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This is a very simple question that I got confused. Is Riemannian metric a symmetric 2-tensor or symmetric 2-tensor field? Wikipedia says that it is a (0,2) tensor but my book says it is a tensor field. Are these things the same? What am I missing? (By the way, if it is a tensor field, I do not know what a symmetric tensor field is.)

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up vote 10 down vote accepted

It's a symmetric 2 tensor field(*) on the manifold (let's say, $M$), and a symmetric 2 tensor when restricted to to the tangent space $T_pM$ for some $p \in M$. But people are often sloppy with this distinction.

Symmetry just means that (if the metric is denoted by $g$) $$g(p)(v,w) = g(p)(w,v) \quad \forall p \in M, v,w \in T_pM$$ which in local coordinates is equivalent to the statement that $$g_{ij} dx^i\otimes dx^j = g_{ji} dx^i\otimes dx^j $$


(*) more precisely: "(0,2) tensor field" or "covariant tensor field"

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Is it covariat or contravariant? –  user20353 Dec 25 '11 at 13:14
    
covariant, sorry, this was wrong first. –  user20266 Dec 25 '11 at 13:37
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