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You can search summation convention. It means sum all possible combinations of indices. https://rpubs.com/lizhi1800/Riemann_curvature_tensor_2nd_kind
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https://rpubs.com/lizhi1800/Riemann_curvature_tensor_2nd_kind
Here is a full code treatment of Riemman tensor.
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First kind and second kind are duals, right? One to compute covariant geometry, other one to compute contravariant geometry.
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https://rpubs.com/lizhi1800/Einstein_Notation
Subscript and superscript are for covariant and contravariant components. They are to tell people the summation result is contravariant or covariant.