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