formulas in Riemannian geometry

The aim of this page is to collectfrequently used formulas in Riemannian geometry.

Symbol conventions.

• $g_{ij}$ : the components of themetric tensor;

• ${\mathrm{\Gamma }}_{ijk}={\mathrm{\Gamma }}_{jik}$ : the Christoffel symbols;

• $X_i=X^j{g}_{ij}$, and $Y^i$ : rank 1tensors;

• $T_{ij}=T_i{}^{k}g_{jk}$ : a rank 2 tensor;

• indices $i,j,k,l$ and subscripted versions thereof: componentstaken with respect to a local coordinate frame;

• $x^i$, $y^j$ : systems of local coordinates;

• ${\partial }_i=\frac{\partial }{\partial x^i}$ : local coordinate frame;

• boldfaced symbols: the actual geometric quantity, ratherthan components; e.g. $𝑿=X^i{\partial }_i.$

Formulas for the covariant derivative.

${\partial }_k{g}_{ij}$$={\mathrm{\Gamma }}_{kij}+{\mathrm{\Gamma }}_{kji},$

${\partial }_k{g}^{ij}$$=-(g^{jb}{\mathrm{\Gamma }}_{bk}{}^i+g^{ia}{\mathrm{\Gamma }}_{ak}{}^j),$

${\nabla }_k{g}_{ij}$$=0,$

${\mathrm{\Gamma }}_{ijk}$$=\frac{1}{2}({\partial }_i{g}_{jk}+{\partial }_j{g}_{ik}-{\partial }_k{g}_{ij}),$

${\nabla }_i{X}^j$$={\partial }_i{X}^j+{\mathrm{\Gamma }}_{ik}{}^{j}X^k,$

${\nabla }_{𝑿}𝒀$$=X^i{\nabla }_i{Y}^j{\partial }_j,$

${\nabla }_i{X}_j$$={\partial }_i{X}_j-{\mathrm{\Gamma }}_{ij}{}^{k}X_k,$

${\nabla }_i{T}_{jk}$$={\partial }_i{T}_{jk}-{\mathrm{\Gamma }}_{ij}{}^{l}T_{lk}-{\mathrm{\Gamma }}_{ik}{}^{l}T_{jl},$

${\nabla }_i{T}^j{}_k$$={\partial }_i{T}^j{}_k+{\mathrm{\Gamma }}_{il}{}^{j}T^l{}_k-{\mathrm{\Gamma }}_{ik}{}^{l}T^j{}_l.$

Formulas for geodesics

A geodesic is a curve $c:I\to M$ satisfying

$$\ddot{c}{}^i+{\mathrm{\Gamma }}_{jk}{}^i\dot{c}^j{\dot{c}}^k=0$$