PlanetMath: Levi-Civita connection

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Levi-Civita connection

(Definition)

On any Riemannian manifold $\langle M, g \rangle$ , there is a unique torsion-free affine connection $\nabla$ on the tangent bundle of such that the covariant derivative of the metric tensor is zero, i.e. 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 as follows: For all vector fields $X, Y, Z \in TM$ , one has

$\displaystyle X(\langle Y,Z\rangle )=\langle \nabla_X Y, Z\rangle +\langle Y, \nabla_X Z\rangle$

and

$\displaystyle \nabla_X Y-\nabla_Y X=[X,Y]$

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

$\displaystyle g_{i\ell}\Gamma^i_{jk}=\frac 12\left(\frac{\partial g_{j\ell}}{\p... ...artial g_{k\ell}}{\partial x_j}-\frac{\partial g_{jk}}{\partial x_\ell}\right).$

"Levi-Civita connection" is owned by rspuzio. [ full author list (2) | owner history (1) ]

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Cross-references: local coordinates, connection, vector fields, induced, operation, inner product, terms, metric tensor, covariant derivative, tangent bundle, affine connection, torsion-free, Riemannian manifold
There are 8 references to this entry.

This is version 4 of Levi-Civita connection, born on 2003-10-09, modified 2007-01-12.
Object id is 4765, canonical name is LeviCivitaConnection.
Accessed 6360 times total.

Classification:

AMS MSC:

53B05 (Differential geometry :: Local differential geometry :: Linear and affine connections)

Pending Errata and Addenda

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