On a Riemannian manifold , 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 on the tangent bundle compatible with the metric, is called
the fundamental theorem of
Riemannian geometry .

These properties can be described as follows. Let , , and 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 is the commutator of vector
fields, .
The Levi-Civita connection is torsion-free, meaning

(1)

and is compatible with the metric

(2)

In coordinates, the Levi-Civita connection can be described using the Christoffel symbols of the second kind . In particular, if , then

(3)

or in other words,

(4)

As a connection on the tangent bundle ,
it induces a connection on the dual bundle and on all their module
tensor products . Also, given a submanifold
it restricts to
to give the Levi-Civita connection from the restriction of the metric to .

The Levi-Civita connection can be used to describe many intrinsic geometric objects. For instance, a path is a geodesic iff where is the path's tangent vector .
On a more general path , the equation defines parallel
transport for a vector field along . The second fundamental
form
of a submanifold is given by where is the tangent bundle of
and
is projection onto the normal bundle . The curvature of is given by .

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

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Cite this as:
Rowland, Todd . "Levi-Civita Connection." From MathWorld --A Wolfram Web Resource, created by Eric
W. Weisstein . https://mathworld.wolfram.com/Levi-CivitaConnection.html

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