Generalized 2-symmetric submanifolds

Summary In this note we give two families of generalized 2-symmetric submanifolds. The first family consists of submanifolds of R ^N which are mapped into theirselves by the reflection of R ^N with respect to each first normal space and it is characterized in a differentiable way. The second family consists of submanifolds obtained by introducing a homomorphism between two Lie groups associated with symmetric R -spaces. All these submanifolds are mapped into theirselves by the reflection of R ^N with respect to subspaces of the normal spaces which contain the first normal spaces.     

Generalized Cauchy--Riemann lightlike submanifolds of Kaehler manifolds

Summary We introduce a class of submanifolds, namely, Generalized Cauchy--Riemann (GCR) lightlike submanifolds of indefinite Kaehler manifolds. We show that this new class is an umbrella of invariant (complex), screen real [8] and CR lightlike [6] submanifolds. We study the existence (or non-existence) of this new class in an indefinite space form. Then, we prove characterization theorems on the existence of totally umbilical, irrotational screen real, complex and CR minimal lightlike submanifolds. We also give one example each of a non totally geodesic proper minimal GCR and CR lightlike submanifolds.     

Generalized Cauchy-Riemann lightlike submanifolds of indefinite Sasakian manifolds

We study a class of submanifolds, called Generalized Cauchy-Riemann (GCR) lightlike submanifolds of indefinite Sasakian manifolds as an umbrella of invariant, screen real, contact CR lightlike subcases [8] and real hypersurfaces [9]. We prove existence and non-existence theorems and a characterization theorem on minimal GCR-lightlike submanifolds.     

Semisymmetric submanifolds

This article surveys new work on semisymmetric and k-parallel submanifolds M^m in Eⁿ and Mⁿ(c).Translated from Itogi Nauki i Tekhniki Seriya Problemy Geometrii, Vol. 23, pp. 3–28, 1991.     

Ric-semisymmetric submanifolds

This article surveys results obtained on various classes of submanifolds with the interior geometry of Riccisemisymmetric spaces.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 23, pp. 29–66, 1991.     

Begleitstrahlflächen windschiefer Flächen

Ohne Zusammenfassung     

Reducible Dupin submanifolds

Pinkall's standard constructions for obtaining a Dupin hypersurface W in ^N from a Dupin hypersurface M in ⁿ , N>n, are studied in the context of Lie sphere geometry. It is shown that a compact Dupin hypersurface W in ^N with g distinct principal curvatures at each point is reducible to a compact Dupin hypersurface M in ⁿ if and only if g=2.This research was supported by NSF Grant No. DMS 87-06015.     

Reflections in submanifolds

This is a study of geodesic symmetries of a Riemannian manifold with respect to a submanifold. We analyze the significance of the property to be volume-preserving, isometric or preserving the mean curvature of the tubular neighborhoods.Work supported in part by the National Science Foundation and the Max-Planck-Institut für Mathematik, Bonn.     

Coordinate finite-type submanifolds

A submanifold of a Euclidean space is called a coordinate finite-type submanifold if its coordinate functions are eigenfunctions of . We prove that the compact coordinate finite-type submanifolds are minimal submanifolds of quadratic hypersurfaces of Euclidean spaces. Moreover, we classify the compact coordinate finite-type submanifolds of codimension 2.     

Volume of submanifolds

We propose a geometric definition of the volume of submanifolds using tubular neighborhoods.     

Asymptotic curves of submanifolds

Translated from Matematicheskie Zametki, Vol. 52, No. 5, pp. 3–12, November, 1992.