Levi-Civita connection

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On any Riemannian manifold $\langle M,g\rangle$ , there is a unique torsion-free affine connection $\nabla$ on the tangent bundle of $M$ such that the covariant derivative of the metric tensor $g$ is zero, i.e. $g$ is covariantly constant. This condition can be also be expressed in terms of the inner product operation $\langle,\rangle\colon TM\times TM\to\mathbb{R}$ induced by $g$ as follows: For all vector fields $X,Y,Z\in TM$ , one has

X(\langle Y,Z\rangle)=\langle\nabla_{X}Y,Z\rangle+\langle Y,\nabla_{X}Z\rangle

and

\nabla_{X}Y-\nabla_{Y}X=[X,Y]

This connection is called the Levi-Civita connection.

In local coordinates $\{x_{1},\ldots,x_{n}\}$ , the Christoffel symbols $\Gamma^{i}_{{jk}}$ are determined by

g_{{i\ell}}\Gamma^{i}_{{jk}}=\frac{1}{2}\left(\frac{\partial g_{{j\ell}}}{% \partial x_{k}}+\frac{\partial g_{{k\ell}}}{\partial x_{j}}-\frac{\partial g_{% {jk}}}{\partial x_{\ell}}\right).

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