Leopold Verstraelen’s research while affiliated with Leuven University College and other places

A proposal is made for what may well be the most elementary Riemannian spaces which are homogeneous but not isotropic. In other words: a proposal is made for what may well be the nicest symmetric spaces beyond the real space forms, that is, beyond the Riemannian spaces which are homogeneous and isotropic. The above qualification of ‘’nicest symmetric spaces” finds a justification in that, together with the real space forms, these spaces are most natural with respect to the importance in human vision of our ability to readily recognise conformal things and in that these spaces are most natural with respect to what inWeyl’s view is symmetry in Riemannian geometry. Following his suggestion to remove the real space forms’ isotropy condition, the quasi space forms thus introduced do offer a metrical, local geometrical solution to the geometrical space form problem as posed by Thurston in his 1979 Princeton Lecture Notes on ‘’The Geometry and Topology of 3- manifolds”. Roughly speaking, quasi space forms are the Riemannian manifolds of dimension greater than or equal to 3, which are not real space forms but which admit two orthogonally complementary distributions such that at all points all the 2-planes that in the tangent spaces there are situated in a same position relative to these distributions do have the same sectional curvatures.

From the basic geometry of submanifolds will be recalled what are the extrinsic principal tangential directions, (first studied by Camille Jordan in the 18seventies), and what are the principal first normal directions, (first studied by Kostadin Trencevski in the 19nineties), and what are their corresponding Casorati curvatures. For reasons of simplicity of exposition only, hereafter this will merely be done explicitly in the case of arbitrary submanifolds in Euclidean spaces. Then, for the special case of Lagrangian submanifolds in complex Euclidean spaces, the natural relationships between these distinguished tangential and normal directions and their corresponding curvatures will be established.

From the basic geometry of submanifolds will be recalled what are the extrinsic principal tangential directions, (first studied by Camille Jordan in the 18seventies), and what are the principal first normal directions, (first studied by Kostadin Trenčevski in the 19nineties), and what are their corresponding Casorati curvatures. For reasons of simplicity of exposition only, hereafter this will merely be done explicitly in the case of arbitrary submanifolds in Euclidean spaces. Then, for the special case of Lagrangian submanifolds in complex Euclidean spaces, the natural relationships between these distinguished tangential and normal directions and their corresponding curvatures will be established.

In this paper, we prove some inequalities in terms of the normalized δ -Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant) of statistical submanifolds in holomorphic statistical manifolds with constant holomorphic sectional curvature. Moreover, we study the equality cases of such inequalities. An example on these submanifolds is presented.

In this article, we consider statistical submanifolds of Kenmotsu statistical manifolds of constant ϕ-sectional curvature. For such submanifold, we investigate curvature properties. We establish some inequalities involving the normalized δ-Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant). Moreover, we prove that the equality cases of the inequalities hold if and only if the imbedding curvature tensors h and h∗ of the submanifold (associated with the dual connections) satisfy h=−h∗, i.e., the submanifold is totally geodesic with respect to the Levi–Civita connection.

The main purpose of the present paper is to well define Minkowskian angles and pseudo-angles between the two null directions and between a null direction and any non-null direction, respectively. Moreover, in a kind of way that will be tried to be made clear at the end of the paper, these new sorts of angles and pseudo-angles can similarly to the previously known angles be seen as (combinations of)Minkowskian lengths of arcs on aMinkowskian unit circle togetherwithMinkowskian pseudo-lengths of parts of the straight null lines.

We investigate properties of 4-dimensional warped product manifolds satisfying a particular set of curvature conditions. As an application, we obtain a generalization of a pseudosymmetric property for Ricci-flat warped product spacetimes which was established previously in some special cases, including the Schwarzschild metric.

Torse-forming vector fields introduced by K. Yano are natural extension of concurrent and concircular vector fields. Such vector fields have many nice applications to geometry and mathematical physics. In this paper we establish a link between rotational hypersurfaces and torse-forming vector fields. More precisely, up to minor details, our main result states that, for a hypersurface M of E^{n+1} with n ≥ 3, the tangential component x^T of the position vector field x of M is a proper torse-forming vector field on M if and only if M is contained in a rotational hypersurface whose axis of rotation contains the origin.

In the present paper, we give a similar spectral variational theory for closed curves in the Euclidean 3–space E3, considering deformations in the direction of the principal normal vector field. Similarly as in the planar case, the closed Euclidean space curves satisfying the corresponding variational minimal principle are characterized by their curvature being a function of finite Chen type; their torsion remains completely free.

An attempt is made to present the geometrical meanings of the notions of parallel and of semi parallel and of pseudo parallel submanifolds, in analogy with the geometrical meanings of the notions of (locally) symmetric and semi symmetric and pseudo symmetric manifolds. In a way, the present article may thus be seen as an extrinsic companion of the article \Natural Intrinsic Geometrical Symmetries" in SIGMA's 2009 Special Issue Elie Cartan and Di�erential Geometry".