Lieven Vanhecke

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.

Lorentz manifolds modelled on a Lorentz symmetric space

Abstract We give examples of Lorentz manifolds modelled on an indecomposable Lorentz symmetric space which are geodesically complete and not locally homogeneous.

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.

Curvature invariants, differential operators and local homogeneity

We first prove that a Riemannian manifold (M, g) with globally constant additive Weyl invariants is locally homogeneous. Then we use this result to show that a manifold (M, g) whose Laplacian commutes with all invariant differential operators is a locally homogeneous space.

Geodesics in weakly symmetric spaces

We prove that any maximal geodesic in a weakly symmetric space is an orbit of a one-parameter group of isometries of that space.

Two natural generalizations of locally symmetric spaces

Abstract One studies two classes of Riemannian manifolds which extend the class of locally symmetric spaces: manifolds all of whose Jacobi operators Rγ have constant eigenvalues ( C -spaces) or parallel eigenspaces ( B -spaces) along geodesics γ. One gives several examples, derives equivalent characterizations and treats classifications for the two- and the three-dimensional case.

Examples of minimal unit vector fields

We provide a series of examples of Riemannian manifoldsequipped with a minimal unit vector field.

Examples of curvature homogeneous Lorentz metrics

Examples of a three- and a four-dimensional Lorentz manifold are presented which are curvature homogeneous up to order one, without being locally homogeneous, in contrast to the situation in the Riemannian case, where a curvature homogeneity up to order one implies local homogeneity in the three- and four-dimensional cases. It is further shown that these manifolds satisfy the property that all scalar curvature invariants vanish identically, i.e. are those of a flat Lorentz manifold. As an immediate consequence, we also obtain examples of Lorentz manifolds whose curvature invariants are all constant, but which are not locally homogeneous, again in contrast to the Riemannian case where such manifolds are always locally homogeneous.

Homogeneity on three-dimensional contact metric manifolds

We study ball-homogeneity, curvature homogeneity, natural reductivity, conformal flatness and ϕ-symmetry for three-dimensional contact metric manifolds. Several classification results are given.

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.