Let (M, I, J,K) be a compact hypercomplex manifold admitting a special kind of quaternionic-Hermitian metric called an HKT-metric. Assume that the canonical bundle of (M, I) is trivial as a holomorphic line bundle. We show that the holonomy of canonical torsion-free connection called Obata connection onM is contained in SL(n,H). In Appendix we apply these arguments to compact nilmanifolds which...

In this paper, we introduce a new entropy-like invariant, named Hausdorff metric entropy, for finitely generated semigroups acting on compact metric spaces from a set-valued view and study its properties. We establish the relation between Hausdorff metric entropy and topological entropy of a semigroup defined by Bis. Some examples with positive or zero Hausdorff metric entropy are given. Moreov...

We show that the total space of any affine C-bundle over CP with negative degree admits an ALE scalar-flat Kähler metric. Here the degree of an affine bundle means the negative of the self-intersection number of the section at infinity in a natural compactification of the bundle, and so for line bundles it agrees with the usual notion of the degree.

Let $(M^{m}, g)$ be a Riemannian manifold and $TM$ its tangent bundle equipped with deformed Sasaki metric. In this paper, firstly we investigate all forms of curvature tensors (Riemannian tensor, Ricci curvature, sectional scalar curvature). Secondly, study the geometry unit metric, where presented formulas Levi-Civita connection also

The Dirac equation for massive free electrically neutral spin 1/2 particles in a gravitation field is considered. The secondary quantization procedure is applied to it and the Hilbert space of multiparticle quantum states is constructed. 1. The Dirac equation and its current. Let M be a space-time manifold. It is a four-dimensional orientable manifold equipped with a pseudo-EuclideanMinkowski-t...

Throughout this paper we work over the base field C. The basic problem in algebraic geometry treated here is the filling-in problem (or degeneration problem). Given a smooth projective family X× → ∆×, we would like to know when we can fill in a reasonably nice special fiber X0 to form a projective family X → ∆, perhaps up to a finite base change on ∆×. For example, when can X0 be smooth? When c...

The objective of this paper is to explore the complete lifts a quarter-symmetric metric connection from Sasakian manifold its tangent bundle. A relationship between Riemannian and bundle was established. Some theorems on curvature tensor projective with respect were proved. Finally, locally ?-symmetric manifolds studied.

We prove that the indices of fibered-cusp and d-Dirac operators on a spin manifold with fibered boundary coincide if the associated family of Dirac operators on the fibers of the boundary is invertible. This answers a question raised by Piazza. Under this invertibility assumption, our method yields an index formula for the Dirac operator of horn-cone and of fibered horn metrics. Let X be a comp...

The Fefferman construction associates to a manifold carrying a CR–structure a conformal structure on a sphere bundle over the manifold. There are some analogs to this construction, with one giving a Lie contact structure, a refinement of the contact bundle on the bundle of rays in the cotangent bundle of a manifold with a conformal metric. Since these structures are parabolic geometries, these ...

We prove that the indices of fibered-cusp and d-Dirac operators on a spin manifold with fibered boundary coincide if the associated family of Dirac operators on the fibers of the boundary is invertible. This answers a question raised by Piazza. Under this invertibility assumption, our method yields an index formula for the Dirac operator of horn-cone and of fibered horn metrics. Let X be a comp...