We establish an identity for closed hyperbolic surfaces whose terms depend on the dilogarithms of the lengths of simple closed geodesics in all 3-holed spheres and 1-holed tori in the surface.

Systematically generalizing planar geometric algorithms to manifold domains is of fundamental importance in computer aided design field. This paper proposes a novel theoretic framework, <i>geometric structure</i>, to conquer this problem. In order to discover the intrinsic geometric structures of general surfaces, we developed a theoretic rigorous and practical efficient method, <i>Discrete Variational Ricci flow</i>.

We introduce a combinatorial curvature flow for piecewise constant curvature metrics on compact triangulated 3-manifolds with boundary consisting of surfaces of negative Euler characteristic. The flow tends to find the complete hyperbolic metric with totally geodesic boundary on a manifold. Some of the basic properties of the combinatorial flow are established. The most important one is that the evolution of the combinatorial curvature satisfies a combinatorial heat equation. It implies that the total curvature decreases along the flow. The local convergence of the flow to the hyperbolic metric is also established if the triangulation is isotopic to a totally geodesic triangulation.

Let M be the interior of a compact 3-manifold with boundary, and let M be an ideal triangulation of M This paper describes necessary and sufficient conditions for the existence of angle structures, semi-angle structures and generalised angle structures on M respectively in terms of a generalised Euler characteristic function on the solution space of the normal surface theory of M This extends previous work of Kang and Rubinstein, and is itself generalised to a more general setting for 3-dimensional pseudo-manifolds.

Suppose M is a m-dimensional submanifold without umbilic points in the (m+ p)-dimensional unit sphere S m+ p. Four basic invariants of M m under the Mbius transformation group of S m+ p are a symmetric positive definite 2-form g called the Mbius metric, a section B of the normal bundle called the Mbius second fundamental form, a 1-form called the Mbius form, and a symmetric (0, 2) tensor A called the Blaschke tensor. In the Mbius geometry of submanifolds, the most important examples of Mbius minimal submanifolds (also called Willmore submanifolds) are Willmore tori and Veronese submanifolds. In this paper, several fundamental inequalities of the Mbius geometry of submanifolds are established and the Mbius characterizations of Willmore tori and Veronese submanifolds are presented by using Mbius invariants.