Levi-Civita connection
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In Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free Riemannian connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric (or pseudo-Riemannian metric). The fundamental theorem of Riemannian geometry states that there is unique connection which satisfies these properties.
In the theory of Riemannian and pseudo-Riemannian manifolds the term covariant derivative is often used for the Levi-Civita connection. The components of this connection with respect to a system of local coordinates are called Christoffel symbols.
Formal definition
Let [(M,g)] be a Riemannian manifold (or pseudo-Riemannian manifold) then an affine connection [\nabla] is a Levi-Civita connection if it satisfies the following conditions
- Preserves metric, i.e., for any vector fields [X], [Y], [Z] we have [Xg(Y,Z)=g(\nabla_X Y,Z)+g(Y,\nabla_X Z)], where [Xg(Y,Z)] denotes the derivative of function [g(Y,Z)] along vector field [X].
- Torsion-free, i.e., for any vector fields [X] and [Y] we have [\nabla_XY-\nabla_YX=[X,Y]], where [[X,Y]] are the Lie brackets for vector fields [X] and [Y].
Derivative along curve
Levi-Civita connection defines also a derivative along curves, usually denoted by [D].
Given a smooth curve [\gamma] on [(M,g)] and a vector field [V] on [\gamma] its derivative is defined by
- [D_tV=\nabla_V.]
See also
External links
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