Curvature of a connection

Definition

Given data

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

Definition part

The curvature of $\mathrm{\nabla }$ is defined as the map:

Failed to parse (syntax error): {\displaystyle R(X,Y) = \nabla_X \circ \nabla_Y - \nabla_Y \circ \nabla_X - \nabla_{{X,Y]}}

where $X,Y\in \mathrm{\Gamma }Notethat<math>R(X,Y)$ itself outputs a linear map $\mathrm{\Gamma }(E)\to \mathrm{\Gamma }(E)$. We can thus write this as:

$R(X,Y)Z={\mathrm{\nabla }}_X({\mathrm{\nabla }}_{Y}Z)-{\mathrm{\nabla }}_Y({\mathrm{\nabla }}_{X}Z)-{\mathrm{\nabla }}_{[X,Y]}Z$

In the linear case

In the special case where $E=TM$, we have that $X,Y,Z\in \mathrm{\Gamma }(TM)$. 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.