The Berezin quantization on a simply connected homogeneous Kähler manifold, which is considered as a phase space for a dynam-ical system, enables a description of the quantal system in a (finite-dimensional) Hilbert space of holomorphic functions corresponding to generalized coherent states. The Lie algebra associated with the manifold symmetry group is given in terms of first-order differentia...

In 1974, Berezin proposed a quantum theory for dynamical systems having a Kähler manifold as their phase space. The system states were represented by holomorphic functions on the manifold. For any homogeneous Kähler manifold, the Lie algebra of its group of motions may be represented either by holomorphic differential operators (“quantum theory”), or by functions on the manifold with Poisson br...

Suppose M a compact manifold which admits an Einstein metric g which is Kähler with respect to some complex structure J . Is every other Einstein metric h on M also Kähler-Einstein? If the complex dimension of (M,J) is ≥ 3, the answer is generally no; for example, CP3 admits both the FubiniStudy metric, which is Kähler-Einstein, and a non-Kähler Einstein metric [2] obtained by appropriately squ...

This is largely an expository paper and dedicated to my friend J. Cheeger for his 65th birthday. The purpose of this paper is to discuss some of my works on the existence of Kähler-Einstein metrics on Fano manifolds and some related topics. I will describe a program I have been following for the last twenty years. It includes some of my results and speculations which were scattered in my previo...

We prove that a minimal isometric immersion of a Kähler-Einstein or homogeneous Kähler manifold into an Euclidean space must be totally geodesic. As an application we show that an open subset of the real hyperbolic plane RH2 cannot be minimally immersed into the Euclidean space. As another application we prove that if an irreducible Kähler manifold is minimally immersed in an Euclidean space th...

We classify flat strict nearly Kähler manifolds with (necessarily) indefinite metric. Any such manifold is locally the product of a flat pseudo-Kähler factor of maximal dimension and a strict flat nearly Kähler manifold of split signature (2m, 2m) with m ≥ 3. Moreover, the geometry of the second factor is encoded in a complex three-form ζ ∈ Λ3(Cm)∗. The first nontrivial example occurs in dimens...

Okumura gave a necessary and sufficient condition for an oriented real hypersurface of a Kähler manifold to be a contact metric manifold with respect to the naturally induced almost contact metric structure. In this paper, we discuss an oriented hypersurface of a quasi Kähler manifold and give a necessary and sufficient condition for such a hypersurface to be a quasi contact metric manifold wit...

In this paper, we introduce and examine invariant semi-invariant lightlike submanifolds of a poly-Norden semi-Riemannian manifold. Also, obtain some examples such types study the conditions for both integrability totally geodesic foliation description distributions.