A Characterization for a (2, 0)-Geodesic Affine Immersion

We give a characterization for a (2, 0)-geodesic affine immersion to an affine space by using its index of relative nullity. Especially, we prove a cylinder theorem for such a hypersurface. We also show a cylinder theorem for a (2, 0)-geodesic isometric immersion from a Kähler manifold and anti-Kähler manifold as a corollary.     

Affine Kähler hypersurfaces satisfying certain conditions on the curvature tensor

The notion of affine Kähler immersions has been recently introduced by Nomizu-Pinkall-Podestà ([N-Pi-Po]). This work is aimed at giving some results towards the classification of non degenerate affine Kähler hypersurfaces with symmetric and parallel Ricci tensor; this problem generalizes the classical results due to Nomizu-Smyth ([N-S]) in the theory of Kählerian hypersurfaces. In a second section we deal with the case of “semisymmetric” affine Kähler immersions, when the curvature tensor R satisfies R · R = 0 and the Ricci tensor is symmetric, providing a complete classification; for affine Kähler curves we prove that the conditions above are actually equivalent to saying that the immersion is isometric for a suitable Kähler metric in C².     

Kähler–Einstein submanifolds of the infinite dimensional projective space

This paper consists of two main results. In the first one we describe all Kähler immersions of a bounded symmetric domain into the infinite dimensional complex projective space in terms of the Wallach set of the domain. In the second one we exhibit an example of complete and non-homogeneous Kähler–Einstein metric with negative scalar curvature which admits a Kähler immersion into the infinite dimensional complex projective space.     

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.     

Trajectories on real hypersurfaces of type (B) in a complex hyperbolic space are not of order 2

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. Contrary to real hypersurfaces of type (A), on real hypersurfaces of type (B) in a complex hyperbolic space we show that non-geodesic trajectories under Sasakian magnetic fields, which are constant multiples of the natural closed 2-form, are not curves of order 2.     

On the groups of c-projective transformations of complete Kähler manifolds

We show that for any complete connected Kähler manifold, the index of the group of complex affine transformations in the group of c-projective transformations is at most two unless the Kähler manifold is isometric to complex projective space equipped with a positive constant multiple of the Fubini–Study metric. This establishes a stronger version of the recently proved Yano–Obata conjecture for complete Kähler manifolds.     

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.     

Compact indefinite almost Kähler Einstein manifolds

In the present article, we extend the integral formula on a compact almost Kähler manifold with positive-definite metric to the one on a compact indefinite almost Kähler manifold and give its applications for some special indefinite almost Kähler Einstein manifolds taking the related problems to the indefinite analogy of Goldberg conjecture into consideration.     

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.     

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 \(S^2\).     

Complex points of minimal surfaces in almost Kähler manifolds

If (N, ω, J, g) is an almost Kähler manifold andM is a branched minimal immersion which is not aJ-holomorphic curve, we show that the complex tangents are isolated and that each has a negative index, which extends the results in the Kähler case by S. S. Chern and J. Wolfson [2] and S. Webster [7] to almost Kähler manifolds. As an application, we get lower estimates for the genus of embedded minimal surfaces in almost Kähler manifolds. The proofs of these results are based on the well-known Cartan’s moving frame methods as in [2, 7]. In our case, we must compute the torsion of the almost complex structures and find a useful representation of torsion. Finally, we prove that the minimal surfaces in complex projective plane with any almost complex structure is aJ-holomorphic curve if it is homologous to the complex line.     

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.     

An integral invariant from the view point of locally conformally Kähler geometry

In this article we study an integral invariant which obstructs the existence on a compact complex manifold of a volume form with the determinant of its Ricci form proportional to itself, in particular obstructs the existence of a Kähler-Einstein metric, and has been studied since 1980s. We study this invariant from the view point of locally conformally Kähler geometry. We first see that we can define an integral invariant for coverings of compact complex manifolds with automorphic volume forms. This situation typically occurs for locally conformally Kähler manifolds. Secondly, we see that this invariant coincides with the former one. We also show that the invariant vanishes for any compact Vaisman manifold.     

Sasakian structures on CR-manifolds

A contact manifold M can be defined as a quotient of a symplectic manifold X by a proper, free action of \(\mathbb{R}\), with the symplectic form homogeneous of degree 2. If X is also Kähler, and its metric is homogeneous of degree 2, M is called Sasakian. A Sasakian manifold is realized naturally as a level set of a Kähler potential on a complex manifold, hence it is equipped with a pseudoconvex CR-structure. We show that any Sasakian manifold M is CR-diffeomorphic to an S ¹-bundle of unit vectors in a positive line bundle on a projective Kähler orbifold. This induces an embedding of M into an algebraic cone C. We show that this embedding is uniquely defined by the CR-structure. Additionally, we classify the Sasakian metrics on an odd-dimensional sphere equipped with a standard CR-structure.     

K¨ahler Finsler Metrics Are Actually Strongly K¨ahler

In this paper, the Kahler conditions of the Chern-Finsler connection in complex Finsler geometry are studied, and it is proved that Kahler Finsler metrics are actually strongly Kahler.     

Simple root flows for Hitchin representations

On a Kähler manifold there is a clear connection between the complex geometry and underlying Riemannian geometry. In some ways, this can be used to characterize the Kähler condition. While such a link is not so obvious in the non-Kähler setting, one can seek to understand extensions of these characterizations to general Hermitian manifolds. This idea has been the subject of much study from the cohomological side, however, the focus here is to address such a question from the perspective of curvature relationships. In particular, on compact manifolds the Kähler condition is characterized by the relationship that the Chern scalar curvature is equal to half the Riemannian scalar curvature. What we study here is the existence, or lack thereof, of non-Kähler Hermitian metrics for which a more general proportionality relationship between these scalar curvatures holds.     

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 .     

Pseudo-holomorphic curves in nearly Kähler manifolds

We study pseudo-holomorphic curves in general nearly Kähler manifolds. For that purpose, we first introduce the fundamental equations of submanifold geometry in terms of the characteristic connection of the nearly Kähler structure. Then we classify pseudo-holomorphic curves with parallel second fundamental form in Chern-flat nearly Kähler manifolds. Moreover, we give a new Simons type identity. As an application of this identity, we show that the closed pseudo-holomorphic curves in Chern-flat nearly Kähler manifolds with a second fundamental form of controlled growth are totally geodesic.     

On quasi-Sasakian hypersurfaces of Kähler manifolds

We establish several conditions which are necessary for a quasi-Sasakian hypersurface of a Kähler manifold to be minimal.