It is well-known that the αG(M)-invariant introduced by Tian plays an important role in the study of the existence of Kähler-Einstein metrics on complex manifolds with positive first Chern class ([T1], [T2], [TY]). Based on the estimate of αG(M)-invariant, Tian in 1990 proved that any complex surface with c1(M) > 0 always admits a Kähler-Einstein metric except in two cases CP2#1CP2 and CP2#2CP2...

The aim of this paper is to introduce and justify a possible generalization the classic Bach field equations on four-dimensional smooth manifold [Formula: see text] in presence text], given by map with source target another Riemannian manifold. Those are characterized vanishing two times covariant, symmetric, traceless conformally invariant tensor field, called text]-Bach tensor, that absence r...

There is a ( ) 4' 4 UU × -bundle on four-dimensional square root Lorentz manifold. Then Pati-Salam model in curved space-time (Lagrangian) and gravity theory are constructed manifold based self-parallel transportation principle. An explicit formulation of Sheaf quantization this shown. superposition principle construct linear space space-time. The transition amplitude path integral given which ...

We study a variational problem whose critical point determines the Reeb vector field for a Sasaki–Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein–Hilbert action, restricted to a space of Sasakian metrics on a link L in a Calabi–Yau cone X, is the volume functional, which in fact is a function...

We show that any contact form whose Fefferman metric admits a nonzero parallel vector field is pseudo-Einstein of constant pseudohermitian scalar curvature. As an application we compute the curvature groups H(C(M),Γ) of the Fefferman space C(M) of a strictly pseudoconvex real hypersurface M ⊂ C. 1. Statement of results Let M be a strictly pseudoconvex CR manifold of CR dimension n and θ a conta...

The Riemannian submersions of a CR-hypersurface M of a Kaehler-Einstein man-ifold˜M are studied. If M is an extrinsic CR-hypersurface of˜M, then it is shown that the base space of the submersion is also a Kaehler-Einstein manifold. 1. Introduction. The study of the Riemannian submersions π : M → B was initiated by O'Neill [14] and Gray [9]. This theory was very much developed in the last thirty...

It is shown that the Hermitian-symmetric space CP1 × CP1 × CP1 and the flag manifold F1,2 endowed with any left invariant metric admit no compatible integrable almost complex structures (even locally) different from the invariant ones. As an application it is proved that any stable harmonic immersion from F1,2 equipped with an invariant metric into an irreducible Hermitian symmetric space of co...

In this paper we prove the existence of Einstein metrics, actually SasakianEinstein metrics, on nontrivial rational homology spheres in all odd dimensions greater than 3. It appears as though little is known about the existence of Einstein metrics on rational homology spheres, and the known ones are typically homogeneous. The are two exception known to the authors. Both involve Sasakian geometr...

In this paper, we prove that Kähler-Ricci flow converges to a Kähler-Einstein metric (or a Kähler-Ricci soliton) in the sense of Cheeger-Gromov as long as an initial Kähler metric is very closed to gKE (or gKS) if a compact Kähler manifold with c1(M) > 0 admits a Kähler Einstein metric gKE (or a Kähler-Ricci soliton gKS). The result improves Main Theorem in [TZ3] in the sense of stability of Kä...