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This is a list of [[formula]]s encountered in [[Riemannian geometry]].
Hello. Let me introduce the author. Her title is Emilia Shroyer but it's not the most feminine name out there. California is our birth place. Doing ceramics is what my family members and I appreciate. Managing individuals is his occupation.<br><br>My website: [http://chatbook.biz/blogs/post/922 at home std testing]
 
==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]]

Revision as of 06:43, 14 February 2014

Hello. Let me introduce the author. Her title is Emilia Shroyer but it's not the most feminine name out there. California is our birth place. Doing ceramics is what my family members and I appreciate. Managing individuals is his occupation.

My website: at home std testing