formulas in Riemannian geometry

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

Symbol conventions.

  • gij : the components of the metric tensor;

  • Γijk=Γjik : the Christoffel symbolsMathworldPlanetmathPlanetmath;

  • Xi=Xjgij, and Yi : rank 1 tensors;

  • Tij=Tigjkk : a rank 2 tensor;

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

  • xi, yj : systems of local coordinates;

  • i=xi : local coordinate frame;

  • boldfaced symbols: the actual geometric quantity, rather than components; e.g. 𝑿=Xii.

Formulas for the covariant derivative.

kgij =Γkij+Γkji,
kgij =-(gjbΓbk+igiaΓak)j,
kgij =0,
Γijk =12(igjk+jgik-kgij),
iXj =iXj+ΓikXkj,
𝑿𝒀 =XiiYjj,
iXj =iXj-ΓijXkk,
iTjk =iTjk-ΓijTlkl-ΓikTjll,
iTjk =iTj+kΓilTlj-kΓikTjl.l

Formulas for geodesics

A geodesic is a curve c:IM satisfying

Title formulas in Riemannian geometry
Canonical name FormulasInRiemannianGeometry
Date of creation 2013-03-22 15:32:55
Last modified on 2013-03-22 15:32:55
Owner matte (1858)
Last modified by matte (1858)
Numerical id 8
Author matte (1858)
Entry type Definition
Classification msc 53B21
Classification msc 53B20