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

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.

We develop a unifed theory to study geometry of manifolds with different holonomy groups. They are classified by (1) real, complex, quaternion or octonion number (in the appropriate cases) and (2) being special or not. Specialty is an orientation with respect to the corresponding normed algebra A. For example, special Riemannian A-manifolds are oriented Riemannian, Calabi-Yau, hyperk¨ahler and G2-manifolds respectively. For vector bundles over such manifolds, we introduce (special) A-connections. They include holomorphic, Hermitian Yang-Mills, Anti-Self-Dual and Donaldson-Thomas connections. Similarly we introduce (special) 1/2 A-Lagrangian submanifolds as maximally real submanifolds. They include (special) Lagrangian, complex Lagrangian, Cayley and (co-)associative submanifolds. We also discuss geometric dualities from this viewpoint: Fourier transformations on A-geometry for flat tori and a conjectural SYZ mirror transformation from (special) A-geometry to (special) 1/2 A-Lagrangian geometry on mirror special A-manifolds.

On a Fano manifold, we prove that the KhlerRicci flow starting from a Khler metric in the anti-canonical class which is sufficiently close to a KhlerEinstein metric must converge in a polynomial rate to a KhlerEinstein metric. The convergence cannot happen in general if we study the flow on the level of Khler potentials. Instead we exploit the interpretation of the Ricci flow as the gradient flow of Perelman's functional. This involves modifying the Ricci flow by a canonical family of gauges. In particular, the complex structure of the limit could be different in general. The main technical ingredient is a Lojasiewicz type inequality for Perelman's functional near a critical point.

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.