Inner product computations on manifolds

[FuzzyFungi]

Hi there! I have a pretty basic question about how to compute an inner product $$\left\langle\omega, X\right\rangle$$ on a manifold.

I understand that, if both arguments are vectors (or vector fields) and we're in euclidean space, the computation is exactly as if I were doing a dot product. However, if we're in a manifold (Lets say... On the surface of a unit sphere in $$\Re^3$$) how would the computation be done?

What if the first argument is a 1-form?

From what I've read, I've found lots of helpful information concerning properties of inner products, their usefulness as metrics, and nice identities with them. But when it comes to finding the value of one, I am lost.

Thanks!

 

[crazyjimbo]

My understanding has always been that the inner product of a 1-form and a vector is given by $$\langle \omega, X \rangle = \omega(X)$$

For the inner product of two vectors, we first have to convert the first vector to a 1-form. If $$X$$ is a vector then the corresponding 1-form or covector is denoted $$X^\flat$$ and is given by $$X^\flat(Y) = g(X,Y)$$. If the components of $$X$$ in some coordinate system are $$X^i$$ then the components of $$X^\flat$$ will be $$X_j = g_{ij}X^i$$. I.e. $$X^\flat = X_j dx^j = g_{ij}X^i dx^j$$.

So $$\langle X,Y \rangle := \langle X^\flat,Y \rangle = X^\flat(Y) = g(X,Y)$$ and g is really our inner product of vectors.

I hope that makes sense