Prove $Index_{p}(\bigtriangledown f)$= "dimension of sourcing flow lines from p" ,where p is a critical point.
Attempt
Near $ p \in Cr(f) $ in some coord. $ f(x) - f(p) = $ $\sum_j x_j^2 - \sum_k x_k^2\ $. Then by choosing a Riemannian metric on $ M $ coinciding with the standard Euclidean near $ p \in Cr(f) $ the gradient flow splits orbits near $ Cr(f) $ into stable $S_p$ and unstable $US_p\ $, i.e. running into $ p \in Cr(f) $ as $t \to -\infty $ or, repectively as $t \to +\infty\ $. Then $S_p$ and $US_p\ $ are manifolds near $p $ and $T_p S_p$ and $T_p US_p$ are the linear subspaces of negative and, respectively, positive eigenvalues of $ Hess_p(f)\ $.