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Corollary 4.3. Let \( F \) be a spray on \( X \), and let \( {x}_{0} \in X \) . There exists an open neighborhood \( U \) of \( {x}_{0} \), and an open neighborhood \( V \) of \( {0}_{{x}_{0}} \) in \( {TX} \) satisfying the following condition. For every \( x \in U \) and \( v \in V \cap {T}_{x}X \) , there exists a u... | \[ {\alpha }_{v} : \left( {-2,2}\right) \rightarrow X \] such that \[ {\alpha }_{v}\left( 0\right) = x\;\text{ and }\;{\alpha }_{v}^{\prime }\left( 0\right) = v. \] Observe that in a chart, we may pick \( V \) as a product \[ V = U \times {V}_{2}\left( 0\right) \subset U \times \mathbf{E} \] where \( {V}_{2}\left( 0\ri... | Yes |
Theorem 5.1. Let \( Y \) be of class \( {C}^{p}\left( {p \geqq 3}\right) \) and admit partitions of unity. Let \( X \) be a closed submanifold. Then there exists a tubular neighborhood of \( X \) in \( Y \), of class \( {C}^{p - 2} \) . | Proof. Consider the exact sequence of tangent bundles:\n\n\[ 0 \rightarrow T\left( X\right) \rightarrow T\left( Y\right) \mid X \rightarrow N\left( X\right) \rightarrow 0. \]\n\nWe know that this sequence splits, and thus there exists some splitting\n\n\[ T\left( Y\right) \mid X = T\left( X\right) \oplus N\left( X\righ... | Yes |
Proposition 6.1. Let \( X \) be a manifold. Let \( \pi : E \rightarrow X \) and \( {\pi }_{1} : {E}_{1} \rightarrow X \) be two vector bundles over \( X \). Let\n\n\[ f : E \rightarrow {E}_{1} \]\n\nbe a tubular neighborhood of \( X \) in \( {E}_{1} \) (identifying \( X \) with its zero section in \( {E}_{1} \) ). Then... | Proof. We define \( F \) by the formula\n\n\[ {F}_{t}\left( e\right) = {t}^{-1}f\left( {te}\right) \]\n\nfor \( t \neq 0 \) and \( e \in E \). Then \( {F}_{t} \) is an embedding since it is composed of embeddings (the scalar multiplications by \( t,{t}^{-1} \) are in fact VB-isomorphism).\n\nWe must investigate what ha... | Yes |
Theorem 6.2. Let \( X \) be a submanifold of \( Y \) . Let\n\n\[ \pi : E \rightarrow X\;\text{ and }\;{\pi }_{1} : {E}_{1} \rightarrow X \]\n\nbe two vector bundles, and assume that \( E \) is compressible. Let \( f : E \rightarrow Y \) and \( g : {E}_{1} \rightarrow Y \) be two tubular neighborhoods of \( X \) in \( Y... | Proof. We observe that \( f\left( E\right) \) and \( g\left( {E}_{1}\right) \) are open neighborhoods of \( X \) in \( Y \) . Let \( U = {f}^{-1}\left( {f\left( E\right) \cap g\left( {E}_{1}\right) }\right) \) and let \( \varphi : E \rightarrow U \) be a compression. Let \( \psi \) be the composite map\n\n\[ E\overset{... | Yes |
Proposition 1.1. There exists a unique function \( {\xi \varphi } \) on \( U \) of class \( {C}^{p - 1} \) such that\n\n\[ \left( {\xi \varphi }\right) \left( x\right) = \left( {{T}_{x}\varphi }\right) \xi \left( x\right) \]\n\nIf \( U \) is open in the Banach space \( \mathbf{E} \) and \( \xi \) denotes the local repr... | Proof. The first formula certainly defines a mapping of \( U \) into \( \mathbf{R} \) . The local formula defines a \( {C}^{p - 1} \) -morphism on \( U \) . It follows at once from the definitions that the first formula expresses invariantly in terms of the tangent bundle the same mapping as the second. Thus it allows ... | Yes |
Proposition 1.2. Let \( X \) be a manifold and \( U \) open in \( X \) . Let \( \xi \) be a vector field over \( X \) . If \( {\partial }_{\xi } = 0 \), then \( \xi \left( x\right) = 0 \) for all \( x \in U \) . Each \( {\partial }_{\xi } \) is a derivation of \( {\mathrm{{Fu}}}^{p}\left( U\right) \) into \( {\mathrm{{... | Proof. Suppose \( \xi \left( x\right) \neq 0 \) for some \( x \) . We work with the local representations, and take \( \varphi \) to be a continuous linear map of \( \mathbf{E} \) into \( \mathbf{R} \) such that \( \varphi \left( {\xi \left( x\right) }\right) \neq 0 \), by Hahn-Banach. Then \( {\varphi }^{\prime }\left... | No |
Proposition 1.3. Let \( \xi ,\eta \) be two vector fields of class \( {C}^{p - 1} \) on \( X \) . Then there exists a unique vector field \( \left\lbrack {\xi ,\eta }\right\rbrack \) of class \( {C}^{p - 2} \) such that for each open set \( U \) and function \( \varphi \) on \( U \) we have\n\n\[ \left\lbrack {\xi ,\et... | Proof. By Proposition 1.2, any vector field having the desired effect on functions is uniquely determined. We check that the local formula gives us this effect locally. Differentiating formally, we have (using the law for the derivative of a product):\n\n\[ {\left( \eta \varphi \right) }^{\prime }\xi - {\left( \xi \var... | Yes |
Corollary 1.4. The bracket \( \left\lbrack {\xi ,\eta }\right\rbrack \) is bilinear in both arguments, we have \( \left\lbrack {\xi ,\eta }\right\rbrack = - \left\lbrack {\eta ,\xi }\right\rbrack \), and Jacobi’s identity\n\n\[ \left\lbrack {\xi ,\left\lbrack {\eta ,\zeta }\right\rbrack }\right\rbrack = \left\lbrack {\... | Proof. The first two assertions are obvious. The third comes from the definition of the bracket. We apply the vector field on the left of the equality to a function \( \varphi \) . All the terms cancel out (the reader will write it out as well or better than the author). The last two formulas are immediate. | No |
Theorem 1.5. Let \( \xi ,\eta \) be vector fields on \( X \), and assume that \( \left\lbrack {\xi ,\eta }\right\rbrack = 0 \) . Let \( \alpha \) and \( \beta \) be the flows for \( \xi \) and \( \eta \) respectively. Then for real values \( t \) , s we have\n\n\[{\alpha }_{t} \circ {\beta }_{s} = {\beta }_{s} \circ {\... | Proof. For a fixed value of \( t \), the two curves in \( s \) given by the right-and left-hand side of the last formula have the same initial condition, namely \( {\alpha }_{t}\left( x\right) \) . The curve on the right\n\n\[ s \mapsto \beta \left( {s,\alpha \left( {t, x}\right) }\right) \]\nis by definition the integ... | Yes |
Proposition 3.1. Let \( {x}_{0} \) be a point of \( X \) and \( \omega \) an r-form on \( X \) . If\n\n\[ \n\\left\\langle {\\omega ,{\\xi }_{1} \\times \\cdots \\times {\\xi }_{r}}\\right\\rangle \\left( {x}_{0}\\right)\n\]\n\nis equal to 0 for all vector fields \( {\xi }_{1},\\ldots ,{\\xi }_{r} \) at \( {x}_{0} \) (... | Proof. Considering things locally in terms of their local representations, we see that if \( \\omega \\left( {x}_{0}\\right) \) is not 0, then it does not vanish at some \( r \) -tuple of vectors \( \\left( {{v}_{1},\\ldots ,{v}_{r}}\\right) \) . We can take vector fields at \( {x}_{0} \) which take on these values at ... | Yes |
Proposition 3.2. Let \( \omega \) be an r-form of class \( {C}^{p - 1} \) on \( X \) . Then there exists a unique \( \left( {r + 1}\right) \) -form \( {d\omega } \) on \( X \) of class \( {C}^{p - 2} \) such that, for any open set \( U \) of \( X \) and vector fields \( {\xi }_{0},\ldots ,{\xi }_{r} \) on \( U \) we ha... | Proof. As before, we observe that the local formula defines a differential form. If we can prove that it gives the same thing as the first formulas, which is expressed invariantly, then we can globalize it, and we are done. Let us denote by \( {S}_{1} \) and \( {S}_{2} \) the two sums occurring in the invariant express... | Yes |
EXD 1. \( d\left( {\omega \land \psi }\right) = {d\omega } \land \psi + {\left( -1\right) }^{\deg \left( \omega \right) }\omega \land {d\psi } \) . | Proof. This is a simple formal exercise in the use of the local formula for the local representation of the exterior derivative. We leave it to the reader. | No |
The map \( d \) is linear, and satisfies\n\n\[ d\left( {\omega \land \psi }\right) = {d\omega } \land \psi + {\left( -1\right) }^{r}\omega \land {d\psi } \]\n\nif \( r = \deg \omega \) . The map \( d \) is uniquely determined by these properties, and by the fact that for a function \( f \), we have \( {df} = {f}^{\prim... | The linearity of \( d \) is obvious. Hence it suffices to prove the formula for decomposable forms. We note that for any function \( f \) we have\n\n\[ d\left( {f\omega }\right) = {df} \land \omega + {fd\omega }. \]\n\nIndeed, if \( \omega \) is a function \( g \), then from the derivative of a product we get \( d\left... | Yes |
Proposition 3.5. Let \( \omega \) be a form of class \( {C}^{2} \) . Then \( {dd\omega } = 0 \) . | Proof. If \( f \) is a function, then\n\n\[ \n{df}\left( x\right) = \mathop{\sum }\limits_{{j = 1}}^{n}\frac{\partial f}{\partial {x}_{j}}d{x}_{j} \n\] \n\nand \n\n\[ \n{ddf}\left( x\right) = \mathop{\sum }\limits_{{j = 1}}^{n}\mathop{\sum }\limits_{{k = 1}}^{n}\frac{{\partial }^{2}f}{\partial {x}_{k}\partial {x}_{j}}d... | Yes |
Property 2. If \( \omega \) is a differential form on \( Y \), then\n\n\[ d{f}^{ * }\left( \omega \right) = {f}^{ * }\left( {d\omega }\right) \] | We shall give the proof of Property 2 in the finite dimensional case and leave the general case to the reader.\n\nFor a form of degree 1 , say\n\n\[ \omega \left( y\right) = g\left( y\right) d{y}_{1} \]\n\nwith \( {y}_{1} = {f}_{1}\left( x\right) \), we find\n\n\[ \left( {{f}^{ * }{d\omega }}\right) \left( x\right) = \... | No |
Property 2. If \( \omega \) is a differential form on \( Y \), then\n\n\[ d{f}^{ * }\left( \omega \right) = {f}^{ * }\left( {d\omega }\right) \] | The verifications are all easy, and even trivial, except possibly for Property 2. We shall give the proof of Property 2 in the finite dimensional case and leave the general case to the reader.\n\nFor a form of degree 1 , say\n\n\[ \omega \left( y\right) = g\left( y\right) d{y}_{1} \]\n\nwith \( {y}_{1} = {f}_{1}\left( ... | No |
Property 4. If \( f : X \rightarrow Y \) is a morphism, and \( g \) is a function on \( Y \), then\n\n\[ d\left( {g \circ f}\right) = {f}^{ * }\left( {dg}\right) \]\n\nand at a point \( x \in X \), the value of this 1 -form is given by\n\n\[ {T}_{f\left( x\right) }g \circ {T}_{x}f = {\left( dg\right) }_{x} \circ {T}_{x... | The verifications are all easy, and even trivial, except possibly for Property 2. We shall give the proof of Property 2 in the finite dimensional case and leave the general case to the reader.\n\nFor a form of degree 1 , say\n\n\[ \omega \left( y\right) = g\left( y\right) d{y}_{1} \]\n\nwith \( {y}_{1} = {f}_{1}\left( ... | No |
If \( \omega \left( y\right) = g\left( y\right) d{y}_{{j}_{1}} \land \cdots \land d{y}_{{j}_{s}} \) is a differential form on \( V \), then \( {f}^{ * }\omega = \left( {g \circ f}\right) d{f}_{{j}_{1}} \land \cdots \land d{f}_{{j}_{s}}. \) | Indeed, we have for \( x \in U \) : \( \left( {{f}^{ * }\omega }\right) \left( x\right) = g\left( {f\left( x\right) }\right) \left( {{\mu }_{{j}_{1}} \circ {f}^{\prime }\left( x\right) }\right) \land \cdots \land \left( {{\mu }_{{j}_{s}} \circ {f}^{\prime }\left( x\right) }\right) \) and \( {f}_{j}^{\prime }\left( x\ri... | Yes |
Example 3. Let \( U, V \) be both open sets in \( n \) -space, and let \( f : U \rightarrow V \) be a \( {C}^{p} \) map. If\n\n\[ \n\omega \left( y\right) = g\left( y\right) d{y}_{1} \land \cdots \land d{y}_{n} \n\]\n\nwhere \( {y}_{j} = {f}_{j}\left( x\right) \) is the \( j \) -th coordinate of \( y \), then\n\n\[ \nd... | and consequently, expanding out the alternating product according to the usual multilinear and alternating rules, we find that\n\n\[ \n{f}^{ * }\omega \left( x\right) = g\left( {f\left( x\right) }\right) {\Delta }_{f}\left( x\right) d{x}_{1} \land \cdots \land d{x}_{n}, \n\]\n\nwhere \( {\Delta }_{f} \) is the determin... | Yes |
Proposition 5.1. Let \( \\xi \) be a vector field and \( \\omega \) a differential form of degree \( r \\geqq 1 \) . The Lie derivative \( {\\mathcal{L}}_{\\xi } \) is a derivation, in the sense that\n\n\[ \n{\\mathcal{L}}_{\\xi }\\left( {\\omega \\left( {{\\xi }_{1},\\ldots ,{\\xi }_{r}}\\right) }\\right) = \\left( {{... | Proof. The proof is routine using the definitions. The first assertion is obvious by the definition of the pull back of a form. For the local expression we actually derive more, namely we derive a local expression for \( {\\alpha }_{t}^{ * }\\omega \) and \( \\frac{d}{dt}{\\alpha }_{t}^{ * }\\omega \) which are charact... | Yes |
Proposition 5.2. Let \( {\xi }_{t} \) be a time-dependent vector field, \( \alpha \) its flow, and let \( \omega \) be a differential form. Then\n\n\[ \frac{d}{dt}\left( {{\alpha }_{t}^{ * }\omega }\right) = {\alpha }_{t}^{ * }\left( {{\mathcal{L}}_{{\xi }_{t}}\omega }\right) \;\text{ or }\;\frac{d}{dt}\left( {{\alpha ... | Proof. Proposition 5.1 gives us a local expression for \( \left( {{\mathcal{L}}_{{\xi }_{t}}\omega }\right) \left( y\right) \), replacing \( x \) by \( y \) because we shall now put \( y = \alpha \left( {t, x}\right) \) . On the other hand, from (1) in the proof of Proposition 5.1, we obtain\n\n\[ {\alpha }_{t}^{ * }\l... | No |
Proposition 6.2. Let \( \omega \) be such that \( {d\omega } = 0 \) . Let \( \alpha \) be the flow of \( {\xi }_{\omega } \) . Then \( {\alpha }_{t}^{ * }\Omega = \Omega \) for all \( t \) (in the domain of the flow). | Proof. By Proposition 5.2,\n\n\[ \frac{d}{dt}{\alpha }_{t}^{ * }\Omega = {\alpha }_{t}^{ * }{\mathcal{L}}_{{\xi }_{\omega }}\Omega = 0\;\text{ by }\Omega \mathbf{2}. \]\n\nHence \( {\alpha }_{t}^{ * }\Omega \) is constant, equal to \( {\alpha }_{0}^{ * }\Omega = \Omega \), as was to be shown. | Yes |
Proposition 6.3. If \( {\xi }_{df} \cdot h = 0 \) then \( {\xi }_{dh} \cdot f = 0 \) . | This is immediate from the antisymmetry of the Poisson bracket. It is interpreted as conservation of momentum in the physical theory of Hamiltonian mechanics, when one deals with the canonical 2-form on the cotangent bundle, to be defined in the next section. | No |
Proposition 7.1. This map defines a 1-form on \( {T}^{ \vee }\left( X\right) \) . Let \( X = U \) be open in \( \mathbf{E} \) and\n\n\[ \n{T}^{ \vee }\left( U\right) = U \times {\mathbf{E}}^{ \vee },\;T\left( {{T}^{ \vee }\left( U\right) }\right) = \left( {U \times {\mathbf{E}}^{ \vee }}\right) \times \left( {\mathbf{E... | Proof. We observe that the projection \( \pi : U \times {\mathbf{E}}^{ \vee } \rightarrow U \) is linear, and hence that its derivative at each point is constant, equal to the projection on the first factor. Our formula is then an immediate consequence of the definition. The local formula shows that \( \theta \) is in ... | Yes |
Proposition 7.2. Let \( U \) be open in \( \mathbf{E} \), and let \( \Omega \) be the local representation of the canonical 2-form on \( {T}^{ \vee }U = U \times {\mathbf{E}}^{ \vee } \) . Let \( \left( {x,\lambda }\right) \in U \times {\mathbf{E}}^{ \vee } \) . Let \( \left( {{u}_{1},{\omega }_{1}}\right) \) and \( \l... | Proof. We observe that \( \theta \) is linear, and thus that \( {\theta }^{\prime } \) is constant. We then apply the local formula for the exterior derivative, given in Proposition 3.2. Our assertion becomes obvious. | No |
Theorem 8.1 (Darboux Theorem). Let \( \mathbf{E} \) be a self-dual Banach space. Let\n\n\[ \omega : U \rightarrow {L}_{a}^{2}\left( \mathbf{E}\right) \]\n\nbe a non-singular closed 2 -form on an open set of \( \mathbf{E} \), and let \( {x}_{0} \in U \) . Then \( \omega \) is locally isomorphic at \( {x}_{0} \) to the c... | Proof. Let \( {\omega }_{0} = \omega \left( {x}_{0}\right) \), and let\n\n\[ {\omega }_{t} = {\omega }_{0} + t\left( {\omega - {\omega }_{0}}\right) ,\;0 \leqq t \leqq 1. \]\n\nWe wish to find a time-dependent vector field \( {\xi }_{t} \) locally at 0 such that if \( \alpha \) denotes its flow, then\n\n\[ {\alpha }_{t... | Yes |
Theorem 1.2. Let \( U, V \) be open subsets of Banach spaces \( \mathbf{E},\mathbf{F} \) respectively. Let\n\n\[ f : U \times V \rightarrow L\left( {\mathbf{E},\mathbf{F}}\right) \]\n\nbe a \( {C}^{r} \) -morphism \( \left( {r \geqq 1}\right) \). Assume that if\n\n\[ {\xi }_{1},{\eta }_{1} : U \times V \rightarrow \mat... | We shall prove Theorem 1.2 in \( §3 \) . We now indicate how Theorem 1.1 follows from it. We denote by \( {\alpha }_{y} \) the map \( {\alpha }_{y}\left( x\right) = \alpha \left( {x, y}\right) \), viewed as a map of \( {U}_{0} \) into \( V \). Then our differential equation can be written\n\n\[ D{\alpha }_{y}\left( x\r... | No |
Proposition 2.1. Let \( U, V \) be open sets in Banach spaces \( \mathbf{E},\mathbf{F} \) respectively. Let \( J \) be an open interval of \( \mathbf{R} \) containing 0, and let\n\n\[ g : J \times U \times V \rightarrow \mathbf{F} \]\n\nbe a morphism of class \( {C}^{r}\left( {r \geqq 1}\right) \). Let \( \left( {{x}_{... | Proof. This follows from the existence and uniqueness of local flows, by considering the ordinary vector field on \( U \times V \)\n\n\[ G : J \times U \times V \rightarrow \mathbf{E} \times \mathbf{F} \]\n\ngiven by \( G\left( {t, x, y}\right) = \left( {0, g\left( {t, x, y}\right) }\right) \). If \( B\left( {t, x, y}\... | Yes |
Proposition 2.2. Let notation be as in Proposition 2.1, and with \( y \) fixed, let \( \beta \left( {t, x}\right) = \beta \left( {t, x, y}\right) \) . Then \( {D}_{2}\beta \left( {t, x}\right) \) satisfies the differential equation\n\n\[ \n{D}_{1}{D}_{2}\beta \left( {t, x}\right) \cdot v = {D}_{2}g\left( {t, x,\beta \l... | Proof. Here again, we consider the vector field as in the proof of Proposition 2.1, and apply the formula for the differential equation satisfied by \( {D}_{2}\beta \) as in Chapter IV,§1. | No |
Theorem 4.1. Let \( Y, Z \) be integral submanifolds of \( X \) for the subbundle \( F \) of \( {TX} \), passing through a point \( {x}_{0} \) . Then there exists an open neighborhood \( U \) of \( {x}_{0} \) in \( X \), such that\n\n\[ Y \cap U = Z \cap U \] | Proof. Let \( U \) be an open neighborhood of \( {x}_{0} \) in \( X \) such that we have a chart\n\n\[ U \rightarrow V \times W \]\n\nwith\n\n\[ {x}_{0} \mapsto \left( {{y}_{0},{w}_{0}}\right) \]\n\nand \( Y \) corresponds to all points \( \left( {y,{w}_{0}}\right), y \in V \) . In other words, \( Y \) corresponds to a... | Yes |
Theorem 4.2. Let \( F \) be an integrable tangent subbundle over \( X \) . If\n\n\[ f : Y \rightarrow X \]\n\nis a morphism such that \( {Tf} : {TY} \rightarrow {TX} \) maps \( {TY} \) into \( F \), then the\n\ninduced map\n\[ {f}_{F} : Y \rightarrow {X}_{F} \]\n\n(same values as \( f \) but viewed as a map into the ne... | Proof. Using the local product structure as in the proof of the local uniqueness Theorem 4.1, we see at once that \( {f}_{F} \) is a morphism. In other words, locally, \( f \) maps a neighborhood of each point of \( Y \) into a sub-manifold of \( X \) which is tangent to \( F \) . If in addition \( f \) is an injective... | Yes |
Let \( {X}_{F}\left( {x}_{0}\right) \) be the connected component of \( {X}_{F} \) containing a point \( {x}_{0} \) . If \( f : Y \rightarrow X \) is an integral manifold for \( F \) passing through \( {x}_{0} \), and \( Y \) is connected, then there exists a unique morphism\n\n\[ \nh : Y \rightarrow {X}_{F}\left( {x}_... | Proof. Clear from the preceding discussion. | No |
Proposition 5.1. Let \( \xi ,\eta \) be left invariant vector fields on \( G \) . Then \( \left\lbrack {\xi ,\eta }\right\rbrack \) is also left invariant. | Proof. This follows from the general functorial formula\n\n\[ \n{\tau }_{ * }^{x}\left\lbrack {\xi ,\eta }\right\rbrack = \left\lbrack {{\tau }_{ * }^{x}\xi ,{\tau }_{ * }^{x}\eta }\right\rbrack = \left\lbrack {\xi ,\eta }\right\rbrack \n\] | Yes |
Theorem 5.2. Let \( G \) be a Lie group, \( \mathfrak{h} \) a Lie subalgebra of \( \mathfrak{l}\left( G\right) \), and let \( F \) be the corresponding left invariant subbundle of \( {TG} \). Then \( F \) is integrable. | Proof. I owe the proof to Alan Weinstein. It is based on the following lemma.\n\nLemma 5.3. Let \( X \) be a manifold, let \( \xi ,\eta \) be vector fields at a point \( {x}_{0} \), and let \( F \) be a subbundle of \( {TX} \). If \( \xi \left( {x}_{0}\right) = 0 \) and \( \xi \) is contained in \( F \), then \( \left\... | Yes |
Lemma 5.3. Let \( X \) be a manifold, let \( \xi ,\eta \) be vector fields at a point \( {x}_{0} \) , and let \( F \) be a subbundle of \( {TX} \) . If \( \xi \left( {x}_{0}\right) = 0 \) and \( \xi \) is contained in \( F \) , then \( \left\lbrack {\xi ,\eta }\right\rbrack \left( {x}_{0}\right) \in F \) . | Proof. We can deal with the local representations, such that \( X = U \) is open in \( \mathbf{E} \), and \( F \) corresponds to a factor, that is\n\n\[ \n{TX} = U \times {\mathbf{F}}_{1} \times {\mathbf{F}}_{2}\;\text{ and }\;F = U \times {\mathbf{F}}_{1}.\n\]\n\nWe may also assume without loss of generality that \( {... | Yes |
Theorem 5.4. Let \( G \) be a Lie group, let \( \mathfrak{h} \) be a Lie subalgebra of \( \mathfrak{l}\left( G\right) \) , and let \( F \) be its associated invariant subbundle. Let\n\n\[ j : H \rightarrow G \]\n\nbe the maximal connected integral manifold of \( F \) passing through \( e \) . Then \( H \) is a subgroup... | Proof. Let \( x \in H \) . The M-isomorphism \( {\tau }^{x} \) induces a VB-isomorphism of \( F \) onto itself, in other words, \( F \) is invariant under \( {\tau }_{ * }^{x} \) . Furthermore, since \( H \) passes through \( e \), and \( {xe} \) lies in \( H \), it follows that \( j : H \rightarrow G \) is also the ma... | Yes |
Proposition 1.1. Let \( X \) be a manifold admitting partitions of unity. Let \( \pi : E \rightarrow X \) be a vector bundle whose fibers are Hilbertable vector spaces. Then \( \pi \) admits a Riemannian metric. | Proof. Find a partition of unity \( \left\{ {{U}_{i},{\varphi }_{i}}\right\} \) such that \( \pi \mid {U}_{i} \) is trivial, that is such that we have a trivialization\n\n\[{\pi }_{i} : {\pi }^{-1}\left( {U}_{i}\right) \rightarrow {U}_{i} \times \mathbf{E}\]\n\n(working over a connected component of \( X \), so that we... | Yes |
For all operators \( A \), the series\n\n\[ \exp \left( A\right) = I + A + \frac{{A}^{2}}{2!} + \cdots \]\n\nconverges. If \( A \) commutes with \( B \), then\n\n\[ \exp \left( {A + B}\right) = \exp \left( A\right) \exp \left( B\right) \] | Proof. Standard. | No |
If \( A \) is symmetric (resp. skew-symmetric), then \( \exp \left( A\right) \) is symmetric positive definite (resp. Hilbertian). If \( A \) is toplinear automorphism sufficiently close to \( I \) and is positive definite symmetric (resp. Hilbertian), then \( \log \left( A\right) \) is symmetric (resp. skew-symmetric)... | The proofs are straightforward. As an example, let us carry out the proof of the last statement. Suppose \( A \) is Hilbertian and sufficiently close to \( I \) . Then \( {A}^{ * }A = I \) and \( {A}^{ * } = {A}^{-1} \) . Then\n\n\[ \log {\left( A\right) }^{ * } = \frac{\left( {A}^{ * } - I\right) }{1} + \cdots \]\n\n\... | No |
Proposition 2.4. The exponential map gives a \( {C}^{\infty } \) -isomorphism from the space \( \operatorname{Sym}\left( \mathbf{E}\right) \) of symmetric endomorphisms of \( \mathbf{E} \) and the space \( \operatorname{Pos}\left( \mathbf{E}\right) \) of symmetric positive definite automorphisms of \( \mathbf{E} \) . | Proof. We must construct its inverse, and for this we use the spectral theorem. Given \( A \), symmetric positive definite, the analytic function \( \log t \) is defined on the spectrum of \( A \), and thus \( \log A \) is symmetric. One verifies immediately that it is the inverse of the exponential function (which can... | No |
The manifold of toplinear automorphisms of the Hilbert space \( \mathbf{E} \) is \( {C}^{\infty } \) -isomorphic to the product of the Hilbert automorphisms and the positive definite symmetric automorphisms, under the mapping\n\n\[ \operatorname{Hilb}\left( \mathbf{E}\right) \times \operatorname{Pos}\left( \mathbf{E}\r... | Proof. Our map is induced by a continuous bilinear map of\n\n\[ L\left( {\mathbf{E},\mathbf{E}}\right) \times L\left( {\mathbf{E},\mathbf{E}}\right) \]\ninto \( L\left( {\mathbf{E},\mathbf{E}}\right) \) and so is \( {C}^{\infty } \) . We must construct an inverse, or in other words express any given toplinear automorph... | Yes |
Theorem 3.1. Let \( \pi \) be a vector bundle over a manifold \( X \), and assume that the fibers of \( \pi \) are all toplinearly isomorphic to a Hilbert space \( \mathbf{E} \). Then the above map, from reductions of \( \pi \) to the Hilbert group, into the Riemannian metrics, is a bijection. | Proof. Suppose that we are given an ordinary VB-trivialization \( \left\{ \left( {{U}_{i},{\tau }_{i}}\right) \right\} \) of \( \pi \). We must construct an HB-trivialization. For each \( i \), let \( {g}_{i} \) be the Riemannian metric on \( {U}_{i} \times \mathbf{E} \) transported from \( {\pi }^{-1}\left( {U}_{i}\ri... | Yes |
Proposition 4.1. Let \( X \) be a manifold and \( \pi : E \rightarrow X \) a Hilbert bundle. Let \( \sigma : X \rightarrow \mathbf{R} \) be a morphism such that \( \sigma \left( x\right) > 0 \) for all \( x \) . Then the mapping \[ w \rightarrow \frac{\sigma \left( {\pi w}\right) w}{{\left( 1 + {\left| w\right| }^{2}\r... | Proof. Obvious. The inverse mapping is constructed in the obvious way. | No |
Corollary 4.2. Let \( X \) be a manifold admitting partitions of unity, and let \( \pi : E \rightarrow X \) be a Hilbert bundle over \( X \) . Then \( E \) is compressible. | Proof. Let \( Z \) be an open neighborhood of the zero section. For each \( x \in X \), there exists an open neighborhood \( {V}_{x} \) and a number \( {a}_{x} > 0 \) such that the vectors in \( {\pi }^{-1}\left( {V}_{x}\right) \) which are of length \( < {a}_{x} \) lie in \( Z \) . We can find a partition of unity \( ... | Yes |
Proposition 4.3. Let \( X \) be a manifold. Let \( \pi : E \rightarrow X \) and \( {\pi }_{1} : {E}_{1} \rightarrow X \) be two Hilbert bundles over \( X \) . Let\n\n\[ \lambda : E \rightarrow {E}_{1} \]\n\nbe a VB-isomorphism. Then there exists an isotopy of VB-isomorphisms\n\n\[ {\lambda }_{t} : E \rightarrow {E}_{1}... | Proof. We find reductions of \( E \) and \( {E}_{1} \) to the Hilbert group, with Hilbert trivializations \( \left\{ \left( {{U}_{i},{\tau }_{i}}\right) \right\} \) for \( E \) and \( \left\{ \left( {{U}_{i},{\rho }_{i}}\right) \right\} \) for \( {E}_{1} \) . We can then factor \( {\rho }_{i}\lambda {\tau }_{i}^{-1} \)... | Yes |
Theorem 4.4. Let \( X \) be a submanifold of \( Y \) . Let \( \pi : E \rightarrow X \) and \( {\pi }_{1} : {E}_{1} \rightarrow X \) be two Hilbert bundles. Assume that \( E \) is compressible. Let \( f : E \rightarrow Y \) and \( g : {E}_{1} \rightarrow Y \) be two tubular neighborhoods of \( X \) in \( Y \) . Then the... | Proof. From Theorem 6.2 of Chapter IV, we know already that there exists a VB-isomorphism \( \lambda \) such that \( f \approx {g\lambda } \) . Using the preceding proposition, we know that \( \lambda \approx \mu \) where \( \mu \) is a HB-isomorphism. Thus \( {g\lambda } \approx {g\mu } \) and by transitivity, \( f \a... | No |
Theorem 5.1. Let \( f \) be a \( {C}^{p + 2} \) function defined on an open neighborhood of 0 in the Hilbert space \( \mathbf{E} \), with \( p \geqq 1 \) . Assume that \( f\left( 0\right) = 0 \), and that 0 is a non-degenerate critical point of \( f \) . Then there exists a local \( {C}^{p} \) - isomorphism at 0, say \... | Proof. We may assume that \( U \) is a ball around 0 . We have\n\n\[ f\left( x\right) = f\left( x\right) - f\left( 0\right) = {\int }_{0}^{1}{Df}\left( {tx}\right) {xdt} \]\n\nand applying the same formula to \( {Df} \) instead of \( f \), we get\n\n\[ f\left( x\right) = {\int }_{0}^{1}{\int }_{0}^{1}{D}^{2}f\left( {st... | Yes |
Theorem 5.2. Let \( A : U \rightarrow \operatorname{Sym}\left( \mathbf{E}\right) \) be a \( {C}^{p} \) map of \( U \) into the open set of invertible symmetric operators on \( \mathbf{E} \) . Then there exists a \( {C}^{p} \) isomorphism of an open subset \( {U}_{1} \) containing 0, of the form\n\n\[ \varphi \left( x\r... | Proof. We seek a map \( C \) such that\n\n\[ C{\left( x\right) }^{ * }A\left( 0\right) C\left( x\right) = A\left( x\right) . \]\n\nIf we let \( B\left( x\right) = A{\left( 0\right) }^{-1}A\left( x\right) \), then \( B\left( x\right) \) is close to the identity \( I \) for small \( x \) . The square root function has a ... | Yes |
Corollary 5.3. Let \( f \) be a \( {C}^{p + 2} \) function near 0 on the Hilbert space \( \mathbf{E} \) , such that 0 is a non-degenerate critical point. Then there exists a local \( {C}^{p} \) -isomorphism \( \psi \) at 0, and an orthogonal decomposition \( \mathbf{E} = \mathbf{F} + {\mathbf{F}}^{ \bot } \) , such tha... | Proof. On a space where \( A \) is positive definite, we can always make the toplinear isomorphism \( x \mapsto {A}^{1/2}x \) to get the quadratic form to become the given hermitian product \( \langle \rangle \), and similarly on a space where \( A \) is negative definite. In general, we use the spectral theorem to dec... | No |
Proposition 7.2. In the chart \( U \), let \( f = \left( {{f}_{1},{f}_{2}}\right) : U \times \mathbf{E} \rightarrow \mathbf{E} \times \mathbf{E} \) represent \( F \) . Then \( {f}_{2}\left( {x, v}\right) \) is the unique vector such that for all \( {w}_{1} \in \mathbf{E} \) we have:\n\n\[ \left\langle {{f}_{2}\left( {x... | From this one sees that \( {f}_{2} \) is homogeneous of degree 2 in the second variable \( v \), in other words that it represents a spray. This concludes the proof of Theorem 7.1. | No |
Proposition 1.1. Let \( \left\{ {{\xi }_{1},\ldots ,{\xi }_{n}}\right\} \) be a frame of vector fields. Let \( \left\{ {{\lambda }_{1},\ldots ,{\lambda }_{n}}\right\} \) be the dual frame of 1 -forms \( \left( {\text{so}{\lambda }_{i}\left( {\xi }_{j}\right) = {\delta }_{ij}}\right) \) . For any form \( \omega \in {\ma... | Proof. Let \( {d}^{\prime }\omega = \sum {\lambda }_{i} \land {D}_{{\xi }_{i}}\omega \) . Then \( {d}^{\prime } \) defines an anti-derivation of the alternating algebra of forms, that is if \( \psi \in {\mathcal{A}}^{q}\left( x\right) \) for any \( q \), then\n\n\[ \n{d}^{\prime }\left( {\omega \land \psi }\right) = \l... | Yes |
Proposition 2.2. Let \( \omega \in \Gamma {L}^{r}\left( {{TX},\mathbf{R}}\right) \) or \( \Gamma {L}^{r}\left( {{TX},{TX}}\right) \) . Let \( \xi \) , \( {\eta }_{1},\ldots ,{\eta }_{r} \) be vector fields over \( X \) . If \( \omega \in \Gamma {L}^{r}\left( {{TX},\mathbf{R}}\right) \), then in a chart \( U \) we have ... | Proof. This comes directly from the definitions in \( §1 \) . Observe that in applying the definitions, the sum\n\n\[ \n\mathop{\sum }\limits_{{j = 1}}^{r}{\omega }_{U}\left( {{\eta }_{1U},\ldots ,{\eta }_{1U}^{\prime } \cdot \xi ,\ldots ,{\eta }_{rU}}\right) \n\]\n\noccurs twice, once with \( \mathrm{a} + \) sign and ... | Yes |
Lemma 2.3. Let \( E, F \) be vector bundles over \( X \), with \( E \) finite dimensional and \( X \) admitting cut off functions. Let\n\n\[ H : {\Gamma E} \rightarrow {\Gamma F} \]\n\nbe a linear map which is \( \mathrm{{Fu}}\left( X\right) \) -linear, that is \( H\left( {\varphi \xi }\right) = {\varphi H}\left( \xi \... | Proof. It suffices to prove that if \( \xi \left( {x}_{0}\right) = 0 \) then \( H\left( \xi \right) \left( {x}_{0}\right) = 0 \) . There exists a cut off function \( \varphi \) near \( {x}_{0} \) by assumption, so we may give the proof locally. By assumption, there exists a finite number of sections \( {e}_{1},\ldots ,... | Yes |
Theorem 3.1. There exists a unique linear map\n\n\\[ \n{D}_{{\alpha }^{\prime }} : \\operatorname{Lift}\\left( \\alpha \\right) \\rightarrow \\operatorname{Lift}\\left( \\alpha \\right) \n\\]\nwhich in a chart \\( U \\) has the expression\n\n\\[ \n{\\left( {D}_{{\alpha }^{\prime }}\\gamma \\right) }_{U}\\left( t\\right... | Proof of Theorem 3.1. The proof is entirely analogous to the proof for Theorem 2.1, using the local representation of the bilinear map \\( {B}_{U} \\) associated with a spray in charts. We have to verify that the formula of Theorem 3.1 transforms in the proper way under a change of charts, i.e. under an isomorphism \\(... | Yes |
Corollary 3.2. Let \( \eta \) be a vector field and suppose \( \gamma \left( t\right) = \eta \left( {\alpha \left( t\right) }\right), t \in J \) . Let \( \xi \) be a vector field on \( X \) such that \( {\alpha }^{\prime }\left( {t}_{0}\right) = \xi \left( {\alpha \left( {t}_{0}\right) }\right) \) for some \( {t}_{0} \... | Proof. Immediate from the chain rule and the local representation of Theorem 3.1. | No |
Theorem 3.3. Let \( \alpha : J \rightarrow X \) be a \( {C}^{2} \) curve in \( X \) . Let \( {t}_{0} \in J \) . Given \( v \in {T}_{\alpha \left( {t}_{0}\right) }X \), there exists a unique lift \( {\gamma }_{v} : J \rightarrow {TX} \) which is \( \alpha \) -paralled and such that \( {\gamma }_{v}\left( {t}_{0}\right) ... | Proof. The existence and uniqueness simply comes from the existence and uniqueness of solutions of differential equations. Note that from the linearity of the equation, the integral curve \( \gamma \) is defined on the whole interval of definition \( J \) by Proposition 1.9 of Chapter IV. | Yes |
Theorem 3.4. Fix \( {t}_{0} \in J \) . For \( t \in J \) define the map\n\n\[ \n{P}_{{t}_{0},\alpha }^{t} = {P}^{t} : {T}_{\alpha \left( {t}_{0}\right) }X \rightarrow {T}_{\alpha \left( t\right) }X\;\text{ by }\;{P}^{t}\left( v\right) = \gamma \left( {t, v}\right) ,\n\]\n\nwhere \( t \mapsto \gamma \left( {t, v}\right)... | Proof. We must verify that\n\n\( {P}^{t}\left( {sv}\right) = s{P}^{t}\left( v\right) \) and \( {P}^{t}\left( {v + w}\right) = {P}^{t}\left( v\right) + {P}^{t}\left( w\right) \; \) for \( s \in \mathbf{R} \) and \( v, w \in {T}_{x}X.\)\n\nBut these properties follow at once from the linearity of the differential equatio... | Yes |
Proposition 3.5 (Local Expression). Let \( \omega = {\omega }_{U},{\eta }_{j} = {\eta }_{jU} \) etc. represent the respective objects in a chart \( U \), omitting the subscript \( U \) to simplify the notation. Then\n\n\[ \n\left( {{D}_{{\alpha }^{\prime }}\omega }\right) \left( {{\eta }_{1},\ldots ,{\eta }_{r}}\right)... | This comes from the definition at the end of \( §1 \), and the fact that the ordinary derivative\n\n\[ \n{\left( {\omega }_{U}\left( {\eta }_{1U},\ldots ,{\eta }_{rU}\right) \right) }^{\prime } \n\]\n\nin the chart is obtained by the Leibniz rule (suppressing the index \( U \) )\n\n\[ \n{\left( \omega \left( {\eta }_{1... | Yes |
Let \( E = {TX} \) or \( \mathbf{R} \) as above. Let \( \Omega : X \rightarrow {L}^{r}\left( {{TX}, E}\right) \) be a section (so a tensor field), and let \( \omega \left( t\right) = \Omega \left( {\alpha \left( t\right) }\right), t \in J \) . Let \( {t}_{0} \in J \) . Let \( \xi \) be a vector field such that \( {\alp... | \[ \left( {{D}_{{\alpha }^{\prime }}\omega }\right) \left( {t}_{0}\right) = \left( {{D}_{\xi }\Omega }\right) \left( {\alpha \left( {t}_{0}\right) }\right) \] Proof. Immediate from the chain rule and the local representation formula. | No |
Theorem 3.8. Let the notation be as in Theorem 3.7. For \( t \in J \) define the map\n\n\[ \n{P}_{{t}_{0},\alpha }^{t} = {P}_{\alpha }^{t} : {L}^{r}\left( {{T}_{\alpha \left( {t}_{0}\right) }X,{E}_{\alpha \left( {t}_{0}\right) }}\right) \rightarrow {L}^{r}\left( {{T}_{\alpha \left( t\right) }X,{E}_{\alpha \left( t\righ... | Proof. This follows at once from the linearity of the differential equation satisfied by \( \gamma \), and the uniqueness theorem for its solutions with given initial conditions. | Yes |
Theorem 4.1. Let \( \left( {X, g}\right) \) be a pseudo Riemannian manifold. There exists a unique covariant derivative \( D \) such that for all vector fields \( \xi ,\eta ,\zeta \) we have\n\nMD 1.\n\[ \n{D}_{\xi }\langle \eta ,\zeta {\rangle }_{g} = {\left\langle {D}_{\xi }\eta ,\zeta \right\rangle }_{g} + {\left\la... | Proof. For the uniqueness, we shall express \( {\left\langle {D}_{\xi }\eta ,\zeta \right\rangle }_{g} \) entirely in terms of operations which do not involve the derivative \( D \) . To do this, we write down the first defining property of a connection for a cyclic permutation of the three variables:\n\n\[ \n\xi \lang... | Yes |
Theorem 4.2. Let \( \left( {X, g}\right) \) be a pseudo Riemannian manifold. There exists a unique spray on \( X \) satisfying the following two equivalent conditions.\n\nMS 1. In a chart \( U \), the associated bilinear map \( {B}_{U} \) satisfies the following formula for all \( v, w, z \in \mathbf{E} \) :\n\n\[ - 2\... | Proof. First observe that \( {B}_{U} \) as defined by the formula is symmetric in \( \left( {v, w}\right) \) . The symmetry is built in the sum of the first two terms, and to see that the third term is symmetric, one differentiates with respect to \( x \) the formula\n\n\[ \langle g\left( x\right) z, v\rangle = \langle... | Yes |
Theorem 4.3. Let \( \alpha : J \rightarrow X \) be a \( {C}^{2} \) curve in a Riemannian manifold \( \left( {X, g}\right) \) . For the metric derivative, and curves \( \gamma ,\zeta \in \operatorname{Lift}\left( {\alpha ,{TX}}\right) \), we have the formula\n\n\[ \langle \gamma ,\zeta {\rangle }_{g}^{\prime } = {\left\... | Proof. The formula is proved in the same way that the computation proving Theorem 3.1 was parallel to the computation proving Theorem 2.1 (giving the behavior under changes of charts). From the formula, if \( {D}_{{\alpha }^{\prime }}\gamma = {D}_{{\alpha }^{\prime }}\zeta = 0 \), it follows that \( \langle \gamma ,\ze... | Yes |
Corollary 4.4. Let \( \varphi \) be a \( {C}^{2} \) function on \( X \) . Let \( \alpha \) be a geodesic for the metric spray. Then\n\n\[ \n{\left( \varphi \circ \alpha \right) }^{\prime \prime } = {\left\langle {D}_{{\alpha }^{\prime }}\left( \operatorname{grad}\varphi \right) \circ \alpha ,{\alpha }^{\prime }\right\r... | Proof. Taking the first derivative of \( \varphi \circ \alpha \) yields\n\n\[ \n{\left( \varphi \circ \alpha \right) }^{\prime }\left( t\right) = \left( {d\varphi }\right) \left( {\alpha \left( t\right) }\right) {\alpha }^{\prime }\left( t\right) = {\left\langle \left( \operatorname{grad}\varphi \right) \left( \alpha \... | No |
Proposition 5.1. The map \( G \) is a local isomorphism at \( \left( {{x}_{0},0}\right) \) . | Proof. The Jacobian matrix of \( G \) in a chart is given immediately from Chapter IV, Theorem 4.1 by\n\n\[ \left( \begin{matrix} \mathrm{{id}} & \mathrm{{id}} \\ 0 & \mathrm{{id}} \end{matrix}\right) \]\n\nwhich is invertible. The inverse mapping theorem concludes the proof. | Yes |
Given \( {x}_{0} \in X \) . Let \( V \) be an open neighborhood of \( \left( {{x}_{0},0}\right) \) in \( {TX} \) such that \( G \) induces an isomorphism of \( V \) with its image, and in a chart, for some \( \epsilon > 0 \) ,\n\n\[ V = {U}_{0} \times \mathbf{E}\left( \epsilon \right) \]\n\nLet \( W \) be a neighborhoo... | The properties are merely an application of the definitions and Proposition 5.1. | No |
Lemma 5.3. We have the rules on lifts of \( \sigma \) to \( {TX} \) : (a) \( {D}_{1}{\partial }_{2} = {D}_{2}{\partial }_{1} \) ; and (b) \( {\partial }_{2}{\left\langle {\partial }_{1}\sigma ,{\partial }_{1}\sigma \right\rangle }_{g} = 2{\left\langle {D}_{1}{\partial }_{2}\sigma ,{\partial }_{1}\sigma \right\rangle }_... | Proof. Let \( {\sigma }_{U} \) represent \( \sigma \) in a chart. Then from Theorem 3.1, \[ {D}_{1}{\partial }_{2}{\sigma }_{U} = {\partial }_{1}{\partial }_{2}{\sigma }_{U} - {B}_{U}\left( {{\sigma }_{U};{\partial }_{1}{\sigma }_{U},{\partial }_{2}{\sigma }_{U}}\right) . \] Since \( {B}_{U} \) is symmetric in the last... | Yes |
Theorem 5.4. Let \( t \mapsto u\left( t\right) \) be a curve in \( {\mathbf{S}}_{g}\left( 1\right) \) . Let \( 0 \leqq r \leqq b \) where \( b \) is such that the points \( {ru}\left( t\right) \) are in the domain of the exponential \( {\exp }_{x} \) .\n\nDefine\n\n\[\n\sigma \left( {r, t}\right) = {\exp }_{x}\left( {{... | Proof. This is immediate since parallel translation is an isometry by Theorem 4.3. | No |
Corollary 5.5. Assume \( \left( {X, g}\right) \) Riemannian. Let \( v \in {T}_{x}X \) . Suppose \( \parallel v{\parallel }_{q} = r \), with \( r > 0 \) . Also suppose the segment \( \{ {tv}\} \left( {0 \leqq t \leqq 1}\right) \) is contained in the domain of the exponential. Let \( \alpha \left( t\right) = {\exp }_{x}\... | Proof. Special case of the length formula in Theorem 5.4, followed by an integration to get the length. | No |
Lemma 5.6. Let \( X \) be pseudo Riemannian. Let \( \sigma : {J}_{1} \times {J}_{2} \rightarrow X \) be a \( {C}^{2} \) map. For each \( t \in {J}_{2} \) let \( {\alpha }_{t}\left( s\right) = \sigma \left( {s, t}\right) \) . Assume that each \( {\alpha }_{t} \) is a geodesic, and that \( {\alpha }_{t}^{\prime 2} \) is ... | Proof. Let \( D \) be the metric derivative. Then \( {D}_{1}{\partial }_{1}\sigma = 0 \) because for a geodesic \( \alpha \), we know that the metric derivative has the property that \( {D}_{{\alpha }^{\prime }}{\alpha }^{\prime } = 0 \) . Thus we get\n\n\[ \n{\partial }_{1}{\left\langle {\partial }_{1}\sigma ,{\partia... | Yes |
Theorem 5.7. Let \( \\left( {X, g}\\right) \) be pseudo Riemannian. Let \( {x}_{0} \\in X \) and let \( W \) be a small open neighborhood of \( {x}_{0} \), selected as in Corollary 5.2, with \( \\epsilon \) sufficiently small. Let \( x \\in W \) . Then the geodesics through \( x \) are orthogonal to the image of \( {\\... | Proof. For \( \\epsilon \) sufficiently small positive, the exponential map is defined on \( {\\mathbf{S}}_{g}\\left( r\\right) \) for \( 0 < r \\leqq \\epsilon \), and as we have seen, the level sets \( {\\mathbf{S}}_{g}\\left( r\\right) \) are submanifolds of \( X \) . Then our assertion amounts to proving that for e... | Yes |
Lemma 5.9. Given \( x \in X \), there exists \( c > 0 \) such that if \( r < c \), and if \( \alpha \) is a geodesic in \( X \), tangent to \( {S}_{g}\left( {x, r}\right) \) at \( y = \alpha \left( {t}_{0}\right) \), then \( \alpha \left( t\right) \) lies outside \( {S}_{g}\left( {x, r}\right) \) for \( t \neq {t}_{0} ... | Proof. We pick \( c \) such that the exponential map \( {\exp }_{x} \) is a differential isomorphism on \( {\mathbf{B}}_{g}\left( {{0}_{x}, r}\right) \) for all \( r < c \) and preserves distances on rays from \( {0}_{x} \) to \( v \in {T}_{x}X \) with \( \parallel v{\parallel }_{q} = r \) . Without loss of generality,... | Yes |
Lemma 6.1. For a piecewise \( {C}^{1} \) curve \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow U\left( x\right) - \{ x\} \) as above, we have the inequality\n\n\[ L\left( \gamma \right) \geqq \left| {r\left( b\right) - r\left( a\right) }\right| . \]\n\nEquality holds only if the function \( t \mapsto r\left( t... | Proof. Let \( \sigma \left( {r, t}\right) = {\exp }_{x}\left( {{ru}\left( t\right) }\right) \) . Then \( \gamma \left( t\right) = \sigma \left( {r\left( t\right), t}\right) \) . We have\n\n\[ {\gamma }^{\prime }\left( t\right) = \frac{d\gamma }{dt} = \frac{\partial \sigma }{\partial r}{r}^{\prime }\left( t\right) + \fr... | Yes |
Theorem 6.2. Let \( \left( {V, W}\right) \) constitute a normal neighborhood of a point \( {x}_{0} \in X \) . Let \( \alpha : \left\lbrack {0,1}\right\rbrack \rightarrow V \) be the geodesic (up to reparametrization) in \( V \) joining two points of \( W \) (namely \( \alpha \left( 0\right) \) and \( \alpha \left( 1\ri... | Proof. Let \( x, y \in W \) and let \( y = {\exp }_{x}\left( {ru}\right) \) with \( 0 < r < \epsilon \), and \( \parallel u{\parallel }_{q} = 1 \) . Then for \( \delta > 0 \) and \( 0 < \delta < r \) the path \( \gamma \) contains a segment joining the shell \( {\mathrm{{Sh}}}_{g}\left( {x,\delta }\right) \) with the s... | Yes |
Corollary 6.3. Let \( \alpha : \left\lbrack {0,1}\right\rbrack \rightarrow X \) be a piecewise \( {C}^{1} \) path, parametrized by are length. If \( L\left( \alpha \right) \leqq L\left( \gamma \right) \) for all paths from \( \alpha \left( 0\right) \) to \( \alpha \left( 1\right) \) in \( X \), then \( \alpha \) is a g... | Proof. We can find a partition of \( \left\lbrack {0,1}\right\rbrack \) such that the image under \( \alpha \) of each small interval in the partition is contained in some neighborhood \( W \) as in the theorem, and its length is small so the image of the segment is contained in a normal neighborhood. By Theorem 6.2, t... | No |
Theorem 6.4. Let \( \left( {X, g}\right) \) be a Riemannian manifold and let \( x \in X \) . There exists \( c > 0 \) such that for all \( r < c \) the map \( {\exp }_{x} \) is defined on \( {\mathbf{B}}_{g}\left( {{0}_{x}, c}\right) \), gives a differential isomorphism\n\n\[ \n{\exp }_{x} : {\mathbf{B}}_{g}\left( {{0}... | Proof. Immediate from Corollary 5.5 and Theorem 6.2. | No |
Proposition 6.5. Each condition implies the next, i.e.\n\n## \(\text{COM}1 \Rightarrow \text{COM}2 \Rightarrow \text{COM}3 \Rightarrow \text{COM 4.}\) | Proof. Assume COM 1. Let \( \alpha : J \rightarrow X \) be a geodesic parametrized by arc length on some interval, and take \( J \) to be maximal in \( \mathbf{R} \) . By the existence and uniqueness theorem for differential equations, \( J \) is open in \( \mathbf{R} \) , and it will suffice to prove that \( J \) is c... | Yes |
Theorem 6.6 (Hopf-Rinow). Assume that \( \left( {X, g}\right) \) is finite dimensional connected geodesically complete at a point \( p \), that is, \( {\exp }_{p} \) is defined on \( {T}_{p}X \) . Then any point in \( X \) can be joined to \( p \) by a minimal geodesic. | Proof. I follow here the variation of the proof given in [Mi 63]. Let \( y \) be a point with \( p \neq y \) . Let \( W \) be a normal neighborhood of \( p \) containing the image of a small ball under the exponential map \( {\exp }_{p} \) . Let \( r = \) \( \operatorname{dist}\left( {p, y}\right) \), and let \( \delta... | No |
Corollary 6.7. In the finite dimensional case the four completeness conditions COM 1 through COM 4 are equivalent to a fifth:\n\nCOM 5. A closed \( {\operatorname{dist}}_{g} \) -bounded subset of \( X \) is compact. | Proof. Assume COM 4 with \( {\exp }_{{x}_{0}} \) defined on \( {T}_{{x}_{0}}X \) . Let \( S \) be closed and bounded in \( X \) . Without loss of generality, we may assume \( {x}_{0} \in S \) . Let \( b \) be a bound for the diameter of \( S \) . Then by Theorem 6.6 (Hopf-Rinow), every point of \( S \) can be joined to... | Yes |
Lemma 6.8. Let \( f : Y \rightarrow X \) be a \( {C}^{1} \) map between Riemannian manifolds \( \left( {Y, h}\right) \) and \( \left( {X, g}\right) \) . Assume that there is a constant \( C > 0 \) such that for all \( y \in Y \) and \( w \in {T}_{y}Y \) we have\n\n\[ \parallel {Tf}\left( y\right) w{\parallel }_{g} \geq... | Proof. We have\n\n\[ {L}_{g}\left( {f \circ \gamma }\right) = {\int }_{a}^{b}{\begin{Vmatrix}{\left( f \circ \gamma \right) }^{\prime }\left( t\right) \end{Vmatrix}}_{g}{dt} = {\int }_{a}^{b}{\begin{Vmatrix}Tf\left( \gamma \left( t\right) \right) {\gamma }^{\prime }\left( t\right) \end{Vmatrix}}_{g}{dt} \]\n\n\[ \geqq ... | Yes |
Proposition 1.1. There exists a unique tensor field \( R \), section of \( {L}^{3}\left( {{TX},{TX}}\right) \), i.e. arising from the functor \( \mathbf{E} \mapsto {L}^{3}\left( {\mathbf{E},\mathbf{E}}\right) \) (continuous trilinear maps of \( \mathbf{E} \) into itself) such that for all vector fields \( \xi ,\eta ,\z... | Proof. The expression on the right-hand side gives a well-defined vector field on \( X \) . To show that this association comes from a tensor field, we can compute in a chart. To do this, we use the local expression for the covariant derivative given in Theorem 2.1 of Chapter VIII. So for the rest of the argument, \( \... | Yes |
\[ R\left( {v, w}\right) = - R\left( {w, v}\right) \text{ (skew-symmetry). } \] | Proof. The first relation is obvious from the definition. | No |
Proposition 1.4. On a pseudo Riemannian manifold, the Riemann tensor satisfies all the above four properties. Furthermore, RIEM 4 follows from RIEM 1, 2, 3. | Proof. Properties RIEM 1 and RIEM 3 have been proved in Proposition 1.3. Property RIEM 2 amounts to proving that \( R\left( {v, w, z, z}\right) = 0 \) for all \( v, w, z \) ; or in terms of vector fields, \( R\left( {\xi ,\eta ,\zeta ,\zeta }\right) = 0 \) . We will need to differentiate. Since all the terms with deriv... | Yes |
The canonical 2-tensor determines the Riemann tensor. Or similarly, if the canonical tensor \( R \) satisfies\n\n\[ R\left( {v, w, v, w}\right) = 0\;\text{ for all }\;v, w, \]\nthen \( R = 0 \) . | Proof. Say we prove the second assertion first. From RIEM 4, which implies that \( R\left( {v, w, v, z}\right) \) is symmetric in \( \left( {w, z}\right) \), if \( R\left( {v, w, v, w}\right) = 0 \) for all \( v, w \) then \( R\left( {v, w, v, z}\right) = 0 \) for all \( v, w, z \) . From the alternating properties of ... | Yes |
Proposition 1.6. Let \( \\left\\{ {{\\xi }_{1},\\ldots ,{\\xi }_{n}}\\right\\} \) be an orthonormal frame on an open set. Then for vector fields \( \\xi ,\\eta \) we have\n\n\[ \n{\\operatorname{Sc}}_{R}\\left( {\\xi ,\\eta }\\right) = \\mathop{\\sum }\\limits_{{i = 1}}^{n}R\\left( {\\xi ,{\\xi }_{i},\\eta ,{\\xi }_{i}... | Proof. This is immediate from the definition of the trace of an endomorphism of a finite dimensional vector space. | No |
Theorem 2.1. Let \( \left( {X, g}\right) \) be pseudo Riemannian, let \( \alpha : \left\lbrack {a, b}\right\rbrack \rightarrow X \) be a geodesic. Given vectors \( z, w \in {T}_{\alpha \left( a\right) }X \), there exists a unique Jacobi lift \( \eta = {\eta }_{z, w} \) of \( \alpha \) to \( {TX} \) such that\n\n\[ \eta... | Proof. One verifies at once that \( {\eta }_{v}\left( 0\right) = 0 \), and since \( {D}_{{\alpha }^{\prime }}{\alpha }^{\prime } = 0 \), we also\n\nhave\n\n\[ {D}_{{\alpha }^{\prime }}{\eta }_{v}\left( t\right) = {\alpha }^{\prime }\left( t\right) \;\text{ and }\;{D}_{{\alpha }^{\prime }}^{2}{\eta }_{v} = 0 = R\left( {... | No |
Proposition 2.2. Let \( \left( {X, g}\right) \) be pseudo Riemannian. Let \( \alpha : \left\lbrack {a, b}\right\rbrack \rightarrow X \) be a geodesic, and let \( \eta \) be a Jacobi lift of \( \alpha \) . Then there are numbers \( c \) , \( d \) such that\n\n\[{\left\langle \eta ,{\alpha }^{\prime }\right\rangle }_{g}\... | Proof. Using the metric derivative, and \( {D}_{{\alpha }^{\prime }}{\alpha }^{\prime } = 0 \) since \( \alpha \) is a geodesic, we find that \( \partial {\left\langle \eta ,{\alpha }^{\prime }\right\rangle }_{g} = {\left\langle {D}_{{\alpha }^{\prime }}\eta ,{\alpha }^{\prime }\right\rangle }_{g} \), and then\n\n\[{\p... | Yes |
Proposition 2.3. As above, let \( {\alpha }^{\prime }\left( 0\right) = v \) . Write \( w = {cv} + {w}_{1} \) with \( {\left\langle {w}_{1}, v\right\rangle }_{g} = 0 \) . Then \( {\eta }_{w} \) has the decomposition\n\n\[ \n{\eta }_{w} = c{\eta }_{v} + {\eta }_{{w}_{1}},\;\text{ also written }\;{\eta }_{w}\left( t\right... | Proof. Immediate from Proposition 2.2. | No |
Proposition 2.4. Notation as in Proposition 2.3, we have an orthogonal decomposition\n\n\\[ \n{D}_{{\alpha }^{\prime }}{\eta }_{w} = c{D}_{{\alpha }^{\prime }}{\eta }_{v} + {D}_{{\alpha }^{\prime }}{\eta }_{{w}_{1}}\\;\\text{ also written }\\;{D}_{{\alpha }^{\prime }}{\eta }_{w}\\left( t\\right) = c{\alpha }^{\prime }\... | Proof. For the first assertion, we take the derivative and use Proposition 2.3 to get\n\n\\[ \n0 = \\partial {\\left\\langle {\\eta }_{{w}_{1}},{\\alpha }^{\prime }\\right\\rangle }_{g} = {\\left\\langle {D}_{{\alpha }^{\prime }}{\\eta }_{{w}_{1}},{\\alpha }^{\prime }\\right\\rangle }_{g}.\n\\]\n\nFor the second, we th... | Yes |
Lemma 2.5. Assume \( \left( {X, g}\right) \) Riemannian. Let \( \eta \) be a Jacobi lift of \( \alpha \) . Let \( f\left( t\right) = \parallel \eta \left( t\right) \parallel \) . Then at those values of \( t > 0 \) such that \( \eta \left( t\right) \neq 0 \), we have\n\n\[ \n{f}^{\prime \prime } = \frac{1}{\parallel \e... | Proof. Straightforward calculus, using the covariant derivative. The first derivative \( {f}^{\prime } \) is given by\n\n\[ \n{f}^{\prime } = {\left( {\eta }^{2}\right) }^{-1/2}{\left\langle \eta ,{D}_{{\alpha }^{\prime }}\eta \right\rangle }_{g} = \frac{1}{\parallel \eta \parallel }{\left\langle \eta ,{D}_{{\alpha }^{... | Yes |
Proposition 2.6. Let \( \alpha : \left\lbrack {0, b}\right\rbrack \rightarrow X \) be a geodesic. Let \( w \in {T}_{\alpha \left( 0\right) }X \) , \( w \neq 0 \) . Let \( {\eta }_{w} = {\eta }_{0, w} = \eta \) be the unique Jacobi lift satisfying\n\n\[ \n{\eta }_{w}\left( 0\right) = 0\;\text{ and }\;{D}_{{\alpha }^{\pr... | Proof. Let \( h\left( t\right) = \parallel \eta \left( t\right) \parallel - \parallel w\parallel t \) for \( 0 \leqq t \leqq b \) . Then \( h \) is continuous, \( h\left( 0\right) = 0 \), and by Lemma 2.5, \( {h}^{\prime \prime } = {f}^{\prime \prime } \geqq 0 \) whenever \( \eta \left( t\right) \neq 0 \) . One cannot ... | Yes |
Lemma 2.7. Let \( \sigma : {J}_{1} \times {J}_{2} \rightarrow X \) be a \( {C}^{2} \) map. Then on lifts of \( \sigma \) to the tangent bundle, we have the equality of operators | Proof. The formula can be verified in a chart. It follows directly from the definitions, especially using the local expression of Proposition 1.2. | No |
Proposition 2.8. Let \( \sigma : \left\lbrack {a, b}\right\rbrack \times J \rightarrow X \) be a variation of a geodesic \( \alpha \) through geodesics. Let\n\n\[ \eta \left( s\right) = {\partial }_{2}\sigma \left( {s,0}\right) \]\n\nThen \( \eta \) is a Jacobi lift of \( \alpha \), said to come from \( \sigma \) or as... | Proof. Given \( \sigma \), we have\n\n\( {D}_{1}^{2}{\partial }_{2}\sigma = {D}_{1}{D}_{1}{\partial }_{2}\sigma = {D}_{1}{D}_{2}{\partial }_{1}\sigma \; \) by Lemma 5.3 of Chapter VIII\n\n\[ = {D}_{2}{D}_{1}{\partial }_{1}\sigma + R\left( {{\partial }_{1}\sigma ,{\partial }_{2}\sigma }\right) {\partial }_{1}\sigma \tex... | Yes |
Theorem 2.9 (Variation at the Beginning Point). Let \( \alpha \) be a geodesic in \( X \) with initial value \( \alpha \left( 0\right) = x \) . Let \( z, w \in {T}_{x}X \) . Let \( \beta \) be a curve such that\n\n\[ \beta \left( 0\right) = \alpha \left( 0\right) \;\text{ and }\;{\beta }^{\prime }\left( 0\right) = z. \... | Proof. The stated values for \( {\alpha }_{t}\left( 0\right) \) and \( {\alpha }_{t}^{\prime }\left( 0\right) \) are immediate. Then from the definition of parallel translation,\n\n\( \left( *\right) \)\n\n\[ \zeta \left( 0\right) = {\alpha }^{\prime }\left( 0\right) \;\text{ and }\;{D}_{{\beta }^{\prime }}\zeta \left(... | Yes |
Proposition 2.10. Assume that the curvature is 0 , or equivalently that the Riemann tensor \( R \) is identically 0 . Then for all \( w \in {T}_{x}X \) we have \[ {\eta }_{w}\left( t\right) = t{\gamma }_{w}\left( t\right) \] | Proof. The two curves \( t \mapsto {\eta }_{w}\left( t\right) \) and \( t \mapsto t{\gamma }_{w}\left( t\right) \) have the same initial conditions. Also they satisfy the same differential equation, namely \[ {D}_{{\alpha }^{\prime }}^{2}{\eta }_{w} = 0\;\text{ and }\;{D}_{{\alpha }^{\prime }}^{2}\left( {t{\gamma }_{w}... | Yes |
Proposition 2.11. Assume that \( \left( {X, g}\right) \) has constant curvature -1 . Then the Jacobi differential equation has the form\n\n(1)\n\n\[ \n{D}_{{\alpha }^{\prime }}^{2}{\eta }_{w} = {\eta }_{w} - {\left\langle {\eta }_{w},{\alpha }^{\prime }\right\rangle }_{g}{\alpha }^{\prime } \n\]\n\nFurthermore, if we o... | Proof. The orthogonalization of Jacobi lifts comes from Proposition 2.3 , so we want to identify the orthogonal components of the Jacobi lift of \( {\alpha }_{v} \) with scalar multiples of parallel translation. It suffices to do so when \( w = v \) and \( w = u \bot v \) separately. The example following Theorem 2.1 a... | No |
Proposition 2.12. Assume \( X \) has constant curvature +1. Let \( x \in X \) . Then the same formulas hold as in Proposition 2.11, except for a minus sign on one side in formula (1), and with \( \sinh t \) replaced by \( \sin t \) in formula (2). | Proof. The arguments are the same. Using \( \sin t \) instead of \( \sinh t \) just guarantees that the differential equation\n\n\[ \n{D}_{{\alpha }^{\prime }}^{2}{\eta }_{u} = - {\eta }_{u}\n\]\n\nis satisfied, with the minus sign. | No |
Theorem 3.1. Let \( x \in X \) and \( v \in {T}_{x} \) . Let \( \alpha \) (defined on an open interval containing 0) be the geodesic such that \( \alpha \left( 0\right) = x \) and \( {\alpha }^{\prime }\left( 0\right) = v \) . Let \( w \in {T}_{x} \) and let \( {\eta }_{w} = {\eta }_{0, w} \) be the Jacobi lift of \( \... | Proof. The curve \( {\sigma }_{t} \) is a geodesic for each \( t \), and\n\n\[ \n{\sigma }_{0}\left( s\right) = {\exp }_{x}\left( {sv}\right) = \alpha \left( s\right)\n\]\n\nso \( \sigma \) is a variation of \( \alpha \) through geodesics. Let \( \eta \left( s\right) = {\partial }_{2}\sigma \left( {s,0}\right) \) . The... | Yes |
Proposition 3.2 (Gauss Lemma, Global). Let \( \left( {X, g}\right) \) be pseudo Riemannian. Let \( x \in X \) and \( v \in {T}_{x}X \) . Let the exponential map \( r \mapsto {\exp }_{x}\left( {rv}\right) \) be defined on an open interval \( J \) . Then for all \( w \in {T}_{x}X \) we have\n\n\[{\left\langle T{\exp }_{x... | Proof. Immediate from Theorem 3.1 and the orthogonalization of Proposition 2.3. | No |
Proposition 3.3. Let \( y = {\exp }_{x}\left( {ru}\right) \) be in a normal chart at \( x \) as above, with the unit vector \( u \) . Let \( \alpha \left( s\right) = {\exp }_{x}\left( {su}\right) \), and let \( \left\{ {\alpha }_{t}\right\} \) be the variation of \( \alpha \) at its end point \( y \) in the direction o... | Proof. First note the uniqueness. If there is another Jacobi lift having the last stated property, then the difference vanishes at 0 and \( r \), and by Theorem 3.1 this difference must be 0 since the exponential map is assumed to be an isomorphism from a ball to its image, which contains \( y = \exp \left( {ru}\right)... | Yes |
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