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Given a Riemannian manfiold $M$ with metric $g=(g_{i,j})$. Let $T=T_{A,B,C,\dots}^{a,b,c,\dots}$ be a tensor on $M$. I would like to compute for example $T_{A,B,C,\dots}^{a,n,c,\dots}g_{n,m}$. We know that we should sum this over $n$ and get another tensor, but where should I put the index $m$, $$ T_{A,B,C,\dots,m}^{a,n,c,\dots} \ \ \text{or} \ \ \ T_{A,m,B,C,\dots}^{a,n,c,\dots}? $$ and why should it be so?

I have been confused by contraction of this kind of tensors. Thanks you for your help.

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Good question. I think the answer depends on what you are reading. I propose the following:

$$ T_{ij} = g_{ik}T^{k}_{ \ \ \ j} =g_{jl}T_{i}^{ \ \ l}= g_{ik}g_{jl}T^{kl} $$

In other words, avoid writing indices in the same space both up and down, unless you have no intention of raising and lowering said indices. This is one fix you can use in your work. One nice thing to read about these sort of convention/notation issues is Gravitation by Misner Thorn and Wheeler.

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Your answer makes perfect sense. Thank you! –  M. K. Sep 10 '12 at 6:00

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