Recently, we researched certain twisted product CR-submanifolds in a Kaehler manifold and some inequalities of the second fundamental form of this submanifold ([11]). In this talk, we consider two kind twisted product CR-submanifolds in a locally conformal Kaehler manifold. In these submanifolds, we give a inequality of the second fundamental form (See Theorems 5.3 and 5.4) and consider the equ...

It is very hard to find computable examples of geodesics in the space of Kaehler metrics. It seems that they come in two types: (i) geodesics of toric Kaehler metrics, the topic of Lecture 2; (iii) geodesics coming from special test configurations and Hele-Shaw flows. Besides Donaldson’s construction (reviewed in §1) there are not many examples of geodesic rays. Almost the only other explicit e...

Warped product manifolds provide excellent setting to model space-time near black holes or bodies with large gravitational force (cf. [1], [2], [14]). Recently, results are published exploring the existence (or non-existence) of warped product submanifolds in Kaehlerian and contact settings (cf. [6], [17], [20]). To continue the sequel, we have considered warped product submanifolds of nearly K...

In this paper, we show that the twistor space J (R2n+2) on Euclidean space R2n+2 is a Kaehler manifold and the twistor space J (S2n) of the sphere S2n can be realized as a Kaehler submanifold in J (R2n+2). Then we show that an orthogonal almost complex structure Jf on S 2n is integrable if and only if the corresponding section f : S2n → J (S2n) is holomorphic. Hence there is no orthogonal integ...

In recent years, there are many progress made in Kähler geometry. In particular, the topics related to the problems of the existence and uniqueness of extremal Kähler metrics, as well as obstructions to the existence of such metrics in general Kähler manifold. In this talk, we will report some recent developments in this direction. In particular, we will discuss the progress recently obtained i...

I Berger, M., "Sur quelques varigtes riemanniennes compactes d'Einstein," Compt. Rend., 260, 1554-1557 (1965). 2 Bishop, R. L., and R. J. Crittenden, Geometry of Manifolds (New York: Academic Press, 1964). 3Bishop, R. L., and S. I. Goldberg, "On the topology of positively curved Kaehler manifolds," T6hoku Math. J., 15, 359-364 (1963). 4 Bishop, R. L., and S. I. Goldberg, "On the second cohomolo...

In this paper, we show that the twistor space J (R2n+2) on Euclidean space R2n+2 is a Kaehler manifold and the orthogonal twistor space J̃ (S2n) of the sphere S2n is a Kaehler submanifold of J (R2n+2). Then we show that an orthogonal almost complex structure Jf on S 2n is integrable if and only if the corresponding section f : S2n → J̃ (S2n) is holomorphic. These shows there is no integrable orth...

In this paper, we show that the twistor space J (R2n+2) on Euclidean space R2n+2 is a Kaehler manifold and the orthogonal twistor space J̃ (S2n) of the sphere S2n is a Kaehler submanifold of J (R2n+2). Then we show that an orthogonal almost complex structure Jf on S 2n is integrable if and only if the corresponding section f : S2n → J̃ (S2n) is holomorphic. These shows there is no integrable orth...

In this study, notion of pluriharmonic map applied onto conformal bi-slant Riemannian maps from a Kaehler manifold to examine its geometric properties. Such that, relations between map, horizontally homothetic and totally geodesic were obtained.