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Projections, Hessian, Invariant: finding kappa1, kappa2

Lots of the time, we want to know the principal curvature magnitudes kappa1 and kappa2 When expressed in the coordinate frame of the principal curvature directions, G is just all zeros except for kappa1 and kappa2 along the diagonal.

But we don't know the principal curvature directions, so we can't just pluck out those two elements and be done with it. So we use two matrix invariants, the trace and the frobenious norm, which are the same regardless of the basis in which G is expressed. The trace is the sum of diagional elements, the frobenius norm is the l2 norm of the matrix considered as a big 9-vector.

So these two invariants will show up in our ...