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Diffgeom β

Curvature of a connection

This curvature is also sometimes known as the Riemann curvature tensor. However, the latter term is usually reserved for situations where we have a linear connection, in particular, the Riemann curvature tensor arising from the Levi-Civita connection for a Riemannian or pseudo-Riemannian manifold

Contents

Definition

Given data

Definition part

The curvature of \nabla is defined as the map:

R(X,Y) = \nabla_X \circ \nabla_Y - \nabla_Y \circ \nabla_X - \nabla_{[X,Y]}

where X, Y \in \Gamma

Note that R(X,Y) itself outputs a linear map \Gamma(E) \to \Gamma(E). We can thus write this as:

R(X,Y)Z = \nabla_X (\nabla_Y Z) - \nabla_Y (\nabla_X Z) - \nabla_{[X,Y]}Z

In local coordinates

Further information: curvature matrix of a connection

Consider a system of local coordinate charts for M such that the vector bundle E is trivial on each chart. For any connection \nabla, we can write a matrix that, in local coordinates, describes the curvature of \nabla. This matrix is sometimes denoted as \Omega, and is defined by:

\Omega := d\omega + \omega \wedge \omega

Here, \omega is a matrix of connection forms.

In the linear case

In the special case where E = TM (the case of a linear connection) we get that X,Y, Z \in \Gamma(TM). We can thus think of this map as a (1,3)-tensor because it takes as input three vector fields and outputs one vector field.

Properties

Tensoriality

Further information: Curvature is tensorial

The curvature is tensorial in all three arguments. This is best proved by proving C^\infty-linearity in all arguments, via a computation.

Antisymmetry

Further information: Curvature is antisymmetric in first two variables

We have the following identity:

R(X,Y) = -R(Y,X)