Leopold Verstraelen

Ruled surfaces of finite type

We show that a ruled surface of finite type in a Euclidean space is a cylinder on a curve of finite type or a helicoid in Euclidean 3-space.

On totally real 3-dimensional submanifolds of the nearly Kaehler 6-sphere

Let M be a compact 3-dimensional totally real submanifold of the nearly Kaehler 6-dimensional unit sphere. Let K be the sectional curvature function of M. Then, if K > 1/16, M is a totally geodesic submanifold (and K 1).

Characterizations of Riemannian space forms, Einstein spaces and conformally flat spaces

In a recent paper the first author introduced two sequences of Riemannian invariants on a Riemannian manifold M, denoted respectively by 6(nl,... ,nk) and 5(ni,... ,nk), which trivially satisfy 6(ni,... ,nk) > 6(n,... , nk). In this article, we completely determine the Riemannian manifolds satisfying the condition 6(nfl,... , nk) = S(nl,... ,nk). By applying the notions of these 6-invariants, we establish new characterizations of Einstein and conformally flat spaces; thus generalizing two well-known results of Singer-Thorpe and of Kulkarni.

The normal curvature of totally real submanifolds of S^6(1)

We prove the pointwise inequality 0 > p + p L — 1 involving the normalized scalar curvature p and normal scalar curvature p 1 of a totally real 3-dimensional sub- manifold of the nearly Kaehler 6-sphere. Further we classify submanifolds realizing the equality in this inequality.