Certain conditions for a Riemannian manifold to be isometric with a sphere:Dedicated to Professor Kentaro Yano on his fiftieth birthday

Integral formulas in Riemannian geometry

The present volume is the written version of the series of lectures the author delivered at the Catholic University of Leuven, Belgium during the period of June-July, 1990. The main purpose of these talks is to present some of author's work and also his joint works with Professor T. Nagano and Professor Y. Tazawa of Japan, Professor P. F. Leung of Singapore and Professor J. M. Morvan of France on geometry of slant submanifolds and its related subjects in a systematical way.

Geometry of Slant Submanifolds

TOTAL MEAN CURVATURE AND SUBMANIFOLDS OF FINITE TYPE (World Scientific Series in Pure Mathematics—Volume 1) By Bang-Yen Chen: pp. 352. £15.70. (Published by World Scientific Publishing, distributed by John Wiley & Sons Ltd, 1984.)

We define and study both bi-slant and semi-slant submanifolds of an almost contact metric manifold and, in particular, of a Sasakian manifold We prove a characterization theorem for semi-slant submanifolds and we obtain integrability conditions for the distributions which are involved in the definition of such submanifolds We also study an interesting particular class of semi-slant submanifolds

Semi{slant submanifolds of a sasakian manifold

In this article we study surfaces in \( {\Bbb S}^2\times {\Bbb R} \) for which the unit normal makes a constant angle with the \( {\Bbb R} \)-direction. We give a complete classification for surfaces satisfying this simple geometric condition.

Constant angle surfaces in \( {\Bbb S}^2\times {\Bbb R} \)

On almost complex curves in the nearly kähler 6-sphere

We give several ways of constructing new nondegenerate affine hypersurfaces in R nn+1 starting from some given proper or improper affine spheres. The resulting hypersurfaces can be forced to have nice properties, if the original affine spheres are chosen appropriately. In this way we obtain new examples of nondegenerate locally homogeneous affine hypersurfaces and of proper and improper affine spheres. We also discuss the properties of the affine metric of the hypersurface, such as the curvature or completeness. Finally we discuss the relation with projective transformations.

Calabi-type composition of affine spheres

In a previous paper, B.-Y. Chen defined a Riemannian invariant δ by subtracting from the scalar curvature at every point of a Riemannian manifold the smallest sectional curvature at that point, and proved, for a submanifold of a real space form, a sharp inequality between δ and the mean curvature function. In this paper, we extend this inequality to totally real submanifolds of a complex space form. As a consequence, we obtain a metric obstruction for a Riemannian manifold Mn to admit a minimal totally real (i.e. Lagrangian) immersion into a complex space form of complex dimension n. Next we investigate three-dimensional submanifolds of the complex projective space ℂP3 which realise the equality in the inequality mentioned above. In particular, we construct and characterise a totally real minimal immersion of S3 in ℂP3.

An exotic totally real minimal immersion of S3 in ℂP3 and its characterisation

We give a complete classification of locally strongly convex affine hypersurfaces ofR n+1 with parallel cubic form with respect to the Levi-Civita connection of the affine Berwald-Blaschke metric. It turns out that all such affine hypersurfaces are quadrics or can be obtained by applying repeatedly the Calabi product construction of hyperbolic affine hyperspheres, using as building blocks either the hyperboloid, or the standard immersion of one of the symmetric spaces SL(m;R)=SO(m), SL(m;C)=SU(m), SU � 2m � =Sp(m), or E6(−26)=F4.

/pdf/locally-strongly-convex-affine-hypersurfaces-with-parallel-2hw55nhded.pdf

Luc Vrancken

Papers

Constant angle surfaces in \( {\Bbb S}^2\times {\Bbb R} \)

On almost complex curves in the nearly kähler 6-sphere

Calabi-type composition of affine spheres

An exotic totally real minimal immersion of S3 in ℂP3 and its characterisation

Locally strongly convex affine hypersurfaces with parallel cubic form