Inthisarticle,weprovideasufficientandnecessaryconditiononparabolic isometries of positive translation lengths on complete visibility CAT(0) spaces. One of the consequences is that each parabolic isometry of a complete simply connected visibility manifold of nonpositive sectional curvature has zero translation length. Applications on the geometry of open negatively curved manifolds will also be discussed.

We derive large time upper bounds for heat kernels on vector bundles of differential forms on a class of non-compact Riemannian manifolds under certain curvature conditions.

In this paper, using connections between Clifford-Wolf isometries and Killing vector fields of constant length on a given Riemannian manifold, we classify simply connected Clifford-Wolf homogeneous Riemannian manifolds. We also get the classification of complete simply connected Riemannian manifolds with the Killing property defined and studied previously by J.E. D’Atri and H.K. Nickerson. In the last part of the paper we study properties of Clifford-Killing spaces, that is, real vector spaces of Killing vector fields of constant length, on odd-dimensional round spheres,and discuss numerous connections between these spaces and various classical objects.

Wedge product on deRham complex of a Riemannian manifold M can be pulled back to H^∗(M) via explicit homotopy, constructed using Green’s operator, to give higher product structures. We prove Fukaya’s conjecture which suggests that Witten deformation of these higher product structures have semiclassical limits as operators defined by counting gradient flow trees with respect to Morse functions, which generalizes the remarkable Witten deformation of deRham differential from a statement concerning homology to one concerning real homotopy type of M. Various applications of this conjecture to mirror symmetry are also suggested by Fukaya.

In this paper, we classify unweighted graphs satisfying the curvature dimension condition $\CD (0,\infty)$ whose girth are at least five.