On the Geometry of Submersions

Subject of present paper is the geometry of foliation defined by submersions on complete Riemannian manifold. It is proven foliation defined by Riemannian submersion on the complete manifold of zero sectional curvature is total geodesic foliation with isometric leaves. Also it is shown level surfaces of metric function are conformally equivalent.

GCM Procedure

This chapter describes the general covariant modulation (GCM) procedure in detail. It considers an axially symmetric polarized spacetime region R foliated by two functions (u, s) such that: on R, (u, s) defines an outgoing geodesic foliation as in section 2.2.4. The chapter then outlines the elliptic Hodge lemma. It also looks at the deformations of S surfaces, frame transformations, and the existence of GCM spheres. It recalls the transformation formulas recorded in Proposition 2.90, before rewriting a subset of these transformations in a more useful form. In the proof of existence and uniqueness of GCMS, one needs, in addition to the equations derived so far, an equation for the average of α‎. Finally, the chapter discusses the construction of GCM hypersurfaces.

The weighted mixed curvature of a foliated manifold

We introduce the weighted mixed curvature of an almost product (e.g. foliated) Riemannian manifold equipped with a vector field. We define several qth Ricci type curvatures, which interpolate between the weighed sectional and Ricci curvatures. New concepts of the ?mixed-curvature-dimension condition? and ?synthetic dimension of a distribution? allow us to renew the estimate of the diameter of a compact Riemannian foliation and splitting results for almost product manifolds of nonnegative/nonpositive weighted mixed scalar curvature. We also study the Toponogov?s type conjecture on dimension of a totally geodesic foliation with positive weighted mixed sectional curvature.

MINIMAL GEODESIC FOLIATION ON T2 IN CASE OF VANISHING TOPOLOGICAL ENTROPY

On a Riemannian 2-torus (T2, g) we study the geodesic flow in the case of low complexity described by zero topological entropy. We show that this assumption implies a nearly integrable behavior. In our previous paper [12] we already obtained that the asymptotic direction and therefore also the rotation number exists for all geodesics. In this paper we show that for all r ∈ ℝ ∪ {∞} the universal cover ℝ2 is foliated by minimal geodesics of rotation number r. For irrational r ∈ ℝ all geodesics are minimal, for rational r ∈ ℝ ∪ {∞} all geodesics stay in strips between neighboring minimal axes. In such a strip the minimal geodesics are asymptotic to the neighboring minimal axes and generate two foliations.