( A \), then \( \omega \) is exact on \( A \). | Proof. We apply the preceding theorem. Let \( \mathbf{p} \) be a point with respect to which \( A \) is star-convex. Let \( h : A \rightarrow A \) be the identity map and let \( g : A \rightarrow A \) be the constant map carrying each point to the point \( \mathbf{p} \). Then \( g \) and \( h \) are differentiably homo... | Yes |
Theorem 39.4. Let \( A \) be a star-convex open set in \( {\mathbb{R}}^{n} \) . Let \( \omega \) be a closed \( k \) - form on \( A \) . If \( k > 1 \), and if \( \eta \) and \( {\eta }_{0} \) are two \( k - 1 \) forms on \( A \) with \( {d\eta } = \omega = \) \( d{\eta }_{0} \), then\n\n\[ \eta = {\eta }_{0} + {d\thet... | Proof. Since \( d\left( {\eta - {\eta }_{0}}\right) = 0 \), the form \( \eta - {\eta }_{0} \) is a closed form on \( A \) . By the Poincaré lemma, it is exact. A similar comment applies to the form \( f - {f}_{0} \) . | Yes |
Theorem 40.1 (Homotopy equivalence theorem). Let \( A \) and \( B \) be open sets in \( {\mathbb{R}}^{n} \) and \( {\mathbb{R}}^{m} \), respectively. Let \( g : A \rightarrow B \) and \( h : B \rightarrow A \) be \( {C}^{\infty } \) maps. If \( g \circ h : B \rightarrow B \) is differentiably homotopic to the identity ... | Proof. If \( \eta \) is a closed \( k \) -form on \( A \), for \( k \geq 0 \), then Theorem 39.2 implies that\n\n\[ \n{\left( h \circ g\right) }^{ * }\eta - {\left( {i}_{A}\right) }^{ * }\eta \n\]\n\n is exact. Then the induced maps of the deRham groups satisfy the equation\n\n\[ \n{g}^{ * }\left( {{h}^{ * }\left( {\{ ... | Yes |
Lemma 40.2. Let \( U \) and \( V \) be open sets in \( {\mathbb{R}}^{n} \) ; let \( X = U \cup V \) ; and suppose \( A = U \cap V \) is non-empty. Then there exists a \( {C}^{\infty } \) function \( \phi : X \rightarrow \left\lbrack {0,1}\right\rbrack \) such that \( \phi \) is identically 0 in a neighborhood of \( U -... | Proof. See Figure 40.1. Let \( \left\{ {\phi }_{i}\right\} \) be a partition of unity on \( X \) dominated by the open covering \( \{ U, V\} \) . Let \( {S}_{i} = \) Support \( {\phi }_{i} \) for each \( i \) . Divide the index set of the collection \( \left\{ {\phi }_{i}\right\} \) into two disjoint subsets \( J \) an... | Yes |
Theorem 40.4. Let \( n \geq 1 \) . Then\n\n\[ \dim {H}^{k}\left( {{\mathbb{R}}^{n} - 0}\right) = \left\{ \begin{array}{ll} 0 & \text{ for }k \neq n - 1, \\ 1 & \text{ for }k = n - 1. \end{array}\right. \] | Proof. Step 1. We prove the theorem for \( n = 1 \) . Let \( A = {\mathbb{R}}^{1} - 0 \) ; write \( A = {A}_{0} \) \( \cup {A}_{1} \), where \( {A}_{0} \) consists of the negative reals and \( {A}_{1} \) consists of the positive reals. If \( \omega \) is a closed \( k \) -form in \( A \), with \( k > 0 \), then \( \ome... | Yes |
Theorem 41.1. Let \( M \) be a compact differentiable manifold. Given a covering of \( M \) by coordinate patches, there exist functions \( {\phi }_{i} : M \rightarrow \mathbb{R} \) of class \( {C}^{\infty } \) for \( i = 1,\ldots ,\ell \), such that:\n\n(1) \( {\phi }_{i}\left( p\right) \geq 0 \) for each \( p \in M \... | Proof. Given \( p \in M \), choose a coordinate patch \( \alpha : U \rightarrow V \) about \( p \) . Let \( \alpha \) (x) \( = p \) ; choose a non-negative \( {C}^{\infty } \) function \( f : U \rightarrow \mathbb{R} \) whose support is compact and is contained in \( U \), such that \( f \) is positive at the point \( ... | Yes |
Theorem 41.2 (Stokes’ theorem). Let \( M \) be a compact, oriented differentiable k-manifold. Let \( \omega \) be a \( k - 1 \) form on M. If \( \partial M \) is non-empty, give \( \partial M \) the induced orientation; then\n\n\[{\int }_{M}{d\omega } = {\int }_{\partial M}\omega\]\n\nIf \( \partial M \) is empty, then... | Proof. The proof given earlier goes through verbatim. Since all the computations were carried out by working within coordinate patches, no changes are necessary. The special conventions involved when \( k = 1 \) and \( \partial M \) is a 0-manifold are handled exactly as before. | Yes |
Lemma 2.1 (Polarization Identity). Suppose \( \langle \cdot , \cdot \rangle \) is an inner product on a vector space \( V \) . Then for all \( v, w \in V \) , | - Exercise 2.2. Prove the preceding lemma. | No |
Proposition 2.3 (Gram-Schmidt Algorithm). Let \( V \) be an \( n \) -dimensional inner product space, and suppose \( \left( {{v}_{1},\ldots ,{v}_{n}}\right) \) is any ordered basis for \( V \) . Then there is an orthonormal ordered basis \( \left( {{b}_{1},\ldots ,{b}_{n}}\right) \) satisfying the following conditions:... | Proof. The basis vectors \( {b}_{1},\ldots ,{b}_{n} \) are defined recursively by\n\n\[ {b}_{1} = \frac{{v}_{1}}{\left| {v}_{1}\right| } \]\n\n\[ {b}_{j} = \frac{{v}_{j} - \mathop{\sum }\limits_{{i = 1}}^{{j - 1}}\left\langle {{v}_{j},{b}_{i}}\right\rangle {b}_{i}}{\left| {v}_{j} - \mathop{\sum }\limits_{{i = 1}}^{{j -... | Yes |
Proposition 2.4. Every smooth manifold admits a Riemannian metric. | Exercise 2.5. Use a partition of unity to prove the preceding proposition. | No |
Example 2.6 (The Euclidean Metric). The Euclidean metric is the Riemannian metric \( \bar{g} \) on \( {\mathbb{R}}^{n} \) whose value at each \( x \in {\mathbb{R}}^{n} \) is just the usual dot product on \( {T}_{x}{\mathbb{R}}^{n} \) under the natural identification \( {T}_{x}{\mathbb{R}}^{n} \cong {\mathbb{R}}^{n} \) ... | \[ \langle v, w{\rangle }_{\bar{g}} = \mathop{\sum }\limits_{{i = 1}}^{n}{v}^{i}{w}^{i} \] | Yes |
Proposition 2.8 (Existence of Orthonormal Frames). Let \( \\left( {M, g}\\right) \) be a Riemannian \( n \) -manifold with or without boundary. If \( \\left( {X}_{j}\\right) \) is any smooth local frame for TM over an open subset \( U \\subseteq M \), then there is a smooth orthonormal frame \( \\left( {E}_{j}\\right) ... | each \( p \\in U \), we obtain an ordered \( n \) -tuple of rough orthonormal vector fields \( \\left( {{E}_{1},\\ldots ,{E}_{n}}\\right) \) over \( U \) satisfying the span conditions. Because the vectors whose norms appear in the denominators of (2.5)-(2.6) are nowhere vanishing, those formulas show that each vector ... | Yes |
Proposition 2.16 (The Normal Bundle). If \( \widetilde{M} \) is a Riemannian m-manifold and \( M \subseteq \widetilde{M} \) is an immersed or embedded \( n \) -dimensional submanifold with or without boundary, then \( {NM} \) is a smooth rank- \( \left( {m - n}\right) \) vector subbundle of the ambient tangent bundle \... | Proof. Given any point \( p \in M \), Theorem A. 16 shows that there is a neighborhood \( U \) of \( p \) in \( M \) that is embedded in \( \widetilde{M} \), and then Proposition 2.14 shows that there is a smooth orthonormal frame \( \left( {{E}_{1},\ldots ,{E}_{m}}\right) \) that is adapted to \( U \) on some neighbor... | Yes |
If \( U \subseteq {\mathbb{R}}^{n} \) is an open set and \( f : U \rightarrow \mathbb{R} \) is a smooth function, then the graph of \( f \) is the subset \( \Gamma \left( f\right) = \) \( \{ \left( {x, f\left( x\right) }\right) : x \in U\} \subseteq {\mathbb{R}}^{n + 1} \), which is an embedded submanifold of dimension... | \[ {X}^{ * }\bar{g} = {X}^{ * }\left( {{\left( d{x}^{1}\right) }^{2} + \cdots + {\left( d{x}^{n + 1}\right) }^{2}}\right) = {\left( d{u}^{1}\right) }^{2} + \cdots + {\left( d{u}^{n}\right) }^{2} + d{f}^{2}. \] | Yes |
Proposition 2.25 (Properties of Horizontal Vector Fields). Let \( \widetilde{M} \) and \( M \) be smooth manifolds, let \( \pi : \widetilde{M} \rightarrow M \) be a smooth submersion, and let \( \widetilde{g} \) be a Riemannian metric on \( \widetilde{M} \) .\n\n(a) Every smooth vector field \( W \) on \( \widetilde{M}... | Proof. Let \( p \in \widetilde{M} \) be arbitrary. Because \( \pi \) is a smooth submersion, the rank theorem (Theorem A.15) shows that there exist smooth coordinate charts \( \left( {\widetilde{U},\left( {x}^{i}\right) }\right) \) centered at \( p \) and \( \left( {U,\left( {u}^{j}\right) }\right) \) centered at \( \p... | Yes |
Theorem 2.28. Let \( \left( {\widetilde{M},\widetilde{g}}\right) \) be a Riemannian manifold, let \( \pi : \widetilde{M} \rightarrow M \) be a surjective smooth submersion, and let \( G \) be a group acting on \( \widetilde{M} \) . If the action is isometric, vertical, and transitive on fibers, then there is a unique R... | ## Proof. Problem 2-6. | No |
Corollary 2.29. Suppose \( \left( {\widetilde{M},\widetilde{g}}\right) \) is a Riemannian manifold, and \( G \) is a Lie group acting smoothly, freely, properly, and isometrically on \( \widetilde{M} \) . Then the orbit space \( M = \) \( \widetilde{M}/G \) has a unique smooth manifold structure and Riemannian metric s... | Proof. Under the given hypotheses, the quotient manifold theorem (Thm. C.17) shows that \( M \) has a unique smooth manifold structure such that the quotient map \( \pi : \widetilde{M} \rightarrow M \) is a smooth submersion. It follows easily from the definitions in that case that the given action of \( G \) on \( \wi... | Yes |
Example 2.30 (The Fubini-Study Metric). Let \( n \) be a positive integer, and consider the complex projective space \( {\mathbb{{CP}}}^{n} \) defined in Example C.19. That example shows that the map \( \pi : {\mathbb{C}}^{n + 1} \smallsetminus \{ 0\} \rightarrow \mathbb{C}{\mathbb{P}}^{n} \) sending each point in \( {... | \[ v\left( z\right) = \frac{z}{\left| z\right| } \] Since dividing an element of \( {\mathbb{C}}^{n + 1} \) by a nonzero scalar does not change its span, it follows that \( p \circ v = \pi \) . Therefore, if we set \( \widetilde{\sigma } = v \circ \sigma \), we have \( p \circ \widetilde{\sigma } = p \circ v \circ \sig... | No |
Proposition 2.31. Suppose \( \pi : \widetilde{M} \rightarrow M \) is a smooth normal covering map, and \( \widetilde{g} \) is any metric on \( \widetilde{M} \) that is invariant under all covering automorphisms. Then there is a unique metric \( g \) on \( M \) such that \( \pi \) is a Riemannian covering. | Proof. Proposition A. 49 shows that \( \pi \) is a surjective smooth submersion. The automorphism group acts vertically by definition, and Proposition C. 21 shows that it acts transitively on fibers when the covering is normal. It then follows from Theorem 2.28 that there is a unique metric \( g \) on \( M \) such that... | Yes |
Proposition 2.32. Suppose \( \left( {\widetilde{M},\widetilde{g}}\right) \) is a Riemannian manifold, and \( \Gamma \) is a discrete Lie group acting smoothly, freely, properly, and isometrically on \( \widetilde{M} \) . Then \( \widetilde{M}/\Gamma \) has a unique Riemannian metric such that the quotient map \( \pi : ... | Proof. Proposition C. 23 shows that \( \pi \) is a smooth normal covering map, and Proposition 2.31 shows that \( M = \widetilde{M}/\Gamma \) has a unique Riemannian metric such that \( \pi \) is a Riemannian covering. | Yes |
Corollary 2.33. Suppose \( \left( {M, g}\right) \) and \( \left( {\widetilde{M},\widetilde{g}}\right) \) are connected Riemannian manifolds, \( \pi : \widetilde{M} \rightarrow M \) is a normal Riemannian covering map, and \( \Gamma = {\operatorname{Aut}}_{\pi }\left( \widetilde{M}\right) \) . Then \( M \) is isometric ... | Proof. Proposition C. 20 shows that with the discrete topology, \( \Gamma \) is a discrete Lie group acting smoothly, freely, and properly on \( \widetilde{M} \), and then Proposition C. 23 shows that \( \widetilde{M}/\Gamma \) is a smooth manifold and the quotient map \( q : \widetilde{M} \rightarrow \widetilde{M}/\Ga... | Yes |
The two-element group \( \Gamma = \{ \pm 1\} \) acts smoothly, freely, properly, and isometrically on \( {\mathbb{S}}^{n} \) by multiplication. | Example C. 24 shows that the quotient space is diffeomorphic to the real projective space \( {\mathbb{{RP}}}^{n} \) and the quotient map \( q : {\mathbb{S}}^{n} \rightarrow \mathbb{R}{\mathbb{P}}^{n} \) is a smooth normal covering map. Because the action is isometric, Proposition 2.32 shows that there is a unique metri... | Yes |
Proposition 2.37. Suppose \( \left( {M, g}\right) \) is a Riemannian manifold, \( f \in {C}^{\infty }\left( M\right) \), and \( \mathcal{R} \subseteq M \) is the set of regular points of \( f \) . For each \( c \in \mathbb{R} \), the set \( {M}_{c} = {f}^{-1}\left( c\right) \cap \mathcal{R} \) , if nonempty, is an embe... | ## Proof. Problem 2-9. | No |
Proposition 2.40 (Inner Products of Tensors). Let \( \left( {M, g}\right) \) be an \( n \) -dimensional Riemannian manifold with or without boundary. There is a unique smooth fiber metric on each tensor bundle \( {T}^{\left( k, l\right) }{TM} \) with the property that if \( {\alpha }_{1},\ldots ,{\alpha }_{k + l} \) , ... | Proof. Problem 2-11. | No |
Proposition 2.41 (The Riemannian Volume Form). Let \( \left( {M, g}\right) \) be an oriented Riemannian \( n \) -manifold with or without boundary. There is a unique \( n \) -form \( d{V}_{g} \) on \( M \), called the Riemannian volume form, characterized by any one of the following three equivalent properties:\n\n(a) ... | Proof. Problem 2-12. | No |
Proposition 2.43. Suppose \( M \) is a hypersurface in an oriented Riemannian manifold \( \left( {\widetilde{M},\widetilde{g}}\right) \) and \( g \) is the induced metric on \( M \) . Then \( M \) is orientable if and only if there exists a global unit normal vector field \( N \) for \( M \), and in that case the volum... | Proof. Problem 2-13. | No |
Proposition 2.44 (The Riemannian Density). If \( \\left( {M, g}\\right) \) is any Riemannian manifold, then there is a unique smooth positive density \( \\mu \) on \( M \), called the Riemannian density, with the property that\n\n\[\\mu \\left( {{E}_{1},\\ldots ,{E}_{n}}\\right) = 1\]\n\n(2.18)\n\nfor every local ortho... | Exercise 2.45. Prove this proposition by showing that \( \\mu \) can be defined in terms of any\n\nlocal orthonormal frame by\n\n\[\\mu = \\left| {{\\varepsilon }^{1} \\land \\cdots \\land {\\varepsilon }^{n}}\\right|\] | No |
Proposition 2.46. Let \( \left( {M, g}\right) \) be a Riemannian manifold with or without boundary, and let \( \left( {x}^{i}\right) \) be any smooth local coordinates on an open set \( U \subseteq M \) . The coordinate representations of the divergence and Laplacian are as follows:\n\n\[ \operatorname{div}\left( {{X}^... | Proof. Problem 2-21. | No |
(a) Every regular curve in \( M \) has a unit-speed forward reparametrization. | Proof. Suppose \( \gamma : I \rightarrow M \) is a regular curve. Choose an arbitrary \( {t}_{0} \in I \), and define \( s : I \rightarrow \mathbb{R} \) by\n\n\[ s\left( t\right) = {\int }_{{t}_{0}}^{t}{\left| {\gamma }^{\prime }\left( u\right) \right| }_{g}{du} \]\n\nSince \( {s}^{\prime }\left( t\right) = {\left| {\g... | Yes |
Proposition 2.50. If \( M \) is a connected smooth manifold (with or without boundary), then any two points of \( M \) can be joined by an admissible curve. | Proof. Let \( p, q \in M \) be arbitrary. Since a connected manifold is path-connected, \( p \) and \( q \) can be connected by a continuous path \( c : \left\lbrack {a, b}\right\rbrack \rightarrow M \) . By compactness, there is a partition of \( \left\lbrack {a, b}\right\rbrack \) such that \( c\left( \left\lbrack {{... | Yes |
Lemma 2.53. Let \( g \) be a Riemannian metric on an open subset \( W \subseteq {\mathbb{R}}^{n} \) or \( {\mathbb{R}}_{ + }^{n} \), and let \( \bar{g} \) denote the Euclidean metric on \( W \) . For every compact subset \( K \subseteq W \), there are positive constants \( c, C \) such that for all \( x \in K \) and al... | Proof. Define a continuous function \( F : {TW} \rightarrow \mathbb{R} \) by \( F\left( {x, v}\right) = {\left| v\right| }_{g} = {g}_{x}{\left( v, v\right) }^{1/2} \) for \( x \in W \) and \( v \in {T}_{x}{\mathbb{R}}^{n} \) . Let \( K \subseteq W \) be any compact subset, and define \( L \subseteq T{\mathbb{R}}^{n} \)... | Yes |
Lemma 2.54. Let \( \left( {M, g}\right) \) be a Riemannian manifold with or without boundary and let \( {d}_{g} \) be its Riemannian distance function. Suppose \( U \) is an open subset of \( M \) and \( p \in U \) . Then \( p \) has a coordinate neighborhood \( V \subseteq U \) with the property that there are positiv... | Proof. Let \( W \) be any neighborhood of \( p \) contained in \( U \) on which there exist smooth coordinates \( \left( {x}^{i}\right) \) . Using these coordinates, we can identify \( W \) with an open subset of \( {\mathbb{R}}^{n} \) or \( {\mathbb{R}}_{ + }^{n} \) . Let \( K \) be a compact subset of \( W \) contain... | Yes |
Theorem 2.55 (Riemannian Manifolds as Metric Spaces). Let \( \left( {M, g}\right) \) be a connected Riemannian manifold with or without boundary. With the distance function \( {d}_{g}, M \) is a metric space whose metric topology is the same as the given manifold topology. | Proof. It is immediate from the definition of \( {d}_{g} \) that \( {d}_{g}\left( {p, q}\right) = {d}_{g}\left( {q, p}\right) \geq 0 \) and \( {d}_{g}\left( {p, p}\right) = 0 \) . On the other hand, suppose \( p, q \in M \) are distinct. Let \( U \subseteq M \) be an open set that contains \( p \) but not \( q \), and ... | Yes |
Lemma 2.62 (Completion of Nondegenerate Bases). Suppose \( \left( {V, q}\right) \) is an \( n \) - dimensional scalar product space, and \( \left( {{v}_{1},\ldots ,{v}_{k}}\right) \) is a nondegenerate \( k \) -tuple in \( V \) with \( 0 \leq k < n \) . Then there exist vectors \( {v}_{k + 1},\ldots ,{v}_{n} \in V \) s... | Proof. Let \( S = \operatorname{span}\left( {{v}_{1},\ldots ,{v}_{k}}\right) \subseteq V \) . Because \( k < n,{S}^{ \bot } \) is a nontrivial subspace of \( V \), and Lemma 2.60 shows that \( {S}^{ \bot } \) is nondegenerate and \( V = S \oplus {S}^{ \bot } \) . By the nondegeneracy of \( {S}^{ \bot } \), there must b... | Yes |
Proposition 2.63 (Gram-Schmidt Algorithm for Scalar Products). Suppose \( \left( {V, q}\right) \) is an \( n \) -dimensional scalar product space. If \( \left( {v}_{i}\right) \) is a nondegenerate basis for \( V \), then there is an orthonormal basis \( \left( {b}_{i}\right) \) with the property that \( \operatorname{s... | Proof. As in the positive definite case, the basis \( \left( {b}_{i}\right) \) is constructed recursively, starting with \( {b}_{1} = {v}_{1}/\left| {v}_{1}\right| \) and noting that the assumption that \( {v}_{1} \) spans a nondegenerate subspace ensures that \( \left| {v}_{1}\right| \neq 0 \) . At the inductive step,... | Yes |
Corollary 2.64. Suppose \( \left( {V, q}\right) \) is an \( n \) -dimensional scalar product space. There is a basis \( \left( {\beta }^{i}\right) \) for \( {V}^{ * } \) with respect to which \( q \) has the expression\n\n\[ q = {\left( {\beta }^{1}\right) }^{2} + \cdots + {\left( {\beta }^{r}\right) }^{2} - {\left( {\... | Proof. Let \( \left( {b}_{i}\right) \) be an orthonormal basis for \( V \), and let \( \left( {\beta }^{i}\right) \) be the dual basis for \( {V}^{ * } \) . A computation shows that \( q \) has a basis expression of the form (2.23), but perhaps with the positive and negative terms in a different order. Reordering the b... | Yes |
Proposition 2.65 (Sylvester’s Law of Inertia). Suppose \( \left( {V, q}\right) \) is a finite-dimensional scalar product space. If \( q \) has the representation (2.23) in some basis, then the number \( r \) is the maximum dimension among all subspaces on which the restriction of \( q \) is positive definite, and thus ... | Proof. Problem 2-33. | No |
Proposition 2.66 (Orthonormal Frames for Pseudo-Riemannian Manifolds). Let \( \left( {M, g}\right) \) be a pseudo-Riemannian manifold. For each \( p \in M \), there exists a smooth orthonormal frame on a neighborhood of \( p \) in \( M \) . | Exercise 2.67. Prove the preceding proposition. | No |
Theorem 2.69. A smooth manifold \( M \) admits a Lorentz metric if and only if it admits a rank-1 tangent distribution (i.e., a rank-1 subbundle of TM). | Proof. Problem 2-34. | No |
Proposition 2.70. Suppose \( \left( {\widetilde{M},\widetilde{g}}\right) \) is a pseudo-Riemannian manifold of signature \( \left( {r, s}\right) \). Let \( M \) be a smooth hypersurface in \( \widetilde{M} \), and let \( \iota : M \hookrightarrow \widetilde{M} \) be the inclusion map. Then the pullback tensor field \( ... | Proof. Given \( p \in M \), Lemma 2.60 shows that \( {T}_{p}M \) is a nondegenerate subspace of \( {T}_{p}\widetilde{M} \) if and only if the one-dimensional subspace \( {\left( {T}_{p}M\right) }^{ \bot } = {N}_{p}M \) is nondegenerate, which is the case if and only if every nonzero \( v \in {N}_{p}M \) satisfies \( \w... | Yes |
Corollary 2.71. Suppose \( \left( {\widetilde{M},\widetilde{g}}\right) \) is a pseudo-Riemannian manifold of signature \( \left( {r, s}\right), f \in {C}^{\infty }\left( \widetilde{M}\right) \), and \( M = {f}^{-1}\left( c\right) \) for some \( c \in \mathbb{R} \) . If \( \widetilde{g}\left( {\operatorname{grad}f,\oper... | Proof. Problem 2-35. | No |
Proposition 2.72 (Pseudo-Riemannian Adapted Orthonormal Frames). Suppose \( \left( {\widetilde{M},\widetilde{g}}\right) \) is a pseudo-Riemannian manifold, and \( M \subseteq \widetilde{M} \) is an embedded pseudo-Riemannian or Riemannian submanifold. For each \( p \in M \), there exists a smooth orthonormal frame on a... | Proof. Write \( m = \dim \widetilde{M} \) and \( n = \dim M \), and let \( p \in M \) be arbitrary. Proposition 2.66 shows that there is a smooth orthonormal frame \( \left( {{E}_{1},\ldots ,{E}_{n}}\right) \) for \( M \) on some neighborhood of \( p \) in \( M \) . Then by Lemma 2.62, we can find vectors \( {v}_{n + 1... | Yes |
Proposition 3.1. Let \( \left( {M, g}\right) \) be a Riemannian manifold.\n\n(a) If \( M \) is isotropic at one point and it is homogeneous, then it is isotropic.\n\n(b) If \( M \) is frame-homogeneous, then it is homogeneous and isotropic. | Proof. Problem 3-3. | No |
Proposition 3.2. The group \( \mathrm{O}\left( {n + 1}\right) \) acts transitively on \( \mathrm{O}\left( {{\mathbb{S}}^{n}\left( R\right) }\right) \), and thus each round sphere is frame-homogeneous. | Proof. It suffices to show that given any \( p \in {\mathbb{S}}^{n}\left( R\right) \) and any orthonormal basis \( \left( {b}_{i}\right) \) for \( {T}_{p}{\mathbb{S}}^{n}\left( R\right) \), there is an orthogonal map that takes the \ | No |
Proposition 3.5. Stereographic projection is a conformal diffeomorphism between \( {\mathbb{S}}^{n}\left( R\right) \smallsetminus \{ N\} \) and \( {\mathbb{R}}^{n} \) . | Proof. The inverse map \( {\sigma }^{-1} \) is a smooth parametrization of \( {\mathbb{S}}^{n}\left( R\right) \smallsetminus \{ N\} \), so we can use it to compute the pullback metric. Using the usual technique of substitution to compute pullbacks, we obtain the following coordinate representation of \( {g}_{R} \) in s... | Yes |
Corollary 3.6. Each sphere with a round metric is locally conformally flat. | Proof. Stereographic projection gives a conformal equivalence between a neighborhood of any point except the north pole and Euclidean space; applying a suitable rotation and then stereographic projection (or stereographic projection from the south pole), we get such an equivalence for a neighborhood of the north pole a... | Yes |
Corollary 3.8. Each hyperbolic space is locally conformally flat. | Proof. In either the Poincaré ball model or the half-space model, the identity map gives a global conformal equivalence with an open subset of Euclidean space. | No |
Proposition 3.9. The group \( {\mathrm{O}}^{ + }\left( {n,1}\right) \) acts transitively on \( \mathrm{O}\left( {{\mathbb{H}}^{n}\left( R\right) }\right) \), and therefore \( {\mathbb{H}}^{n}\left( R\right) \) is frame-homogeneous. | Proof. The argument is entirely analogous to the proof of Proposition 3.2, so we give only a sketch. Suppose \( p \in {\mathbb{H}}^{n}\left( R\right) \) and \( \left( {b}_{i}\right) \) is an orthonormal basis for \( {T}_{p}{\mathbb{H}}^{n}\left( R\right) \) . Identifying \( p \in {\mathbb{R}}^{n,1} \) with an element o... | Yes |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.