Connections

Connection

A connection on a vector bundle over smooth manifold is a map , such that the following hold:

• -linearity in the first argument: for any ;
• -linearity in the second argument: ;
• Leibniz rule: for any .

An important special case of a connection is the affine connection or covariant derivative:

Affine Connection

Let be a smooth manifold. An affine connection on is a connection on the tangent bundle. Explicitly, it is a map , such that the following hold:

• Linearity in the first argument: ;
• Leibniz rule: for any .

Affine connection is also called the covariant derivative.

Section

A section of the vector bundle is called parallel (with respect to a connection ) if for all .

Def Covariant Derivative of Real Functions Given a point of the manifold , a real function on the manifold and a tangent vector , the covariant derivative of at along , denoted , is a scalar that represents the principal part of the change in the value of when the argument of is changed infinitesimally in the same direction as the displacement vector . Formally, there is a differentiable curve such that and , and the covariant derivative of at is defined by When is a vector field on , the covariate derivative is the function that assigns each point a scaler . Notice that this derivative exists without a definition of connection. So we simply write instead of .

Connection Coefficients

Christoffel Symbols of The Second Kind

Suppose we are working in a chart for , with corresponding coordinate tangent vectors , and write for . The Christoffel symbols (of the second kind) or connection coefficients of a connection with respect to this chart are defined by

Proposition

The connection is determined on by the connection coefficients.

Proof Write . Then we have for any vector ,

Parallel Transport

Parallel

Let be a manifold with an affine connection Then a vector field is said to be parallel if for any vector field , .

Remark

Intuitively, parallel vector fields have all their derivatives equal to zero and are therefore in some sense constant.

Parallel Transport

If is a smooth map parameterized by an interval and , then a vector field along is called the parallel transport of along if it satisfies the following conditions:

• for all ;
• .

Geodesic

A curve on a smooth manifold with an affine connection is a geodesic if

Natural Dual Connection

Dual Connection

Given a connection on a vector bundle over manifold , the dual connection on the dual bundle is naturally defined by the following equation: for all .