On holomorphic isometric immersions of nonhomogeneous Kähler–Einstein manifolds into the infinite dimensional complex projective space

In the paper, we study the existence of holomorphic isometric immersions from nonhomogeneous Kähler–Einstein manifolds into infinite dimensional complex projective space. It can also be regarded as an application of explicit solutions of complex Monge–Ampère equations on some pseudoconvex domains.     

Holomorphic maps into Grassmann manifolds (harmonic maps into Grassmann manifolds III)

Annals of Global Analysis and Geometry - A well-known Calabi’s rigidity theorem on holomorphic isometric immersions into the complex projective space is generalized to the case that the...     

Complex analytic curves and maximal surfaces

Maximal immersions of a surfaceM ² inton-dimensional Lorentz space which are isometric to a fixed holomorphic mapping ofM ² into complex Lorentz space are determined. The main tool is an adaption of Calabi's Rigidity Theorem. Such an adaption is necessary because of the existence of degenerate hyperplanes in complex Lorentz space.Partially supported by a grant from Wellesley College.     

不定度量空间型到不定度量空间型的等距浸入和孤立子理论

我们利用孤立子理论得到了构造有相同指标的不定度量空间型到不定度量空间型的等距浸入的方法．     

On the fundamental theorem for non-degenerate complex affine hypersurface immersions

Two geometric versions of the fundamental theorem for non-degenerate complex affine hypersurface immersions are proved. We consider non-degenerate complex affine hypersurface immersions with complex transversal connection form (or equivalently, with holomorphic normalization) and prove that the conormal map is a holomorphic map. These considerations inspired the definitions of complex semi-compatible and complex semi-conjugate connections. This allows us to formulate the integrability conditions of the fundamental theorem, on one hand in terms of the induced connection, which has to be complex semi-compatible and dualH-projective flat, and on the other hand, in terms of its semi-conjugate connection, which has to beH-projective flat. Using this results, we formulate the conditions of the fundamental theorem in terms of anyH-projective flat complex affine connection.Research partially supported by Contract MM 413/1994 with the Ministry of Science and Education of Bulgaria and by Contract 219/1994 with the University of Sofia St. Kl. Ohridcki.     

On real local isometric immersions of {\mathbb{C}Q^2_c} into {\mathbb{C} P^{3}}

We investigate real local isometric immersions of Kähler manifolds ${\mathbb{C}Q^2_c}$ of constant holomorphic curvature 4c into complex projective 3-space. Our main result is that the standard embedding of ${\mathbb{C}P^2}$ into ${\mathbb{C}P^3}$ has strong rigidity under the class of local isometric transformations. We also prove that there are no local isometric immersions of ${\mathbb{C}Q^2_c}$ into ${\mathbb{C}P^3}$ when they have different holomorphic curvature. An important method used is a study of the relationship between the complex structure of any locally isometric immersed ${\mathbb{C}Q^2_c}$ and the complex structure of the ambient space ${\mathbb{C}P^3}$ .     

Some explicit examples of Hamiltonian minimal Lagrangian submanifolds in complex space forms

In this paper, we find some new explicit examples of Hamiltonian minimal Lagrangian submanifolds among the Lagrangian isometric immersions of a real space form in a complex space form.     

Orbits of SU(2)-representations and minimal isometric immersions

In contrast to all known examples, we show that in the case of minimal isometric immersions of into the smallest target dimension is almost never achieved by an -equivariant immersion. We also give new criteria for linear rigidity of a fixed minimal isometric immersion of into . The minimal isometric immersions arising from irreducible SU(2)-representations are linearly rigid within the moduli space of SU(2)-equivariant immersions. Hence the question arose whether they are still linearly rigid within the full moduli space. We show that this is false by using our new criteria to construct an explicit SU(2)-equivariant immersion which is not linearly rigid. Various authors [GT], [To3], [W1] have shown that minimal isometric immersions of higher isotropy order play an important role in the study of the moduli space of all minimal isometric immersions of into . Using a new necessary and sufficient condition for immersions of isotropy order , we derive a general existence theorem of such immersions. Received: 13 May 1999 / in final form: 13 July 1999     

Codimension two Kähler submanifolds of space forms

In this article we study isometric immersions from Kähler manifolds whose (1, 1) part of the second fundamental form is parallel, theppmc isometric immersions. When the domain is a Riemann surface these immersions are precisely those with parallel mean curvature. P. J. Ryan has classified the Kähler manifolds that admit isometric immersions, as real hypersurfaces, in space forms. We classify the codimension twoppmc isometric immersions into space forms.     

Characterizations of Null Holomorphic Sectional Curvature of G C R-Lightlike Submanifolds of Indefinite Nearly K ¨ahler Manifolds

We obtain the expressions for sectional curvature, holomorphic sectional curvature and holomorphic bisectional curvature of a GCR-lightlike submanifold of an indefinite nearly K ¨ahler manifold and obtain characterization theorems for holo-morphic sectional and holomorphic bisectional curvature. We also establish a condi-tion for a GCR-lightlike submanifold of an indefinite complex space form to be a null holomorphically flat.     

Isotropic ppmc immersions

Isometric immersions with parallel pluri-mean curvature (“ppmc”) in euclidean n-space generalize constant mean curvature (“cmc”) surfaces to higher dimensional Kähler submanifolds. Like cmc surfaces they allow a one-parameter family of isometric deformations rotating the second fundamental form at each point. If these deformations are trivial the ppmc immersions are called isotropic. Our main result drastically restricts the intrinsic geometry of such a submanifold: Locally, it must be a symmetric space or a Riemannian product unless the immersion is holomorphic or a superminimal surface in a sphere. We can give a precise classification if the codimension is less than 7. The main idea of the proof is to show that the tangent holonomy is restricted and to apply the Berger-Simons holonomy theorem.     

Complex affine distributions

Geometry of affine immersions is the study of hypersurfaces that are invariant under affine transformations. As with the hypersurface theory on the Euclidean space, an affine immersion can induce a torsion-free affine connection and a (pseudo)-Riemannian metric on the hypersurface. Moreover, an affine immersion can induce a statistical manifold, which plays a central role in information geometry. Recently, a statistical manifold with a complex structure is actively studied since it connects information geometry and Kähler geometry. However, a holomorphic complex affine immersion cannot induce such a statistical manifold with a Kähler structure. In this paper, we introduce complex affine distributions, which are non-integrable generalizations of complex affine immersions. We then present the fundamental theorem for a complex affine distribution, and show that a complex affine distribution can induce a statistical manifold with a Kähler structure.     

Twistor holomorphic Lagrangian surfaces in the complex projective and hyperbolic planes

We completely classify all the twistor holomorphic Lagrangian immersions in the complex projective plane ², i.e. those Lagrangian immersions such that their twistor lifts to the twistor space over ² are holomorphic. This classification provides a one-parameter family of examples of Lagrangian spheres in ².Research partially supported by a DGICYT grant No. PB91-0731.     

On holomorphic immersions into K(a)hler manifolds of constant holomorphic sectional curvature

We study holomorphic immersions f: X → M from a complex manifold X into a Kahler manifold of constant holomorphic sectional curvature M, i.e. a complex hyperbolic space form, a complex Euclidean space form, or the complex projective space equipped with the Fubini-Study metric. For X compact we show that the tangent sequence splits holomorphically if and only if f is a totally geodesic immersion. For X not necessarily compact we relate an intrinsic cohomological invariant p(X) on X, viz. the invariant defined by Gunning measuring the obstruction to the existence of holomorphic projective connections, to an extrinsic cohomological invariant v(f)measuring the obstruction to the holomorphic splitting of the tangent sequence. The two invariants p(X) and v(f) are related by a linear map on cohomology groups induced by the second fundamental form.In some cases, especially when X is a complex surface and M is of complex dimension 4, under the assumption that X admits a holomorphic projective connection we obtain a sufficient condition for the holomorphic splitting of the tangent sequence in terms of the second fundamental form.     

On an invariant on isometric immersions into spaces of constant sectional curvature

In the present paper, we give an invariant on isometric immersions into spaces of constant sectional curvature. This invariant is a direct consequence of the Gauss equation and the Codazzi equation of isometric immersions. We apply this invariant on some examples. Further, we apply it to codimension 1 local isometric immersions of 2-step nilpotent Lie groups with arbitrary leftinvariant Riemannian metric into spaces of constant nonpositive sectional curvature. We also consider the more general class, namely, three-dimensional Lie groups G with nontrivial center and with arbitrary left-invariant metric. We show that if the metric of G is not symmetric, then there are no local isometric immersions of G into Q _c ⁴.     

Axioms of spheres in lightlike geometry of submanifolds

We prove that if an indefinite Kaehler manifold \(\bar {M}\) with lightlike submanifolds satisfies the axioms of holomorphic 2r-spheres, axioms of holomorphic 2r-planes, axioms of transversal r-spheres and axioms of transversal r-planes, then it is an indefinite complex space form.     

Bochner and conformal flatness on normal complex contact metric manifolds

We will prove that normal complex contact metric manifolds that are Bochner flat must have constant holomorphic sectional curvature 4 and be Kähler. If they are also complete and simply connected, they must be isometric to the odd-dimensional complex projective space \({{\mathbb{C}P^{2n+1}}}\)(4) with the Fubini-Study metric. On the other hand, it is not possible for normal complex contact metric manifolds to be conformally flat.     

Holomorphic centroaffine immersions and the Lelieuvre correspondence

The rigidity and intrinsic characterization of holomorphic centroaffine immersions are given. We also obtain a method to construct nondegenerate holomorphic affine hypersurfaces from centroaffine immersions and metrics satisfying some conditions. Mathematics subject classification: 53A15.     

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 \(S^2\).     

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.