In this paper, we produce an elementary approach to Thurston's theory of measured laminations on compact surfaces with non-empty boundary. We show that the theory can be derived from a simple inequality for geometric intersection numbers between arcs inside an octagon.

Fix a number $g>1$, let $S$ be a close surface of genus $g$ and $\Teich(S)$ be the Teichm\"uller space of $S$ endowed with the Weil-Petersson metric. In this paper we show that the Riemannian sectional curvature operator of $\Teich(S)$ is non-positive definite. As an application we show that any twist harmonic map from rank-one hyperbolic spaces $H_{Q,m}=Sp(m,1)/Sp(m)\cdot Sp(1)$ or $H_{O,2}=F_{4}^{-20}/SO(9)$ into $\Teich(S)$ is a constant.

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.

Let G={h_t|t∈ℝ} be a continuous flow on a connected n-manifold M. The flow G is said to be strongly reversible by an involution τ if h_{−t}=τh_tτ for all t∈ℝ, and it is said to be periodic if h_s= identity for some s∈ℝ^∗. A closed subset K of M is called a global section for G if every orbit G(x) intersects K in exactly one point. In this paper, we study how the two properties “strongly reversible” and “has a global section” are related. In particular, we show that if G is periodic and strongly reversible by a reflection, then G has a global section.

We propose an approach to find constant curvature metrics on triangulated closed 3-manifolds using a finite dimensional variational method whose energy function is the volume. The concept of an angle structure on a tetrahedron and on a triangulated closed 3-manifold is introduced following the work of Casson, Murakami and Rivin. It is proved by A. Kitaev and the author that any closed 3-manifold has a triangulation supporting an angle structure. The moduli space of all angle structures on a triangulated 3-manifold is a bounded open convex polytope in a Euclidean space. The volume of an angle structure is defined. Both the angle structure and the volume are natural generalizations of tetrahedra in the constant sectional curvature spaces and their volume. It is shown that the volume functional can be extended continuously to the compact closure of the moduli space. In particular, the maximum point of the