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Theorem 4.6. Fix a point \( p \in X \) . For all vector fields \( \xi ,\eta ,\zeta \in {\mathfrak{m}}_{p} \), we have\n\n\[ R\left( {\xi ,\eta }\right) \zeta \left( p\right) = {D}_{\zeta }\left\lbrack {\xi ,\eta }\right\rbrack \left( p\right) = \left\lbrack {\zeta ,\left\lbrack {\xi ,\eta }\right\rbrack }\right\rbrack ... | Proof. By Kill 3, using \( {D}_{\eta }\zeta \left( p\right) = 0 \) and \( {D}_{\zeta }\xi \left( p\right) = 0 \), we get\n\n\[ {D}_{\eta }{D}_{\zeta }\xi \left( p\right) + R\left( {\xi ,\eta }\right) \zeta \left( p\right) = 0, \]\n\n\[ {D}_{\zeta }{D}_{\xi }\eta \left( p\right) + R\left( {\eta ,\zeta }\right) \xi \left... | Yes |
Lemma 5.1. Let \( x, y \in X \) . Assume that \( {\exp }_{x} : {T}_{x} \rightarrow X \) is surjective. Given a linear isomorphism \( L : {T}_{x} \rightarrow {T}_{y} \), there is at most one \( D \) - automorphism \( f : X \rightarrow X \) such that \( f\left( x\right) = y \) and \( {T}_{x}f = L \) . | Proof. A D-automorphism \( f \) maps geodesics to geodesics, and a geodesic is uniquely determined by its initial conditions, namely the value at 0 and the derivative at 0 . Thus the condition that \( {\exp }_{x} \) is surjective is just what is needed to determine \( f \) globally on \( X \) from its initial condition... | Yes |
Proposition 5.2. Suppose \( X \) has a symmetry at every point \( x \in X \) . Then\n\n\( X \) is geodesically complete, that is \( {\exp }_{x} \) is defined on \( {T}_{x} \) for all \( x \) . | Proof. Let \( \alpha : \left\lbrack {0, c}\right\rbrack \rightarrow X \) be a geodesic, defined on a finite interval. Let \( x = \alpha \left( c\right) \) . Then \( {T}_{x}{\sigma }_{x} \) maps \( - {\alpha }^{\prime }\left( c\right) \) to \( {\alpha }^{\prime }\left( c\right) \) . But \( {\sigma }_{x} \) being a \( D ... | Yes |
Proposition 5.3. Let \( x, y \in X \) . Let \( \alpha \) be a non-constant geodesic such that \( \alpha \left( c\right) = x \) and \( \alpha \left( b\right) = y \) . Then\n\n\[ \n{T}_{y}{\sigma }_{x} = - {P}_{b,\alpha }^{{2c} - b}\;\text{ on }\;{T}_{y}X.\n\] | Proof. Let \( v \in {T}_{\alpha \left( c\right) }X \) be a tangent vector as above. By the remarks at the beginning of this section, \( \left( {T{\sigma }_{\alpha \left( c\right) }}\right) \left( {\gamma \left( {t, v}\right) }\right) \) is parallel translation of \( \left( {T\sigma }\right) \left( v\right) \) along \( ... | Yes |
Proposition 5.4. Let \( {P}_{t,\alpha }^{t + s} : {T}_{\alpha \left( t\right) } \rightarrow {T}_{\alpha \left( {t + s}\right) } \) be parallel translation. Then\n\n\[ \n{T}_{\alpha \left( t\right) }{\tau }_{\alpha, s} = {P}_{t,\alpha }^{t + s} \n\]\n\nIn particular, for \( v \in {T}_{\alpha \left( 0\right) } \), we hav... | Proof. This is immediate from Proposition 5.3, using the chain rule for the tangent map of a composite mapping, and PAR 1. | Yes |
Proposition 5.5. Let \( \alpha \) be a non-constant geodesic.\n\n(i) Then \( \left\{ {\tau }_{\alpha, s}\right\} \) is the flow of a Killing field, i.e. it is a one-parameter group of D-automorphisms. In other words, if for \( x \in X \) we define\n\n\[ \n{\xi }_{\alpha }\left( x\right) = {\partial }_{1}{\tau }_{\alpha... | Proof. We first show that \( {\tau }_{\alpha, s + t} = {\tau }_{\alpha, s} \circ {\tau }_{\alpha, t} \) for all \( s, t \in \mathbf{R} \) . Both sides are \( D \) -automorphisms. By Lemma 5.1 it suffices to show that they coincide at one point and that their tangent maps coincide at this point. We can select the point ... | Yes |
Proposition 5.6. Let \( \alpha ,\beta \) be non-constant geodesics with\n\n\[ \alpha \left( 0\right) = \beta \left( 0\right) = p. \]\n\nLet \( {\alpha }^{\prime }\left( 0\right) = w \) . Let \( {\tau }_{\alpha } \) be translation along \( \alpha \) as above, and let\n\n\[ \eta \left( t\right) = {\partial }_{1}{\tau }_{... | Proof. That \( \eta \left( 0\right) = {\alpha }^{\prime }\left( 0\right) = w \) comes from Proposition 5.5(ii), so we next have to show that \( {D}_{{\beta }^{\prime }}\eta \left( 0\right) = 0 \) . Let \( v = {\beta }^{\prime }\left( 0\right) \) . By Proposition 5.4, we know that \( {T}_{p}{\tau }_{\alpha, s} = {P}_{0,... | Yes |
Corollary 5.7. Let \( \alpha \) be a non-constant geodesic, and put \( \alpha \left( 0\right) = p \) . Then\n\n\( {\xi }_{\alpha } \in {\mathfrak{m}}_{p} \) . | Proof. Special case of Proposition 5.6, because given \( v \in {T}_{p}X \) we can find a geodesic \( \beta \) such that \( \beta \left( 0\right) = p \) and \( {\beta }^{\prime }\left( 0\right) = v \) . | Yes |
Theorem 5.8. The Killing sequence is exact, and is split by the map \( v \mapsto {\xi }_{v} \) . The map \( \xi \mapsto \xi \left( p\right) \) thus induces an isomorphism\n\n\[ \n{\mathfrak{m}}_{p}\overset{ \approx }{ \rightarrow }{T}_{p}X \n\]\n\nof \( {\mathfrak{m}}_{p} \) with the tangent space at \( p \) . We have ... | Proof. That \( {\mathfrak{h}}_{p} \) is the kernel of \( \xi \mapsto \xi \left( p\right) \) comes from the definition of \( {\mathfrak{h}}_{p} \) . The map is surjective, because at the given point \( p \) we can find a geodesic \( \alpha \) such that \( \alpha \left( 0\right) = p \) and \( {\alpha }^{\prime }\left( 0\... | Yes |
Proposition 6.1. Let \( \Omega : X \rightarrow {L}^{3}\left( {{TX},{TX}}\right) \) be a trilinear tensor field on a D-manifold \( X \) . Then \( {D}_{\xi }\Omega = 0 \) for all \( \xi \) if and only if parallel translation commutes with \( \Omega \), that is for every geodesic \( \alpha \), \[ {P}_{a,\alpha }^{b} \circ... | Proof. If \( {D}_{\xi }\Omega = 0 \) for all vector fields \( \xi \), then the commutation comes directly from the definition of \( {D}_{\xi }\Omega = 0 \), and, say, the local expression as in Chapter VIII,3.5,3.6, and 3.7. Conversely, for a trilinear tensor field \( \Omega \) and a geodesic \( \alpha \), we have \[ \... | Yes |
Proposition 6.2. Let \( \\left( {X, D}\\right) \) be a symmetric space. Then for all vector fields \( \\xi \) we have\n\n\[ \n{D}_{\\xi }R = 0\\text{.}\n\]\n\nIn other words, the Riemann tensor is parallel. | Proof. At a given point \( x \), we compute \( {T}_{x}{\\sigma }_{x} \) applied to \( \\left( {{D}_{u}R}\\right) \\left( {v, w, z}\\right) \) in two ways, with vectors \( u, v, w, z \\in {T}_{x} \) . First,\n\n\[ \n{T}_{x}{\\sigma }_{x} \\cdot \\left( {{D}_{u}R}\\right) \\left( {v, w, z}\\right) = - \\left( {{D}_{u}R}\... | Yes |
Proposition 6.3. Let \( \\left( {X, D}\\right) \) be a D-manifold. Let \( \\alpha \) be a geodesic, \( \\alpha \\left( 0\\right) = x,{\\alpha }^{\\prime }\\left( 0\\right) = u \\neq 0 \) . Let \( \\eta \) be the Jacobi lift of \( \\alpha \) with initial conditions\n\n\[\\eta \\left( 0\\right) = {v}_{0}\\;\\text{ and }\... | Proof. Let \( {\\eta }_{1}\\left( t\\right) = {P}_{0,\\alpha }^{t}A\\left( t\\right) \) . Trivially, \( {\\eta }_{1}\\left( 0\\right) = {v}_{0} \) . By Chapter IX, Proposition 5.1, we also see that \( {D}_{{\\alpha }^{\\prime }}{\\eta }_{1}\\left( 0\\right) = {v}_{1} \) because \( {D}_{{\\alpha }^{\\prime }}{P}_{0,\\al... | Yes |
Theorem 1.1. Let \( {\zeta }_{X},{v}_{X} \) be extensions of \( \zeta, v \) to \( X \) . The covariant derivatives of \( {\zeta }_{X} \) and \( {v}_{X} \) on \( Y \) can be expressed in the form\n\n\[ \n{D}_{\eta }^{X}{\zeta }_{X} = {D}_{\eta }^{Y}\zeta + {h}_{12}\left( {\eta ,\zeta }\right) \n\]\n\n\[ \n{D}_{\eta }^{X... | We may then define an operator\n\n(1)\n\n\[ \n{H}_{\eta } : {\Gamma TY} \rightarrow {\Gamma NY}\;\text{ by the condition }\;{H}_{\eta }\left( \zeta \right) = {h}_{12}\left( {\eta ,\zeta }\right) , \n\]\n\nand then\n\n(2)\n\n\[ \n{h}_{21}\left( {\eta, v}\right) = - {}^{t}{H}_{\eta }\left( v\right) \n\]\n\nAs usual, the ... | Yes |
Proposition 1.2. Let \( \eta ,\zeta \) be vector fields on \( Y \) . Let \( {\zeta }_{X} \) be a vector field on \( X \) extending \( \zeta \) locally one some open set. Then on \( Y \), \[ {\operatorname{pr}}_{TY}{D}_{\eta }^{X}{\zeta }_{X} = {D}_{\eta }^{Y}\zeta \] | Proof. Let \( {\nabla }_{\eta }{\zeta }_{X} = {\operatorname{pr}}_{TY}{D}_{\eta }^{X}{\zeta }_{X} \) . Let \( {\eta }_{X} \) be an extension of \( \eta \) to a vector field on an open set in \( X \) . Then for \( x \in Y \) in this open set we have \[ {\left\lbrack {\eta }_{X},{\zeta }_{X}\right\rbrack }_{X}\left( x\ri... | Yes |
Proposition 1.3. Let \( x \in Y \) . Let \( v, w \in {T}_{x}Y \) . Let \( \eta ,\zeta \) be sections of \( {TY} \) on a neighborhood of \( x \) such that \( \eta \left( x\right) = v \) and \( \zeta \left( x\right) = w \) . Let \( {\eta }_{X} \) and \( {\zeta }_{X} \) be extensions of \( \eta ,\zeta \) to local vector f... | Proof. By definition of the covariant derivative,\n\n\[{D}_{\eta }^{X}{\zeta }_{X} - {D}_{\zeta }^{X}{\eta }_{X} = \left\lbrack {{\eta }_{X},{\zeta }_{X}}\right\rbrack = \left\lbrack {\eta ,\zeta }\right\rbrack \;\text{ at }x.\]\n\nBut \( \eta ,\zeta \) being sections of \( {TY} \), so is \( \left\lbrack {\eta ,\zeta }... | Yes |
Corollary 1.4. The submanifold \( Y \) is totally geodesic if and only if its second fundamental form is 0 at every point. | Proof. The condition that a curve \( \alpha \) is a geodesic is that \( {D}_{{\alpha }^{\prime }}{\alpha }^{\prime } = 0 \) . Suppose \( Y \) is totally geodesic. Let \( \alpha \) be a geodesic in \( Y \) with \( \alpha \left( 0\right) = x \) and \( {\alpha }^{\prime }\left( 0\right) = v \in {T}_{x}Y \) . Then by assum... | Yes |
Theorem 1.5. Let \( \eta ,\xi \) be vector fields on \( Y \) and let \( \mu \) be a normal field on \( Y \) . Then on \( Y \), \[ \left\langle {{D}_{\eta }^{X}{\mu }_{X},\xi }\right\rangle = \left\langle {{h}_{21}\left( {\eta ,\mu }\right) ,\xi }\right\rangle = \left\langle {\mu , - {h}_{12}\left( {\eta ,\xi }\right) }... | Proof. We take \( {D}_{\eta }^{X} \) (Lie derivative) of \( \left\langle {{\xi }_{X},{\mu }_{X}}\right\rangle \), evaluated at a point of \( Y \) . The scalar product is taken in \( {TX} \), of course. To find the derivative at a point \( x \in Y \), one may differentiate along any curve passing through that point, suc... | Yes |
Proposition 1.6. Let \( {\mu }_{X} \) be an extension of a normal field to \( X \). Then \( {\operatorname{pr}}_{NY}{D}_{\eta }^{X}{\mu }_{X} \) is independent of this extension, so we may denote\n\n\[ \n{\nabla }_{\eta }\mu = {\operatorname{pr}}_{NY}{D}_{\eta }^{X}{\mu }_{X} \n\]\n\nFurthermore, \( \nabla \) is a metr... | Proof. We prove the metric formula first. By definition of the covariant derivative on \( X \), we know that on \( Y \), for any normal field \( v \), \n\n\[ \n\eta \cdot \left\langle {{\mu }_{X},{v}_{X}}\right\rangle = \left\langle {{D}_{\eta }^{X}{\mu }_{X},{v}_{X}}\right\rangle + \left\langle {{\mu }_{X},{D}_{\eta }... | Yes |
Proposition 2.1. Let \( {f}_{X} \) be an extension of \( f \) to \( X \) . Let \( \xi ,\eta \) be vector fields on \( Y \) . Then on \( Y \), we have\n\n\[ \n{D}_{X}^{2}{f}_{X}\left( {\xi ,\eta }\right) = {D}_{Y}^{2}f\left( {\xi ,\eta }\right) - {h}_{12}\left( {\xi ,\eta }\right) \cdot {f}_{X}, \n\]\n\nwhere \( {h}_{12... | Proof. We have\n\n\[ \n{D}_{X}^{2}{f}_{X}\left( {{\xi }_{X},{\xi }_{X}}\right) = \xi \cdot \eta \cdot f - \left( {{D}_{\xi }^{X}{\eta }_{X}}\right) \cdot {f}_{X} \n\]\n\nBy Theorem 1.1, at points of \( Y \) we have\n\n\[ \n{D}_{\xi }^{X}{\eta }_{X} = \left( {{D}_{\xi }^{Y}\eta }\right) + {h}_{12}\left( {\xi ,\eta }\rig... | No |
Proposition 2.2. Let \( {f}_{X} \) be the normal extension of \( f \) to a tubular neighborhood of \( Y \) . Then for vector fields \( \xi ,\eta \) on \( Y \), we have\n\n\[ \n{D}_{X}^{2}{f}_{X}\left( {\xi ,\eta }\right) = {D}_{Y}^{2}f\left( {\xi ,\eta }\right) \n\] | Proof. This is immediate, because if \( v \) is a normal vector field on \( Y \) , then \( \left( {v \cdot {f}_{X}}\right) \left( x\right) = 0 \) for \( x \in Y \), immediately from the definitions. Indeed, the Lie derivative may be taken along a geodesic from \( x \), along which \( f \) is constant, so its Lie deriva... | No |
Proposition 2.3. Let \( v \) be a normal field on \( Y \) . Let \( f \) be a function on \( Y \) and \( {f}_{X} \) its normal extension to a tubular neighborhood of \( Y \) . Then on \( Y \) , \[ {D}_{X}^{2}{f}_{X}\left( {v, v}\right) = 0 \] | Proof. Let \( {v}_{X} \) be any extension of \( v \) to a neighborhood of a point \( {x}_{0} \) in in \( Y \) . Then at \( {x}_{0} \) , \[ {D}_{X}^{2}{f}_{X}\left( {v, v}\right) = {v}_{X} \cdot {v}_{X} \cdot {f}_{X} - \left( {{D}_{v\left( {x}_{0}\right) }{v}_{X}}\right) \cdot {f}_{X}. \] We select a suitable extension ... | Yes |
Theorem 2.4. Let \( X \) be a finite dimensional Riemannian manifold, and let \( Y \) be a submanifold. Let \( f \) be a function on \( Y \) and let \( {f}_{X} \) be its normal extension to \( X \) . Then on \( Y \) , \[ {\Delta }_{Y}f = {\Delta }_{X}{f}_{X} \] | Proof. Let \( \left\{ {{\xi }_{1},\ldots ,{\xi }_{p}}\right\} \) be an orthonormal frame of vector fields locally on \( Y \), and let \( \left\{ {{v}_{1},\ldots ,{v}_{q}}\right\} \) be an orthonormal frame of normal fields. Together they form an orthonormal frame of sections of \( {TX} \) restricted to \( Y \) . Then a... | Yes |
Proposition 2.6. Let \( X \) be a Riemannian manifold, \( Y \) a submanifold, and \( {x}_{0} \in Y \) . Let \( {W}_{0} \) be the normal submanifold to \( Y \) at \( {x}_{0} \) . Let \( f \) be a function on \( X \), and let \( w \in {N}_{{x}_{0}}Y \) . Then \( {D}_{X}^{2}f\left( {w, w}\right) \) depends only on the res... | Proof. By the Killing definition of the second tensorial derivative (Chapter XIII, §1) and Corollary 3.2 of Chapter VIII, §3 we may compute this derivative along the geodesic, that is\n\n\[ {D}_{X}^{2}f\left( {w, w}\right) = \left( {{D}_{{\alpha }^{\prime }}{D}_{{\alpha }^{\prime }}f}\right) \left( {x}_{0}\right) - \le... | Yes |
Proposition 2.7. Suppose \( X \) finite dimensional. Let \( Y \) be a submanifold, and \( {x}_{0} \in Y \) . Let \( {W}_{0} \) be the normal submanifold of \( Y \) at \( {x}_{0} \) . Let \( f \) be a function on \( X \) . Then\n\n\[ \n\left( {\left( {\operatorname{tr}{h}_{12}}\right) \cdot f}\right) \left( {x}_{0}\righ... | Proof. Immediate from (4) and Proposition 2.6, using \( w = {w}_{j} = {v}_{j}\left( {x}_{0}\right) \) and \( v = {v}_{i} = {\xi }_{i}\left( {x}_{0}\right) \) . | No |
Proposition 2.8. Suppose \( X \) finite dimensional. Let \( \pi : X \rightarrow Z \) be a submersion. Then\n\n\[{\Delta }_{X} = {\Delta }_{V,\pi } + {\Delta }_{T,\pi }\] | Proof. This is just a reformulation of Proposition 2.7, taking the previous definitions into account. | No |
Lemma 2.9. The above map \( \varphi \) is a local isomorphism at \( \left( {{x}_{0},0}\right) \) . Its differential at this point is in fact the identity. | Proof. This is a routine verification left to the reader. Note that the tangent space of \( {V}_{0} \times {W}_{0}^{\prime } \) at the point is precisely \( {T}_{{x}_{0}}Y \times {N}_{{x}_{0}}Y \), which we identify with \( {T}_{{x}_{0}}X \) . The second statement about the differential implies the first about the loca... | No |
Lemma 2.9. The above map \( \varphi \) is a local isomorphism at \( \left( {{x}_{0},0}\right) \) . Its differential at this point is in fact the identity. | Proof. This is a routine verification left to the reader. Note that the tangent space of \( {V}_{0} \times {W}_{0}^{\prime } \) at the point is precisely \( {T}_{{x}_{0}}Y \times {N}_{{x}_{0}}Y \), which we identify with \( {T}_{{x}_{0}}X \) . The second statement about the differential implies the first about the loca... | No |
Lemma 2.10. Under a regular action by \( H \), the map \[ \varphi : {V}_{0} \times {W}_{0} \rightarrow X\;\text{given by}\;\left( {h, x}\right) \mapsto {hx} \] is a local isomorphism at the origin \( \left( {e,{x}_{0}}\right) \). | Proof. This is a simple exercise in computing the differential of the map at the origin, and showing that it is the identity. | No |
Lemma 2.10. Under a regular action by \( H \), the map\n\n\[ \n\varphi : {V}_{0} \times {W}_{0} \rightarrow X\;\text{given by}\;\left( {h, x}\right) \mapsto {hx} \n\] \n\nis a local isomorphism at the origin \( \left( {e,{x}_{0}}\right) \) . | Proof. This is a simple exercise in computing the differential of the map at the origin, and showing that it is the identity. | No |
Lemma 3.1. Let \( x \in {Y}_{\pi \left( x\right) } \) be a point in a fiber. Let \( f \) be a function on Z. Then for \( w \in {N}_{x}{Y}_{\pi \left( x\right) } \) we have\n\n\[ \left( {{D}_{w}{\pi }^{ * }f}\right) \left( x\right) = \left( {D{\pi }_{*w}f}\right) \left( {\pi \left( x\right) }\right) \]\n\nor in differen... | Proof. One may prove the formulas in a chart, in which case both merely come from the chain rule\n\n\[ {\left( f \circ \pi \right) }^{\prime }\left( x\right) = {f}^{\prime }\left( {\pi \left( x\right) }\right) {T\pi }\left( x\right) \]\n\napplied to any vector in \( {T}_{x}X = {T}_{x}{Y}_{\pi \left( x\right) } + {N}_{x... | Yes |
Lemma 3.1. Let \( x \in {Y}_{\pi \left( x\right) } \) be a point in a fiber. Let \( f \) be a function on Z. Then for \( w \in {N}_{x}{Y}_{\pi \left( x\right) } \) we have\n\n\[ \left( {{D}_{w}{\pi }^{ * }f}\right) \left( x\right) = \left( {D{\pi }_{*w}f}\right) \left( {\pi \left( x\right) }\right) \]\n\nor in differen... | Proof. One may prove the formulas in a chart, in which case both merely come from the chain rule\n\n\[ {\left( f \circ \pi \right) }^{\prime }\left( x\right) = {f}^{\prime }\left( {\pi \left( x\right) }\right) {T\pi }\left( x\right) \]\n\napplied to any vector in \( {T}_{x}X = {T}_{x}{Y}_{\pi \left( x\right) } + {N}_{x... | Yes |
Proposition 3.2. Let \( \mu, v \) be vector fields on \( Z \), and \( {\mu }_{X},{v}_{X} \) their horizontal liftings to \( X \) . Then\n\n\[ \n{\operatorname{pr}}_{E}\left( {{D}_{{\mu }_{X}}{v}_{X}}\right) = {\left( {D}_{\mu }v\right) }_{X} \n\]\n\nor equivalently, for every horizontal field \( {\lambda }_{X} \), \n\n... | Proof. The expression \( \left\langle {{D}_{{\mu }_{X}}{v}_{X},{\lambda }_{X}}\right\rangle \) coming from (3) involves only the Lie derivative, scalar product of vector fields and brackets. The scalar product is preserved under lifting, by definition of a Riemannian submersion. Formula (1) gives the preservation of th... | No |
Proposition 3.3. Let \( \mu, v,\lambda ,\zeta \) be vector fields on \( Z \) . Then\n\n\[{\mu }_{X} \cdot \left\langle {{D}_{{v}_{X}}{\lambda }_{X},{\zeta }_{X}}\right\rangle = {\pi }^{ * }\left( {\mu \cdot \left\langle {{D}_{v}\lambda ,\zeta }\right\rangle }\right) . | Proof. Again, direct consequence of (3) and Proposition 3.2. | No |
Proposition 3.4. Let \( \mu, v \) be vector fields on \( Z \) . Then\n\n\[ \n{D}_{{\mu }_{X}}{v}_{X} = \frac{1}{2}{\left\lbrack {\mu }_{X},{v}_{X}\right\rbrack }^{V} + {\left( {D}_{\mu }v\right) }_{X}.\n\] | Proof. The horizontal component was already determined in Proposition 3.2, which gives the second term on the right of the equation. As for the vertical component, we use (3) with a vertical field \( \xi \) . Since \( \left\langle {{\mu }_{X},{v}_{X}}\right\rangle = \langle \mu, v\rangle \), if \( \xi \) is vertical, w... | Yes |
Proposition 3.5. Let \( \alpha : \left\lbrack {a, b}\right\rbrack \rightarrow Z \) be a curve such that \( {\alpha }^{\prime }\left( t\right) \neq 0 \) for all \( t \) . (i) Let \( y \in {Y}_{\alpha \left( a\right) } \) . There exists a unique lifting \( A = {A}_{y} \) of \( \alpha \) in \( X \) which is horizontal, i.... | Proof. The existence and uniqueness of the lifting are elementary, at the level of the existence and uniqueness of solutions of a differential equation. We give the details. The global assertion is a consequence of local existence and uniqueness, so we may suppose that there is a vector field \( v \) locally on \( Z \)... | Yes |
Proposition 3.6. Let \( \\xi \) be a vertical field. Then\n\n\[ \n\\left\\langle {{D}_{\\xi }{\\mu }_{X},{v}_{X}}\\right\\rangle = - \\frac{1}{2}\\left\\langle {{\\left\\lbrack {\\mu }_{X},{v}_{X}\\right\\rbrack }^{V},\\xi }\\right\\rangle \n\] | Proof. By the metric derivative formula (3) and Proposition 3.4, we\n\nobtain\n\n\[ \n\\left\\langle {{D}_{\\xi }{\\mu }_{X},{v}_{X}}\\right\\rangle = \\left\\langle {{D}_{{\\mu }_{X}}\\xi ,{v}_{X}}\\right\\rangle + \\left\\langle {\\left\\lbrack {\\xi ,{\\mu }_{X}}\\right\\rbrack ,{v}_{X}}\\right\\rangle \n\]\n\n\[ \n... | Yes |
Proposition 4.1. Let \( \xi ,\eta \) be vertical fields on \( X \) . Then for every function \( f \) on \( Z \), we have\n\n\[ \left( {{D}_{X}^{2}{\pi }^{ * }f}\right) \left( {\xi ,\eta }\right) = - {h}_{12}\left( {\xi ,\eta }\right) \cdot {\pi }^{ * }f \]\n\n\[ = - {\pi }_{ * }{h}_{12}\left( {\xi ,\eta }\right) \cdot ... | Proof. We have\n\n\[ {D}^{2}{\pi }^{ * }f\left( {\xi ,\eta }\right) = \xi \cdot \left( {\eta \cdot {\pi }^{ * }f}\right) - \left( {{D}_{\xi }\eta }\right) \cdot {\pi }^{ * }f \]\n\n\[ = - \left( {{D}_{\xi }\eta }\right) \cdot {\pi }^{ * }f \]\n\nbecause \( \eta \cdot {\pi }^{ * }f = 0 \) since \( {\pi }^{ * }f \) is co... | Yes |
Proposition 4.3. Let \( \mu, v \) be vector fields on \( Z \), with horizontal liftings \( {\mu }_{X},{v}_{X} \) . Then\n\n\[ \n{D}_{X}^{2}{\pi }^{ * }f\left( {{\mu }_{X},{v}_{X}}\right) = {D}_{Z}^{2}f\left( {\mu, v}\right) .\n\] | Proof. We have\n\n\[ \n{D}_{Z}^{2}f\left( {\mu, v}\right) = \mu \cdot v \cdot f - \left( {{D}_{\mu }v}\right) \cdot f,\n\]\n\nand the similar expression on \( X \) with subscript \( X \) . The vertical component of \( {D}_{{\mu }_{X}}{v}_{X} \) annihilates \( {\pi }^{ * }f \) because \( {\pi }^{ * }f \) is constant on ... | Yes |
Theorem 4.4. Assume that \( X \), and hence \( Z \), are finite dimensional. Then for all functions \( f \) on \( Z \) we have \[ {\Delta }_{X}{\pi }^{ * }f = {\pi }^{ * }{\Delta }_{Z}f + \left( {\operatorname{tr}{h}_{12}}\right) \cdot {\pi }^{ * }f. \] | Proof. Let \( \left\{ {{\xi }_{1},\ldots ,{\xi }_{p}}\right\} \) be an orthonormal frame of local sections of the vertical bundle \( F \), and let \( \left\{ {{\mu }_{1},\ldots ,{\mu }_{q}}\right\} \) be an orthonormal frame of sections on \( Z \) . Let \( \left\{ {{\mu }_{1X},\ldots ,{\mu }_{qX}}\right\} \) be their l... | Yes |
Theorem 5.1 (Gauss Equation). For \( {v}_{i}\left( {i = 1,2,3,4}\right) \) in \( {T}_{x}Y \), we have\n\n\[ \n{R}_{X}\left( {{v}_{1},{v}_{2},{v}_{3},{v}_{4}}\right) = {R}_{Y}\left( {{v}_{1},{v}_{2},{v}_{3},{v}_{4}}\right) \n\]\n\n\[ \n+ \left\langle {{h}_{12}\left( {{v}_{2},{v}_{3}}\right) ,{h}_{12}\left( {{v}_{1} \cdo... | Proof. The proof is routine, and forced. We have by Theorem 1.1, or SFF 2 in \( §1 \), on \( Y \) :\n\n\[ \n{D}_{\eta }^{Y}\zeta = {D}_{\eta }^{X}{\zeta }_{X} + {h}_{12}\left( {\eta ,\zeta }\right) \n\]\n\nso iterating,\n\n\[ \n{D}_{\xi }^{Y}{D}_{\eta }^{Y}\zeta = {\operatorname{pr}}_{TY}\left( {{D}_{\xi }^{X}{D}_{\eta... | Yes |
Theorem 5.2 (Codazzi Equation). For vector fields \( \xi ,\eta ,\zeta \) on \( Y \) , \[ {\operatorname{pr}}_{NY}{R}_{X}\left( {\xi ,\eta ,\zeta }\right) = \left( {{\nabla }_{\xi }{h}_{12}}\right) \left( {\eta ,\zeta }\right) - \left( {{\nabla }_{\eta }{h}_{12}}\right) \left( {\xi ,\zeta }\right) . \] | Proof. We start again with \[ {D}_{\eta }^{X}{\zeta }_{X} = {D}_{\eta }^{Y}\zeta + {h}_{12}\left( {\eta ,\zeta }\right) \] so \[ {D}_{\xi }^{X}{D}_{\eta }^{X}{\zeta }_{X} = {D}_{\xi }^{X}\left( {{\left( {D}_{\eta }^{Y}\zeta \right) }_{X} + {D}^{X}{\left( {h}_{12}\left( \eta ,\zeta \right) \right) }_{X}}\right. \] \[ = ... | Yes |
Theorem 5.3 (Ricci Equation). We have\n\n\[ \n{R}_{X}\left( {\xi ,\eta ,\mu, v}\right) = {R}_{NY}\left( {\xi ,\eta ,\mu, v}\right) - \left\langle {\left\lbrack {{S}_{\mu },{S}_{v}}\right\rbrack \xi ,\eta }\right\rangle .\n\] | Proof. More of the same type of computation. We use (6) in \( §1 \) twice to get\n\n\[ \n{R}_{X}\left( {\xi ,\eta }\right) \mu = {D}_{\xi }^{X}{D}_{\eta }^{X}\mu - {D}_{\eta }^{X}{D}_{\xi }^{X}\mu - {D}_{\left\lbrack \xi ,\eta \right\rbrack }^{X}\mu \n\]\n\n\[ \n= {R}_{NY}\left( {\xi ,\eta }\right) + {S}_{{\nabla }_{\x... | Yes |
Theorem 6.1. Let \( \mu, v,\lambda ,\zeta \) be vector fields on \( Z \) . Then\n\n\[ \n{R}_{X}\left( {{\mu }_{X},{v}_{X},{\lambda }_{X},{\zeta }_{X}}\right) = {R}_{Z}\left( {\mu, v,\lambda ,\zeta }\right) + {V}_{R}\left( {{\mu }_{X},{v}_{X},{\lambda }_{X},{\zeta }_{X}}\right) \n\]\n\nwhere \( {V}_{R} \) denotes the ve... | Proof. The Riemann tensor involves second derivatives, but all the formulas needed to perform the iteration easily have been proved in \( §3 \) . So we forge ahead. First, by Propositions 3.3, 3.5, and 3.6, we find\n\n(1)\n\[ \n\left\langle {{D}_{{\mu }_{X}}{D}_{{v}_{X}}{\lambda }_{X},{\zeta }_{X}}\right\rangle = {\mu ... | Yes |
Corollary 6.2. For the tensor \( {R}_{2} \) such that \( {R}_{2}\left( {v, w}\right) = R\left( {v, w, v, w}\right) \), we get\n\n\[ \n{R}_{2X}\left( {{\mu }_{X},{v}_{X}}\right) = {R}_{2Z}\left( {\mu, v}\right) + \frac{3}{4}{\begin{Vmatrix}\left\lbrack {\mu }_{X},{v}_{X}\right\rbrack \end{Vmatrix}}^{2}.\n\] \n\nIn parti... | Proof. This is immediate from the definition and Theorem 6.1. | No |
Proposition 1.1. Let \( \Omega = {\operatorname{vol}}_{g} \) . Then for all n-tuples of vectors \( \left\{ {{v}_{1},\ldots ,{v}_{n}}\right\} \) and \( \left\{ {{w}_{1},\ldots ,{w}_{n}}\right\} \) in \( V \), we have\n\n\[ \Omega \left( {{v}_{1},\ldots ,{v}_{n}}\right) \Omega \left( {{w}_{1},\ldots ,{w}_{n}}\right) = \d... | Proof. The determinant on the right side of the first formula is multilinear and alternating in each \( n \) -tuple \( \left\{ {{v}_{1},\ldots ,{v}_{n}}\right\} \) and \( \left\{ {{w}_{1},\ldots ,{w}_{n}}\right\} \).\n\nHence there exists a number \( c \in \mathbf{R} \) such that\n\n\[ \det {\left\langle {v}_{i},{w}_{j... | Yes |
Proposition 1.3. For functions \( \varphi ,\psi \) we have\n\n\[ \mathbf{\Delta }\left( {\varphi \psi }\right) = \varphi \mathbf{\Delta }\psi + \psi \mathbf{\Delta }\varphi - 2\langle {d\varphi },{d\psi }{\rangle }_{g}. \] | Proof. The routine gives:\n\n\[ \mathbf{\Delta }\left( {\varphi \psi }\right) = {d}^{ * }d\left( {\varphi \psi }\right) = {d}^{ * }\left( {{\psi d\varphi } + {\varphi d\psi }}\right) \]\n\n\[ = - \operatorname{div}\left( {\psi {\xi }_{d\varphi }}\right) - \operatorname{div}\left( {\varphi {\xi }_{d\psi }}\right) \]\n\n... | Yes |
Corollary 1.4. Let \( \delta \) be a positive function. Then\n\n\[ \mathbf{\Delta } - \left\lbrack {\operatorname{gr}\log \delta }\right\rbrack = {\delta }^{-1/2}\mathbf{\Delta } \circ {\delta }^{1/2} - {\delta }^{-1/2}\mathbf{\Delta }\left( {\delta }^{1/2}\right) . \] | Proof. For a function \( \psi \), by Proposition 1.3,\n\n\[ \left( {\mathbf{\Delta } \circ {\delta }^{1/2}}\right) \psi = \mathbf{\Delta }\left( {{\delta }^{1/2}\psi }\right) \]\n\n\[ = {\delta }^{1/2}\mathbf{\Delta }\psi + \psi \mathbf{\Delta }\left( {\delta }^{1/2}\right) - 2\left( {\operatorname{gr}{\delta }^{1/2}}\... | Yes |
Proposition 1.5.\n\n\[ \n{\\operatorname{div}}_{\\Omega }\\xi = {\\delta }^{-1}\\sum {\\partial }_{i}\\left( {\\delta {\\varphi }_{i}}\\right) \n\]\n\n\[ \n= \\sum {\\partial }_{i}{\\varphi }_{i} + \\sum \\left( {{\\partial }_{i}\\log \\delta }\\right) {\\varphi }_{i} \n\]\n\nIn matrix form,\n\n\[ \n{\\operatorname{div... | Proof. We have\n\n\[ \n\\left( {\\Omega \\circ \\xi }\\right) \\left( {{u}_{1},\\ldots ,{\\widehat{u}}_{i},\\ldots ,{u}_{n}}\\right) = \\Omega \\left( {\\xi ,{u}_{1},\\ldots ,{\\widehat{u}}_{i},\\ldots ,{u}_{n}}\\right) \n\]\n\n\[ \n= {\\left( -1\\right) }^{i - 1}\\Omega \\left( {{u}_{1},\\ldots ,\\xi ,\\ldots ,{u}_{n}... | Yes |
Proposition 1.6. Let \( \operatorname{gr}\left( \psi \right) = \sum {\varphi }_{i}{u}_{i} \) . Let \( g\left( x\right) \) be the \( n \times n \) matrix representing the metric at a point \( x \) . Then the coordinate vector of \( \operatorname{gr}\left( \psi \right) \) is\n\n\[ \Phi = \left( \begin{matrix} {\varphi }_... | Proof. By definition,\n\n\[ {\left\langle \operatorname{gr}\left( \psi \right) ,{u}_{j}\right\rangle }_{g} = \left( {d\psi }\right) \left( {u}_{j}\right) = {\partial }_{j}\psi \]\n\nThe left side is equal to \( \left\langle {\operatorname{gr}\left( \psi \right), g\left( x\right) {u}_{j}}\right\rangle \) at a point \( x... | Yes |
Proposition 1.7. Let \( f \) and \( \psi \) be function, and let \( \operatorname{gr}\left( \psi \right) = \sum {\varphi }_{j}{u}_{j} \) as in Proposition 1.6. Then \[ \operatorname{gr}\left( \psi \right) \cdot f = \mathop{\sum }\limits_{{j = 1}}^{n}\left( {{\partial }_{j}f}\right) {\varphi }_{j} \] | Proof. Since \( {u}_{j} \cdot f = {\partial }_{j}f \), the formula is clear. | No |
Proposition 1.6. Then\n\n\\[ \n\\operatorname{gr}\\left( \\psi \\right) \\cdot f = \\mathop{\\sum }\\limits_{{j = 1}}^{n}\\left( {{\\partial }_{j}f}\\right) {\\varphi }_{j} \n\\] | Proof. Since \\( {u}_{j} \\cdot f = {\\partial }_{j}f \\), the formula is clear. | No |
Proposition 1.8. On an open set of \( {\mathbf{R}}^{n} \), with metric matrix \( g,\delta = \) \( {\left( \det g\right) }^{1/2} \), and Laplacian \( {\mathbf{\Delta }}_{g} \), we have\n\n\[ - {\mathbf{\Delta }}_{g} = {\operatorname{div}}_{g}{\operatorname{gr}}_{g} = {}^{t}{\mathbf{D}}_{g}{g}^{-1}\partial \]\n\n\[ = {\d... | Here, \( {\mathbf{D}}_{g} \) abbreviates \( {\mathbf{D}}_{{\Omega }_{g}} \), and \( {\operatorname{div}}_{g} \) abbreviates \( {\operatorname{div}}_{{\Omega }_{g}} \). Putting all the indices in, we get\n\n(1)\n\n\[ - {\mathbf{\Delta }}_{g}f = {\delta }^{-1}\mathop{\sum }\limits_{i}{\partial }_{i}\left( {\delta \mathop... | Yes |
Theorem 1.9. Let \( X \) be a Riemannian manifold. Then the Laplacian determines the metric, i.e. if two Riemannian metrics have the same Laplacian, they are equal. If \( F : X \rightarrow Y \) is a differential isomorphism of Riemannian manifolds, and \( F \) maps \( {\mathbf{\Delta }}_{X} \) on \( {\mathbf{\Delta }}_... | Note that the second statement about the differential isomorphism is just a piece of functorial abstract nonsense, in light of the first statement. Indeed, \( F \) maps the metric \( {g}_{X} \) to a metric \( {F}_{ * }{g}_{X} \) on \( Y \), and similarly for the Laplacian. By assumption, \( {F}_{ * }{\mathbf{\Delta }}_... | Yes |
Theorem 2.1. Let \( D \) be the metric covariant derivative. Then\n\n\[ \n{D}_{\xi }{\operatorname{vol}}_{g} = 0 \n\]\n\nfor all vector fields \( \xi \) . | Proof. Let \( \Omega = {\operatorname{vol}}_{g} \) be the Riemannian volume form. If \( \left\{ {{\xi }_{1},\ldots ,{\xi }_{n}}\right\} \) is an orthonormal frame, then \( \Omega = \pm {\xi }_{1}^{ \vee } \land \cdots \land {\xi }_{n}^{ \vee } \) and \( \langle \Omega ,\Omega {\rangle }_{g} = 1 \) . Taking the Lie deri... | Yes |
Theorem 2.2. Let \( {\xi }_{1},\ldots ,{\xi }_{n} \) be an orthonormal frame of vector fields, and let \( \xi \) be a vector field. Then\n\n\[ \n\operatorname{div}\xi = \mathop{\sum }\limits_{{i = 1}}^{n}{\left\langle {D}_{{\xi }_{i}}\xi ,{\xi }_{i}\right\rangle }_{g} = \operatorname{tr}\left( {D\xi }\right)\n\]\n\nIn ... | Proof. Let \( \Omega = {\operatorname{vol}}_{g} \) be the volume form. By COVD 6 of Chapter VIII, §1, and Proposition 2.1, we get\n\n\[ \nd\left( {\Omega \circ \xi }\right) \left( {{\xi }_{1},\ldots ,{\xi }_{n}}\right) = \mathop{\sum }\limits_{{i = 1}}^{n}{\left( -1\right) }^{i - 1}{D}_{{\xi }_{i}}\left( {\Omega \circ ... | Yes |
Corollary 2.4. Let \( {\xi }_{1},\ldots ,{\xi }_{n} \) be an orthonormal frame as in Theorem 2.2. Let \( \varphi \) be a function. Then\n\n\[ \n{\Delta \varphi } = - \operatorname{tr}\left( {Dd\varphi }\right) = - \mathop{\sum }\limits_{{i = 1}}^{n}\left\langle {{D}_{{\xi }_{i}}{d\varphi },{\xi }_{i}}\right\rangle = - ... | Proof. The first assertion comes from applying Theorem 2.2 to \( \lambda = {d\varphi } \) . The second assertion then follows by using Corollary 5.6 of Chapter VIII. | No |
Proposition 2.5. Let \( \alpha = {\alpha }_{1} \) be the unique geodesic from \( x \) to \( y \neq x \) , parametrized by arc length, and let \( {e}_{1} = {\alpha }^{\prime }\left( r\right) \in {T}_{y}X \) . Let \( {e}_{2},\ldots ,{e}_{n} \) be unit vectors in \( {T}_{y}X \) such that \( \left\{ {{e}_{1},\ldots ,{e}_{n... | Proof. Let \( {\beta }_{i}\left( {i = 1,\ldots, n}\right) \) be the geodesic from \( y \) such that \[ {\beta }_{i}\left( 0\right) = y\;\text{ and }\;{\beta }_{i}^{\prime }\left( 0\right) = {e}_{i}. \] Observe that \( {\beta }_{1}\left( t\right) = {\alpha }_{1}\left( {r + t}\right) \) for small \( t \), by the uniquene... | Yes |
Proposition 2.6. Let \( \delta = {\left( \det g\right) }^{1/2} \) . For each \( j \) we have\n\n\[ \n{\partial }_{j}\log \delta = - \mathop{\sum }\limits_{k}{B}_{U}\left( {{u}_{j},{u}_{k}}\right) \n\]\n\nand\n\n\[ \n\operatorname{div}\xi = \operatorname{tr}\left( {D\xi }\right) = \mathop{\sum }\limits_{i}{\partial }_{i... | Proof. The second formula for the trace comes from the definition of the trace and the definition of \( {D\xi } \) . The first formula then follows componentwise from Proposition 1.4. This concludes the proof. | No |
Proposition 3.1. Let \( u \) be a unit vector in \( {T}_{x}X \) and let \( \alpha \) be the geodesic parametrized by arc length such that \( \alpha \left( 0\right) = x \) and \( {\alpha }^{\prime }\left( 0\right) = u \) . Put \( u = {w}_{1} \) and let \( \left\{ {u,{w}_{2},\ldots ,{w}_{n}}\right\} \) be a basis of \( {... | Proof. Observe that we may also use \( {\eta }_{1} \), which is such that \( {\eta }_{1}\left( t\right) = t{\alpha }^{\prime }\left( t\right) \) . The equality between the two expressions on the right of the equality sign follows from Proposition 1.1. Let \( f = {\exp }_{x} \) . Then for any vectors \( {w}_{1},\ldots ,... | Yes |
Corollary 3.2. If in Proposition 3.1 all the vectors \( {w}_{i} \) are unit vectors \( {u}_{i} \) such that \( \left\{ {{u}_{1},\ldots ,{u}_{n}}\right\} \) is an orthonormal basis of \( {T}_{x}X \), and \( u = {u}_{1} \), then we have simply\n\n\[ \n{r}^{n - 1}J\left( {r, u}\right) = \mathop{\det }\limits^{{1/2}}{\left... | Proof. By Corollary 3.2, \( J\left( {r, u}\right) \) is \( \mathop{\det }\limits^{{1/2}}{\left\langle {\eta }_{i}\left( r\right) /r,{\eta }_{j}\left( r\right) /r\right\rangle }_{q} \) with the determinant taken for \( i, j = 1,\ldots, n \) or \( i, j = 2,\ldots, n \) . Using the asymptotic expansion of Chapter IX, Prop... | Yes |
Again with an orthonormal basis \( \\left\\{ {{u}_{1},\\ldots ,{u}_{n}}\\right\\} \) of \( {T}_{x}X \), let \( u = {u}_{1} \) and\n\n\\[ \n\\operatorname{Ric}\\left( {u, u}\\right) = \\mathop{\\sum }\\limits_{{i = 2}}^{n}{R}_{2}\\left( {u,{u}_{i}}\\right)\n\\]\n\nThen\n\n\\[ \n{\\exp }_{x}^{ * }{\\operatorname{vol}}_{g... | Proof. By Corollary 3.2, \( J\\left( {r, u}\\right) \) is \( \\mathop{\\det }\\limits^{{1/2}}{\\left\\langle {\\eta }_{i}\\left( r\\right) /r,{\\eta }_{j}\\left( r\\right) /r\\right\\rangle }_{q} \) with the determinant taken for \( i, j = 1,\\ldots, n \) or \( i, j = 2,\\ldots, n \) . Using the asymptotic expansion of... | Yes |
Let \( {\exp }_{x} : {\mathbf{B}}_{x} \rightarrow X \) be the normal chart in \( X \) as at the beginning of the section, and \( y = {\exp }_{x}\left( {ru}\right) \) with \( {ru} \in {\mathbf{B}}_{x} \), and some unit vector \( u \) . Let \( \alpha \left( s\right) = {\exp }_{x}\left( {su}\right) \) and let \( {e}_{1} =... | Proof. In the present case, \( {D}_{{\alpha }^{\prime }}{\eta }_{i}\left( 0\right) = {w}_{i} \) is whatever it is, but we observe that the determinant \( \det \left( {{w}_{2},\ldots ,{w}_{n}}\right) \) is constant, so disappears in taking the logarithmic derivative of the expression in Proposition 3.1. We also observe ... | Yes |
Lemma 3.5. Let \( A = \left( {{A}^{1},\ldots ,{A}^{m}}\right) \) be a non-singular \( m \times m \) matrix over a field, where \( {A}^{1},\ldots ,{A}^{m} \) are the columns of \( A \) . Let \( B = \left( {{B}^{1},\ldots ,{B}^{m}}\right) \) be any \( m \times m \) matrix over the field. Then\n\n\[ \mathop{\sum }\limits_... | Proof. Let \( X = \left( {x}_{ij}\right) \) be the matrix such that\n\n\[ {x}_{1i}{A}^{1} + \cdots + {x}_{mi}{A}^{m} = {B}^{i}\;\text{ for }\;i = 1,\ldots, m. \]\n\nBy Cramer's rule,\n\n\[ {x}_{ii}\det \left( A\right) = \det \left( {{A}^{1},\ldots ,{B}^{i},\ldots ,{A}^{m}}\right) . \]\n\nBut \( {AX} = B \) so \( X = {A... | Yes |
Corollary 3.6. Let \( \varphi \) be a \( {C}^{2} \) function on a normal ball centered at the point \( x \in X \) . Suppose that \( \varphi \) depends only on the \( g \) -distance \( r \) from \( x \), say \( \varphi \left( y\right) = f\left( {r\left( y\right) }\right) \) . Let \( y = \exp \left( {ru}\right) \), with ... | Proof. Combine Corollary 3.4 with Proposition 2.5. | No |
Lemma 3.7. Let \( u \) be a unit vector in \( {T}_{x}X \) . Let \( \varphi \) be a \( {C}^{2} \) function on a normal ball centered at \( x \), and define the function \( {f}_{u} \) by\n\n\[ \n{f}_{u}\left( r\right) = \varphi \left( {{\exp }_{x}\left( {ru}\right) }\right) .\n\]\n\nThen\n\n\[ \n{f}_{u}^{\prime }\left( r... | Proof. Let \( y = {\exp }_{x}\left( {ru}\right) \) with some unit vector \( u \in {T}_{x}X \) . Let \( \alpha \) be the geodesic defined by \( \alpha \left( t\right) = {\exp }_{x}\left( {tu}\right) \) . Then\n\n\[ \n{f}_{u}^{\prime }\left( r\right) = \left( {T\varphi }\right) \left( y\right) T{\exp }_{x}\left( {ru}\rig... | Yes |
Theorem 3.8. Let \( \varphi \) be a \( {C}^{2} \) function on a normal ball centered at the point \( x \in X \) . Let \( {S}_{r}\left( x\right) \) for \( r > 0 \) be the Riemannian sphere of radius \( r \) centered at \( x \), and contained in the ball. Let \( {\mathbf{\Delta }}_{S} \) denoted the Laplacian on \( S = {... | Proof. We apply Proposition 2.5 of Chapter XIV, which decomposes the Laplacian into a tangential part relative to a submanifold, which we now take to be the sphere \( Y = S \) ; and a transversal part. The tangential part gives precisely the term \( {\mathbf{\Delta }}_{S}{\varphi }_{S} \) at \( y \) . For the transvers... | No |
For \( \varphi \in {L}_{a}^{r}\left( T\right) \), and \( v,{v}_{1},\ldots ,{v}_{n - r + 1} \in T \), we have\n\n\[ \left( {\varphi \circ v}\right) \land {v}_{1}^{ \vee } \land \cdots \land {v}_{n - r + 1}^{ \vee } \]\n\n\[ = \mathop{\sum }\limits_{{i = 1}}^{{n - r + 1}}{\left( -1\right) }^{r + i}\left\langle {{v}^{ \ve... | Proof. The basic formalism of forms tells us that the contraction with a vector is an anti-derivation on the algebra of forms (Chapter V, §5, CON 3). Since \( \varphi \land {v}_{1}^{ \vee } \land \cdots \land {v}_{n - r + 1}^{ \vee } \) has degree \( n + 1 \) and so is equal to 0 , we find\n\n\[ 0 = \left( {\varphi \la... | Yes |
Proposition 4.2. Let \( \\left\\{ {{v}_{1},\\ldots ,{v}_{n}}\\right\\} \) be an orthonormal basis of \( T \) . Let \( {\\omega }_{1},\\ldots ,{\\omega }_{n} \) be the dual basis of 1 -forms. Let \( I = \\left( {{i}_{1},\\ldots ,{i}_{r}}\\right) \) with \( {i}_{1} < \\cdots < {i}_{r} \) and let \( J = \\left( {{j}_{1},\... | Proof. Directly from the definition of \( \\Omega = {\\Omega }_{g} \) we have that\n\n\[ \n{\\Omega }_{g} = {\\omega }_{1} \\land \\cdots \\land {\\omega }_{n} = {v}_{1}^{ \\vee } \\land \\cdots \\land {v}_{n}^{ \\vee }.\n\]\nAt first, let \( J \) be an arbitrary sequence of \( n - r \) indices among \( \\left( {1,\\ld... | Yes |
Proposition 4.3. The star operation commutes with every \( {D}_{\xi } \), i.e. for any vector field \( \xi \) and \( \varphi \in {\mathcal{A}}^{r}\left( X\right) \), we have\n\n\[ \n* {D}_{\xi }\varphi = {D}_{\xi } * \varphi .\n\] | Proof. For 0-forms (functions) and \( n \) -forms (functions times the volume form) the assertion is immediate by using Proposition 2.1, to the effect that \( {D}_{\xi }{\operatorname{vol}}_{g} = 0 \) . Now let \( \varphi \in \Gamma {L}_{a}^{r}\left( {TX}\right) \) . Then:\n\n\[ \n\left( {{D}_{\xi } * \varphi }\right) ... | Yes |
Proposition 4.4. For \( \varphi ,\psi \in {\mathcal{A}}^{r}\left( X\right) \) we have\n\n\[ \n{d\varphi } \land * \psi = \varphi \land \left( {*{d}^{ * }\psi }\right) + d\left( {\varphi \land * \psi }\right) .\n\] | Proof. Immediate from the definition of \( {d}^{ * },\mathbf{S}\mathbf{6} \), and the basic formula for \( d \) of a wedge product (a graded derivation). | No |
Proposition 4.5. Let \( {\xi }_{1},\ldots ,{\xi }_{n} \) be a frame of vector fields, and let \( {\xi }_{1}^{\prime },\ldots ,{\xi }_{n}^{\prime } \) be the dual frame, that is \( {\left\langle {\xi }_{i}^{\prime },{\xi }_{j}\right\rangle }_{g} = {\delta }_{ij} \) . Then for any form \( \varphi \in {\mathcal{A}}^{r}\le... | Proof. Proposition 1.1. of Chapter VIII gives us an expression for \( d\left( {*\varphi }\right) \) in terms of the frame. The dual frame is such that \( {\lambda }_{i}^{ \vee } = {\xi }_{i}^{\prime } \) . Then the formula of Proposition 4.4 is an immediate consequence of \( \mathbf{S} \) 5. | No |
Proposition 4.6. Let \( {\xi }_{1},\ldots ,{\xi }_{n} \) be an orthonormal frame. As an operator on 1 -forms, \( \Delta : {\mathcal{A}}^{1}\left( X\right) \rightarrow {\mathcal{A}}^{1}\left( X\right) \) is given by\n\n\[ \Delta = - \sum {D}_{{\xi }_{i}}^{2} - \text{ Ric. } \]\n\nWritten in terms of the variables, this ... | Proof. By Proposition 4.5, we have\n\n\[ {d}^{ * }\lambda = - \sum \left( {{D}_{{\xi }_{i}}\lambda }\right) \left( {\xi }_{i}\right) \]\n\nand so by a general formula on covariant derivatives we get a value for \( d{d}^{ * }\lambda \), namely\n\n\[ \left\langle {d{d}^{ * }\lambda ,\xi }\right\rangle = - \mathop{\sum }\... | Yes |
Theorem 5.1. Under the above two Hodge conditions, we have\n\n\[ \n{\mathbf{H}}^{ \bot } = {DA} + {D}^{ * }A \n\]\n\nand an orthogonal decomposition\n\n\[ \nA = \mathbf{H} \bot \mathbf{\Delta }A = \mathbf{H} \bot {DA} \bot {D}^{ * }A. \n\]\n\nThe restriction of \( \mathbf{\Delta } \) to \( {\mathbf{H}}^{ \bot } \) is i... | Proof. By orthogonalization and \( \mathbf{H}\mathbf{2} \), given \( u \in A \) we have\n\n\[ \nu = \mathbf{H}u + \mathbf{\Delta }v = \mathbf{H}u + D{D}^{ * }v + {D}^{ * }{Dv} \]\n\nwith some \( v \in A \). Hence \( A \) is contained in \( \mathbf{H} + {DA} + {D}^{ * }A \), so we get equality. Furthermore\n\n\[ \langle... | Yes |
Proposition 5.3. Under these assumptions, \( D = S{D}^{ * }S \) and \( \mathbf{H},\mathbf{\Delta }, G \) commute with \( S \) . | Proof. We give the proof when \( n \) is even for simplicity. For \( u \in {A}^{p} \), we have:\n\n\[ S{D}^{ * }{Su} = - {S}^{2}D{S}^{2}u = - {S}^{2}D{\left( -1\right) }^{p}u \]\n\n\[ = - {\left( -1\right) }^{p}{\left( -1\right) }^{p + 1}{Du} \]\n\n\[ = {Du} \]\n\nso \( D = S{D}^{ * }S \) .\n\nFor the commutation of \(... | Yes |
Lemma 6.1. There is a canonical isomorphism\n\n\[ \land ^{p}{T}_{y}^{ \vee } \otimes \land ^{q}{T}_{z}^{ \vee } \rightarrow \land ^{n}{T}_{x}^{ \vee } \]\n\ndefined as follows. For \( \omega \in \mathop{\bigwedge }\limits^{q}{T}_{z}^{ \vee } \) and \( \eta \in \mathop{\bigwedge }\limits^{p}{T}_{y}^{ \vee } \), let \( \... | Proof. Routine algebraic verification. The above lemma is sometimes stated in the form\n\n\[ \det \left( {T}_{x}^{ \vee }\right) = \det \left( {T}_{y}^{ \vee }\right) \otimes \det \left( {T}_{z}^{ \vee }\right) . \] | No |
Lemma 6.2. Let \( \pi : X \rightarrow Z \) be a submersion. Suppose \( X \) is orientable. Then every fiber \( {Y}_{z} \) is orientable. If \( \Omega \) and \( \omega \) are volume forms on \( X, Z \) respectively, then there exists a p-form \( \widetilde{\eta } \) on \( X \) whose restriction to each fiber \( {Y}_{\pi... | Proof. The orientability comes from the existence of the family of forms \( \left\{ {\eta }_{v}\right\} \), which is verified to be \( {C}^{\infty } \) in terms of coordinates. The local existence of \( \widetilde{\eta } \) is immediate. The global existence follows by using a partition of unity. | No |
Lemma 6.3. Under the above assumptions, let \( {\Omega }_{x} \) and \( {\Omega }_{z} \) be metric volume forms on \( {T}_{x} \) and \( {T}_{z} \) (so they determine an orientation). Then one of the possible (up to sign) metric volume forms \( {\Omega }_{y} \) on \( {T}_{y} \) satisfies the relation\n\n\[ \n{\Omega }_{x... | Proof. Let \( \left\{ {{e}_{1},\ldots ,{e}_{p}}\right\} \) be an orthonormal basis for \( {T}_{y} \), and \( \left\{ {{e}_{p + 1},\ldots ,{e}_{p + q}}\right\} \) an orthonormal basis for \( {T}_{v}^{ \bot } \) . Together they form an orthonormal basis for \( {T}_{x} \) . The metric dual bases \( \left\{ {{e}_{1}^{ \vee... | Yes |
Proposition 6.4. Let \( \pi : X \rightarrow Z \) be a Riemannian submersion. Suppose \( X, Z \) oriented, so \( {Y}_{z} \) is oriented for each \( z \) . Let \( {\Omega }_{X},{\Omega }_{Z} \) be the Riemannian volume forms on \( X, Z \) respectively. Then for each \( z \in Z \), the Riemannian volume form \( {\Omega }_... | The relation of Proposition 6.4 is punctual. However, the individual volume forms on the fibers locally are the restriction of a form on an open set of \( X \) itself. Indeed, if \( \left\{ {{\xi }_{1},\ldots ,{\xi }_{p}}\right\} \) is an orthonormal frame of vertical vector fields on \( X \), suitably oriented, then\n... | Yes |
Proposition 6.5. Let \( \pi : X \rightarrow Z \) be a Riemannian submersion. Let \( {\Omega }_{X} \) and \( {\Omega }_{Z} \) be Riemannian volume forms on \( X, Z \) respectively. Let \( v \) be a vector field on \( Z \), and \( {v}_{X} \) its horizontal lift to \( X \) . Abbreviate \( {\operatorname{div}}_{X} \) for \... | Proof. The first formula comes from definition DIV 2 of the divergence, and the fact that the Lie derivative is a derivation for the wedge product, by Chapter V, Proposition 5.3, LIE 2, namely\n\n\[ {\mathcal{L}}_{{v}_{X}}\left( {\Omega }_{X}\right) = {\mathcal{L}}_{{v}_{X}}{\Omega }_{Y} \land {\Omega }_{Z} + {\Omega }... | No |
Proposition 7.1. The exponential commutes with conjugation, namely for \( v \in {T}_{e}G \), we have\n\n\[ \exp {\mathbf{c}}_{\text{Lie }}\left( x\right) v = {\mathbf{c}}_{x}\left( {\exp v}\right) = x\exp \left( v\right) {x}^{-1}. \]\n | Proof. This is actually a special case of the general fact that if \( f : G \rightarrow {G}^{\prime } \) is a Lie group homomorphism, and \( v \in {T}_{e}G \), then\n\n\[ f\left( {\exp v}\right) = \exp \left( {{Tf}\left( e\right) v}\right) . \]\n\nWe apply this formula to \( f = {\mathbf{c}}_{x} \). As to the general f... | Yes |
Proposition 7.2. We have \( \chi \left( a\right) = \det {\mathbf{c}}_{\text{Lie }}\left( a\right) \) for \( a \in G \) . | Proof. We use \( {\mathbf{c}}_{a} = {L}_{a} \circ {R}_{a}^{-1} \), and abbreviate \( {\mathbf{c}}_{a}V = \det {\mathbf{c}}_{\text{Lie }}\left( a\right) V \) . Then for \( V \neq 0 \) ,\n\n\[ \n\Omega \left( V\right) = {\left( {\mathbf{c}}_{a}\Omega \right) }_{e}\left( {{\mathbf{c}}_{a}V}\right) = {\left( {R}_{{a}^{-1}}... | Yes |
Proposition 7.3. Let \( \Omega \) be a left invariant volume form on \( G \) . Then \( {\chi \Omega } \) is right invariant, i.e. is a right Haar form. | Proof. We have\n\n\[ \n{R}_{a}\left( {\chi \Omega }\right) = {R}_{a}\left( \chi \right) {R}_{a}\left( \Omega \right) = \chi \left( {a}^{-1}\right) {\chi \chi }\left( a\right) \Omega = {\chi \Omega }, \n\] \n\nthus proving the proposition. | Yes |
Proposition 7.4. For \( h \in H \), we have\n\n\[ \text{det}T{\mathbf{c}}_{h}\left( {e}_{G}\right) = \det T{\mathbf{c}}_{h}\left( {e}_{G/H}\right) \cdot \det T{\mathbf{c}}_{h}\left( {e}_{H}\right) \text{.} \] | Proof. Let \( {T}_{y} = {T}_{y}Y,{T}_{x} = {T}_{y}X \) and \( {T}_{z} = {T}_{\pi \left( y\right) }Z \), so we have the exact\n\nsequence\n\n\[ 0 \rightarrow {T}_{y} \rightarrow {T}_{x} \rightarrow {T}_{z} \rightarrow 0. \]\n\nThe map \( f \) induces tangent linear maps on each of those spaces, and we denote these by \(... | Yes |
Proposition 7.5. Let \( X \) be a homogeneous space for \( G \) . If \( X \) is strictly unimodular, then there exists a left G-invariant volume form on \( X \), unique up to a constant multiple. | Proof. We want to define the invariant form on \( G/H \) by translating a given volume form \( {\omega }_{e} \) on \( {T}_{e}\left( {G/H}\right) \) . On \( G/H \), the left translation \( {L}_{h} \) is induced by conjugation \( {\mathbf{c}}_{h} \) on \( G \) . By Proposition 7.4 and the hypothesis, we have\n\n\[ \text{... | Yes |
Proposition 8.1. Suppose there is a section \( \sigma : Z \rightarrow X \) of a homogeneously fibered submersion. Define\n\n\[ \gamma : H \times Z \rightarrow X\;\text{ by }\;\gamma \left( {h, z}\right) = {h\sigma }\left( z\right) . \]\n\nThen \( \gamma \) is a submersion. | Proof. The tangent map \( {T\gamma }\left( {h, z}\right) \) is a surjective homomorphism of tangent spaces at each point. In fact, if we let \( {\gamma }_{h}\left( {\sigma \left( z\right) }\right) = h\left( {\sigma z}\right) = \gamma \left( {h, z}\right) \) , then \( T{\gamma }_{h}\left( {\sigma \left( z\right) }\right... | Yes |
Theorem 8.2 (Wu). Let \( \pi : X \rightarrow Z \) be a metrically homogeneously fibered submersion. For any two points \( x, y \in X \), the isotropy groups \( {H}_{x} \) , \( {H}_{y} \) are conjugate in \( H \) . In fact, let \( x, y \) be points of \( X \) which can be joined by the horizontal lift of a curve in \( Z... | Proof. We recall that the horizontal lift was defined in Chapter XIV, §3. Suppose first that \( x, y \) can be joined by a horizontal lift \( A \) . Let \( h \in {H}_{x} \) . Since \( H \) acts isometrically on \( X, h \circ A \) is the unique horizontal lift from \( {hx} = x \) to \( {hy} \) . But \( h \circ A \) has ... | Yes |
Theorem 8.3. Let \( \pi : X \rightarrow Z \) be a metrically homogeneously fibered strictly unimodular submersion. Let \( v \) be a vector field on \( Z \). Then the Haar form \( \Psi \) is \( {v}_{X} \) -constant over the fibers. If \( \delta \) is the Riemannian Haar density, then\n\n\[{\operatorname{div}}_{X}\left( ... | Proof. Let \( \alpha \) be an integral curve of \( v \) in \( Z \) and let \( A \) be its horizontal lift, so \( {v}_{X} \) restricts to \( {A}^{\prime } \) on the curve. By Theorem 8.2, the flow \( {F}_{t} \) gives a homogeneous space isomorphism \( {Y}_{\alpha \left( 0\right) } \rightarrow {Y}_{\alpha \left( t\right)... | Yes |
Theorem 8.4 (Helgason). Let \( \pi : X \rightarrow Z \) be a Riemannian submersion metrically homogeneously fibered, and unimodular. Let \( \delta \) be the Riemannian Haar density. Let \( {\mathbf{\Delta }}_{X},{\mathbf{\Delta }}_{Z} \) be the Laplacians. Then for a function \( \psi \) on \( Z \), we have\n\n\[{\mathb... | Proof. All the work has been done, and the statement merely puts together Proposition 6.5 via Theorem 8.3, and the definition of the Laplacian as minus the divergence of the gradient. | No |
We cannot define \[ {\pi }_{ * } : \mathrm{{DO}}\left( X\right) \rightarrow \mathrm{{DO}}\left( Z\right) \] in general, but we can define \( {\pi }_{ * } \) in a natural way on a subset of \( \operatorname{DO}\left( X\right) \). | Indeed, an element of the group \( H \) acting on \( X \) also acts on any object functorially associated with \( X \), especially on \( \operatorname{DO}\left( X\right) \). By definition, given \( h \in H \), let \( \left\lbrack h\right\rbrack D \) for \( D \in \mathrm{{DO}}\left( X\right) \) be defined by \[ \left( {... | Yes |
Lemma 1.1. Let \( A \) have measure 0 in \( {\mathbf{R}}^{n} \) and let \( f : A \rightarrow {\mathbf{R}}^{n} \) satisfy a Lipschitz condition. Then \( f\left( A\right) \) has measure 0 . | Proof. Let \( C \) be a Lipschitz constant for \( f \) . Let \( \left\{ {R}_{j}\right\} \) be a sequence of cubes covering \( A \) such that \( \sum \mu \left( {R}_{j}\right) < \epsilon \) . Let \( {r}_{j} \) be the length of the side of \( {R}_{j} \) . Then for each \( j \) we see that \( f\left( {A \cap {S}_{j}}\righ... | Yes |
Lemma 1.2. Let \( U \) be open in \( {\mathbf{R}}^{n} \) and let \( f : U \rightarrow {\mathbf{R}}^{n} \) be a \( {C}^{1} \) map. Let\n\n\( Z \) be a set of measure 0 in \( U \) . Then \( f\left( Z\right) \) has measure 0 . | Proof. For each \( x \in U \) there exists a rectangle \( {R}_{x} \) contained in \( U \) such that the family \( \left\{ {R}_{x}^{0}\right\} \) of interiors covers \( Z \) . Since \( U \) is separable, there exists a denumerable subfamily covering \( Z \), say \( \left\{ {R}_{j}\right\} \) . It suffices to prove that ... | Yes |
Lemma 1.3. Let \( A \) be a subset of \( {\mathbf{R}}^{m} \) . Assume that \( m < n \) . Let\n\n\[ f : A \rightarrow {\mathbf{R}}^{n} \]\n\nsatisfy a Lipschitz condition. Then \( f\left( A\right) \) has measure 0 . | Proof. We view \( {\mathbf{R}}^{m} \) as embedded in \( {\mathbf{R}}^{n} \) on the space of the first \( m \) coordinates. Then \( {\mathbf{R}}^{m} \) has measure 0 in \( {\mathbf{R}}^{n} \), so that \( A \) has also \( n \) - dimensional measure 0 . Lemma 1.3 is therefore a consequence of Lemma 1.1. | No |
Corollary 2.2. Let \( S \) be the unit cube spanned by the unit vectors in \( {\mathbf{R}}^{n} \). Let \( \lambda : {\mathbf{R}}^{n} \rightarrow {\mathbf{R}}^{n} \) be a linear map. Then \[ \operatorname{Vol}\lambda \left( S\right) = \left| {\operatorname{Det}\left( \lambda \right) }\right| \] | Proof. If \( {v}_{1},\ldots ,{v}_{n} \) are the images of \( {e}_{1},\ldots ,{e}_{n} \) under \( \lambda \), then \( \lambda \left( S\right) \) is the block spanned by \( {v}_{1},\ldots ,{v}_{n} \). If we represent \( \lambda \) by the matrix \( A = \left( {a}_{ij}\right) \), then \[ {v}_{i} = {a}_{1i}{e}_{1} + \cdots ... | Yes |
Corollary 2.3. If \( R \) is any rectangle in \( {\mathbf{R}}^{n} \) and \( \lambda : {\mathbf{R}}^{n} \rightarrow {\mathbf{R}}^{n} \) is a linear map, then\n\n\[ \operatorname{Vol}\lambda \left( R\right) = \left| {\operatorname{Det}\left( \lambda \right) }\right| \operatorname{Vol}\left( R\right) \] | Proof. After a translation, we can assume that the rectangle is a block. If \( R = {\lambda }_{1}\left( S\right) \) where \( S \) is the unit cube, then\n\n\[ \lambda \left( R\right) = \lambda \circ {\lambda }_{1}\left( S\right) \]\n\nwhence by Corollary 2.2,\n\n\[ \operatorname{Vol}\lambda \left( R\right) = \left| {\o... | Yes |
Corollary 2.5. If \( g \) is continuous on \( f\left( R\right) \), then\n\n\[{\int }_{f\left( R\right) }{gd\mu } = {\int }_{R}\left( {g \circ f}\right) \left| {\Delta }_{f}\right| {d\mu }.\] | Proof. The functions \( g \) and \( \left( {g \circ f}\right) \left| {\Delta }_{f}\right| \) are uniformly continuous on \( f\left( R\right) \) and \( R \) respectively. Let us take a partition of \( R \) and let \( \left\{ {S}_{j}\right\} \) be the subrectangles of this partition. If \( \delta \) is the maximum length... | Yes |
Corollary 2.7. Let \( U \) be open in \( {\mathbf{R}}^{n} \) and let \( f : U \rightarrow {\mathbf{R}}^{n} \) be a \( {C}^{1} \) map. Let \( A \) be a measurable subset of \( U \) such that the boundary of \( A \) has measure 0, and such that \( f \) is \( {C}^{1} \) invertible on the interior of \( A \) . Let \( g \) ... | Proof. Let \( {U}_{0} \) be the interior of \( A \) . The sets \( f\left( A\right) \) and \( f\left( {U}_{0}\right) \) differ only by a set of measure 0, namely \( f\left( {\partial A}\right) \) . Also the sets \( A,{U}_{0} \) differ only by a set of measure 0 . Consequently we can replace the domains of integration \(... | Yes |
Lemma 4.1. Let \( \lambda : {C}_{c}\left( X\right) \rightarrow \mathbf{C} \) be a positive linear map. Then \( \lambda \) is bounded on \( {C}_{K}\left( X\right) \) for any compact \( K \) . | Proof. By the corollary of Urysohn's lemma, there exists a continuous real function \( g \geqq 0 \) on \( X \) which is 1 on \( K \) has compact support. If \( f \in {C}_{K}\left( X\right) \), let \( b = \parallel f\parallel \) . Say \( f \) is real. Then \( {bg} \pm f \geqq 0 \), whence \[ \lambda \left( {bg}\right) \... | Yes |
Lemma 4.2. Let \( \left\{ {W}_{\alpha }\right\} \) be an open covering of \( X \) . For each index \( \alpha \), let \( {\lambda }_{\alpha } \) be a functional on \( {C}_{c}\left( {W}_{\alpha }\right) \) . Assume that for each pair of indices \( \alpha ,\beta \) the functionals \( {\lambda }_{\alpha } \) and \( {\lambd... | Proof. Let \( f \in {C}_{c}\left( X\right) \) and let \( K \) be the support of \( f \) . Let \( \left\{ {h}_{i}\right\} \) be a partition of unity over \( K \) subordinated to a covering of \( K \) by a finite number of the open sets \( {W}_{\alpha } \) . Then each \( {h}_{i}f \) has support in some \( {W}_{\alpha \le... | Yes |
Theorem 4.3. Let \( \\dim X = n \) and let \( \\omega \) be an \( n \) -form on \( X \) of class \( {C}^{0} \) , that is continuous. Then there exists a unique positive functional \( \\lambda \) on \( {C}_{c}\\left( X\\right) \) having the following property. If \( \\left( {U,\\varphi }\\right) \) is a chart and\n\n\\[... | Proof. The integral in (1) defines a positive functional on \( {C}_{c}\\left( U\\right) \) . The change of variables formula shows that if \( \\left( {U,\\varphi }\\right) \) and \( \\left( {V,\\psi }\\right) \) are two charts, and if \( g \) has support in \( U \\cap V \), then the value of the functional is independe... | Yes |
Theorem 4.4. Let \( \dim X = n \) and assume that \( X \) is oriented. Let \( \omega \) be an \( n \) -form on \( X \) of class \( {C}^{0} \) . Then there exists a unique functional \( \lambda \) on \( {C}_{c}\left( X\right) \) having the following property. If \( \left( {U,\varphi }\right) \) is an oriented chart and\... | Proof. Since the Jacobian determinant of transition maps belonging to oriented charts is positive, we see that Theorem 4.4 follows like Theorem 4.3 from the change of variables formula (in which the absolute value sign now becomes unnecessary) and the existence of partitions of unity. | No |
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