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I don't understand the difference between tensor and tensor field.

I'm learning from Barret O'neill's Semi-Riemann Geometry and here are the definitions:

If $A:(V^*)^r \times V^s\to K$ transformation is $K$-multilinear then $A$ is a tensor on $V$.

$M$ is a manifold, $\mathfrak{X}(M)$ is the vector fields' set that is an $F(M)$-module.

If $A$ is a tensor on $\mathfrak{X}(M)$ then we say $A$ is a tensor field on $M$.

I did not understand the last sentence. What is the difference between a tensor and tensor field?

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Usually, a tensor field of a manifold $M$ is an assignment of a tensor to each point of $M$. Just like a vector field of $M$ gives you a vector at a particular point of $M$. – Lemon Jan 4 '13 at 14:23
I got it.I have another question.We said A is a tensor on V but how did we say A is a tensor on M.Shouldn't it be a tensor on X(M)? – Serkan Yaray Jan 4 '13 at 14:29
Where is it said that $A$ is a tensor on $M$? O'Neill's book says precisely that $A$ is a tensor on $\mathfrak{X}(M)$ and equivalently $A$ is a tensor field on $M$. – Willie Wong Jan 4 '13 at 14:31
Oh pardon you are right.I wrote it wrong. – Serkan Yaray Jan 4 '13 at 14:37
up vote 8 down vote accepted

The difference is all in your head. Literally.

The difference in calling the same object $A$ a "tensor over $\mathfrak{X}(M)$" as opposed to "a tensor field over $M$" is that the former emphasizes the fact that we have an algebraic object: a tensor over some module, while the latter emphasizes the fact that underlying the module there is some manifold and geometry is going on there.

Calling something a tensor field instead of a tensor forces you to remember that $\mathfrak{X}(M)$ is not just some arbitrary module, but that its elements can be identified with smooth sections of the tangent bundle of some manifold. These additional structures are occasionally useful.

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Oh it finally has been much helpful.Thanks for your help. – Serkan Yaray Jan 4 '13 at 15:27

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