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.

In this paper, we extend the sharp lower bounds of spectal gap, due to Chen- Wang [12, 13], Bakry-Qian [6] and Andrews-Clutterbuck [5], from smooth Riemannian manifolds to general metric measure spaces with Riemannian curvature-dimension condition RCD(K;N).

In this paper, we will study the (linear) geometric analysis on metric measure spaces. We will establish a local Li-Yau’s estimate for weak solutions of the heat equation and prove a sharp Yau’s gradient gradient for harmonic functions on metric measure spaces, under the Riemannian curvature-dimension condition RCD(K; N).

If an (n + 2)-dimensional Lorentzian manifold is indecomposable, but non-irreducible, then its holonomy algebra is contained in the parabolic algebra (R⊕so(n)).Rn. We show that its projection onto so(n) is the holonomy algebra of a Riemannian manifold. This leads to a classification of Lorentzian holonomy groups and implies that the holonomy group of an indecomposable Lorentzian spin manifold with parallel spinor equals to G . Rn where G is a product of SU(p), Sp(q), G2 or Spin(7).

A piecewise flat manifold is a triangulated manifold given a geometry by specifying edge lengths (lengths of 1-simplices) and specifying that all simplices are Euclidean. We consider the variation of angles of piecewise flat manifolds as the geometry varies in a particular way, which we call a conformal variation. This variation generalizes variations within the class of circles with fixed intersection angles (such as circle packings) as well as other formulations of conformal variation of piecewise flat manifolds previously suggested. We describe the angle derivatives of the angles in two-and three-dimensional piecewise flat manifolds, giving rise to formulas for the derivatives of curvatures. The formulas for derivatives of curvature resemble the formulas for the change of scalar curvature under a conformal variation of Riemannian metric. They allow us to explicitly describe the variation of certain curvature functionals, including Regge’s formulation of the Einstein-Hilbert functional (total scalar curvature), and to consider convexity of these functionals. They also allow us to prove rigidity theorems for certain analogues of constant curvature and Einstein manifolds in the piecewise flat setting.