Fumiko Ohtsuka

Total excess and Tits metric for piecewise Riemannian 2-manifolds

Abstract A piecewise Riemannian 2 -manifold is a combinatorial 2-manifold with a triangulation such that each 2-simplex is a geodesic triangle of some Riemannian 2-manifold. In this paper, we study the total excess e ( X ) of a simply connected nonpositively curved piecewise Riemannian 2-manifold X in connection with the Tits metric on the boundary at infinity X (∞).

On manifolds having some restricted ideal boundaries

The theory of the ideal boundary has arisen as a method of a study of noncompact manifolds. In this paper we shall investigate properties of manifolds whose ideal boundaries satisfy some conditions related to the Tits metric. In Section 2 we shall consider in what conditions the Tits topology is equivalent to the sphere topology and in Section 3 the property of points at infinity which characterize the R-factor. Furthermore, in Section 4 we shall give a sufficient condition for a projective map Φ to be isometríc.

Total excess on length surfaces

Abstract. We study the space of directions on a length space and examine examples having particular spaces of directions. Then we generalize the notion of total excess on length spaces satisfying some suitable conditions, which we call good surfaces. For good surfaces we generalize the Euler characteristic, and prove the generalized Gauss-Bonnet Theorem and other relations between the total excess and the Euler characteristic. Furthermore, we see that the Gaussian curvature can be defined almost everywhere on a good surface with non-positive total excess.