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:

$R(X,Y)=\nabla _{X}\circ \nabla _{Y}-\nabla _{Y}\circ \nabla _{X}-\nabla _{[X,Y]}=-\left(\nabla _{Y}\circ \nabla _{X}-\nabla _{X}\circ \nabla _{Y}-\nabla _{[Y,X]}\right)=-R(Y,X)$.