## Tensor Fields and Multilinear Maps

A tensor field over a manifold gives us a tensor at each point . And we know that can be considered as a multilinear map. Specifically, if is a tensor field of type , then we find

which we can interpret as a multilinear map:

where multilinearity means that is linear in each variable separately.

As we let vary over , we can interpret as defining a function which takes vector fields and covector fields and gives a function. Explicitly:

And, in particular, this function is multilinear over . That is,

And a similar calculation holds for any of the other variables, vector or covector.

So each tensor field gives us a multilinear function , and this multilinearity is not only true over but over as well.

Conversely, let be an -multilinear function. If it’s also linear over in each variable, then it “lives locally”. That is, if and then

and so at each there is some tensor so that is a tensor field.

This is as distinguished from things like differential operators — , for instance — which fail both sides. On the one side, we can calculate

which picks up an extra term. It’s -linear but not -linear. On the other side, the value of this function at doesn’t just depend on the value of at , but on how changes through . That is, this operator does not “live locally”, and is not a tensor field.

To prove this assertion, it will suffice to deal with the case where takes a single vector variable , and we only need to verify that if then . Let be a chart around , and write

where by assumption each . We let be a neighborhood of whose closure is contained in . We know we can find a smooth bump function supported in and with on .

Now we define vector fields on and on . Similarly we define on and on . Then we can write

and thus

as asserted.

[...] property onto the pile. Notice, though, how this condition is different from the way we said that tensor fields live locally. In this case we need to know that vanishes in a whole neighborhood, not just at [...]

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