The principal curvature of a 2D (m=2) manifold in a 3D (n=3) ambient Euclidean space, is given by the eigenvalues of the second fundamental form (or the Hessian matrix) $\Pi \in \Re^{m \times m}$ at each point of the surface. The principal directions are the corresponding eigenvectors.
I'm looking for the generalization of this for higher dimensions of both the manifold and its ambient space (i.e., both $m$ and $n$). According to wikipedia the eigenvectors of the second fundamental form of a hypersurface can give us the principal directions, and therefore the generalization is straight-forward.
However, this seems to hold only if $n = m+1$, because otherwise the hypersurface has a normal hyperplane rather than a normal vector, and the second fundamental form will be a 3D array $\Pi \in \Re^{m \times m \times (n-m)}$, rather than a matrix. In this case, what is the generalization of principal (directions of) curvatures, and how do we calculate them?