Levi-Civita Connection

On a Riemannian manifold M, there is a canonical connection called the Levi-Civita connection (pronounced lē-vē shi-vit-e), sometimes also known as the Riemannian connection or covariant derivative. As a connection on the tangent bundle, it provides a well-defined method for differentiating vector fields, forms, or any other kind of tensor. The theorem asserting the existence of the Levi-Civita connection, which is the unique torsion-free connection del on the tangent bundle TM compatible with the metric, is called the fundamental theorem of Riemannian geometry.

These properties can be described as follows. Let X, Y, and Z be any vector fields, and <,> denote the metric. Recall that vector fields act as derivation algebras on the ring of smooth functions by the directional derivative, and that this action extends to an action on vector fields. The notation [X,Y] is the commutator of vector fields, XY-YX. The Levi-Civita connection is torsion-free, meaning

 del _XY-del _YX=[X,Y],

and is compatible with the metric

 X<Y,Z>=<del _XY,Z>+<Y,del _XZ>.

In coordinates, the Levi-Civita connection can be described using the Christoffel symbols of the second kind Gamma_(i,j)^k. In particular, if e_i=partial/partialx_i, then

 g_(kl)Gamma_(i,j)^l=<del _(e_i)e_j,e_k>,

or in other words,

 del _(e_i)e_j=sum_(k)Gamma_(i,j)^ke_k.

As a connection on the tangent bundle TM, it induces a connection on the dual bundle T^*M and on all their module tensor products TM^k tensor TM^(*l). Also, given a submanifold N it restricts to TN to give the Levi-Civita connection from the restriction of the metric to N.

The Levi-Civita connection can be used to describe many intrinsic geometric objects. For instance, a path c:R->M is a geodesic iff del _(c^.(t))c^.(t)=0 where c^. is the path's tangent vector. On a more general path c, the equation del _(c^.(t))v(t)=0 defines parallel transport for a vector field v along c. The second fundamental form II of a submanifold N is given by pi_Q degreesdel _(TN) where TN is the tangent bundle of N and pi_Q is projection onto the normal bundle Q. The curvature of M is given by del  degreesdel .

See also

Christoffel Symbol, Connection, Covariant Derivative, Curvature, Fundamental Theorem of Riemannian Geometry, Geodesic, Principal Bundle, Riemannian Manifold, Riemannian Metric

This entry contributed by Todd Rowland

Explore with Wolfram|Alpha

Cite this as:

Rowland, Todd. "Levi-Civita Connection." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

Subject classifications