Minimal surfaces in a complex hyperquadric Q 2

In this paper, we discuss minimal surfaces in a complex hyperquadric Q ₂. It is proved that every minimal surface of constant Kähler angle in Q ₂ is holomorphic, anti-holomorphic, or totally real. We also prove that minimal two-spheres in Q ₂ with either constant curvature or parallel second fundamental form must be totally geodesic.     

Generalized hermann theorems and conformal holomorphic submersions

It is shown that if a Kähler manifold admits a holomorphic Riemann submersion, then this manifold is locally reducible. Hermann's well-known theorems are generalized to conformal and holomorphic submersions. A method for constructing Kähler fiber spaces with holomorphic conformal (non-Riemannian) projection and totally geodesic isomorphic fibers is suggested. The method allows us to construct complete, including compact, Kähler fiber spaces of the specified type.     

Minimal surfaces in symmetric spaces with parallel second fundamental form

In this paper, we study geometry of isometric minimal immersions of Riemannian surfaces in a symmetric space by moving frames and prove that the Gaussian curvature must be constant if the immersion is of parallel second fundamental form. In particular, when the surface is \(S^2\), we discuss the special case and obtain a necessary and sufficient condition such that its second fundamental form is parallel. We also consider isometric minimal two-spheres immersed in complex two-dimensional Kähler symmetric spaces with parallel second fundamental form, and prove that the immersion is totally geodesic with constant Kähler angle if it is neither holomorphic nor anti-holomorphic with Kähler angle \(\alpha \ne 0\) (resp. \(\alpha \ne \pi \)) everywhere on \(S^2\).     

Kähler tubes of constant radial holomorphic sectional curvature

We determine (up to holomorphic isometries) the family of Kähler tubes, around totally geodesic complex submanifolds, of constant radial holomorphic sectional curvature when the centreP of the tube is either simply connected or a complex hypersurface withH ¹ (P, R)=0. In the last case, these tubes have the topology of tubular neighbourhoods of the zero section of the complex lines bundles over symplectic manifolds (when they are Kähler) of the Kostant-Souriau prequantization.     

Transformations of locally conformally Kähler manifolds

We consider several transformation groups of a locally conformally Kähler manifold and discuss their inter-relations. Among other results, we prove that all conformal vector fields on a compact Vaisman manifold which is neither locally conformally hyperkähler nor a diagonal Hopf manifold are Killing, holomorphic and that all affine vector fields with respect to the minimal Weyl connection of a locally conformally Kähler manifold which is neither Weyl-reducible nor locally conformally hyperkähler are holomorphic and conformal.     

Kähler manifolds of quasi-constant holomorphic sectional curvatures

The Kähler manifolds of quasi-constant holomorphic sectional curvatures are introduced as Kähler manifolds with complex distribution of codimension two, whose holomorphic sectional curvature only depends on the corresponding point and the geometric angle, associated with the section. A curvature identity characterizing such manifolds is found. The biconformal group of transformations whose elements transform Kähler metrics into Kähler ones is introduced and biconformal tensor invariants are obtained. This makes it possible to classify the manifolds under consideration locally. The class of locally biconformal flat Kähler metrics is shown to be exactly the class of Kähler metrics whose potential function is only a function of the distance from the origin in ? ⁿ . Finally we show that any rotational even dimensional hypersurface carries locally a natural Kähler structure which is of quasi-constant holomorphic sectional curvatures.     

Surfaces with zero mean curvature vector in neutral 4-manifolds

Space-like surfaces and time-like surfaces with zero mean curvature vector in oriented neutral 4-manifolds are isotropic and compatible with the orientations of the spaces if and only if their lifts to the space-like and the time-like twistor spaces respectively are horizontal. In neutral Kähler surfaces and paraKähler surfaces, complex curves and paracomplex curves respectively are such surfaces and characterized by one additional condition. In neutral 4-dimensional space forms, the holomorphic quartic differentials defined on such surfaces vanish. There exist time-like surfaces with zero mean curvature vector and zero holomorphic quartic differential which are not compatible with the orientations of the spaces and the conformal Gauss maps of time-like surfaces of Willmore type and their analogues give such surfaces.     

Kähler manifolds with homothetic foliation by curves

The aim of this paper is to classify compact, simply connected Kähler manifolds which admit totally geodesic, holomorphic complex homothetic foliations by curves.     

Almost contact Kähler manifolds of constant holomorphic sectional curvature

We introduce the notion of an almost contact Kähler structure. We also define the holomorphic sectional curvature of the distribution of an almost contact Kähler structure with respect to an interior metric connection and establish relations between the φ-sectional curvature of an almost contact Kähler manifold and the holomorphic sectional curvature of the distribution of an almost contact Kähler structure.     

Rigidity of Riemannian foliations with complex leaves on kähler manifolds

We study Riemannian foliations with complex leaves on Kähler manifolds. The tensor T, the obstruction to the foliation be totally geodesic, is interpreted as a holomorphic section of a certain vector bundle. This enables us to give classification results when the manifold is compact.     

Trajectories for Sasakian magnetic fields on real hypersurfaces of type (B) in a complex hyperbolic space

On a real hypersurface in a Kähler manifold we can consider a natural closed 2-form associated with the almost contact metric structure induced by Kähler structure. We treat trajectories under magnetic fields which are constant multiples of this 2-form. We consider a condition for them to be also curves of order 2 on tubes around totally geodesic real hyperbolic spaces in a complex hyperbolic space.     

Totally geodesic immersions of Kähler manifolds and Kähler Frenet curves

In a given Kähler manifold (M,J) we introduce the notion of Kähler Frenet curves, which is closely related to the complex structure J of M. Using the notion of such curves, we characterize totally geodesic Kähler immersions of M into an ambient Kähler manifold and totally geodesic immersions of M into an ambient real space form of constant sectional curvature .     

Totally Real Minimal Surfaces with Non-Circular Ellipse of Curvature in the Nearly Kahler S6

In [2] we discussed almost complex curves in the nearly KählerS⁶. These are surfaces with constant Kähler angle 0 or and, as a consequence of this, are also minimal and have circularellipse of curvature. We also considered minimal immersionswith constant Kähler angle not equal to 0 or , but withellipse of curvature a circle. We showed that these are linearlyfull in a totally geodesic S⁵ in S⁶ and that (in the simplyconnected case) each belongs to the S¹-family of horizontallifts of a totally real (non-totally geodesic) minimal surfacein CP². Indeed, every element of such an S¹-family has constantKähler angle and in each family all constant Kählerangles occur. In particular, every minimal immersion with constantKähler angle and ellipse of curvature a circle is obtainedby rotating an almost complex curve which is linearly full ina totally geodesic S⁵.     

Schwarz lemma from a Kähler manifold into a complex Finsler manifold

Science China Mathematics - Suppose that M is a complete Kähler manifold such that its holomorphic sectional curvature is bounded from below by a constant and its radial sectional curvature is...     

Associated families of pluriharmonic maps and isotropy

Like minimal surface immersions in 3-space, pluriharmonic maps into symmetric spaces allow a one-parameter family of isometric deformations rotating the differential (“associated family”); in fact, pluriharmonic maps are characterized by this property. We give a geometric proof of this fact and investigate the “isotropic” case where this family is constant. It turns out that isotropic pluriharmonic maps arise from certain holomorphic maps into flag manifolds. Further, we also consider higher dimensional generalizations of constant mean curvature surfaces which are Kähler submanifolds with parallel (1,1) part of their soecond fundamental form; under certain restrictions there are also characterized by having some kind of (“weak”) associated family. Examples where this family is constant arise from extrinsic Kähler symmetric spaces.     

On the Space of Kähler Potentials

We consider the geodesic equation for the generalized Kähler potential with only mixed second derivatives bounded. We show that given two such generalized Kähler potentials, there is a unique geodesic segment such that for each point on the geodesic, the generalized Kähler potential has uniformly bounded mixed second derivatives (in manifold directions). This generalizes a fundamental theorem of Chen (2000) on the space of Kähler potentials.© 2014 Wiley Periodicals, Inc.     

Pluriclosed Manifolds with Constant Holomorphic Sectional Curvature

Acta Mathematica Sinica, English Series - A long-standing conjecture in complex geometry says that a compact Hermitian manifold with constant holomorphic sectional curvature must be Kähler...     

On Kähler manifolds with harmonic Bochner curvature tensor

We prove that every irreducible Kähler manifold with harmonic Bochner curvature tensor and constant scalar curvature is Kähler–Einstein and that every irreducible compact Kähler manifold with harmonic Bochner curvature tensor and negative semi-definite Ricci tensor is Kähler–Einstein.     

The Comparison Theorem for Bergman and Kobayashi Metrics on Cartan-Hartogs Domain of the Third Type

In this paper, we give the holomorphic sectional curvature under invariant Kähler metric on a Cartan-Hartogs domain of the third type Y _III (N,q,K) and construct an invariant Kähler metric, which is complete and not less than the Bergman metric, such that its holomorphic sectional curvature is bounded above by a negative constant. Hence we obtain a comparison theorem for the Bergman and Kobayashi metrics on Y _III (N,q,K).