We prove theorems on the structure of the fundamental group of a compact riemannian manifold of non-positive curvature. In particular, a conjecture of J. Wolf [<i>J. Differential Geometry</i>, 2, 421-446 (1968)] is proved.
Unit tangent bundle of a surface carries various information of tangent vector fields on that surface. For 2-spheres (ie genus-zero closed surfaces), the unit tangent bundle is a closed 3-manifold that has non-trivial topology and cannot be embedded in R3. Therefore it cannot be constructed by existing mesh generation algorithms directly. This work aims at the first discrete construction of unit tangent bundles over 2-spheres using tetrahedral meshes. We propose a two-stage algorithm for the construction, which starts from constructing two local bundles and then combines them into a global bundle.
For compact Riemannian manifolds all of whose geodesics are closed (aka Zoll manifolds) one can define the determinant of a
zeroth order pseudodifferential operator by mimicking Szego’s definition of this determinant for the operator: multiplication by a
bounded function, on the Hilbert space of square-integrable functions on the circle. In this paper we prove that the non-local
contribution to this determinant can be computed in terms of a much simpler “zeta-regularized” determinant.