The theory of non-positively curved spaces and groups is tremendously powerful and has enormous consequences for the groups and spaces involved. Nevertheless, our ability to construct examples to which the theory can be applied has been severely limited by an inability to test – in real time – whether a random finite piecewise Euclidean complex is non-positively curved. In this article I focus ...

I. Introduction-a quick historical survey of geodesic flows on negatively curved spaces. II. Preliminaries on Riemannian manifolds A. Riemannian metric and Riemannian volume element B. Levi Civita connection and covariant differentiation along curves C. Parallel translation of vectors along curves D. Curvature E. Geodesics and geodesic flow F. Riemannian exponential map and Jacobi vector fields...

Non-negatively Curved Cohomogeneity One Manifolds Chenxu He Prof. Wolfgang Ziller, Advisor A Riemannian manifold M is called cohomogeneity one if it admits an isometric action by a compact Lie group G and the orbit space is one dimension. Many new examples of non-negatively curved manifolds were discovered recently in this category. However not every cohomogeneity one manifold carries an invari...

Gradient flows for energy functionals have been studied extensively in the past. Well known examples are the heat flow or the mean curvature flow. To make sense of the term gradient an inner product structure is assumed. One works on a Hilbert space, or on the tangent space to a manifold, for example. However, it is possible to do without an inner product. The domain of the energy functionals c...

In this paper, we show that an aspherical compact graph manifold is nonpositively curved if and only if its fundamental group virtually embeds into a right-angled Artin group. As a consequence, nonpositively curved graph manifolds have linear fundamental groups.

Let M denote a complete simply connected Riemannian manifold with all sectional curvatures ≥ 1. The purpose of this paper is to prove that when M has conjugate radius at least π/2, its injectivity radius and conjugate radius coincide. Metric characterizations of compact rank one symmetric spaces are given as applications.

By refining an idea of Farrell, we present a sufficient condition in terms the Jiang subgroup for vanishing signature and Hirzebruch's $\chi_y$-genus on compact smooth K\"{a}hler manifolds respectively. Along this line show that non-positively curved manifold vanishes when center its fundamental group is non-trivial, which partially answers question Farrell. Moreover, latter case all Chern numb...