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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 ) {\displaystyle R(X,Y)=-R(Y,X)}
The proof is based on the fact that [ X , Y ] = − [ Y , X ] {\displaystyle [X,Y]=-[Y,X]}
We have:
R ( X , Y ) = ∇ X ∘ ∇ Y − ∇ Y ∘ ∇ X − ∇ [ X , Y ] = − ( ∇ Y ∘ ∇ X − ∇ X ∘ ∇ Y − ∇ [ Y , X ] ) = − R ( Y , X ) {\displaystyle 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)} .
![{\displaystyle [X,Y]=-[Y,X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c433e248c0c7e659ef3f2d9df64d8c7505630bc)
![{\displaystyle 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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e391d3fb7a3e4a9c386d2832cbac92a27b1c01d)
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