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