It is isn't it? This is a scalar (a 1-1 matrix).
Therefore it is equal to its transpose, and whenever is symmetric.
, is an inner product, where and the are the basis vectors for , the tangent space of a manifold at a point .
So I need to prove that it is positive definite, but I can't see why this holds. I suppose the point to use would be that elements on the diagonal are all (strictly) positive, but I don't see how to use this.