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Theorem 3.2 \( {s}^{-1/2}\vartheta \left( {1/s}\right) = \vartheta \left( s\right) \) whenever \( s > 0 \) . | The proof of this identity consists of a simple application of the Poisson summation formula to the pair\n\n\[ f\left( x\right) = {e}^{-{\pi s}{x}^{2}}\;\text{ and }\;\widehat{f}\left( \xi \right) = {s}^{-1/2}{e}^{-\pi {\xi }^{2}/s}. \] | Yes |
Theorem 3.3 The heat kernel on the circle is the periodization of the heat kernel on the real line: | \[ {H}_{t}\left( x\right) = \mathop{\sum }\limits_{{n = - \infty }}^{\infty }{\mathcal{H}}_{t}\left( {x + n}\right) \] | Yes |
Corollary 3.4 The kernel \( {H}_{t}\left( x\right) \) is a good kernel for \( t \rightarrow 0 \) . | Proof. We already observed that \( {\int }_{\left| x\right| \leq 1/2}{H}_{t}\left( x\right) {dx} = 1 \) . Now note that \( {H}_{t} \geq 0 \), which is immediate from the above formula since \( {\mathcal{H}}_{t} \geq 0 \) . Finally, we claim that when \( \left| x\right| \leq 1/2 \) ,\n\n\[ \n{H}_{t}\left( x\right) = {\m... | Yes |
Theorem 3.5 \( {P}_{r}\left( {2\pi x}\right) = \mathop{\sum }\limits_{{n \in \mathbb{Z}}}{\mathcal{P}}_{y}\left( {x + n}\right) \) where \( r = {e}^{-{2\pi y}} \) . | This is again an immediate corollary of the Poisson summation formula applied to \( f\left( x\right) = {\mathcal{P}}_{y}\left( x\right) \) and \( \widehat{f}\left( \xi \right) = {e}^{-{2\pi }\left| \xi \right| y} \) . Of course, here we use the Poisson summation formula under the assumptions that \( f \) and \( \wideha... | Yes |
Theorem 4.1 Suppose \( \psi \) is a function in \( \mathcal{S}\left( \mathbb{R}\right) \) which satisfies the normalizing condition \( {\int }_{-\infty }^{\infty }{\left| \psi \left( x\right) \right| }^{2}{dx} = 1 \) . Then\n\n\[ \left( {{\int }_{-\infty }^{\infty }{x}^{2}{\left| \psi \left( x\right) \right| }^{2}{dx}}... | Proof. The second inequality actually follows from the first by replacing \( \psi \left( x\right) \) by \( {e}^{-{2\pi ix}{\xi }_{0}}\psi \left( {x + {x}_{0}}\right) \) and changing variables. To prove the first inequality, we argue as follows. Beginning with our normalizing assumption \( \int {\left| \psi \right| }^{2... | Yes |
Proposition 2.1 Let \( f \in \mathcal{S}\left( {\mathbb{R}}^{d}\right) \). (i) \( f\left( {x + h}\right) \rightarrow \widehat{f}\left( \xi \right) {e}^{{2\pi i\xi } \cdot h} \) whenever \( h \in {\mathbb{R}}^{d} \). (ii) \( f\left( x\right) {e}^{-{2\pi ixh}} \rightarrow \widehat{f}\left( {\xi + h}\right) \) whenever \(... | The first five properties are proved in the same way as in the one-dimensional case. To verify the last property, simply change variables \( y = {Rx} \) in the integral. Then, recall that \( \left| {\det \left( R\right) }\right| = 1 \), and \( {R}^{-1}y \cdot \xi = y \cdot {R\xi } \), because \( R \) is a rotation. | Yes |
Corollary 2.3 The Fourier transform of a radial function is radial. | This follows at once from property (vi) in the last proposition. Indeed, the condition \( f\left( {Rx}\right) = f\left( x\right) \) for all \( R \) implies that \( \widehat{f}\left( {R\xi }\right) = \widehat{f}\left( \xi \right) \) for all \( R \), thus \( \widehat{f} \) is radial whenever \( f \) is. | Yes |
A solution of the Cauchy problem for the wave equation is\n\n\[ u\left( {x, t}\right) = {\int }_{{\mathbb{R}}^{d}}\left\lbrack {\widehat{f}\left( \xi \right) \cos \left( {{2\pi }\left| \xi \right| t}\right) + \widehat{g}\left( \xi \right) \frac{\sin \left( {{2\pi }\left| \xi \right| t}\right) }{{2\pi }\left| \xi \right... | Proof. We first verify that \( u \) solves the wave equation. This is straightforward once we note that we can differentiate in \( x \) and \( t \) under the integral sign (because \( f \) and \( g \) are both Schwartz functions) and therefore \( u \) is at least \( {C}^{2} \). On the one hand we differentiate the expo... | Yes |
Theorem 3.2 If \( u \) is the solution of the wave equation given by formula (3), then \( E\left( t\right) \) is conserved, that is,\n\n\[ E\left( t\right) = E\left( 0\right) ,\;\text{ for all }t \in \mathbb{R}. \] | The proof requires the following lemma.\n\nLemma 3.3 Suppose a and \( b \) are complex numbers and \( \alpha \) is real. Then\n\n\[ {\left| a\cos \alpha + b\sin \alpha \right| }^{2} + {\left| -a\sin \alpha + b\cos \alpha \right| }^{2} = {\left| a\right| }^{2} + {\left| b\right| }^{2}. \]\n\nThis follows directly becaus... | Yes |
Lemma 3.3 Suppose a and \( b \) are complex numbers and \( \alpha \) is real. Then\n\n\[ \n{\left| a\cos \alpha + b\sin \alpha \right| }^{2} + {\left| -a\sin \alpha + b\cos \alpha \right| }^{2} = {\left| a\right| }^{2} + {\left| b\right| }^{2}.\n\] | This follows directly because \( {e}_{1} = \left( {\cos \alpha ,\sin \alpha }\right) \) and \( {e}_{2} = \left( {-\sin \alpha ,\cos \alpha }\right) \) are a pair of orthonormal vectors, hence with \( Z = \left( {a, b}\right) \in {\mathbb{C}}^{2} \), we have\n\n\[ \n{\left| Z\right| }^{2} = {\left| Z \cdot {e}_{1}\right... | Yes |
Lemma 3.4 If \( f \in \mathcal{S}\left( {\mathbb{R}}^{3}\right) \) and \( t \) is fixed, then \( {M}_{t}\left( f\right) \in \mathcal{S}\left( {\mathbb{R}}^{3}\right) \) . Moreover, \( {M}_{t}\left( f\right) \) is indefinitely differentiable in \( t \), and each \( t \) -derivative also belongs to \( \mathcal{S}\left( {... | Proof. Let \( F\left( x\right) = {M}_{t}\left( f\right) \left( x\right) \) . To show that \( F \) is rapidly decreasing, start with the inequality \( \left| {f\left( x\right) }\right| \leq {A}_{N}/\left( {1 + {\left| x\right| }^{N}}\right) \) which holds for every fixed \( N \geq 0 \) . As a simple consequence, wheneve... | Yes |
Lemma 3.5 \( \frac{1}{4\pi }{\int }_{{S}^{2}}{e}^{-{2\pi i\xi } \cdot \gamma }{d\sigma }\left( \gamma \right) = \frac{\sin \left( {{2\pi }\left| \xi \right| }\right) }{{2\pi }\left| \xi \right| } \) | Proof. Note that the integral on the left is radial in \( \xi \) . Indeed, if \( R \) is a rotation then\n\n\[ \n{\int }_{{S}^{2}}{e}^{-{2\pi iR}\left( \xi \right) \cdot \gamma }{d\sigma }\left( \gamma \right) = {\int }_{{S}^{2}}{e}^{-{2\pi i\xi } \cdot {R}^{-1}\left( \gamma \right) }{d\sigma }\left( \gamma \right) = {... | Yes |
Theorem 3.6 The solution when \( d = 3 \) of the Cauchy problem for the wave equation\n\n\[ \bigtriangleup u = \frac{{\partial }^{2}u}{\partial {t}^{2}}\;\text{ subject to }\;u\left( {x,0}\right) = f\left( x\right) \;\text{ and }\;\frac{\partial u}{\partial t}\left( {x,0}\right) = g\left( x\right) \]\n\nis given by\n\n... | Proof. Consider first the problem\n\n\[ \bigtriangleup u = \frac{{\partial }^{2}u}{\partial {t}^{2}}\;\text{ subject to }\;u\left( {x,0}\right) = 0\;\text{ and }\;\frac{\partial u}{\partial t}\left( {x,0}\right) = g\left( x\right) . \]\n\nThen by Theorem 3.1, we know that its solution \( {u}_{1} \) is given by\n\n\[ {u... | Yes |
Theorem 3.7 A solution of the Cauchy problem for the wave equation in two dimensions with initial data \( f, g \in \mathcal{S}\left( {\mathbb{R}}^{2}\right) \) is given by\n\n\[ u\left( {x, t}\right) = \frac{\partial }{\partial t}\left( {t{\widetilde{M}}_{t}\left( f\right) \left( x\right) }\right) + t{\widetilde{M}}_{t... | Formally, the identity in the theorem arises as follows. If we start with an initial pair of functions \( f \) and \( g \) in \( \mathcal{S}\left( {\mathbb{R}}^{2}\right) \), we may consider the corresponding functions \( \widetilde{f} \) and \( \widetilde{g} \) on \( {\mathbb{R}}^{3} \) that are merely extensions of \... | Yes |
Proposition 5.1 If \( f \in \mathcal{S}\left( {\mathbb{R}}^{3}\right) \), then for each \( \gamma \) the definition of \( {\int }_{{\mathcal{P}}_{t,\gamma }}f \) is independent of the choice of \( {e}_{1} \) and \( {e}_{2} \) . Moreover\n\n\[{\int }_{-\infty }^{\infty }\left( {{\int }_{{\mathcal{P}}_{t,\gamma }}f}\righ... | Proof. If \( {e}_{1}^{\prime },{e}_{2}^{\prime } \) is another choice of basis vectors so that \( \gamma ,{e}_{1}^{\prime },{e}_{2}^{\prime } \) is orthonormal, consider the rotation \( R \) in \( {\mathbb{R}}^{2} \) which takes \( {e}_{1} \) to \( {e}_{1}^{\prime } \) and \( {e}_{2} \) to \( {e}_{2}^{\prime } \) . Cha... | Yes |
Lemma 5.2 If \( f \in \mathcal{S}\left( {\mathbb{R}}^{3}\right) \), then \( \mathcal{R}\left( f\right) \left( {t,\gamma }\right) \in \mathcal{S}\left( \mathbb{R}\right) \) for each fixed \( \gamma \) . Moreover, \[ \widehat{\mathcal{R}}\left( f\right) \left( {s,\gamma }\right) = \widehat{f}\left( {s\gamma }\right) \] | Proof. Since \( f \in \mathcal{S}\left( {\mathbb{R}}^{3}\right) \), for every positive integer \( N \) there is a constant \( {A}_{N} < \infty \) so that \[ {\left( 1 + \left| t\right| \right) }^{N}{\left( 1 + \left| u\right| \right) }^{N}\left| {f\left( {{t\gamma } + u}\right) }\right| \leq {A}_{N} \] if we recall tha... | Yes |
Corollary 5.3 If \( f, g \in \mathcal{S}\left( {\mathbb{R}}^{3}\right) \) and \( \mathcal{R}\left( f\right) = \mathcal{R}\left( g\right) \), then \( f = g \) . | The proof of the corollary follows from an application of the lemma to the difference \( f - g \) and use of the Fourier inversion theorem. | No |
Theorem 5.4 If \( f \in \mathcal{S}\left( {\mathbb{R}}^{3}\right) \), then\n\n\[ \bigtriangleup \left( {{\mathcal{R}}^{ * }\mathcal{R}\left( f\right) }\right) = - 8{\pi }^{2}f \] | We recall that \( \bigtriangleup = \frac{{\partial }^{2}}{\partial {x}_{1}^{2}} + \frac{{\partial }^{2}}{\partial {x}_{2}^{2}} + \frac{{\partial }^{2}}{\partial {x}_{3}^{2}} \) is the Laplacian.\n\nProof. By our previous lemma, we have\n\n\[ \mathcal{R}\left( f\right) \left( {t,\gamma }\right) = {\int }_{-\infty }^{\in... | Yes |
Lemma 1.1 The family \( \left\{ {{e}_{0},\ldots ,{e}_{N - 1}}\right\} \) is orthogonal. In fact,\n\n\[ \left( {{e}_{m},{e}_{\ell }}\right) = \left\{ \begin{array}{ll} N & \text{ if }m = \ell \\ 0 & \text{ if }m \neq \ell \end{array}\right. \] | Proof. We have\n\n\[ \left( {{e}_{m},{e}_{\ell }}\right) = \mathop{\sum }\limits_{{k = 0}}^{{N - 1}}{\zeta }^{mk}{\zeta }^{-\ell k} = \mathop{\sum }\limits_{{k = 0}}^{{N - 1}}{\zeta }^{\left( {m - \ell }\right) k}. \]\n\nIf \( m = \ell \), each term in the sum is equal to 1, and the sum equals \( N \) . If \( m \neq \e... | Yes |
Theorem 1.2 If \( F \) is a function on \( \mathbb{Z}\left( N\right) \), then\n\n\[ F\left( k\right) = \mathop{\sum }\limits_{{n = 0}}^{{N - 1}}{a}_{n}{e}^{{2\pi ink}/N}. \]\n\nMoreover,\n\n\[ \mathop{\sum }\limits_{{n = 0}}^{{N - 1}}{\left| {a}_{n}\right| }^{2} = \frac{1}{N}\mathop{\sum }\limits_{{k = 0}}^{{N - 1}}{\l... | The proof follows directly from (1) once we observe that\n\n\[ {a}_{n} = \frac{1}{N}\left( {F,{e}_{n}}\right) = \frac{1}{\sqrt{N}}\left( {F,{e}_{n}^{ * }}\right) . \] | Yes |
Lemma 1.4 If we are given \( {\omega }_{2M} = {e}^{-{2\pi i}/\left( {2M}\right) } \), then\n\n\[ \n\# \left( {2M}\right) \leq 2\# \left( M\right) + {8M}.\n\] | Proof. The calculation of \( {\omega }_{2M},\ldots ,{\omega }_{2M}^{2M} \) requires no more than \( {2M} \) operations. Note that in particular we get \( {\omega }_{M} = {e}^{-{2\pi i}/M} = {\omega }_{2M}^{2} \). The main idea is that for any given function \( F \) on \( \mathbb{Z}\left( {2M}\right) \), we consider two... | Yes |
Lemma 2.1 The set \( \widehat{G} \) is an abelian group under multiplication defined \( {by} \)\n\n\[ \left( {{e}_{1} \cdot {e}_{2}}\right) \left( a\right) = {e}_{1}\left( a\right) {e}_{2}\left( a\right) \;\text{ for all }a \in G. \] | The proof of this assertion is straightforward if one observes that the trivial character plays the role of the unit. We call \( \widehat{G} \) the dual group of \( G \) . | No |
Lemma 2.2 Let \( G \) be a finite abelian group, and \( e : G \rightarrow \mathbb{C} - \{ 0\} \) a multiplicative function, namely \( e\left( {a \cdot b}\right) = e\left( a\right) e\left( b\right) \) for all \( a, b \in G \) . Then \( e \) is a character. | Proof. The group \( G \) being finite, the absolute value of \( e\left( a\right) \) is bounded above and below as \( a \) ranges over \( G \) . Since \( \left| {e\left( {b}^{n}\right) }\right| = {\left| e\left( b\right) \right| }^{n} \), we conclude that \( \left| {e\left( b\right) }\right| = 1 \) for all \( b \in G \)... | Yes |
Theorem 2.3 The characters of \( G \) form an orthonormal family with respect to the inner product defined above. | Since \( \left| {e\left( a\right) }\right| = 1 \) for any character, we find that\n\n\[ \left( {e, e}\right) = \frac{1}{\left| G\right| }\mathop{\sum }\limits_{{a \in G}}e\left( a\right) \overline{e\left( a\right) } = \frac{1}{\left| G\right| }\mathop{\sum }\limits_{{a \in G}}{\left| e\left( a\right) \right| }^{2} = 1.... | Yes |
Lemma 2.4 If \( e \) is a non-trivial character of the group \( G \), then \( \mathop{\sum }\limits_{{a \in G}}e\left( a\right) = 0. \) | Proof. Choose \( b \in G \) such that \( e\left( b\right) \neq 1 \) . Then we have\n\n\[ e\left( b\right) \mathop{\sum }\limits_{{a \in G}}e\left( a\right) = \mathop{\sum }\limits_{{a \in G}}e\left( b\right) e\left( a\right) = \mathop{\sum }\limits_{{a \in G}}e\left( {ab}\right) = \mathop{\sum }\limits_{{a \in G}}e\lef... | Yes |
Lemma 2.6 Suppose \( \\left\\{ {{T}_{1},\\ldots ,{T}_{k}}\\right\\} \) is a commuting family of unitary transformations on the finite-dimensional inner product space \( V \) ; that is,\n\n\[ \n{T}_{i}{T}_{j} = {T}_{j}{T}_{i}\\;\\text{ for all }i, j.\n\]\n\nThen \( {T}_{1},\\ldots ,{T}_{k} \) are simultaneously diagonal... | Proof. We use induction on \( k \) . The case \( k = 1 \) is simply the spectral theorem. Suppose that the lemma is true for any family of \( k - 1 \) commuting unitary transformations. The spectral theorem applied to \( {T}_{k} \) says that \( V \) is the direct sum of its eigenspaces\n\n\[ \nV = {V}_{{\\lambda }_{1}}... | Yes |
Theorem 2.7 Let \( G \) be a finite abelian group. The characters of \( G \) form an orthonormal basis for the vector space \( V \) of functions on \( G \) equipped with the inner product\n\n\[ \n\\left( {f, g}\\right) = \\frac{1}{\\left| G\\right| }\\mathop{\\sum }\\limits_{{a \\in G}}f\\left( a\\right) \\overline{g\\... | Null | No |
Theorem 2.8 If \( f \) is a function on \( G \), then \( \parallel f{\parallel }^{2} = \mathop{\sum }\limits_{{e \in \widehat{G}}}{\left| \widehat{f}\left( e\right) \right| }^{2} \) . | Proof. Since the characters of \( G \) form an orthonormal basis for the vector space \( V \), and \( \left( {f, e}\right) = \widehat{f}\left( e\right) \), we have that\n\n\[ \parallel f{\parallel }^{2} = \left( {f, f}\right) = \mathop{\sum }\limits_{{e \in \widehat{G}}}\left( {f, e}\right) \overline{\widehat{f}\left( ... | Yes |
Theorem 1.1 (Euclid’s algorithm) For any integers \( a \) and \( b \) with \( b > 0 \), there exist unique integers \( q \) and \( r \) with \( 0 \leq r < b \) such that\n\n\[ a = {qb} + r. \] | Proof. First we prove the existence of \( q \) and \( r \) . Let \( S \) denote the set of all non-negative integers of the form \( a - {qb} \) with \( q \in \mathbb{Z} \) . This set is non-empty and in fact \( S \) contains arbitrarily large positive integers since \( b \neq 0 \) . Let \( r \) denote the smallest elem... | Yes |
Theorem 1.2 If \( \gcd \left( {a, b}\right) = d \), then there exist integers \( x \) and \( y \) such that\n\n\[ \n{ax} + {by} = d\text{.} \n\] | Proof. Consider the set \( S \) of all positive integers of the form \( {ax} + {by} \) where \( x, y \in \mathbb{Z} \), and let \( s \) be the smallest element in \( S \) . We claim that \( s = \) \( d \) . By construction, there exist integers \( x \) and \( y \) such that\n\n\[ \n{ax} + {by} = s.\n\]\n\nClearly, any ... | Yes |
Corollary 1.3 Two positive integers \( a \) and \( b \) are relatively prime if and only if there exist integers \( x \) and \( y \) such that \( {ax} + {by} = 1 \) . | Proof. If \( a \) and \( b \) are relatively prime, two integers \( x \) and \( y \) with the desired property exist by Theorem 1.2. Conversely, if \( {ax} + {by} = 1 \) holds and \( d \) is positive and divides both \( a \) and \( b \), then \( d \) divides 1, hence \( d = 1 \) . | Yes |
Corollary 1.4 If a and \( c \) are relatively prime and \( c \) divides \( {ab} \), then \( c \) divides \( b \) . In particular, if \( p \) is a prime that does not divide \( a \) and \( p \) divides \( {ab} \), then \( p \) divides \( b \) . | Proof. We can write \( 1 = {ax} + {cy} \), so multiplying by \( b \) we find \( b = \) \( {abx} + {cby} \) . Hence \( c \mid b \) . | Yes |
Corollary 1.5 If \( p \) is prime and \( p \) divides the product \( {a}_{1}\cdots {a}_{r} \), then \( p \) divides \( {a}_{i} \) for some \( i \) . | Proof. By the previous corollary, if \( p \) does not divide \( {a}_{1} \), then \( p \) divides \( {a}_{2}\cdots {a}_{r} \), so eventually \( p \mid {a}_{i} \) . | Yes |
Theorem 1.6 Every positive integer greater than 1 can be factored uniquely into a product of primes. | Proof. First, we show that such a factorization is possible. We do so by proving that the set \( S \) of positive integers \( > 1 \) which do not have a factorization into primes is empty. Arguing by contradiction, we assume that \( S \neq \varnothing \) . Let \( n \) be the smallest element of \( S \) . Since \( n \) ... | Yes |
Theorem 1.7 There are infinitely many primes. | Proof. Suppose not, and denote by \( {p}_{1},\ldots ,{p}_{n} \) the complete set of primes. Define\n\n\[ N = {p}_{1}{p}_{2}\cdots {p}_{n} + 1 \]\n\nSince \( N \) is larger than any \( {p}_{i} \), the integer \( N \) cannot be prime. Therefore, \( N \) is divisible by a prime that belongs to our list. But this is also a... | Yes |
Lemma 1.8 The exponential and logarithm functions satisfy the following properties:\n\n(i) \( {e}^{\log x} = x \) .\n\n(ii) \( \log \left( {1 + x}\right) = x + E\left( x\right) \) where \( \left| {E\left( x\right) }\right| \leq {x}^{2} \) if \( \left| x\right| < 1/2 \) .\n\n(iii) If \( \log \left( {1 + x}\right) = y \)... | Proof. Property (i) is standard. To prove property (ii) we use the power series expansion of \( \log \left( {1 + x}\right) \) for \( \left| x\right| < 1 \), that is,\n\n(2)\n\n\[ \log \left( {1 + x}\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{\left( -1\right) }^{n + 1}}{n}{x}^{n}. \]\n\nThen we have\n\n\[ ... | Yes |
Proposition 1.9 If \( {A}_{n} = 1 + {a}_{n} \) and \( \sum \left| {a}_{n}\right| \) converges, then the product \( \mathop{\prod }\limits_{n}{A}_{n} \) converges, and this product vanishes if and only if one of its factors \( {A}_{n} \) vanishes. Also, if \( {a}_{n} \neq 1 \) for all \( n \), then \( \mathop{\prod }\li... | Proof. If \( \sum \left| {a}_{n}\right| \) converges, then for all large \( n \) we must have \( \left| {a}_{n}\right| < \) \( 1/2 \) . Disregarding finitely many terms if necessary, we may assume that this inequality holds for all \( n \) . Then we may write the partial products as follows:\n\n\[ \mathop{\prod }\limit... | Yes |
Theorem 1.10 For every \( s > 1 \), we have\n\n\[ \zeta \left( s\right) = \mathop{\prod }\limits_{p}\frac{1}{1 - 1/{p}^{s}} \]\n\nwhere the product is taken over all primes. | Proof. Suppose \( M \) and \( N \) are positive integers with \( M > N \). Observe now that any positive integer \( n \leq N \) can be written uniquely as a product of primes, and that each prime must be less than or equal to \( N \) and repeated less than \( M \) times. Therefore\n\n\[ \mathop{\sum }\limits_{{n = 1}}^... | Yes |
Proposition 1.11 The series\n\n\\[ \n\\mathop{\\sum }\\limits_{p}1/p \n\\]\n\n diverges, when the sum is taken over all primes p. | Proof. We take logarithms of both sides of the Euler formula. Since \\( \\log x \\) is continuous, we may write the logarithm of the infinite product as the sum of the logarithms. Therefore, we obtain for \\( s > 1 \\)\n\n\\[ \n- \\mathop{\\sum }\\limits_{p}\\log \\left( {1 - 1/{p}^{s}}\\right) = \\log \\zeta \\left( s... | Yes |
Lemma 2.2 The Dirichlet characters are multiplicative. Moreover, | \[ {\delta }_{\ell }\left( m\right) = \frac{1}{\varphi \left( q\right) }\mathop{\sum }\limits_{\chi }\overline{\chi \left( \ell \right) }\chi \left( m\right) \] where the sum is over all Dirichlet characters. With the above lemma we have taken our first step towards a proof of the theorem, since this lemma shows that \... | No |
Theorem 2.3 If \( \chi \) is a nontrivial Dirichlet character, then the sum\n\n\[ \mathop{\sum }\limits_{p}\frac{\chi \left( p\right) }{{p}^{s}} \]\n\nremains bounded as \( s \rightarrow {1}^{ + } \) . | The proof of Theorem 2.3 requires the introduction of the \( L \) -functions, to which we now turn. | No |
Theorem 2.4 If \( s > 1 \), then\n\n\[ \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{\chi \left( n\right) }{{n}^{s}} = \mathop{\prod }\limits_{p}\frac{1}{\left( 1 - \chi \left( p\right) {p}^{-s}\right) } \]\n\nwhere the product is over all primes. | Null | No |
Proposition 3.1 The logarithm function \( {\log }_{1} \) satisfies the following properties:\n\n(i) If \( \left| z\right| < 1 \), then\n\n\[ \n{e}^{{\log }_{1}\left( \frac{1}{1 - z}\right) } = \frac{1}{1 - z}.\n\]\n\n(ii) If \( \left| z\right| < 1 \), then\n\n\[ \n{\log }_{1}\left( \frac{1}{1 - z}\right) = z + {E}_{1}\... | Proof. To establish the first property, let \( z = r{e}^{i\theta } \) with \( 0 \leq r < 1 \) , and observe that it suffices to show that\n\n(5)\n\n\[ \n\left( {1 - r{e}^{i\theta }}\right) {e}^{\mathop{\sum }\limits_{{k = 1}}^{\infty }{\left( r{e}^{i\theta }\right) }^{k}/k} = 1.\n\]\n\nTo do so, we differentiate the le... | Yes |
Proposition 3.2 If \( \sum \left| {a}_{n}\right| \) converges, and \( {a}_{n} \neq 1 \) for all \( n \), then\n\n\[ \mathop{\prod }\limits_{{n = 1}}^{\infty }\left( \frac{1}{1 - {a}_{n}}\right) \]\n\nconverges. Moreover, this product is non-zero. | Proof. For \( n \) large enough, \( \left| {a}_{n}\right| < 1/2 \), so we may assume without loss of generality that this inequality holds for all \( n \geq 1 \) . Then\n\n\[ \mathop{\prod }\limits_{{n = 1}}^{N}\left( \frac{1}{1 - {a}_{n}}\right) = \mathop{\prod }\limits_{{n = 1}}^{N}{e}^{{\log }_{1}\left( \frac{1}{1 -... | Yes |
Proposition 3.3 Suppose \( {\chi }_{0} \) is the trivial Dirichlet character,\n\n\[ \n{\chi }_{0}\left( n\right) = \left\{ \begin{array}{ll} 1 & \text{ if }n\text{ and }q\text{ are relatively prime,} \\ 0 & \text{ otherwise,} \end{array}\right.\n\]\n\nand \( q = {p}_{1}^{{a}_{1}}\cdots {p}_{N}^{{a}_{N}} \) is the prime... | Proof. The identity follows at once on comparing the Dirichlet and Euler product formulas. The final statement holds because \( \zeta \left( s\right) \rightarrow \infty \) as \( s \rightarrow {1}^{ + } \) | Yes |
Lemma 3.5 If \( \chi \) is a non-trivial Dirichlet character, then\n\n\[ \left| {\mathop{\sum }\limits_{{n = 1}}^{k}\chi \left( n\right) }\right| \leq q,\;\text{ for any }k. \] | Proof. First, we recall that\n\n\[ \mathop{\sum }\limits_{{n = 1}}^{q}\chi \left( n\right) = 0 \]\n\nIn fact, if \( S \) denotes the sum and \( a \in {\mathbb{Z}}^{ * }\left( q\right) \), then the multiplicative property of the Dirichlet character \( \chi \) gives\n\n\[ \chi \left( a\right) S = \sum \chi \left( a\right... | Yes |
Proposition 3.6 If \( s > 1 \), then\n\n\[ \n{e}^{{\log }_{2}L\left( {s,\chi }\right) } = L\left( {s,\chi }\right) \n\]\n\nMoreover\n\n\[ \n{\log }_{2}L\left( {s,\chi }\right) = \mathop{\sum }\limits_{p}{\log }_{1}\left( \frac{1}{1 - \chi \left( p\right) /{p}^{s}}\right) .\n\] | Proof. Differentiating \( {e}^{-{\log }_{2}L\left( {s,\chi }\right) }L\left( {s,\chi }\right) \) with respect to \( s \) gives\n\n\[ \n- \frac{{L}^{\prime }\left( {s,\chi }\right) }{L\left( {s,\chi }\right) }{e}^{-{\log }_{2}L\left( {s,\chi }\right) }L\left( {s,\chi }\right) + {e}^{-{\log }_{2}L\left( {s,\chi }\right) ... | Yes |
Lemma 3.8 If \( s > 1 \), then\n\n\[ \mathop{\prod }\limits_{\chi }L\left( {s,\chi }\right) \geq 1 \]\n\nwhere the product is taken over all Dirichlet characters. In particular the product is real-valued. | Proof. We have shown earlier that for \( s > 1 \)\n\n\[ L\left( {s,\chi }\right) = \exp \left( {\mathop{\sum }\limits_{p}{\log }_{1}\left( \frac{1}{1 - \chi \left( p\right) {p}^{-s}}\right) }\right) .\n\nHence,\n\n\[ \mathop{\prod }\limits_{\chi }L\left( {s,\chi }\right) = \exp \left( {\mathop{\sum }\limits_{\chi }\mat... | Yes |
Proposition 3.10 If \( N \) is a positive integer, then:\n\n(i) \( \mathop{\sum }\limits_{{1 \leq n \leq N}}\frac{1}{n} = {\int }_{1}^{N}\frac{dx}{x} + O\left( 1\right) = \log N + O\left( 1\right) \).\n\n(ii) More precisely, there exists a real number \( \gamma \), called Euler’s constant, so that\n\n\[ \mathop{\sum }\... | Proof. It suffices to establish the more refined estimate given in part (ii). Let\n\n\[ {\gamma }_{n} = \frac{1}{n} - {\int }_{n}^{n + 1}\frac{dx}{x} \]\n\nSince \( 1/x \) is decreasing, we clearly have\n\n\[ 0 \leq {\gamma }_{n} \leq \frac{1}{n} - \frac{1}{n + 1} \leq \frac{1}{{n}^{2}} \]\n\nso the series \( \mathop{\... | Yes |
Proposition 3.11 If \( N \) is a positive integer, then\n\n\[ \mathop{\sum }\limits_{{1 \leq n \leq N}}\frac{1}{{n}^{1/2}} = {\int }_{1}^{N}\frac{dx}{{x}^{1/2}} + {c}^{\prime } + O\left( {1/{N}^{1/2}}\right) \]\n\n\[ = 2{N}^{1/2} + c + O\left( {1/{N}^{1/2}}\right) \text{.} \] | The proof is essentially a repetition of the proof of the previous proposition, this time using the fact that\n\n\[ \left| {\frac{1}{{n}^{1/2}} - \frac{1}{{\left( n + 1\right) }^{1/2}}}\right| \leq \frac{C}{{n}^{3/2}} \]\n\nThis last inequality follows from the mean-value theorem applied to \( f\left( x\right) = {x}^{-... | Yes |
Theorem 3.12 If \( k \) is a positive integer, then\n\n\[ \frac{1}{N}\mathop{\sum }\limits_{{k = 1}}^{N}d\left( k\right) = \log N + O\left( 1\right) \]\n\nMore precisely,\n\n\[ \frac{1}{N}\mathop{\sum }\limits_{{k = 1}}^{N}d\left( k\right) = \log N + \left( {{2\gamma } - 1}\right) + O\left( {1/{N}^{1/2}}\right) ,\]\n\n... | Proof. Let \( {S}_{N} = \mathop{\sum }\limits_{{k = 1}}^{N}d\left( k\right) \) . We observed that summing \( F = 1 \) along hyperbolas gives \( {S}_{N} \) . Summing vertically, we find\n\n\[ {S}_{N} = \mathop{\sum }\limits_{{1 \leq m \leq N}}\mathop{\sum }\limits_{{1 \leq n \leq N/m}}1 \]\n\nBut \( \mathop{\sum }\limit... | Yes |
Proposition 3.13 The following statements are true:\n\n(i) \( {S}_{N} \geq c\log N \) for some constant \( c > 0 \) .\n\n(ii) \( {S}_{N} = 2{N}^{1/2}L\left( {1,\chi }\right) + O\left( 1\right) \) . | It suffices to prove the proposition, since the assumption \( L\left( {1,\chi }\right) = 0 \) would give an immediate contradiction.\n\nWe first sum along hyperbolas. Observe that\n\n\[ \mathop{\sum }\limits_{{{nm} = k}}\frac{\chi \left( n\right) }{{\left( nm\right) }^{1/2}} = \frac{1}{{k}^{1/2}}\mathop{\sum }\limits_{... | Yes |
Lemma 3.14 \( \mathop{\sum }\limits_{{n \mid k}}\chi \left( n\right) \geq \left\{ \begin{array}{ll} 0 & \text{ for all }k \\ 1 & \text{ if }k = {\ell }^{2}\text{ for some }\ell \in \mathbb{Z}. \end{array}\right. \) | The proof of the lemma is simple. If \( k \) is a power of a prime, say \( k = {p}^{a} \), then the divisors of \( k \) are \( 1, p,{p}^{2},\ldots ,{p}^{a} \) and\n\n\[ \mathop{\sum }\limits_{{n \mid k}}\chi \left( n\right) = \chi \left( 1\right) + \chi \left( p\right) + \chi \left( {p}^{2}\right) + \cdots + \chi \left... | Yes |
Lemma 3.15 For all integers \( 0 < a < b \) we have\n\n(i) \( \mathop{\sum }\limits_{{n = a}}^{b}\frac{\chi \left( n\right) }{{n}^{1/2}} = O\left( {a}^{-1/2}\right) \), \n\n(ii) \( \mathop{\sum }\limits_{{n = a}}^{b}\frac{\chi \left( n\right) }{n} = O\left( {a}^{-1}\right) \). | Proof. This argument is similar to the proof of Proposition 3.4; we use summation by parts. Let \( {s}_{n} = \mathop{\sum }\limits_{{1 \leq k \leq n}}\chi \left( k\right) \), and remember that \( \left| {s}_{n}\right| \leq q \) for all \( n \) . Then\n\n\[ \mathop{\sum }\limits_{{n = a}}^{b}\frac{\chi \left( n\right) }... | Yes |
Lemma 1.2 If \( f \) is real-valued integrable on \( \left\lbrack {a, b}\right\rbrack \) and \( \varphi \) is a real-valued continuous function on \( \mathbb{R} \), then \( \varphi \circ f \) is also integrable on \( \left\lbrack {a, b}\right\rbrack \) . | Proof. Let \( \epsilon > 0 \) and remember that \( f \) is bounded, say \( \left| f\right| \leq M \) . Since \( \varphi \) is uniformly continuous on \( \left\lbrack {-M, M}\right\rbrack \) we may choose \( \delta > 0 \) so that if \( s, t \in \left\lbrack {-M, M}\right\rbrack \) and \( \left| {s - t}\right| < \delta \... | Yes |
Proposition 1.3 A bounded monotonic function \( f \) on an interval \( \left\lbrack {a, b}\right\rbrack \) is integrable. | Proof. We may assume without loss of generality that \( a = 0, b = 1 \) , and \( f \) is monotonically increasing. Then, for each \( N \), we choose the uniform partition \( {P}_{N} \) given by \( {x}_{j} = j/N \) for all \( j = 0,\ldots, N \) . If \( {\alpha }_{j} = \) \( f\left( {x}_{j}\right) \), then we have\n\n\[ ... | Yes |
Proposition 1.4 Let \( f \) be a bounded function on the compact interval \( \left\lbrack {a, b}\right\rbrack \) . If \( c \in \left( {a, b}\right) \), and if for all small \( \delta > 0 \) the function \( f \) is integrable on the intervals \( \left\lbrack {a, c - \delta }\right\rbrack \) and \( \left\lbrack {c + \del... | Proof. Suppose \( \left| f\right| \leq M \) and let \( \epsilon > 0 \) . Choose \( \delta > 0 \) (small) so that \( {4\delta M} \leq \epsilon /3 \) . Now let \( {P}_{1} \) and \( {P}_{2} \) be partitions of \( \left\lbrack {a, c - \delta }\right\rbrack \) and \( \lbrack c + \) \( \delta, b\rbrack \) so that for each \(... | Yes |
Lemma 1.6 The union of countably many sets of measure 0 has measure 0. | Proof. Say \( {E}_{1},{E}_{2},\ldots \) are sets of measure 0, and let \( E = { \cup }_{i = 1}^{\infty }{E}_{i} \) . Let \( \epsilon > 0 \), and for each \( i \) choose open interval \( {I}_{i,1},{I}_{i,2},\ldots \) so that\n\n\[ \n{E}_{i} \subset \mathop{\bigcup }\limits_{{k = 1}}^{\infty }{I}_{i, k}\;\text{ and }\;\m... | Yes |
Lemma 1.8 If \( \epsilon > 0 \), then the set \( {A}_{\epsilon } \) is closed and therefore compact. | Proof. The argument is simple. Suppose \( {c}_{n} \in {A}_{\epsilon } \) converges to \( c \) and assume that \( c \notin {A}_{\epsilon } \) . Write \( \operatorname{osc}\left( {f, c}\right) = \epsilon - \delta \) where \( \delta > 0 \) . Select \( r \) so that \( \operatorname{osc}\left( {f, c, r}\right) < \epsilon - ... | Yes |
Theorem 2.1 Let \( f \) be a continuous function defined on a closed rectangle \( R \subset {\mathbb{R}}^{d} \). Suppose \( R = {R}_{1} \times {R}_{2} \) where \( {R}_{1} \subset {\mathbb{R}}^{{d}_{1}} \) and \( {R}_{2} \subset {\mathbb{R}}^{{d}_{2}} \) with \( d = {d}_{1} + {d}_{2} \). If we write \( x = \left( {{x}_{... | Proof. The continuity of \( F \) follows from the uniform continuity of \( f \) on \( R \) and the fact that \[ \left| {F\left( {x}_{1}\right) - F\left( {x}_{1}^{\prime }\right) }\right| \leq {\int }_{{R}_{2}}\left| {f\left( {{x}_{1},{x}_{2}}\right) - f\left( {{x}_{1}^{\prime },{x}_{2}}\right) }\right| d{x}_{2}. \] To ... | Yes |
Theorem 2.2 Suppose \( A \) and \( B \) are compact subsets of \( {\mathbb{R}}^{d} \) and \( g : A \rightarrow B \) is a diffeomorphism of class \( {C}^{1} \) . If \( f \) is continuous on \( B \) , then\n\n\[ \n{\int }_{g\left( A\right) }f\left( x\right) {dx} = {\int }_{A}f\left( {g\left( y\right) }\right) \left| {\de... | The proof of this theorem consists first of an analysis of the special situation when \( g \) is a linear transformation \( L \) . In this case, if \( R \) is a rectangle, then\n\n\[ \n\left| {g\left( R\right) }\right| = \left| {\det \left( L\right) }\right| \left| R\right|\n\]\nwhich explains the term \( \left| {\det ... | No |
Theorem 1.5. (Plane Curve Classification Theorem) Suppose \( \gamma \) and \( \widetilde{\gamma } : \left\lbrack {a, b}\right\rbrack \rightarrow {\mathbf{R}}^{2} \) are smooth, unit speed plane curves with unit normal vector fields \( N \) and \( \widetilde{N} \), and \( {\kappa }_{N}\left( t\right) ,{\kappa }_{\wideti... | Null | No |
Theorem 1.6. (Total Curvature Theorem) If \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow {\mathbf{R}}^{2} \) is a unit speed simple closed curve such that \( \dot{\gamma }\left( a\right) = \dot{\gamma }\left( b\right) \), and \( N \) is the inward-pointing normal, then\n\n\[{\int }_{a}^{b}{\kappa }_{N}\left( ... | The second will be derived as a consequence of a more general result in Chapter 9; the proof of the first is left to Problem 9-6. | No |
Theorem 1.7. (Uniformization Theorem) Every connected 2-manifold is diffeomorphic to a quotient of one of the three constant curvature model surfaces listed above by a discrete group of isometries acting freely and properly discontinuously. Therefore, every connected 2-manifold has a complete Riemannian metric with con... | Null | No |
Theorem 1.8. (Gauss-Bonnet Theorem) Let \( S \) be an oriented compact 2-manifold with a Riemannian metric. Then\n\n\[{\int }_{S}{KdA} = {2\pi \chi }\left( S\right)\]\n\nwhere \( \chi \left( S\right) \) is the Euler characteristic of \( S \) (which is equal to 2 if \( S \) is the sphere, 0 if it is the torus, and 2 -2g... | Null | No |
Theorem 1.9. (Classification of Constant Curvature Metrics) \( A \) complete, connected Riemannian manifold \( M \) with constant sectional curvature is isometric to \( \widetilde{M}/\Gamma \), where \( \widetilde{M} \) is one of the constant curvature model spaces \( {\mathbf{R}}^{n},{\mathbf{S}}_{R}^{n} \), or \( {\m... | Null | No |
Theorem 1.11. (Bonnet) Suppose \( M \) is a complete, connected Riemannian manifold with all sectional curvatures bounded below by a positive constant. Then \( M \) is compact and has a finite fundamental group. | Null | No |
Lemma 2.1. Let \( V \) be a finite-dimensional vector space. There is a natural (basis-independent) isomorphism between \( {T}_{l + 1}^{k}\left( V\right) \) and the space of multilinear maps\n\n\[ \underset{l}{\underbrace{{V}^{ * } \times \cdots \times {V}^{ * }}} \times \underset{k}{\underbrace{V \times \cdots \times ... | Exercise 2.1. Prove Lemma 2.1. [Hint: In the special case \( k = 1, l = 0 \) , consider the map \( \Phi : \operatorname{End}\left( V\right) \rightarrow {T}_{1}^{1}\left( V\right) \) by letting \( {\Phi A} \) be the \( \left( \begin{array}{l} 1 \\ 1 \end{array}\right) \) -tensor defined by \( {\Phi A}\left( {\omega, X}\... | No |
Lemma 2.2. Let \( M \) be a smooth manifold, \( E \) a set, and \( \pi : E \rightarrow M \) a surjective map. Suppose we are given an open covering \( \left\{ {U}_{\alpha }\right\} \) of \( M \) together with bijective maps \( {\varphi }_{\alpha } : {\pi }^{-1}\left( {U}_{\alpha }\right) \rightarrow {U}_{\alpha } \time... | Proof. For each \( p \in M \), let \( {E}_{p} = {\pi }^{-1}\left( p\right) \). If \( p \in {U}_{\alpha } \), observe that the map \( {\left( {\varphi }_{\alpha }\right) }_{p} : {E}_{p} \rightarrow \{ p\} \times {\mathbf{R}}^{k} \) obtained by restricting \( {\varphi }_{\alpha } \) is a bijection. We can define a vector... | Yes |
Lemma 2.3. Let \( F : M \rightarrow E \) be a section of a vector bundle. \( F \) is smooth if and only if the components \( {F}_{{i}_{1}\ldots {i}_{k}}^{{j}_{1}\ldots {j}_{l}} \) of \( F \) in terms of any smooth local frame \( \left\{ {E}_{i}\right\} \) on an open set \( U \in \dot{M} \) depend smoothly on \( p \in U... | Null | No |
Lemma 2.4. (Tensor Characterization Lemma) A map\n\n\\[ \n\\tau : {\\Im }^{1}\\left( M\\right) \\times \\cdots \\times {\\Im }^{1}\\left( M\\right) \\times \\Im \\left( M\\right) \\times \\cdots \\times \\Im \\left( M\\right) \\rightarrow {C}^{\\infty }\\left( M\\right) \n\\]\n\nis induced by a \\( \\left( \\begin{arra... | Null | No |
Lemma 3.1. Let \( g \) be a Riemannian metric on a manifold \( M \) . There is a unique fiber metric on each tensor bundle \( {T}_{l}^{k}M \) with the property that if \( \left( {{E}_{1},\ldots ,{E}_{n}}\right) \) is an orthonormal basis for \( {T}_{p}M \) and \( \left( {{\varphi }^{1},\ldots ,{\varphi }^{n}}\right) \)... | Exercise 3.8. Prove Lemma 3.1 by showing that in any local coordinate system, the required inner product is given by\n\n\[ \langle F, G\rangle = {g}^{{i}_{1}{r}_{1}}\cdots {g}^{{i}_{k}{r}_{k}}{g}_{{j}_{1}{s}_{1}}\cdots {g}_{{j}_{l}{s}_{l}}{F}_{{i}_{1}\ldots {i}_{k}}^{{j}_{1}\ldots {j}_{l}}{G}_{{r}_{1}\ldots {r}_{k}}^{{... | No |
Lemma 3.2. On any oriented Riemannian n-manifold \( \left( {M, g}\right) \), there is a unique \( n \) -form \( {dV} \) satisfying the property that \( {dV}\left( {{E}_{1},\ldots ,{E}_{n}}\right) = 1 \) whenever \( \left( {{E}_{1},\ldots ,{E}_{n}}\right) \) is an oriented orthonormal basis for some tangent space \( {T}... | Exercise 3.9. Prove Lemma 3.2, and show that the expression for \( {dV} \) with respect to any oriented local frame \( \left\{ {E}_{i}\right\} \) is\n\n\[ \n{dV} = \sqrt{\det \left( {g}_{ij}\right) }{\varphi }^{1} \land \cdots \land {\varphi }^{n}, \n\] \n\nwhere \( {g}_{ij} = \left\langle {{E}_{i},{E}_{j}}\right\rangl... | No |
Proposition 3.3. \( O\left( {n + 1}\right) \) acts transitively on orthonormal bases on \( {\mathbf{S}}_{R}^{n} \) . More precisely, given any two points \( p,\widetilde{p} \in {\mathbf{S}}_{R}^{n} \), and orthonormal bases \( \left\{ {E}_{i}\right\} \) for \( {T}_{p}{\mathbf{S}}_{R}^{n} \) and \( \left\{ {\widetilde{E... | Proof. It suffices to show that given any \( p \in {\mathbf{S}}_{R}^{n} \) and any orthonormal basis \( \left\{ {E}_{i}\right\} \) for \( {T}_{p}{\mathbf{S}}_{R}^{n} \), there is an orthogonal map that takes the \ | No |
Lemma 3.4. Stereographic projection is a conformal equivalence between \( {\mathbf{S}}_{R}^{n} - \{ N\} \) and \( {\mathbf{R}}^{n} \) . | Proof. The inverse map \( {\sigma }^{-1} \) is a local parametrization, so we will use it to compute the pullback metric. Consider an arbitrary point \( q \in {\mathbf{R}}^{n} \) and a vector \( V \in {T}_{q}{\mathbf{R}}^{n} \), and compute\n\n\[ \n{\left( {\sigma }^{-1}\right) }^{ * }{\overset{ \circ }{g}}_{R}\left( {... | Yes |
Proposition 3.6. \( {O}_{ + }\left( {n,1}\right) \) acts transitively on the set of orthonormal bases on \( {\mathbf{H}}_{R}^{n} \), and therefore \( {\mathbf{H}}_{R}^{n} \) is homogeneous and isotropic. | Proof. The argument is entirely analogous to the proof of Proposition 3.3, so we give only a sketch. If \( p \in {\mathbf{H}}_{R}^{n} \) and \( \left\{ {E}_{i}\right\} \) is an orthonormal basis for \( {T}_{p}{\mathbf{H}}_{R}^{n} \), an easy computation shows that \( \left\{ {{E}_{1},\ldots ,{E}_{n},{E}_{n + 1} = p/R}\... | Yes |
Lemma 4.1. If \( \nabla \) is a connection in a bundle \( E, X \in \mathcal{T}\left( M\right), Y \in \mathcal{E}\left( M\right) \) , and \( p \in M \), then \( {\left. {\nabla }_{X}Y\right| }_{p} \) depends only on the values of \( X \) and \( Y \) in an arbitrarily small neighborhood of \( p \) . More precisely, if \(... | Proof. First consider \( Y \) . Replacing \( Y \) by \( Y - \widetilde{Y} \), it clearly suffices to show that \( {\left. {\nabla }_{X}Y\right| }_{p} = 0 \) if \( Y \) vanishes on a neighborhood \( U \) of \( p \) .\n\nChoose a bump function \( \varphi \in {C}^{\infty }\left( M\right) \) with support in \( U \) such th... | Yes |
Lemma 4.2. With notation as in Lemma 4.1, \( {\left. {\nabla }_{X}Y\right| }_{p} \) depends only on the values of \( Y \) in a neighborhood of \( p \) and the value of \( X \) at \( p \) . | Proof. By linearity, it suffices to show that \( {\left. {\nabla }_{X}Y\right| }_{p} = 0 \) whenever \( {X}_{p} = \) 0 . Choose a coordinate neighborhood \( U \) of \( p \), and write \( X = {X}^{i}{\partial }_{i} \) in coordinates on \( U \), with \( {X}^{i}\left( p\right) = 0 \) . Then, for any \( Y \in \mathcal{E}\l... | Yes |
Lemma 4.3. Let \( \nabla \) be a linear connection, and let \( X, Y \in \mathcal{T}\left( U\right) \) be expressed in terms of a local frame by \( X = {X}^{i}{E}_{i}, Y = {Y}^{j}{E}_{j} \) . Then\n\n\[ \n{\nabla }_{X}Y = \left( {X{Y}^{k} + {X}^{i}{Y}^{j}{\Gamma }_{ij}^{k}}\right) {E}_{k} \n\]\n\n(4.3) | Proof. Just use the defining rules for a connection and compute:\n\n\[ \n{\nabla }_{X}Y = {\nabla }_{X}\left( {{Y}^{j}{E}_{j}}\right) \n\]\n\n\[ \n= \left( {X{Y}^{j}}\right) {E}_{j} + {Y}^{j}{\nabla }_{{X}^{i}{E}_{i}}{E}_{j} \n\]\n\n\[ \n= \left( {X{Y}^{j}}\right) {E}_{j} + {X}^{i}{Y}^{j}{\nabla }_{{E}_{i}}{E}_{j} \n\]... | Yes |
Lemma 4.4. Suppose \( M \) is a manifold covered by a single coordinate chart. There is a one-to-one correspondence between linear connections on \( M \) and choices of \( {n}^{3} \) smooth functions \( \left\{ {\Gamma }_{ij}^{k}\right\} \) on \( M \), by the rule\n\n\[{\nabla }_{X}Y = \left( {{X}^{i}{\partial }_{i}{Y}... | Proof. Observe that (4.5) is equivalent to (4.3) when \( {E}_{i} = {\partial }_{i} \) is a coordinate frame, so for every connection the functions \( \left\{ {\Gamma }_{ij}^{k}\right\} \) defined by (4.2) satisfy (4.5). On the other hand, given \( \left\{ {\Gamma }_{ij}^{k}\right\} \), it is easy to see by inspection t... | No |
Proposition 4.5. Every manifold admits a linear connection. | Proof. Cover \( M \) with coordinate charts \( \left\{ {U}_{\alpha }\right\} \) ; the preceding lemma guarantees the existence of a connection \( {\nabla }^{\alpha } \) on each \( {U}_{\alpha } \) . Choosing a partition of unity \( \left\{ {\varphi }_{\alpha }\right\} \) subordinate to \( \left\{ {U}_{\alpha }\right\} ... | Yes |
Lemma 4.7. If \( \nabla \) is a linear connection on \( M \), and \( F \in {\mathcal{T}}_{l}^{k}\left( M\right) \), the map \( \nabla F : {\mathcal{T}}^{1}\left( M\right) \times \cdots \times {\mathcal{T}}^{1}\left( M\right) \times \mathcal{T}\left( M\right) \times \cdots \times \mathcal{T}\left( M\right) \rightarrow {... | Proof. This follows immediately from the tensor characterization lemma: \( {\nabla }_{X}F \) is a tensor field, so it is multilinear over \( {C}^{\infty }\left( M\right) \) in its \( k + l \) arguments; and it is linear over \( {C}^{\infty }\left( M\right) \) in \( X \) by definition of a connection. | Yes |
Lemma 4.8. Let \( \nabla \) be a linear connection. The components of the total covariant derivative of a \( \left( \begin{array}{l} k \\ l \end{array}\right) \) -tensor field \( F \) with respect to a coordinate system are given by \[ {F}_{{i}_{1}\ldots {i}_{k};m}^{{j}_{1}\ldots {j}_{l}} = {\partial }_{m}{F}_{{i}_{1}\... | Exercise 4.6. Prove Lemma 4.8. | No |
Lemma 4.9. Let \( \nabla \) be a linear connection on \( M \) . For each curve \( \gamma : I \rightarrow \) \( M,\nabla \) determines a unique operator\n\n\[ \n{D}_{t} : \mathcal{T}\left( \gamma \right) \rightarrow \mathcal{T}\left( \gamma \right)\n\]\n\nsatisfying the following properties:\n\n(a) Linearity over \( \ma... | Proof. First we show uniqueness. Suppose \( {D}_{t} \) is such an operator, and let \( {t}_{0} \in I \) be arbitrary. An argument similar to that of Lemma 4.1 shows that the value of \( {D}_{t}V \) at \( {t}_{0} \) depends only on the values of \( V \) in any interval \( \left( {{t}_{0} - \varepsilon ,{t}_{0} + \vareps... | Yes |
Theorem 4.10. (Existence and Uniqueness of Geodesics) Let \( M \) be a manifold with a linear connection. For any \( p \in M \), any \( V \in {T}_{p}M \), and any \( {t}_{0} \in \mathbf{R} \), there exist an open interval \( I \subset \mathbf{R} \) containing \( {t}_{0} \) and a geodesic \( \gamma : I \rightarrow M \) ... | Proof. Choose coordinates \( \left( {x}^{i}\right) \) on some neighborhood \( U \) of \( p \). From (4.10), a curve \( \gamma : I \rightarrow U \) is a geodesic if and only if its component functions \( \gamma \left( t\right) = \left( {{x}^{1}\left( t\right) ,\ldots ,{x}^{n}\left( t\right) }\right) \) satisfy the geode... | Yes |
Theorem 4.11. (Parallel Translation) Given a curve \( \gamma : I \rightarrow M,{t}_{0} \in \) \( I \), and a vector \( {V}_{0} \in {T}_{\gamma \left( {t}_{0}\right) }M \), there exists a unique parallel vector field \( V \) along \( \gamma \) such that \( V\left( {t}_{0}\right) = {V}_{0} \) . | Proof of Theorem 4.11. First suppose \( \gamma \left( I\right) \) is contained in a single coordinate chart. Then, using formula (4.10), \( V \) is parallel along \( \gamma \) if and only if\n\n\[{\dot{V}}^{k}\left( t\right) = - {V}^{j}\left( t\right) {\dot{\gamma }}^{i}\left( t\right) {\Gamma }_{ij}^{k}\left( {\gamma ... | Yes |
Theorem 4.12. (Existence and Uniqueness for Linear ODEs) Let \( I \subset \mathbf{R} \) be an interval, and for \( 1 \leq j, k \leq n \) let \( {A}_{j}^{k} : I \rightarrow \mathbf{R} \) be arbitrary smooth functions. The linear initial-value problem\n\n\[ \n{\dot{V}}^{k}\left( t\right) = {A}_{j}^{k}\left( t\right) {V}^... | Exercise 4.11. Prove Theorem 4.12, as follows. Consider the vector field \( Y \) on \( I \times {\mathbf{R}}^{n} \) given by\n\n\[ \n{Y}^{0}\left( {{x}^{0},\ldots ,{x}^{n}}\right) = 1 \n\]\n\n\[ \n{Y}^{k}\left( {{x}^{0},\ldots ,{x}^{n}}\right) = {A}_{j}^{k}\left( {x}^{0}\right) {x}^{j},\;k = 1,\ldots, n. \n\]\n\n(a) Sh... | No |
Lemma 5.1. The operator \( {\nabla }^{\top } \) is well defined, and is a connection on \( M \) . | Proof. Since the value of \( {\bar{\nabla }}_{X}Y \) at a point \( p \in M \) depends only on \( {X}_{p} \) , \( {\nabla }_{X}^{\top }Y \) is clearly independent of the choice of vector field extending \( X \) . On the other hand, because of the result of Exercise 4.7, the value of \( {\bar{\nabla }}_{X}Y \) at \( p \)... | Yes |
Lemma 5.2. The following conditions are equivalent for a linear connection \( \nabla \) on a Riemannian manifold:\n\n(a) \( \nabla \) is compatible with \( g \) .\n\n(b) \( \nabla g \equiv 0 \) .\n\n(c) If \( V, W \) are vector fields along any curve \( \gamma \) ,\n\n\[ \n\frac{d}{dt}\langle V, W\rangle = \left\langle... | Null | No |
Lemma 5.3. The tangential connection on an embedded submanifold \( M \subset \) \( {\mathbf{R}}^{n} \) is symmetric. | Exercise 5.3. Prove Lemma 5.3. [Hint: If \( X \) and \( Y \) are vector fields on \( {\mathbf{R}}^{n} \) that are tangent to \( M \) at points of \( M \), so is \( \left\lbrack {X, Y}\right\rbrack \) by Exercise 2.3.] | No |
Theorem 5.4. (Fundamental Lemma of Riemannian Geometry) Let \( \left( {M, g}\right) \) be a Riemannian (or pseudo-Riemannian) manifold. There exists a unique linear connection \( \nabla \) on \( M \) that is compatible with \( g \) and symmetric. | Proof. We prove uniqueness first, by deriving a formula for \( \nabla \) . Suppose, therefore, that \( \nabla \) is such a connection, and let \( X, Y, Z \in \mathcal{T}\left( M\right) \) be arbitrary vector fields. Writing the compatibility equation three times with \( X, Y, Z \) cyclically permuted, we obtain\n\n\[ X... | Yes |
Lemma 5.5. All Riemannian geodesics are constant speed curves. | Proof. Let \( \gamma \) be a Riemannian geodesic. Since \( \dot{\gamma } \) is parallel along \( \gamma \), its length \( \left| \dot{\gamma }\right| = \langle \dot{\gamma },\dot{\gamma }{\rangle }^{1/2} \) is constant by Lemma 5.2(d). | Yes |
Proposition 5.6. (Naturality of the Riemannian Connection) Suppose \( \varphi : \left( {M, g}\right) \rightarrow \left( {\widetilde{M},\widetilde{g}}\right) \) is an isometry.\n\n(a) \( \varphi \) takes the Riemannian connection \( \nabla \) of \( g \) to the Riemannian connection \( \widetilde{\nabla } \) of \( \widet... | Exercise 5.4. Prove Proposition 5.6 as follows. For part (a), define a map\n\n\[{\varphi }^{ * }\widetilde{\nabla } : \mathfrak{T}\left( M\right) \times \mathfrak{T}\left( M\right) \rightarrow \mathfrak{T}\left( M\right)\]\n\nby\n\n\[{\left( {\varphi }^{ * }\widetilde{\nabla }\right) }_{X}Y = {\varphi }_{ * }^{-1}\left... | No |
Proposition 5.7. (Properties of the Exponential Map)\n\n(a) \( \\mathcal{E} \) is an open subset of TM containing the zero section, and each set \( {\\mathcal{E}}_{p} \) is star-shaped with respect to 0 .\n\n(b) For each \( V \\in {TM} \), the geodesic \( {\\gamma }_{V} \) is given by\n\n\[ \n{\\gamma }_{V}\\left( t\\r... | Proof of Proposition 5.7. The rescaling lemma with \( t = 1 \) says precisely that \( \\exp \\left( {cV}\\right) = {\\gamma }_{cV}\\left( 1\\right) = {\\gamma }_{V}\\left( c\\right) \) whenever either side is defined; this is (b). Moreover, if \( V \\in {\\mathcal{E}}_{p} \), by definition \( {\\gamma }_{V} \) is defin... | No |
Lemma 5.8. (Rescaling Lemma) For any \( V \in {TM} \) and \( c, t \in R \) , \[ {\gamma }_{cV}\left( t\right) = {\gamma }_{V}\left( {ct}\right) \] whenever either side is defined. | Proof. It suffices to show that \( {\gamma }_{cV}\left( t\right) \) exists and (5.5) holds whenever the right-hand side is defined, for then the converse statement follows by replacing \( V \) by \( {cV}, t \) by \( {ct} \), and \( c \) by \( 1/c \) . Suppose the domain of \( {\gamma }_{V} \) is the open interval \( I ... | Yes |
Proposition 5.9. (Naturality of the Exponential Map) Suppose that \( \varphi : \left( {M, g}\right) \rightarrow \left( {\widetilde{M},\widetilde{g}}\right) \) is an isometry. Then, for any \( p \in M \), the following diagram commutes:  For any \( p \in M \), there is a neighborhood \( \mathcal{V} \) of the origin in \( {T}_{p}M \) and a neighborhood \( \mathcal{U} \) of \( p \) in \( M \) such that \( {\exp }_{p} : \mathcal{V} \rightarrow \mathcal{U} \) is a diffeomorphism. | Proof. This follows immediately from the inverse function theorem, once we show that \( {\left( {\exp }_{p}\right) }_{ * } \) is invertible at 0 . Since \( {T}_{p}M \) is a vector space, there is a natural identification \( {T}_{0}\left( {{T}_{p}M}\right) = {T}_{p}M \) . Under this identification, we will show that \( ... | Yes |
Proposition 5.11. (Properties of Normal Coordinates) Let \( \left( {\mathcal{U},\left( {x}^{i}\right) }\right) \) be any normal coordinate chart centered at \( p \) .\n\n(a) For any \( V = {V}^{i}{\partial }_{i} \in {T}_{p}M \), the geodesic \( {\gamma }_{V} \) starting at \( p \) with initial velocity vector \( V \) i... | Null | No |
Lemma 6.1. For any curve segment \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow M \), and any reparametrization \( \widetilde{\gamma } \) of \( \gamma, L\left( \gamma \right) = L\left( \widetilde{\gamma }\right) \) . | Exercise 6.1. Prove Lemma 6.1. | No |
Lemma 6.3. (Symmetry Lemma) Let \( \Gamma : \left( {-\varepsilon ,\varepsilon }\right) \times \left\lbrack {a, b}\right\rbrack \rightarrow M \) be an admissible family of curves in a Riemannian (or pseudo-Riemannian) manifold. On any rectangle \( \left( {-\varepsilon ,\varepsilon }\right) \times \left\lbrack {{a}_{i - ... | Proof. This is a local question, so we may compute in coordinates \( \left( {x}^{i}\right) \) around any point \( \Gamma \left( {{s}_{0},{t}_{0}}\right) \) . Writing the components of \( \Gamma \) as \( \Gamma \left( {s, t}\right) = \) \( \left( {{x}^{1}\left( {s, t}\right) ,\ldots ,{x}^{n}\left( {s, t}\right) }\right)... | Yes |
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