For a smooth function    $f:M\to \mathbb{R}$ one usually defines the degeneracy and index of a critical point    $p\in M$ in terms of the eigenvalues of the Hessian matrix    $(\partial^2 f/\partial x_i\partial x_j)$.  

On the other hand, if we have a Riemannian metric    $g$ we can define the Hessian tensor    $H(f,g)=\nabla df$.  From what I understand, one can recover the Hessian matrix from this tensor and define the index and degeneracy of a critical point as above.

My question is: can we cut out the middle man, i.e., is there a natural way to define the index and degeneracy of a critical point from the Hessian tensor without using the Hessian matrix?