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?



Index of a Morse function via the Hessian tensor
The Hessian tensor (defined at a critical point of a function) is a symmetric bilinear form on the tangent space of the manifold at that point. For a symmetric bilinear form on a finite-dimensional vector space (in characteristic different from $2$) there is always a basis such that the corresponding symmetric matrix is diagonal. The diagonal entries are not well-defined independent of diagonalizing basis, but if your field is $\mathbb R$ then the number of positive (or negative) entries is well-defined. You can intrinsically define the index of a nondegenerate critical point as the maximal dimension of a tangent vector subspace on which the Hessian is negative definite.


