Curvature is antisymmetric in first two variables
(Redirected from Antisymmetry of Riemann curvature tensor)
Statement
The Riemann curvature tensor is an alternating tensor, or an antisymmetric tensor, in the first two variables. In other words:
Related facts
Proof
The proof is based on the fact that
![{\displaystyle [X,Y]=-[Y,X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c433e248c0c7e659ef3f2d9df64d8c7505630bc)
We have:
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![{\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)