# principal curvature in high dimensions

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?

• Be careful when you say that the second fundamental form is the Hessian matrix. When you have the graph $z=f(x,y)$, it's still not even correct. You're correct that in higher codimension you get a second fundamental form for each normal (unit) direction. Each of those will have principal directions, but there is no universal principal direction for the entire second fundamental form tensor. – Ted Shifrin Aug 21 '18 at 18:11
• I understand. Since the choice of these normal directions is arbitrary, does this means the notion of principle curvature does not really generalize to cases where $n > m+1$? – self-educator Aug 21 '18 at 18:24
• Yes. That's correct. You can choose an orthonormal basis for the normal space and give $n-m$ different second fundamental form matrices, but there's still no way to diagonalize them simultaneously, in general. – Ted Shifrin Aug 21 '18 at 18:25
• No, not really. The Ricci tensor assigns to unit tangent vector $v$ the average sectional curvature of $2$-planes containing $v$. Each of those, in turn, is the average (over normal directions) of the determinants of the second fundamental form for the $2$-dimensional surface coming from those two tangent vectors. So, Ric does a double average. As far as I can tell, you're not going to see anything about a specific second fundamental form. – Ted Shifrin Aug 25 '18 at 16:33