Johan Deprez

Semi-parallel surfaces in Euclidean space

Semi-parallel immersions are defined as extrinsic analogue for semi-symmetric spaces and as a direct generalization of parallel immersions. Using results of Backes on Euclidean Jordan triple systems, the totally geodesic immersions are shown to be the only minimal semi-parallel immersions into a Euclidean space. Semi-parallel immersions of surfaces into Em are studied and a classification of semi-parallel immersions with pointwise planar normal sections of surfaces in Em is given.

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.