  Principal Component Analysis can be performed over small domains of an
embedded Riemannian manifold in order to relate the covariance analysis of the
underlying point set with the local extrinsic and intrinsic curvature. We show
that the volume of domains on a submanifold of general codimension, determined
by the intersection with higher-dimensional cylinders and balls in the ambient
space, have asymptotic expansions in terms of the mean and scalar curvatures.
Moreover, we propose a generalization of the classical third fundamental form
to general submanifolds and prove that the eigenvalue decomposition of the
covariance matrices of the domains have asymptotic expansions with scale that
contain the curvature information encoded by the traces of this tensor. In the
case of hypersurfaces, this covariance analysis recovers the principal
curvatures and principal directions, which can be used as descriptors at scale
to build up estimators of the second fundamental form, and thus the Riemann
tensor, of general submanifolds.
