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Levi-Civita connection

8 bytes added, 17:59, 6 January 2012
Christoffel symbols
{{further|[[Christoffel symbols of a connection]]}}
The Levi-Civita connection on a manifold <math>M</math> is a map <math>\nabla: \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)</math>. This means that at any point <math>p \in M</math>, it gives a map <math>T_p(M) \times T_p\Gamma(MTM) \to T_p\Gamma(MTM)</math>, which roughly ''differentiates'' one tangent vector along another.
Let <math>\partial_1, \partial_2, \ldots, \partial_n</math> form a basis for the tangent space <math>TM</math>. Then, the Christoffel symbol <math>\Gamma_{ij}^k</math> is the component along <math>e_k</math> of the vector <math>\nabla_{\partial_i}\partial_j</math>.
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.
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