Normal curvature and Principal curvatures

Question

Principal curvatures are the maximum and minimum values of the normal curvature but normal curvature is a measure of the curvature right? How can it have a maximum and a minimum value? Shouldn’t it be just one value? How to intuitively understand this? And how to understand the geodesic curvature and how are these related?

Answer 1

You arre right in principle that curvature is attached to a point. However, both in classcial differential geometry and in Riemannian geometry, a "curvature" is not necessarily a number. One often reduces to numbers in the end (since these are easy to compare), but intially, one has more general objects. In the setting of surfaces, I would say that the basic curvature object is the Weingarten map, which, in a point, is a symmetric endomorphism of the tangent plane at that point. The principal curvatures (which are indeed numbers) are the eigenvalues of this linear map. Using the inner product, the Weingarten map can be converted to a symmetric bilinear form (the second fundamental form) on the tangent space. The usual interpretation of the normal cuvature is as the restriction of the quadratic form defined by this symmetric bilinear form to the unit sphere in the tangent space. As such, it is a smooth function on the unit circle, which (if non-constant) has a unique minumum and a unique maximum. These are easily seen to coincide with the two eigenvalues of the Weingarten map and thus with the principal cuvatures.

The Riemann cuvature in general is a much more complicated object (a trilinear map from three copies of the tangent space to the tangent space with certain symmetries). However, in two dimensions, the space of such maps is one-dimensional, so again you can reduce it to one number. In an appropriate identification that number coincides with the determinant of the Weingarten map, i.e. with the Gauss curvature. Things look very different in higher dimensions.