从复流形到对称空间的多重调和映射

本文首先证明了从复流形到对称空间的多重调和映射空间与扩张提升空间之间在相差一规范变换下存在一一对应,并给出确定的环路群在扩张提升空间的作用,因而也给出多重调和映射空间上的作用.其次,利用环路群及其代数的Iwasawa分解给出从Cn到对称空间的有限型的多重调和映射不同于文[1]中的刻划.     

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.     

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 K?hler 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 K?hler symmetric spaces. Received: 8 July 1997     

A Remark on Harmonic Maps Between Pseudo-Riemannian Spheres

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An intrinsic characterization of isometric pluriharmonic immersions with codimension one

We give an intrinsic characterization of isometric pluriharmonic immersions of Kähler manifolds into semi-Euclidean spaces with real codimension one, which is a generalization of the Ricci-Curbastro theorem.Research partly supported by the Grants-in-Aid for Encouragement of Young Scientists, The Ministry of Education, Science and Culture, Japan.     

A Cylinder Theorem for Isometric Pluriharmonic Immersions

We prove a cylinder theorem for isometric pluriharmonic immersions of complete Kähler manifolds into semi-Euclidean spaces under an assumption concerning the index of relative nullity.     

ON DEGREES OF MINIMAL IMMERSIONS OF SYMMETRIC SPACES INTO SPHERES

This paper considers the degrees of minimal immersions of symmetric spaces intospheres.A practical way to count the degree of a given regular minimal immersion of acompact irreducible symmetric space into sphere S_1~n is given.As an application,degrees ofregular minimal immersions of all rank one compact symmetric spaces into spheres arecounted.     

Isometric immersions of Riemannian Spaces in Euclidean Spaces

Questions of the theory of isometric immersions of Riemannian spaces in Euclidean spaces beginning with the very first results onthis topic and also results on immersions of pseudo-Riemannian spaces in pseudo-Euclidean spaces and applications of the theory of immersions in the general theory of relativity are considered.Translated from Itogi Nauki i Tekhniki, Algebra, Topologiya, Geometriya, Vol. 15, pp. 173–211, 1977.     

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.     

Pseudo-Riemannian almost quaternionic homogeneous spaces with irreducible isotropy

We show that pseudo-Riemannian almost quaternionic homogeneous spaces with index 4 and an \(\mathbb {H}\)-irreducible isotropy group are locally isometric to a pseudo-Riemannian quaternionic Kähler symmetric space if the dimension is at least 16. In dimension 12 we give a non-symmetric example.     

The Fundamental Theorems for Affine Immersionsinto Hyperquadrics and its Applications

 We prove the fundamental theorems for affine immersions into hyperquadrics (including affine spaces) with arbitrary codimension, which are generalizations of those for isometric immersions into space forms. As applications, the fundamental theorems for equiaffine immersions into hyperquadrics with arbitrary codimension are obtained. (Received 10 February 2000)     

Complete classification of spatial surfaces with parallel mean curvature vector in arbitrary non-flat pseudo-Riemannian space forms

Submanifolds with parallel mean curvature vector play important roles in differential geometry, theory of harmonic maps as well as in physics. Spatial surfaces in 4D Lorentzian space forms with parallel mean curvature vector were classified by B. Y. Chen and J. Van der Veken in [9]. Recently, spatial surfaces with parallel mean curvature vector in arbitrary pseudo-Euclidean spaces are also classified in [7]. In this article, we classify spatial surfaces with parallel mean curvature vector in pseudo-Riemannian spheres and pseudo-hyperbolic spaces with arbitrary codimension and arbitrary index. Consequently, we achieve the complete classification of spatial surfaces with parallel mean curvature vector in all pseudo-Riemannian space forms. As an immediate by-product, we obtain the complete classifications of spatial surfaces with parallel mean curvature vector in arbitrary Lorentzian space forms.      

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.     

Locally symmetric immersions

We use reflections with respect to submanifolds and related geometric results to develop, inspired by the work of Ferus and other authors, in a unified way a local theory of extrinsic symmetric immersions and submanifolds in a general analytic Riemannian manifold and in locally symmetric spaces. In particular we treat the case of real and complex space forms and study additional relations with holomorphic and symplectic reflections when the ambient space is almost Hermitian. The global case is also taken into consideration and several examples are given.     

The geometry of closed conformal vector fields on Riemannian spaces

In this paper we examine different aspects of the geometry of closed conformal vector fields on Riemannian manifolds. We begin by getting obstructions to the existence of closed conformal and nonparallel vector fields on complete manifolds with nonpositive Ricci curvature, thus generalizing a theorem of T.K. Pan. Then we explain why it is so difficult to find examples, other than trivial ones, of spaces having at least two closed, conformal and homothetic vector fields. We then focus on isometric immersions, firstly generalizing a theorem of J. Simons on cones with parallel mean curvature to spaces furnished with a closed, Ricci null conformal vector field; then we prove general Bernstein-type theorems for certain complete, not necessarily cmc, hypersurfaces of Riemannian manifolds furnished with closed conformal vector fields. In particular, we obtain a generalization of theorems J. Jellett and A. Barros and P. Sousa for complete cmc radial graphs over finitely punctured geodesic spheres of Riemannian space forms.     

A Parametrization of Isometric Immersions between Hyperbolic Spaces

We parametrize the space of isometric immersions of the hyperbolic n -space into the hyperbolic ( n +1)-space by a family of properly chosen (at most) countable n -tuples of real-valued functions. This is an answer to the problem posed by Nomizu in 1973. We also construct a one-parameter family of deformable isometric immersions from the quotient spaces of the hyperbolic 2-plane modulo and some totally discontinuous subgroups of isometries.     

COMPLEX SYMMETRIC TOEPLITZ OPERATORS ON THE UNIT POLYDISK AND THE UNIT BALL

In this article, we study complex symmetric Toeplitz operators on the Bergman space and the pluriharmonic Bergman space in several variables. Surprisingly, the necessary and sufficient conditions for Toeplitz operators to be complex symmetric on these two spaces with certain conjugations are just the same. Also, some interesting symmetry properties of complex symmetric Toeplitz operators are obtained.