ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

5. Differential Geometry and Global Analysis

1991 MS Classification: 53 Dimitric, Ivko, Pennsylvania State University, USA Low-type submanifolds of non-flat complex space forms We consider submanifolds of a complex projective or a complex hyperbolic space ℂQ^{m}(4c), c = ±1, which are of low type in the (pseudo) Euclidean space of Hermitian matrices H(m + 1) via the imbedding of ℂQ^{m} which identifies a complex line with the projection operator onto it. Finite-type (k-type) submanifolds of Euclidean spaces in the sense of B. Y. Chen are defined as those whose position vectors allow decomposition into sum of finitely many (k) vector eigenfunctions of the Laplacian. We classify 1type submanifolds of ℂQ^{m} as: (1) Canonically imbedded ℂQ^{n} ⊂ ℂQ^{m};

(2) Minimal Lagrangian submanifolds M^{n} of a canonical ℂQ^{n} ⊂ ℂQ^{m};

(3) Geodesic sphere of radius arctan \sqrt{n + 2} of a canonical ℂQ^{(n+1)/2}. We also classify Hopf hypersurfaces of 2-type and characterize those that are mass-symmetric and of 3-type in H(m + 1).