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[parent] formulas in Riemannian geometry (Definition)

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. $\bX = X^i \partial_i.$

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$$ \ddot{c}{}^i + \Gamma_{jk}{}^i\, \dot{c}^j\dot{c}^k = 0$$



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Cross-references: curve, geodesic, frame, local coordinate, indices, tensors, rank, Christoffel symbols, metric tensor, components

This is version 5 of formulas in Riemannian geometry, born on 2005-10-25, modified 2005-10-26.
Object id is 7446, canonical name is FormulasInRiemannianManifold.
Accessed 1459 times total.

Classification:
AMS MSC: 53B20 (Differential geometry :: Local differential geometry :: Local Riemannian geometry)
 53B21 (Differential geometry :: Local differential geometry :: Methods of Riemannian geometry)
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