# formulas in Riemannian geometry

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

## Symbol conventions.

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

• $\Gamma_{ijk}=\Gamma_{jik}$ : the Christoffel symbols;

• $X_{i}=X^{j}g_{ij}$, and $Y^{i}$ : rank 1 tensors;

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

• indices $i,j,k,l$ and subscripted versions thereof: components taken 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, rather than components; e.g. $\boldsymbol{X}=X^{i}\partial_{i}.$

## Formulas for the covariant derivative.

 $\displaystyle\partial_{k}g_{ij}$ $\displaystyle=\Gamma_{kij}+\Gamma_{kji},$ $\displaystyle\partial_{k}g^{ij}$ $\displaystyle=-(g^{jb}\Gamma_{bk}{}^{i}+g^{ia}\Gamma_{ak}{}^{j}),$ $\displaystyle\nabla_{k}g_{ij}$ $\displaystyle=0,$ $\displaystyle\Gamma_{ijk}$ $\displaystyle=\tfrac{1}{2}(\partial_{i}g_{jk}+\partial_{j}g_{ik}-\partial_{k}g% _{ij}),$ $\displaystyle\nabla_{i}X^{j}$ $\displaystyle=\partial_{i}X^{j}+\Gamma_{ik}{}^{j}X^{k},$ $\displaystyle\nabla_{\boldsymbol{X}}\boldsymbol{Y}$ $\displaystyle=X^{i}\,\nabla_{i}Y^{j}\,\partial_{j},$ $\displaystyle\nabla_{i}X_{j}$ $\displaystyle=\partial_{i}X_{j}-\Gamma_{ij}{}^{k}X_{k},$ $\displaystyle\nabla_{i}T_{jk}$ $\displaystyle=\partial_{i}T_{jk}-\Gamma_{ij}{}^{l}T_{lk}-\Gamma_{ik}{}^{l}T_{% jl},$ $\displaystyle\nabla_{i}T^{j}{}_{k}$ $\displaystyle=\partial_{i}T^{j}{}_{k}+\Gamma_{il}{}^{j}T^{l}{}_{k}-\Gamma_{ik}% {}^{l}T^{j}{}_{l}.$

## Formulas for geodesics

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

 $\ddot{c}{}^{i}+\Gamma_{jk}{}^{i}\,\dot{c}^{j}\dot{c}^{k}=0$
Title formulas in Riemannian geometry FormulasInRiemannianGeometry 2013-03-22 15:32:55 2013-03-22 15:32:55 matte (1858) matte (1858) 8 matte (1858) Definition msc 53B21 msc 53B20