|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| This is a list of [[formula]]s encountered in [[Riemannian geometry]].
| | Nice to meet you, my name is [http://Www.channel4embarrassingillnesses.com/men-in-white-coats/genital-herpes/symptoms/ Figures Held] although I don't really like being called like that. His wife [http://www.streaming.iwarrior.net/blog/258268 home std test] doesn't like it the way he does but what he really likes performing is to do aerobics and he's been doing it for quite a whilst. Years in the past we moved to Puerto Rico and my family members loves it. Since she was eighteen she's been operating as a receptionist but her promotion never at home std testing arrives.<br><br>Feel free to visit my blog [http://www.gaysphere.net/blog/212030 at home std test] post; [http://www.uknewsx.com/what-you-want-to-do-when-confronted-with-candida-albicans/ home std test] |
| | |
| ==Christoffel symbols, covariant derivative==
| |
| | |
| | |
| In a smooth [[coordinate chart]], the [[Christoffel symbols]] of the first kind are given by
| |
| | |
| :<math>\Gamma_{kij}=\frac12 \left(
| |
| \frac{\partial}{\partial x^j} g_{ki}
| |
| +\frac{\partial}{\partial x^i} g_{kj}
| |
| -\frac{\partial}{\partial x^k} g_{ij}
| |
| \right)
| |
| =\frac12 \left( g_{ki,j} + g_{kj,i} - g_{ij,k} \right) \,,
| |
| </math>
| |
| | |
| and the Christoffel symbols of the second kind by
| |
| | |
| :<math>\begin{align}
| |
| \Gamma^m{}_{ij} &= g^{mk}\Gamma_{kij}\\
| |
| &=\frac12\, g^{mk} \left(
| |
| \frac{\partial}{\partial x^j} g_{ki}
| |
| +\frac{\partial}{\partial x^i} g_{kj}
| |
| -\frac{\partial}{\partial x^k} g_{ij}
| |
| \right)
| |
| =\frac12\, g^{mk} \left( g_{ki,j} + g_{kj,i} - g_{ij,k} \right) \,.
| |
| \end{align}
| |
| </math>
| |
| | |
| Here <math>g^{ij}</math> is the [[inverse matrix]] to the metric tensor <math>g_{ij}</math>. In other words,
| |
| | |
| :<math>
| |
| \delta^i{}_j = g^{ik}g_{kj}
| |
| </math>
| |
| | |
| and thus
| |
| | |
| :<math>
| |
| n = \delta^i{}_i = g^i{}_i = g^{ij}g_{ij}
| |
| </math>
| |
| | |
| is the dimension of the [[manifold]].
| |
| | |
| Christoffel symbols satisfy the symmetry relation
| |
| | |
| :<math>
| |
| \Gamma^i{}_{jk}=\Gamma^i{}_{kj} \,,
| |
| </math>
| |
| | |
| which is equivalent to the torsion-freeness of the [[Levi-Civita connection]].
| |
| | |
| The contracting relations on the Christoffel symbols are given by
| |
| | |
| :<math>\Gamma^i{}_{ki}=\frac{1}{2} g^{im}\frac{\partial g_{im}}{\partial x^k}=\frac{1}{2g} \frac{\partial g}{\partial x^k} = \frac{\partial \log \sqrt{|g|}}{\partial x^k} \ </math> | |
| | |
| and
| |
| | |
| :<math>g^{k\ell}\Gamma^i{}_{k\ell}=\frac{-1}{\sqrt{|g|}} \;\frac{\partial\left(\sqrt{|g|}\,g^{ik}\right)} {\partial x^k}</math>
| |
| | |
| where |''g''| is the absolute value of the [[determinant]] of the metric tensor <math>g_{ik}\ </math>. These are useful when dealing with divergences and Laplacians (see below).
| |
| | |
| The [[covariant derivative]] of a [[vector field]] with components <math>v^i</math> is given by:
| |
| | |
| :<math>
| |
| v^i {}_{;j}=\nabla_j v^i=\frac{\partial v^i}{\partial x^j}+\Gamma^i{}_{jk}v^k
| |
| </math>
| |
| | |
| and similarly the covariant derivative of a <math>(0,1)</math>-[[tensor field]] with components <math>v_i</math> is given by:
| |
| | |
| :<math>
| |
| v_{i;j}=\nabla_j v_i=\frac{\partial v_i}{\partial x^j}-\Gamma^k{}_{ij} v_k
| |
| </math>
| |
| | |
| For a <math>(2,0)</math>-[[tensor field]] with components <math>v^{ij}</math> this becomes
| |
| | |
| :<math>
| |
| v^{ij}{}_{;k}=\nabla_k v^{ij}=\frac{\partial v^{ij}}{\partial x^k} +\Gamma^i{}_{k\ell}v^{\ell j}+\Gamma^j{}_{k\ell}v^{i\ell}
| |
| </math>
| |
| | |
| and likewise for tensors with more indices.
| |
| | |
| The covariant derivative of a function (scalar) <math>\phi</math> is just its usual differential:
| |
| | |
| :<math>
| |
| \nabla_i \phi=\phi_{;i}=\phi_{,i}=\frac{\partial \phi}{\partial x^i}
| |
| </math>
| |
| | |
| Because the [[Levi-Civita connection]] is metric-compatible, the covariant derivatives of metrics vanish,
| |
| | |
| :<math>
| |
| \nabla_k g_{ij} = \nabla_k g^{ij} = 0
| |
| </math>
| |
| | |
| The [[geodesic]] <math>X(t)</math> starting at the origin with initial speed <math>v^i</math> has Taylor expansion in the chart:
| |
| | |
| :<math>
| |
| X(t)^i=tv^i-\frac{t^2}{2}\Gamma^i{}_{jk}v^jv^k+O(t^3)
| |
| </math>
| |
| | |
| ==Curvature tensors==
| |
| ===Riemann curvature tensor===
| |
| | |
| If one defines the [[Riemann curvature tensor|curvature operator]] as <math>R(U,V)W=\nabla_U \nabla_V W - \nabla_V \nabla_U W -\nabla_{[U,V]}W</math>
| |
| and the coordinate components of the <math>(1,3)</math>-[[Riemann curvature tensor]] by <math>(R(U,V)W)^\ell=R^\ell{}_{ijk}W^iU^jV^k</math>, then these components are given by:
| |
| | |
| :<math>
| |
| R^\ell{}_{ijk}=
| |
| \frac{\partial}{\partial x^j} \Gamma^\ell{}_{ik}-\frac{\partial}{\partial x^k}\Gamma^\ell{}_{ij}
| |
| +\Gamma^\ell{}_{js}\Gamma_{ik}^s-\Gamma^\ell{}_{ks}\Gamma^s{}_{ij}
| |
| </math>
| |
| | |
| Lowering indices with <math>R_{\ell ijk}=g_{\ell s}R^s{}_{ijk}</math> one gets
| |
| | |
| :<math>R_{ik\ell m}=\frac{1}{2}\left(
| |
| \frac{\partial^2g_{im}}{\partial x^k \partial x^\ell}
| |
| + \frac{\partial^2g_{k\ell}}{\partial x^i \partial x^m}
| |
| - \frac{\partial^2g_{i\ell}}{\partial x^k \partial x^m}
| |
| - \frac{\partial^2g_{km}}{\partial x^i \partial x^\ell} \right) | |
| +g_{np} \left(
| |
| \Gamma^n{}_{k\ell} \Gamma^p{}_{im} -
| |
| \Gamma^n{}_{km} \Gamma^p{}_{i\ell} \right).
| |
| \ </math>
| |
| | |
| The symmetries of the tensor are
| |
| | |
| :<math>R_{ik\ell m}=R_{\ell mik}\ </math> and <math>R_{ik\ell m}=-R_{ki\ell m}=-R_{ikm\ell}.\ </math>
| |
| | |
| That is, it is symmetric in the exchange of the first and last pair of indices, and antisymmetric in the flipping of a pair.
| |
| | |
| The cyclic permutation sum (sometimes called first Bianchi identity) is
| |
| | |
| :<math>R_{ik\ell m}+R_{imk\ell}+R_{i\ell mk}=0.\ </math>
| |
| | |
| The (second) '''[[Bianchi identity]]''' is
| |
| | |
| :<math>\nabla_m R^n {}_{ik\ell} + \nabla_\ell R^n {}_{imk} + \nabla_k R^n {}_{i\ell m}=0,\ </math>
| |
| | |
| that is, | |
| | |
| :<math> R^n {}_{ik\ell;m} + R^n {}_{imk;\ell} + R^n {}_{i\ell m;k}=0 \ </math>
| |
| | |
| which amounts to a cyclic permutation sum of the last three indices, leaving the first two fixed.
| |
| | |
| ===Ricci and scalar curvatures===
| |
| | |
| Ricci and scalar curvatures are contractions of the Riemann tensor. They simplify the Riemann tensor, but contain less information.
| |
| | |
| The [[Ricci curvature]] tensor is essentially the unique nontrivial way of contracting the Riemann tensor:
| |
| | |
| :<math> | |
| R_{ij}=R^\ell{}_{i\ell j}=g^{\ell m}R_{i\ell jm}=g^{\ell m}R_{\ell imj}
| |
| =\frac{\partial\Gamma^\ell{}_{ij}}{\partial x^\ell} - \frac{\partial\Gamma^\ell{}_{i\ell}}{\partial x^j} + \Gamma^\ell{}_{ij} \Gamma^m{}_{\ell m} - \Gamma^m{}_{i\ell}\Gamma^\ell_{jm}.\
| |
| </math>
| |
| | |
| The Ricci tensor <math>R_{ij}</math> is symmetric.
| |
| | |
| By the contracting relations on the Christoffel symbols, we have
| |
| | |
| :<math>
| |
| R_{ik}=\frac{\partial\Gamma^\ell{}_{ik}}{\partial x^\ell} - \Gamma^m{}_{i\ell}\Gamma^\ell{}_{km} - \nabla_k\left(\frac{\partial}{\partial x^i}\left(\log\sqrt{|g|}\right)\right).\
| |
| </math>
| |
| | |
| The [[scalar curvature]] is the trace of the Ricci curvature,
| |
| | |
| :<math>
| |
| R=g^{ij}R_{ij}=g^{ij}g^{\ell m}R_{i\ell jm}
| |
| </math>.
| |
| | |
| The "gradient" of the scalar curvature follows from the Bianchi identity ([[Proofs involving Christoffel symbols#Proof 1|proof]]):
| |
| | |
| :<math>\nabla_\ell R^\ell {}_m = {1 \over 2} \nabla_m R, \ </math>
| |
| | |
| that is,
| |
| | |
| :<math> R^\ell {}_{m;\ell} = {1 \over 2} R_{;m}. \ </math>
| |
| | |
| ===Einstein tensor===
| |
| The [[Einstein tensor]] ''G<sup>ab</sup>'' is defined in terms of the Ricci tensor ''R<sup>ab</sup>'' and the Ricci scalar ''R'',
| |
| | |
| :<math> G^{ab} = R^{ab} - {1 \over 2} g^{ab} R \ </math>
| |
| | |
| where ''g'' is the metric tensor.
| |
| | |
| The Einstein tensor is symmetric, with a vanishing divergence ([[Proofs involving Christoffel symbols#Proof 2|proof]]) which is due to the Bianchi identity:
| |
| | |
| :<math> \nabla_a G^{ab} = G^{ab} {}_{;a} = 0. \ </math>
| |
| | |
| ===Weyl tensor===
| |
| The '''[[Weyl tensor]]''' is given by
| |
| | |
| :<math>C_{ik\ell m}=R_{ik\ell m} + \frac{1}{n-2}\left(
| |
| - R_{i\ell}g_{km}
| |
| + R_{im}g_{k\ell}
| |
| + R_{k\ell}g_{im}
| |
| - R_{km}g_{i\ell} \right)
| |
| + \frac{1}{(n-1)(n-2)} R \left(
| |
| g_{i\ell}g_{km} - g_{im}g_{k\ell} \right),\ </math>
| |
| | |
| where <math>n</math> denotes the dimension of the Riemannian manifold.
| |
| | |
| The Weyl tensor satisfies the first (algebraic) Bianchi identity:
| |
| | |
| :<math>C_{ijkl} + C_{kijl} + C_{jkil} = 0 .</math>
| |
| | |
| The Weyl tensor is a symmetric product of alternating 2-forms,
| |
| | |
| :<math> C_{ijkl} = -C_{jikl} \qquad C_{ijkl} = C_{klij} ,</math>
| |
| | |
| just like the Riemann tensor. Moreover, taking the trace over any two indices gives zero,
| |
| | |
| :<math> C^i{}_{jki} = 0 </math>
| |
| | |
| The Weyl tensor vanishes (<math>C=0</math>) if and only if a manifold <math>M</math> of dimension <math>n \geq 4</math> is locally conformally flat. In other words, <math>M</math> can be covered by coordinate systems in which the metric <math>ds^2</math> satisfies
| |
| | |
| :<math>ds^2 = f^2\left(dx_1^2 + dx_2^2 + \ldots dx_n^2\right)</math>
| |
| | |
| This is essentially because <math>C^i{}_{jkl}</math> is invariant under conformal changes.
| |
| | |
| ==Gradient, divergence, Laplace–Beltrami operator==
| |
| | |
| The [[gradient#The gradient on manifolds|gradient]] of a function <math>\phi</math> is obtained by raising the index of the differential <math>\partial_i\phi dx^i</math>, whose components are given by:
| |
| | |
| :<math>\nabla^i \phi=\phi^{;i}=g^{ik}\phi_{;k}=g^{ik}\phi_{,k}=g^{ik}\partial_k \phi=g^{ik}\frac{\partial \phi}{\partial x^k}
| |
| </math>
| |
| | |
| The [[divergence]] of a vector field with components <math>V^m</math> is
| |
|
| |
| :<math>\nabla_m V^m = \frac{\partial V^m}{\partial x^m} + V^k \frac{\partial \log \sqrt{|g|}}{\partial x^k} = \frac{1}{\sqrt{|g|}} \frac{\partial (V^m\sqrt{|g|})}{\partial x^m}.\ </math>
| |
| | |
| The [[Laplace–Beltrami operator]] acting on a function <math>f</math> is given by the divergence of the gradient:
| |
| | |
| :<math>
| |
| \begin{align}
| |
| \Delta f &= \nabla_i \nabla^i f
| |
| = \frac{1}{\sqrt{|g|}} \frac{\partial }{\partial x^j}\left(g^{jk}\sqrt{|g|}\frac{\partial f}{\partial x^k}\right) \\
| |
| &=
| |
| g^{jk}\frac{\partial^2 f}{\partial x^j \partial x^k} + \frac{\partial g^{jk}}{\partial x^j} \frac{\partial
| |
| f}{\partial x^k} + \frac12 g^{jk}g^{il}\frac{\partial g_{il}}{\partial x^j}\frac{\partial f}{\partial x^k}
| |
| = g^{jk}\frac{\partial^2 f}{\partial x^j \partial x^k} - g^{jk}\Gamma^l{}_{jk}\frac{\partial f}{\partial x^l}
| |
| \end{align}
| |
| </math>
| |
| | |
| The divergence of an [[antisymmetric tensor]] field of type <math>(2,0)</math> simplifies to
| |
| | |
| :<math>\nabla_k A^{ik}= \frac{1}{\sqrt{|g|}} \frac{\partial (A^{ik}\sqrt{|g|})}{\partial x^k}.\ </math>
| |
| | |
| The Hessian of a map <math>\phi: M \rightarrow N </math> is given by
| |
| :<math> \left( \nabla \left( d \phi\right) \right) _{ij} ^\gamma= \frac{\partial ^2 \phi ^\gamma}{\partial x^i \partial x^j}- ^M \Gamma ^k{}_{ij} \frac{\partial \phi ^\gamma}{\partial x^k} + ^N \Gamma ^{\gamma}{}_{\alpha \beta} \frac{\partial \phi ^\alpha}{\partial x^i}\frac{\partial \phi ^\beta}{\partial x^j}.</math>
| |
| | |
| ==Kulkarni–Nomizu product==
| |
| | |
| The [[Kulkarni–Nomizu product]] is an important tool for constructing new tensors from existing tensors on a Riemannian manifold. Let <math>h</math> and <math>k</math> be symmetric covariant 2-tensors. In coordinates,
| |
| | |
| :<math>h_{ij} = h_{ji} \qquad \qquad k_{ij} = k_{ji} </math>
| |
| | |
| Then we can multiply these in a sense to get a new covariant 4-tensor, which is often denoted <math> h {~\wedge\!\!\!\!\!\!\bigcirc~} k</math>. The defining formula is
| |
| | |
| <math>\left(h {~\wedge\!\!\!\!\!\!\bigcirc~} k\right)_{ijkl} = h_{ik}k_{jl} + h_{jl}k_{ik} - h_{il}k_{jk} - h_{jk}k_{il}</math>
| |
| | |
| Clearly, the product satisfies
| |
| | |
| :<math>h {~\wedge\!\!\!\!\!\!\bigcirc~} k = k {~\wedge\!\!\!\!\!\!\bigcirc~} h</math>
| |
| | |
| ==In an inertial frame==
| |
| | |
| An orthonormal [[inertial frame]] is a coordinate chart such that, at the origin, one has the relations <math>g_{ij}=\delta_{ij}</math> and <math>\Gamma^i{}_{jk}=0</math> (but these may not hold at other points in the frame). These coordinates are also called normal coordinates.
| |
| In such a frame, the expression for several operators is simpler. Note that the formulae given below are valid ''at the origin of the frame only''.
| |
| | |
| :<math>R_{ik\ell m}=\frac{1}{2}\left(
| |
| \frac{\partial^2g_{im}}{\partial x^k \partial x^\ell}
| |
| + \frac{\partial^2g_{k\ell}}{\partial x^i \partial x^m}
| |
| - \frac{\partial^2g_{i\ell}}{\partial x^k \partial x^m}
| |
| - \frac{\partial^2g_{km}}{\partial x^i \partial x^\ell} \right)
| |
| </math>
| |
| | |
| ==Under a conformal change==
| |
| | |
| Let <math>g</math> be a Riemannian metric on a smooth manifold <math>M</math>, and <math>\varphi</math> a smooth real-valued function on <math>M</math>. Then
| |
| | |
| :<math>\tilde g = e^{2\varphi}g </math>
| |
| | |
| is also a Riemannian metric on <math>M</math>. We say that <math>\tilde g</math> is conformal to <math>g</math>. Evidently, conformality of metrics is an equivalence relation. Here are some formulas for conformal changes in tensors associated with the metric. (Quantities marked with a tilde will be associated with <math>\tilde g</math>, while those unmarked with such will be associated with <math>g</math>.)
| |
| | |
| :<math>\tilde g_{ij} = e^{2\varphi}g_{ij} </math>
| |
| | |
| :<math>\tilde \Gamma^k{}_{ij} = \Gamma^k{}_{ij}+ \delta^k_i\partial_j\varphi + \delta^k_j\partial_i\varphi-g_{ij}\nabla^k\varphi </math> | |
| | |
| Note that the difference between the Christoffel symbols of two different metrics always form the components of a tensor.
| |
| | |
| We can also write this in a coordinate-free manner:
| |
| | |
| :<math>\tilde\nabla_{F_* X}F_* Y = F_*\Bigl( \nabla_X Y + X(\varphi)Y + Y(\varphi) X - g(X,Y)\operatorname{grad}\varphi \Bigr)</math>,
| |
| | |
| (where <math>F:M \to N</math> is the conformal map, i.e.: <math>F^* \tilde g = e^{2\varphi} g</math>, and <math>X,Y</math> are vector fields.)
| |
| | |
| :<math>d\tilde V = e^{n\varphi}dV</math>
| |
| | |
| Here <math>dV</math> is the Riemannian volume element.
| |
| | |
| :<math>\tilde R_{ijkl} = e^{2\varphi}\left( R_{ijkl} - \left[ g {~\wedge\!\!\!\!\!\!\bigcirc~} \left( \nabla\partial\varphi - \partial\varphi\partial\varphi + \frac{1}{2}\|\nabla\varphi\|^2g \right)\right]_{ijkl} \right)</math>
| |
| | |
| Here <math>{~\wedge\!\!\!\!\!\!\bigcirc~}</math> is the Kulkarni–Nomizu product defined earlier in this article. The symbol <math>\partial_k</math> denotes partial derivative, while <math>\nabla_k</math> denotes covariant derivative.
| |
| | |
| :<math>\tilde R_{ij} = R_{ij} - (n-2)\left[ \nabla_i\partial_j \varphi - (\partial_i \varphi)(\partial_j \varphi) \right] + \left( \triangle \varphi - (n-2)\|\nabla \varphi\|^2 \right)g_{ij} </math>
| |
| | |
| Beware that here the Laplacian <math>\triangle </math> is minus the trace of the Hessian on functions,
| |
| | |
| :<math>\triangle f = -\nabla^i\partial_i f</math> | |
| | |
| Thus the operator <math>-\triangle</math> is elliptic because the metric <math>g</math> is Riemannian.
| |
| | |
| :<math>\tilde\triangle f = e^{-2\varphi}\left(\triangle f -(n-2)\nabla^k\varphi\nabla_kf\right)</math>
| |
| | |
| :<math>\tilde R = e^{-2\varphi}\left(R + 2(n-1)\triangle\varphi - (n-2)(n-1)\|\nabla\varphi\|^2\right) </math>
| |
| | |
| If the dimension <math>n > 2</math>, then this simplifies to
| |
| | |
| :<math>\tilde R = e^{-2\varphi}\left[R + \frac{4(n-1)}{(n-2)}e^{-(n-2)\varphi/2}\triangle\left( e^{(n-2)\varphi/2} \right) \right] </math>
| |
| | |
| :<math>\tilde C^i{}_{jkl} = C^i{}_{jkl}</math>
| |
| | |
| We see that the (3,1) Weyl tensor is invariant under conformal changes.
| |
| | |
| Let <math>\omega</math> be a differential <math>p</math>-form. Let <math>*</math> be the Hodge star, and <math>\delta</math> the codifferential. Under a conformal change, these satisfy
| |
| | |
| :<math>\tilde * = e^{(n-2p)\varphi}*</math>
| |
| | |
| :<math>\left[\tilde\delta\omega\right](v_1 , v_2 , \ldots , v_{p-1}) = e^{-2\varphi}\left[ \delta\omega - (n-2p)\omega\left(\nabla\varphi, v_1, v_2, \ldots , v_{p-1}\right) \right]</math>
| |
| | |
| ==See also==
| |
| | |
| *[[Liouville equations]]
| |
| *[[List of formulas in elementary geometry]]
| |
| | |
| [[Category:Riemannian geometry|formulas]]
| |
| [[Category:Mathematics-related lists|Riemannian geometry formulas]]
| |