Eric Boeckx

Riemannian manifolds of conullity two

Definition and early development local structure of semi-symmetric spaces explicit treatment of foliated semi-symmetric spaces curvature homogeneous semi-symmetric spaces asymptotic distributions and algebraic rank three-dimensional Riemannian manifolds of conullity two asymptotically foliated semi-symmetric spaces elliptic semi-symmetric spaces complete foliated semi-symmetric spaces application - local rigidity problems for hypersurfaces with type number two in IR4 three-dimensional Riemannian manifolds of relative conullity two appendix - more about curvature homogeneous spaces.

Harmonic and minimal vector fields on tangent and unit tangent bundles

Abstract We show that the geodesic flow vector field on the unit tangent sphere bundle of a two-point homogeneous space is both minimal and harmonic and determines a harmonic map. For a complex space form, we exhibit additional unit vector fields on the unit tangent sphere bundle with those properties. We find the same results for the corresponding unit vector fields on the pointed tangent bundle. Moreover, the unit normal to the sphere bundles in the pointed tangent bundle of any Riemannian manifold always enjoys those properties.

Unit Tangent Sphere Bundles with Constant Scalar Curvature

As a first step in the search for curvature homogeneous unit tangent sphere bundles we derive necessary and sufficient conditions for a manifold to have a unit tangent sphere bundle with constant scalar curvature. We give complete classifications for low dimensions and for conformally flat manifolds. Further, we determine when the unit tangent sphere bundle is Einstein or Ricci-parallel.

Harmonic and Minimal Radial Vector Fields

We present new examples of harmonic and minimal unit vector fields. These are radial vector fields on tubular neighbourhoods about points and submanifolds in two-point homogeneous spaces and harmonic manifolds, and about characteristic curves in Sasakian space forms.

Non-homogeneous relatives of symmetric spaces

Abstract For every fixed Riemannian symmetric space ( M , g ) we determine explicitly all locally non-homogeneous Riemannian spaces which have, at all points, the same curvature tensor as ( M , g ). For this purpose, we describe explicitly all parabolically foliated semi-symmetric spaces in the sense of Z.I. Szabo.

Energy of Radial Vector Fields on Compact Rank One Symmetric Spaces

We consider the energy (or the total bending) of unit vector fields oncompact Riemannian manifolds for which the set of its singularitiesconsists of a finite number of isolated points and a finite number ofpairwise disjoint closed submanifolds. We determine lower bounds for theenergy of such vector fields on general compact Riemannian manifolds andin particular on compact rank one symmetric spaces. For this last classof spaces, we compute explicit expressions for the total bending whenthe unit vector field is the gradient field of the distance function toa point or to special totally geodesic submanifolds (i.e., for radialunit vector fields around this point or these submanifolds).

Geodesics on the unit tangent bundle

A geodesic γ on the unit tangent sphere bundle T 1 M of a Riemannian manifold ( M, g ), equipped with the Sasaki metric gS , can be considered as a curve x on M together with a unit vector field V along it. We study the curves x . In particular, we investigate for which manifolds ( M, g ) all these curves have constant first curvature κ 1 or have vanishing curvature κ i for some i = 1, 2 or 3.

A Case for Curvature: the Unit Tangent Bundle

In the scientific work of L. Vanhecke, the notion of curvature is never more than a step away, if not studied explicitly. This is only right, since, in the words of R. Osserman, “curvature is the central concept (in differential geometry and, more in particular, in Riemannian geometry), distinguishing the geometrical core of the subject from those aspects that are analytic, algebraic or topological”. The reason for this can be seen as follows: if we equip a differentiable manifold M with a metric g, then its curvature is completely determined. If the metric g has nice properties (e.g., a large group of isometries), then this is reflected in a ‘nice’ curvature; conversely, we can often deduce information about the metric from special properties of the curvature. In some cases, knowledge about the curvature even suffices to completely determine the metric (at least locally). Locally symmetric spaces are the prime example here: they are distinguished from non-symmetric spaces by their parallel curvature and, starting from the curvature, one can reconstruct the manifold and its metric (locally). The curvature information is contained in the Riemannian curvature tensor R. This is an analytic object, a (0, 4)-tensor which is not easy to handle, in general, despite its many symmetries. It is often very difficult to extract the geometrical information which is, as it were, encoded within. For this reason, the famous geometer M. Gromov calls the curvature tensor “a little monster of multilinear algebra whose full geometric meaning remains obscure”.