On a Metric Affine Manifold with Several Orthogonal Complementary Distributions

A Riemannian manifold endowed with k>2 orthogonal complementary distributions (called here an almost multi-product structure) appears in such topics as multiply twisted or warped products and the webs or nets composed of orthogonal foliations. In this article, we define the mixed scalar curvature of an almost multi-product structure endowed with a linear connection, and represent this kind of curvature using fundamental tensors of distributions and the divergence of a geometrically interesting vector field. Using this formula, we prove decomposition and non-existence theorems and integral formulas that generalize results (for k=2) on almost product manifolds with the Levi-Civita connection. Some of our results are illustrated by examples with statistical and semi-symmetric connections.

ROBUST EYE TRACKING IN VIDEO SEQUENCE

In this paper, we present a novel method for eye tracking, in detail describing the eye contour and the visible iris center. Combining the IVT (Incremental Visual Tracking) tracker, the proposed online affine manifold model, in which the sequentially learning shape and texture are modeled in the first stage and noniterative recovering estimation in the second stage, tracks the eye contour in video sequences. After that, an adaptive black round mask is generated to match the visible iris center. Experimental results of eye tracking indicate that our tracker works well in the PC or domestic camera captured image streams with considerable head and eyeball rotation.

Affine Anosov Diffeomorphims of Affine Manifolds

We show that a compact affine manifold endowed with an affine Anosov transformation is finitely covered by a complete affine nilmanifold. This is a partial answer of a conjecture of Franks for affine manifolds.

THE UNIVERSAL COVER OF AN AFFINE THREE-MANIFOLD WITH HOLONOMY OF SHRINKABLE DIMENSION ≤ TWO

An affine manifold is a manifold with an affine structure, i.e. a torsion-free flat affine connection. We show that the universal cover of a closed affine 3-manifold M with holonomy group of shrinkable dimension (or discompacité in French) less than or equal to two is diffeomorphic to R3. Hence, M is irreducible. This follows from two results: (i) a simply connected affine 3-manifold which is 2-convex is diffeomorphic to R3, whose proof using the Morse theory takes most of this paper; and (ii) a closed affine manifold of holonomy of shrinkable dimension less or equal to d is d-convex. To prove (i), we show that 2-convexity is a geometric form of topological incompressibility of level sets. As a consequence, we show that the universal cover of a closed affine three-manifold with parallel volume form is diffeomorphic to R3, a part of the weak Markus conjecture. As applications, we show that the universal cover of a hyperbolic 3-manifold with cone-type singularity of arbitrarily assigned cone-angles along a link removed with the singular locus is diffeomorphic to R3. A fake cell has an affine structure as shown by Gromov. Such a cell must have a concave point at the boundary.

Conditions for an affine manifold with torsion to have a Riemann–Cartan structure

Necessary and sufficient conditions for the local existence of a metric compatible with the affine connection are obtained in terms of the Riemann tensor and its first-order covariant derivatives in a generic affine manifold with torsion. In case these conditions are satisfied, the solutions of the metric are given in terms of integrals and are unique up to a constant scale factor. Some global conditions are also obtained and discussed.