The Unapologetic Mathematician

Mathematics for the interested outsider

Inner Products of Vector Fields

Now that we can define the inner product of two vectors using a metric g, we want to generalize this to apply to vector fields.

This should be pretty straightforward: if v and w are vector fields on an open region U it gives us vectors v_p and w_p at each p\in U. We can hit these pairs with g_p to get g_p(v_p,w_p), which is a real number. Since we get such a number at each point p, this gives us a function g(v,w):U\to\mathbb{R}.

That this g is a bilinear function is clear. In fact we’ve already implied this fact when saying that g is a tensor field. But in what sense is it an inner product? It’s symmetric, since each g_p is, and positive definite as well. To be more explicit: g_p(v_p,v_p)\geq0 with equality if and only if v_p is the zero vector in \mathcal{T}_pM. Thus the function g(v,v) always takes on nonneative values, is zero exactly where v is, and is the zero function if and only if v is the zero vector field.

What about nondegeneracy? This is a little trickier. Given a nonzero vector field, we can find some point p where v_p is nonzero, and we know that there is some w_p such that g_p(v_p,w_p)\neq0. In fact, we can find some region U around p where v is everywhere nonzero, and for each point q\in U we can find a w_q such that g_q(v_q,w_q)\neq0. The question is: can we do this in such a way that w_q is a smooth vector field?

The trick is to pick some coordinate map x on U, shrinking the region if necessary. Then there must be some i such that

\displaystyle g_p\left(v_p,\frac{\partial}{\partial x^i}\bigg\vert_p\right)\neq0

because otherwise g_p would be degenerate. Now we get a smooth function near p:

\displaystyle g\left(v,\frac{\partial}{\partial x^i}\right)

which is nonzero at p, and so must be nonzero in some neighborhood of p. Letting w be this coordinate vector field gives us a vector field that when paired with v using g gives a smooth function that is not identically zero. Thus g is also nonzero, and is worthy of the title “inner product” on the module of vector fields \mathfrak{X}(U) over the ring of smooth functions \mathcal{O}(U).

Notice that we haven’t used the fact that the g_p are positive-definite except in the proof that g is, which means that if g is merely pseudo-Riemannian then g 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.

September 30, 2011 Posted by | Differential Geometry, Geometry | 2 Comments