Difference between revisions of "Levi-Civita connection"

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* The connection is [[defining ingredient::metric connection|metric]], viz <math>X g (Y,Z) = g (\nabla_X Y, Z) + g(Y, \nabla_X Z)</math>
 
* The connection is [[defining ingredient::metric connection|metric]], viz <math>X g (Y,Z) = g (\nabla_X Y, Z) + g(Y, \nabla_X Z)</math>
* The connection is [[defining ingredient::torsion-free linear connection|torsion-free]], viz <math>\nabla_X Y - \nabla_Y X = [X,Y]</math>
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* The connection is [[defining ingredient::torsion-free linear connection|torsion-free]], viz <math>\nabla_X Y - \nabla_Y X = [X,Y]</math>.
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===Definition by formula===
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The formula for the Levi-Civita connection requires us to use the fact that <math>g</math> is nondegenerate. So, instead of directly specifying <math>\nabla_XY</math> for vector fields <math>X</math> and <math>Y</math>, the formula specifies <math>g(\nabla_XY,Z)</math> for <math>X,Y,Z</math> vector fields, as follows:
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<math>g(\nabla_XY,Z) = \frac{Xg(Y,Z) + Yg(Z,X) - Zg(X,Y) + g(Y,[Z,X]) + g(Z,[X,Y]) - g(X,[Y,Z])}{2}</math>.
  
 
==Facts==
 
==Facts==

Revision as of 19:53, 24 July 2009

This lives as an element of: the space of all linear connections, which in turn sits inside the space of all \R-bilinear maps \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)

Definition

Given data

A Riemannian manifold (M,g) (here, M is a differential manifold and g is the additional structure of a Riemannian metric on it).

More generally, we can also look at a pseudo-Riemannian manifold, or a manifold with a pseudo-Riemannian metric: a smoothly varying nondegenerate (not necessarily positive definite) symmetric bilinear form g in each tangent space.

Definition part

A Levi-Civita connection on (M,g) is a linear connection \nabla on M satisfying the following two conditions:

Definition by formula

The formula for the Levi-Civita connection requires us to use the fact that g is nondegenerate. So, instead of directly specifying \nabla_XY for vector fields X and Y, the formula specifies g(\nabla_XY,Z) for X,Y,Z vector fields, as follows:

g(\nabla_XY,Z) = \frac{Xg(Y,Z) + Yg(Z,X) - Zg(X,Y) + g(Y,[Z,X]) + g(Z,[X,Y]) - g(X,[Y,Z])}{2}.

Facts

The Levi-Civita connection is unique

Further information: Levi-Civita connection exists and is unique

The proof combines the metric condition, the torsion-free condition, and the nondegeneracy of g.

Note that the nondegeneracy of g is very important, otherwise knowing the value of the inner product for any three vectors may not necessarily help us in computing the value of nabla_XY

To show that the Levi-Civita connection exists, it suffices to check that the map sending Z to what we propose for g(\nabla_X Y, Z) is actually a linear map.

Christoffel symbols

Further information: Christoffel symbol

The Levi-Civita connection on a manifold M is a map \nabla: \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM). This means that at any point p \in M, it gives a map T_p(M) \times T_p(M) \to T_p(M), which roughly differentiates one tangent vector along another.

Let \partial_1, \partial_2, \ldots, \partial_n form a basis for the tangent space TM. Then, the Christoffel symbol \Gamma_{ij}^k is the component along e_k of the vector \nabla_{\partial_i}\partial_j.

The Christoffel symbols thus give an explicit description of the Levi-Civita connection. Namely, the Levi-Civita connection can be expressed using the Christoffel symbols.