Covariance integral invariants of embedded Riemannian manifolds for manifold learning

2018 Summer.Includes bibliographical references.This thesis develops an effective theoretical foundation for the integral invariant approach to study submanifold geometry via the statistics of the underlying point-set, i.e., Manifold Learning from covariance analysis. We perform Principal Component Analysis over a domain determined by the intersection of an embedded Riemannian manifold with spheres or cylinders of varying scale in ambient space, in order to generalize to arbitrary dimension the relationship between curvature and the eigenvalue decomposition of covariance matrices. In the case of regular curves in general dimension, the covariance eigenvectors converge to the Frenet-Serret frame and the corresponding eigenvalues have ratios that asymptotically determine the generalized curvatures completely, up to a constant that we determine by proving a recursion relation for a certain sequence of Hankel determinants. For hypersurfaces, the eigenvalue decomposition has series expansion given in terms of the dimension and the principal curvatures, where the eigenvectors converge to the Darboux frame of principal and normal directions. In the most general case of embedded Riemannian manifolds, the eigenvalues and limit eigenvectors of the covariance matrices are found to have asymptotic behavior given in terms of the curvature information encoded by the third fundamental form of the manifold, a classical tensor that we generalize to arbitrary dimension, and which is related to the Weingarten map and Ricci operator. These results provide descriptors at scale for the principal curvatures and, in turn, for the second fundamental form and the Riemann curvature tensor of a submanifold, which can serve to perform multi-scale Geometry Processing and Manifold Learning, making use of the advantages of the integral invariant viewpoint when only a discrete sample of points is available

Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group

Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We observe that there exist Apollonian packings which have strong integrality properties, in which all circles in the packing have integer curvatures and rational centers such that (curvature)$\times$(center) is an integer vector. This series of papers explain such properties. A {\em Descartes configuration} is a set of four mutually tangent circles with disjoint interiors. We describe the space of all Descartes configurations using a coordinate system \sM_\DD consisting of those $4 \times 4$ real matrices \bW with \bW^T \bQ_{D} \bW = \bQ_{W} where \bQ_D is the matrix of the Descartes quadratic form $Q_D= x_1^2 + x_2^2+ x_3^2 + x_4^2 -{1/2}(x_1 +x_2 +x_3 + x_4)^2$ and \bQ_W of the quadratic form $Q_W = -8x_1x_2 + 2x_3^2 + 2x_4^2$. There are natural group actions on the parameter space \sM_\DD. We observe that the Descartes configurations in each Apollonian packing form an orbit under a certain finitely generated discrete group, the {\em Apollonian group}. This group consists of $4 \times 4$ integer matrices, and its integrality properties lead to the integrality properties observed in some Apollonian circle packings. We introduce two more related finitely generated groups, the dual Apollonian group and the super-Apollonian group, which have nice geometrically interpretations. We show these groups are hyperbolic Coxeter groups.Comment: 42 pages, 11 figures. Extensively revised version on June 14, 2004. Revised Appendix B and a few changes on July, 2004. Slight revision on March 10, 200

Isoparametric functions and exotic spheres

The first part of the paper is to improve the fundamental theory of isoparametric functions on general Riemannian manifolds. Next we focus our attention on exotic spheres, especially on "exotic" 4-spheres (if exist) and the Gromoll-Meyer sphere. In particular, as one of main results we prove: there exists no properly transnormal function on any exotic 4-sphere if it exists. Furthermore, by projecting an $S^3$-invariant isoparametric function on $Sp(2)$, we construct a properly transnormal but not an isoparametric function on the Gromoll-Meyer sphere with two points as the focal varieties.Comment: 21 pages, to appear in Journal f\"ur die reine und angewandte Mathematik (Crelles Journal