The index of a bilinear form on a vector space $V$ is the dimension of a maximal negative definite subspace of $V$. Let $M$ be a manifold. A fundamental bilinear form $b$ on M is a $C^\infty$ symmetric tensor field of type $(0,2)$ defined on all points of $M$ and which is nondegenerate at every point.
I want to proof that the index of $b$ is constant on each connected component of $M$.
First approach: assume $m$ and $n$ are points in the same connected component of $M$. Then there is a $C^\infty$ curve connecting the two points. I just need to show the index is constant/continuous through the curve. Then the problem reduces to showing that the index is a continuous function of the components of b in the basis induced by the coordinates in each coordinate domain.
I'm not sure how to complete the proof.