关于第二典型联络的Laplace算子

讨论近Hermite流形上第二典型联络的Laplace算子,得到它与几何上通常Laplace算子之间相差一个挠向量场.特别地,得到semi-Khler流形上第二典型联络的Laplace算子与通常Laplace算子是相同的.这推广了WEINKOVE等人在quasi-Khler流形上的这两类算子相等的结果.

A note on R_2-and cR_2-manifolds

The necessary and sufficient conditions for an arbitrary almost Hermitian manifold to be an R 2 or a cR 2 manifold are established. Some examples of non Khlerian and non nearly Khlerian R 2 and cR 2 manifolds are given.

Embeddings of Almost Hermitian Manifold in Almost Hyper Hermitian Manifold and Complex (Hypercomplex) Numbers in Riemannian Geometry

Tubular neighborhoods play an important role in differential topology. We have applied these constructions to geometry of almost Hermitian manifolds. At first, we consider deformations of tensor structures on a normal tubular neighborhood of a submanifold in a Riemannian manifold. Further, an almost hyper Hermitian structure has been constructed on the tangent bundle TM with help of the Riemannian connection of an almost Hermitian structure on a manifold M then, we consider an embedding of the almost Hermitian manifold M in the corresponding normal tubular neighborhood of the null section in the tangent bundle TM equipped with the deformed almost hyper Hermitian structure of the special form. As a result, we have obtained that any Riemannian manifold M of dimension n can be embedded as a totally geodesic submanifold in a Kaehlerian manifold of dimension 2n (Theorem 6) and in a hyper Kaehlerian manifold of dimension 4n (Theorem 7). Such embeddings are “good” from the point of view of Riemannian geometry. They allow solving problems of Riemannian geometry by methods of Kaehlerian geometry (see Section 5 as an example). We can find similar situation in mathematical analysis (real and complex).

Two theorems on cosymplectic hypersurfaces of six-dimensional Hermitian submanifolds of Cayley algebra

Proves that cosymplectic hypersurfaces of six dimensional Hermitian submanifolds of the octave algebra are ruled manifolds and establishes a necessary and sufficient condition for a cosymplectic hypersurface of Hermitian M 6 O  to be a minimal submanifold of M 6.

Note on Almost Hermitian Manifolds

We give a necessary and sufficient condition for an almost Hermitian manifold to be a Kahler manifold. By making use of this condition, we give a new proof of Goldberg＇s theorem.

The U-Kenmotsu Hypersurfaces Axiom and Six-Dimensional Hermitian Submanifolds of Cayley Algebra

Six-dimensional Hermitian submanifolds of Cayley algebra are considered.It is proved that if such a submanifold of the octave algebta complies with the U-Kenmotsu hypersurfaces axiom,then it is Khlerian.

On Kirichenko Tensors of Nearly-Khlerian Manifolds

A short description of structural and virtual Kirichenko tensors that form a complete system of first-order differential-geometrical invariants of an arbitrary almost Hermitian structure is given.A characterization of nearly-Khlerian structures in terms of Kirichenko tensors is also given.

Some Connections in Almost Hermitian Manifold

The idea of this research is to study different types of connections in an almost Hermite manifold. The connection has been established between linear connection and Riemannian connection. Three new linear connections ∇1, ∇2, ∇3 are introduced. The necessary and sufficient condition for ∇1, ∇2, ∇3 to be metric is discussed. A new metric s* (X,Y) has been defined for (Mn,F,g*) and additional properties are discussed. It is also proved that for the quarter symmetric connection ∇ is unique in given manifold. The hessian operator with respect to all connections defined above has also been discussed.

Dirichlet Problem for Hermitian-Einstein Equation over Almost Hermitian Manifold

In this paper, we investigate the Dirichlet problem for Hermitian-Einstein equation on complex vector bundle over almost Hermitian manifold, and we obtain the unique solution of the Dirichlet problem for Hermitian-Einstein equation.

Nonpositively Curved Almost Hermitian Metrics on Product of Compact Almost Complex Manifolds

In this paper, we give a classification of almost Hermitian metrics with nonpositive holo- morphic bisectional curvature on a product of compact almost complex manifolds. This generalizes previous results of Zheng [Ann. of Math. （2）, 137（3）, 671-673 （1993）] and the author [Proc. Amer. Math. Soc., 139（4）, 1469-1472 （2011）].

Hessian Comparison and Spectrum Lower Bound of Almost Hermitian Manifolds

The authors obtain a complex Hessian comparison for almost Hermitian manifolds, which generalizes the Laplacian comparison for almost Hermitian manifolds by Tossati, and a sharp spectrum lower bound for compact quasi Kahler manifolds and a sharp complex Hessian comparison on nearly Kahler manifolds that generalize previous results of Aubin, Li Wang and Tam-Yu.