Paul Verheyen

Characterization and examples of Chen submanifolds

In terms of an operator of J. Simons we give a new characterization for the Chen submanifolds orA-submanifolds of Riemannian manifolds. Also we give a number of non-trivial examples of Chen surfaces and study the impact of a conformal change of the metric of the ambient space on the property of being anA-surface.

Immersions with circular normal sections and normal sections of product immersions

We study immersions with normal sections that are circles in the ambient Euclidean space and formulate lemmas concerning normal sections of product immersions. As applications we determine all parallel immersions with planar normal sections and all immersions with planar normal sections and trivial normal connection.

Intrinsic characterizations for complex hypercylinders and complex hyperspheres

In terms of conditions on the curvature tensors of Riemann-Christoffel, Ricci, Weyl and Bochner we obtain several new characterizations of complex hyperspheres in complex projective spaces, of complex hypercylinders in complex Euclidean spaces and of complex hyperlanes in complex space forms.

Conformally flat C-totally real submanifolds of Sasakian space forms

Using the expression for the Laplacian of the square of the length of the second fundamental form of conformally flat minimal C-totally real submanifolds of a Sasakian space form pinchings for scalar curvature and sectional curvature are obtained which imply that the submanifolds must be totally geodesic.

Totally umbilical submanifolds in irreducible symmetric spaces

A submanifold of a Riemannian manifold is called a totally umbilical submanifold if its first and second fundamental forms are proportional. In this paper we prove the following best possible result. THEOREM. There is no totally umbilical submanifold of codimension less than rank of M in any irreducible symmetric space M.