Local eigenvalue decomposition for embedded Riemannian manifolds

Local 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 local eigenvalue decomposition (EVD) of the covariance matrices have asymptotic expansions that contain the curvature information encoded by the traces of this tensor. This proves the general correspondence between the local EVD integral invariants and differential-geometric curvature for arbitrary embedded Riemannian submanifolds, found so far for curves and hypersurfaces only. Thus, we establish a key theoretical bridge, via covariance matrices at scale, for potential applications in manifold learning relating the statistics of point clouds sampled from Riemannian submanifolds to the underlying geometry.