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:

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