Inner Products of Vector Fields
Now that we can define the inner product of two vectors using a metric , we want to generalize this to apply to vector fields.
This should be pretty straightforward: if and
are vector fields on an open region
it gives us vectors
and
at each
. We can hit these pairs with
to get
, which is a real number. Since we get such a number at each point
, this gives us a function
.
That this is a bilinear function is clear. In fact we’ve already implied this fact when saying that
is a tensor field. But in what sense is it an inner product? It’s symmetric, since each
is, and positive definite as well. To be more explicit:
with equality if and only if
is the zero vector in
. Thus the function
always takes on nonneative values, is zero exactly where
is, and is the zero function if and only if
is the zero vector field.
What about nondegeneracy? This is a little trickier. Given a nonzero vector field, we can find some point where
is nonzero, and we know that there is some
such that
. In fact, we can find some region
around
where
is everywhere nonzero, and for each point
we can find a
such that
. The question is: can we do this in such a way that
is a smooth vector field?
The trick is to pick some coordinate map on
, shrinking the region if necessary. Then there must be some
such that
because otherwise would be degenerate. Now we get a smooth function near
:
which is nonzero at , and so must be nonzero in some neighborhood of
. Letting
be this coordinate vector field gives us a vector field that when paired with
using
gives a smooth function that is not identically zero. Thus
is also nonzero, and is worthy of the title “inner product” on the module of vector fields
over the ring of smooth functions
.
Notice that we haven’t used the fact that the are positive-definite except in the proof that
is, which means that if
is merely pseudo-Riemannian then
is still symmetric and nondegenerate, so it’s still sort of like an inner product, like an symmetric, nondegenerate, but indefinite form is still sort of like an inner product.