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Oxymoron
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Producing the Ricci tensor
On a pseudo-Riemannian manifold we can contract the Riemann curvature tensor to form the Ricci tensor. In this process of contraction we sum over two indices to make a (3-1)-tensor into a (2-0)-tensor. My question is, why must we contract two indices? Why can't we contract one index to form a (3-0)-tensor or a (2-1)-tensor, etc...
On a pseudo-Riemannian manifold we can contract the Riemann curvature tensor to form the Ricci tensor. In this process of contraction we sum over two indices to make a (3-1)-tensor into a (2-0)-tensor. My question is, why must we contract two indices? Why can't we contract one index to form a (3-0)-tensor or a (2-1)-tensor, etc...
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