In this article, we prove a theorem comparing the dihedral angles of simplices in the hyperbolic, spherical and Euclidean geometries.

The notion of a generalized CRF-structure on a smooth manifold was recently introduced and studied by Vaisman (2008)[6]. An important class of generalized CRF-structures on an odd dimensional manifold M consists of CRF-structures having complementary frames of the form , where is a vector field and is a 1-form on M with ()= 1. It turns out that these kinds of CRF-structures give rise to a special class of what we called strong generalized contact structures in Poon and Wade [5]. More precisely, we show that to any CRF-structures with complementary frames of the form , there corresponds a canonical Lie bialgebroid. Finally, we explain the relationship between generalized contact structures and another generalization of the notion of a CauchyRiemann structure on a manifold.

We prove that a pair (X, D) with X Fano and D a smooth anti-canonical divisor is K-unstable for negative angles, and K-semistable for zero angle.

We study the rigidity of polyhedral surfaces using variational principles. The action functionals are derived from the cosine laws. The main focus of this paper is on the cosine law for a nontriangular region bounded by three possibly disjoint geodesics. Several of these cosine laws were first discovered and used by Fenchel and Nielsen. By studying the derivative of the cosine laws, we discover a uniform approach to several variational principles on polyhedral surfaces with or without boundary. As a consequence, the work of Penner, Bobenko and Springborn and Thurston on rigidity of polyhedral surfaces and circle patterns are extended to a very general context.

In this paper, we compute the adiabatic limit of the scalar curvature and prove several vanishing theorems by taking adiabatic limits. As an application, we give a Kastler-Kalau-Walze type theorem for foliations.