Levi-Civita connection

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1 Definition

Let M be a smooth manifold with tangent bundle TM. Let \phi : \Rr^n_o \to M be a local parametrization, defined on some open domain \Rr^n_o\subset\Rr^n, and let \phi_i = \partial_i\phi be the partial derivatives; the vectors \phi_i(u), i=1,\dots,n form a basis of T_{\phi(u)}M for every u\in\Rr^n_o.

A semi-Riemannian metric on a M is a linear bundle map g : S^2TM \to \Rr, sometimes called metric tensor, which is nondegenerate, that is \det g_{ij} \neq 0 where g_{ij} = g(\phi_i,\phi_j). If g is positive definite, g(X,X) > 0 for every nonzero vector field X on M, it is called a Riemannian metric. A (semi-) Riemannian manifold is a smooth manifold M together with a (semi-)Riemannian metric g.

Every semi-Riemannian manifold (M,g) carries a particular affine connection, the Levi-Civita connection. This is a covariant derivative \nabla on the tangent bundle TM with the following two properties:

  1. it has no torsion, T = 0,
  2. the metric is parallel, \nabla g = 0.

The second equation involves the covariant derivative of the metric tensor g : S^2TM \to \Rr which is defined in such a way that applications of tensors to vector fields satisfy the Leibniz product rule:

\displaystyle  \partial_X g(Y,Z) = (\nabla_Xg)(Y,Z) + g(\nabla_XY,Z) + g(Y,\nabla_XZ)

for all X,Y,Z \in \Gamma TM. Thus the rules (1) and (2) can be rephrased as

  1. \nabla_XY - \nabla_YX = [X,Y],
  2. \partial_Xg(Y,Z) = g(\nabla_XY,Z) + g(Y,\nabla_XZ).

for all X,Y,Z \in \Gamma TM. If the semi-Riemannian metric g is fixed, we often write

\displaystyle \langle X,Y \rangle = g(X,Y).

The Levi-Civita connection is uniquely determined by properties (1) and (2) which imply

\begin{matrix} 2\langle\nabla_XY,Z\rangle &=& X\langle Y, Z\rangle + Y\langle Z,X\rangle - Z\langle X,Y\rangle  \\                 && -\langle X,[Y,Z]\rangle + \langle Y,[Z,X]\rangle + \langle Z,[X,Y]\rangle \end{matrix}(1)

for all X,Y,Z \in \Gamma TM (Koszul formula). If we specialize to the coordinate vector fields \phi_i, the Lie bracket terms vanish:

\displaystyle  2\langle\nabla_i\phi_j,\phi_k\rangle = \partial_i g_{jk} + \partial_jg_{ki} - \partial_kg_{ij}.

Denoting the coefficents of \nabla_i\phi_j, the so called Christoffel symbols by \Gamma_{ij}^k,

\nabla_i\phi_j = \sum_l \Gamma_{ij}^l\phi_l\, ,(2)

we obtain the Levi-Civita formula

\Gamma_{ij}^l = {1\over2}\sum_{k}g^{kl}\left(\partial_ig_{jk} + \partial_jg_{ki}- \partial_kg_{ij}\right)(3)

where (g^{kl}) denotes the inverse matrix of the metric coefficient matrix (g_{kl}).

For further information, see [Milnor1963], [Kobayashi&Nomizu1963] and [O'Neill1983].

2 References

3 External links

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