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?