  Integral invariants obtained from Principal Component Analysis on a small
kernel domain of a submanifold encode important geometric information
classically defined in differential-geometric terms. We generalize to
hypersurfaces in any dimension major results known for surfaces in space, which
in turn yield a method to estimate the extrinsic and intrinsic curvature of an
embedded Riemannian submanifold of general codimension. In particular, integral
invariants are defined by the volume, barycenter, and the EVD of the covariance
matrix of the domain. We obtain the asymptotic expansion of such invariants for
a spherical volume component delimited by a hypersurface and for the
hypersurface patch created by ball intersetions, showing that the eigenvalues
and eigenvectors can be used as multi-scale estimators of the principal
curvatures and principal directions. This approach may be interpreted as
performing statistical analysis on the underlying point-set of a submanifold in
order to obtain geometric descriptors at scale with potential applications to
Manifold Learning and Geometry Processing of point clouds.
