List of formulas in Riemannian geometry

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This is a list of formulas encountered in Riemannian geometry.

1 Christoffel symbols, covariant derivative
2 Curvature tensors
3 Gradient, divergence, Laplace-Beltrami operator
4 In an inertial frame

[edit] Christoffel symbols, covariant derivative

In a smooth coordinate chart, the Christoffel symbols are given by:

$\Gamma_{ij}^m=\frac12 g^{km} \left( \frac{\partial}{\partial x^i} g_{kj} +\frac{\partial}{\partial x^j} g_{ik} -\frac{\partial}{\partial x^k} g_{ij} \right)$

One has the symmetry relation

$\Gamma^i_{jk}=\Gamma^i_{kj}$

which amounts to torsion-freeness of the Levi-Civita connection.

The covariant derivative of a vector field with components $v i$ is given by:

$v^i_{;j}=\nabla_j v^i=\frac{\partial v^i}{\partial x^j}+\Gamma^i_{jk}v^k$

and similarly the covariant derivative of a $(0,1)$ -tensor field with components $v i$ is given by:

$v_{i;j}=\nabla_j v_i=\frac{\partial v_i}{\partial x^j}-\Gamma^k_{ij} v_k$

For a $(2,0)$ -tensor field with components $v i j$ this becomes

$v^{ij}_{;k}=\nabla_k v^{ij}=\frac{\partial v^{ij}}{\partial x^k} +\Gamma^i_{k\ell}v^{\ell j}+\Gamma^j_{k\ell}v^{i\ell}$

and likewise for tensors with more indices.

The covariant derivative of a function (scalar) $φ$ is just its usual differential:

$\nabla_i \phi=\phi_{;i}=\phi_{,i}=\frac{\partial \phi}{\partial x^i}$

The geodesic $X (t)$ starting at the origin with initial speed $v i$ has Taylor expansion in the chart:

$X(t)^i=tv^i-\frac{t^2}{2}\Gamma^i_{jk}v^jv^k+O(t^2)$

[edit] Curvature tensors

[edit] Riemann curvature tensor

If one defines the curvature operator as $R(U,V)W=\nabla_U \nabla_V W - \nabla_V \nabla_U W -\nabla_{[U,V]}W$ and the coordinate components of the $(1,3)$ -Riemann curvature tensor by $(R(U,V)W)^\ell={R^\ell}_{ijk}W^iU^jV^k$ , then these components are given by:

${R^\ell}_{ijk}= \frac{\partial}{\partial x^j} \Gamma_{ik}^\ell-\frac{\partial}{\partial x^k}\Gamma_{ij}^\ell +\Gamma_{js}^\ell\Gamma_{ik}^s-\Gamma_{ks}^\ell\Gamma_{ij}^s$

and lowering indices with $R_{\ell ijk}=g_{\ell s}{R^s}_{ijk}$ one gets

$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). \$

The symmetries of the tensor are

$R_{ik\ell m}=R_{\ell mik}\$ and $R_{ik\ell m}=-R_{ki\ell m}=-R_{ikm\ell}.\$

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

$R_{ik\ell m}+R_{imk\ell}+R_{i\ell mk}=0.\$

The (second) Bianchi identity is

$\nabla_m R^n {}_{ik\ell} + \nabla_\ell R^n {}_{imk} + \nabla_k R^n {}_{i\ell m}=0,\$

that is,

$R^n {}_{ik\ell;m} + R^n {}_{imk;\ell} + R^n {}_{i\ell m;k} \$

which amounts to a cyclic permutation sum of the last three indices, leaving the first two fixed.

[edit] Ricci and scalar curvatures

The Ricci curvature tensor is given by:

$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}.\$

The Ricci tensor $R i j$ is symmetric.

The scalar curvature is $R=g^{ij}R_{ij}=g^{ij}g^{\ell m}R_{i\ell jm}$ .

The "divergence" of the scalar curvature follows from the Bianchi identity (proof):

$\nabla_\ell R^\ell {}_m = {1 \over 2} \nabla_m R, \$

that is,

$R^\ell {}_{m;\ell} = {1 \over 2} R_{;m}. \$

[edit] Einstein tensor

The Einstein tensor G^ab is defined in terms of the Ricci tensor R^ab and the Ricci scalar R,

$G^{ab} = R^{ab} - {1 \over 2} g^{ab} R \$

where g is the metric tensor.

The Einstein tensor is symmetric, with a vanishing divergence (proof) which is due to the Bianchi identity:

$\nabla_a G^{ab} = G^{ab} {}_{;a} = 0. \$

[edit] Weyl tensor

The Weyl tensor is given by

$C_{ik\ell m}=R_{ik\ell m} + \frac{1}{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}{6} R \left( g_{i\ell}g_{km} - g_{im}g_{k\ell} \right).\$

[edit] Gradient, divergence, Laplace-Beltrami operator

The gradient of a function $φ$ is obtained by raising the index of the differential $\partial_i\phi$ , that is:

$\nabla^i \phi=\phi^{;i}=g^{ik}\phi_{;k}=g^{ik}\phi_{,k}=g^{ik}\partial_k \phi=g^{ik}\frac{\partial \phi}{\partial x^i}$

The covariant divergence of a vector field with components $V m$ is

$\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}.\$

The Laplace-Beltrami operator acting on a function $f$ is given by the divergence of the gradient:

$\Delta f=\nabla_i \nabla^i f =\frac{1}{\sqrt{\det g}} \frac{\partial }{\partial x^j}\left(g^{jk}\sqrt{\det g}\frac{\partial f}{\partial x^k}\right) = g^{jk}\frac{\partial^2 f}{\partial x^j \partial x^k} + \frac{\partial g^{jk}}{\partial x^j} \frac{\partial f}{\partial x^k} + \frac12 g^{jk}g^{il}\frac{\partial g_{il}}{\partial x^j}\frac{\partial f}{\partial x^k}$

The contracting relations on the Christoffel symbols

$\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} \$

and

$g^{k\ell}\Gamma^i {}_{k\ell}=\frac{-1}{\sqrt{|g|}} \;\frac{\partial\sqrt{|g|}\,g^{ik}} {\partial x^k}$

where |g| is the absolute value of the determinant of the metric tensor $g_{ik}\$ , are useful when dealing with divergences and Laplacians.

The divergence of an antisymmetric tensor field of type $(2,0)$ simplifies to

$\nabla_k A^{ik}= \frac{1}{\sqrt{|g|}} \frac{\partial (A^{ik}\sqrt{|g|})}{\partial x^k}.\$

[edit] In an inertial frame

An orthonormal inertial frame is a coordinate chart such that, at the origin, one has the relations $g i j = δ i j$ and $\Gamma^i_{jk}=0$ (but these may not hold at other points in the frame). 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.