Curvature is antisymmetric in first two variables

Statement

The Riemann curvature tensor is an alternating tensor, or an antisymmetric tensor, in the first two variables. In other words:

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

Related facts

• Curvature is tensorial

Proof

The proof is based on the fact that $[X,Y] = - [Y,X]$

We have:

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