Curvature of a connection

Definition

Given data

• A connected differential manifold $M$
• A vector bundle $E$ over $M$
• A connection $\nabla$ for $E$

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 the linear case

In the special case where $E=TM$, we have 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.

This is the famed Riemann curvature tensor that is important for its algebraic and differential properties.