An open manifold M with nonnegative sectional curvature contains a compact totally geodesic submanifold S called the soul. In his solution of the Cheeger-Gromoll conjecture, G. Perelman showed that the metric projection π : M → S was a C Riemannian submersion which coincided with a map previously constructed by V. Sharafutdinov. In this paper we improve Perelman’s result to show that π is in fa...

In a paper of Menasco and Reid, it is conjectured that there exist no hyperbolic knots in S3 for which the complement contains a closed embedded totally geodesic surface. In this note, we show that one can get ”as close as possible” to a counter-example. Specifically, we construct a sequence of hyperbolic knots {Kn} with complements containing closed embedded essential surfaces having principal...

Let F : Σ n × [0, T) → R n+m be a family of compact immersed sub-manifolds moving by their mean curvature vector. We show the Gauss maps γ : (Σ n , g t) → G(n, m) form a harmonic heat flow with respect to the time-dependent induced metric g t. This provides a more systematic approach to investigate higher codimension mean curvature flows. A direct consequence is any convex function on G(n, m) p...

It is proved that on a real hypersurface $M$ in the complex two-plane Grassmannian $G_2(\mathbb {C}^{m+2})$, structure Lie operator parallel if and only an open part of tube around totally geodesic {C}^{m+1})$ $G_2(\ma

Abstract We provide a meromorphic continuation for Poincaré series counting orthogeodesics of negatively curved surface with totally geodesic boundary, as well arcs linking two points. For the latter series, we show that value at zero coincides inverse Euler characteristic surface.

Local primitives and their spatial relationships are useful in the analysis, recognition and retrieval of document and patent binary images. In this paper, a morphology based approach is proposed to establish the connections between the local primitives found at the optimally detected junction points and end points. The grayscale geodesic dilation is employed as the basic technique by taking a ...

A toric integrable geodesic flow on a manifold M is a completely integrable geodesic flow such that the integrals generate a homogeneous torus action on the punctured cotangent bundle T ∗M \ M . A toric integrable manifold is a manifold which has a toric integrable geodesic flow. Toric integrable manifolds can be characterized as those whose cosphere bundle (the sphere bundle in the cotangent b...

The aim of the present paper is to study holomorphically planar conformal vector fields(HPCV) on almost alpha-cosymplectic (k,m)-spaces. This done assuming various conditions such as i) U pointwise collinear with xi ( in this case integral manifold distribution D totally geodesic or umbilic), ii) M has a constant xi-sectional curvature (under condition (or umbilic) isometric sphere S2n+1(pc) ra...