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 volume functional always exists in the compactification. The main result shows that for a 1-vertex triangulation of a closed 3-manifold if the volume function on the moduli space has a local maximum point, then either the manifold admits a constant curvature Riemannian metric or the manifold contains a non-separating 2-sphere or real projective plane.

We prove that any closed orientable surface may be smoothly embedded in Euclidean 3-space so that when it is illuminated by parallel rays from any direction the shade cast on the surface is connected.

In this paper we give a global version of the Bryant representation of surfaces of constant mean curvature one (cmc-1 surfaces) in hyperbolic space. This allows to set the associated non-abelian period problem in the framework of flat unitary vector bundles on Riemann surfaces. We use this machinery to prove the existence of certain cmc-1 surfaces having prescribed global monodromy.

We analyze the geometry of sub-Finsler Engel manifolds, computing a complete set of local invariants for a large class of these manifolds. We derive geodesic equations for regular geodesics and show that in the symmetric case, the rigid curves are local minimizers. We end by illustrating our results with an example.