Curvature is antisymmetric in first two variables

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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)

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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).