Let (M, I) be an almost complex 6-manifold. The obstruction to integrability of almost complex structure (socalled Nijenhuis tensor) N : Λ0,1(M)−→ Λ(M) maps a 3-dimensional bundle to a 3-dimensional one. We say that Nijenhuis tensor is non-degenerate if it is an isomorphism. An almost complex manifold (M, I) is called nearly Kähler if it admits a Hermitian form ω such that ∇(ω) is totally antis...

This review article has grown out of notes for the three lectures the second author presented during the XXIV-th Winter School of Geometry and Physics in Srni, Czech Republic, in January of 2004. Our purpose is twofold. We want give a brief introduction to some of the techniques we have developed over the last 5 years while, at the same time, we summarize all the known results. We do not give a...

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...

We find complete expanding Kähler-Ricci solitons of two types. The first are cohomogeneity one under the action (2m - 1)-dimensional Heisenberg group or a certain quotient it, for any m ? 2. second type, which generalizes first, reside on ?*-bundles over compact Kähler Ricci-flat manifolds, admit an S1 by isometries, and have ends. Both types local structure stemming from ansatz described in [2...

There are three separate approaches to the challenge of constructing WCW Kähler geometry and spinor structure. The first approach relies on a direct guess of Kähler function. Second approach relies on the construction of Kähler form and metric utilizing the huge symmetries of the geometry needed to guarantee the mathematical existence of Riemann connection. The third approach discussed in this ...

We study surfaces in TN that are area-stationary with respect to a neutral Kähler metric constructed on TN from a riemannian metric g on N. We show that holomorphic curves in TN are area-stationary, while lagrangian surfaces that are area-stationary are also holomorphic and hence totally null. However, in general, area stationary surfaces are not holomorphic. We prove this by constructing count...

In this article, we study Codazzi-couples of an arbitrary connection $\nabla$ with a nondegenerate 2-form $\omega$, isomorphism $L$ on the space derivation $\rho$-commutative algebra $A$, which important examples are almost complex and para-complex structures, metric $g$ that $(g, \omega,L)$ form compatible triple. We statistical structure algebras by classical manner Riemannian manifolds. Then...

We construct explicit left invariant quaternionic contact structures on Lie groups with zero and non-zero torsion, and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of quaternionic contact manifolds not locally quaternionic contact conformal to the quaternionic sphere. We present a left invariant quaternionic contact structure on a seven dimensio...

We construct a Kähler structure (J,Ω, G) on the space L(H) of oriented geodesics of hyperbolic 3-space H and investigate its properties. We prove that (L(H), J) is biholomorphic to P×P−∆, where ∆ is the reflected diagonal, and that the Kähler metric G is of neutral signature, conformally flat and scalar flat. We establish that the identity component of the isometry group of the metric G on L(H)...

I show that the effective action of string compactifications has a structure that can naturally solve the supersymmetric flavour and CP problems. At leading order in the gs and α ′ expansions, the hidden sector factorises. The moduli space splits into two mirror parts that depend on Kähler and complex structure moduli. Holomorphy implies the flavour structure of the Yukawa couplings arises in o...