Stability numbers in K-contact manifolds

The aim of this paper is to study the stability of the characteristic vector field of a compact K-contact manifold with respect to the energy and volume functionals when we consider on the manifold a two-parameter variation of the metric. First of all, we multiply the metric in the direction of the characteristic vector field by a constant and then we change the metric by homotheties. We will study to what extent the results obtained in [V. Borrelli, Stability of the characteristic vector field of a Sasakian manifold, Soochow J. Math. 30 (2004) 283-292. Erratum on the article: Stability of the characteristic vector field of a Sasakian manifold, Soochow J. Math. 32 (2006) 179-180] for Sasakian manifolds are valid for a general K-contact manifold. Finally, as an example, we will study the stability of Hopf vector fields on Berger spheres when we consider homotheties of Berger metrics.     

Harmonic morphisms of warped product type from Einstein manifolds

Weitzenb?ck type identities for harmonic morphisms of warped product type are developed which lead to some necessary conditions for their existence. These necessary conditions are further studied to obtain many nonexistence results for harmonic morphisms of warped product type from Einstein manifolds. Received: 14 March 2006     

Flag transitive planes of dimension four over GF(3)

Two non desarguesian flag transitive planes of order 3⁴ whose Kernel is GF(3) are constructed. These planes are distinct from the planes of the same order contained in the class constructed by Narayana Rao M. L. (Proceedings of American Mathematical Society 39 (1973) 51–56) and Ebert, G.L. and Baker, R. (Enumeration of two dimensional Flag-Transitive planes, Algebras, Groups and Geometries 3 (1985) 248–257). The Flag Transitive group modulo the scalar collineations of these planes is generated by two elements and is of order 328.     

Blow-ups and resolutions of strong Kähler with torsion metrics

On a compact complex manifold we study the behaviour of strong Kähler with torsion (strong KT) structures under small deformations of the complex structure and the problem of extension of a strong KT metric. In this context we obtain the analogous result of Miyaoka extension theorem. Studying the blow-up of a strong KT manifold at a point or along a complex submanifold, we prove that a complex orbifold endowed with a strong KT metric admits a strong KT resolution. In this way we obtain new examples of compact simply-connected strong KT manifolds.     

Second variation of volume and energy of vector fields. Stability of Hopf vector fields

In this paper we compute the Hessian of the volume of unit vector fields at a minimal one. We also find the Hessians of a family of functionals thus generalizing the known results concerning second variation of the energy or total bending. We use them to study the stability of Hopf vector fields on and to show that they are stable for , but that for there is such that for the index is at least . Received May 10, 1999 / Published online April 12, 2001     

Harmonic sections of Riemannian vector bundles, and metrics of Cheeger-Gromoll type

We study harmonic sections of a Riemannian vector bundle E→M when E is equipped with a 2-parameter family of metrics hp,q which includes both the Sasaki and Cheeger-Gromoll metrics. For every k>0 there exists a unique p such that the harmonic sections of the radius-k sphere subbundle are harmonic sections of E with respect to hp,q for all q. In both compact and non-compact cases, Bernstein regions of the (p,q)-plane are identified, where the only harmonic sections of E with respect to hp,q are parallel. Examples are constructed of vector fields which are harmonic sections of E=TM in the case where M is compact and has non-zero Euler characteristic.     

Sub-Finsler geometry in dimension three

We define the notion of sub-Finsler geometry as a natural generalization of sub-Riemannian geometry with applications to optimal control theory. We compute a complete set of local invariants, geodesic equations, and the Jacobi operator for the three-dimensional case and investigate homogeneous examples.     

Harmonicity of sections of sphere bundles

We consider the energy functional on the space of sections of a sphere bundle over a Riemannian manifold equipped with the Sasaki metric and discuss the characterising condition for critical points. Furthermore, we provide a useful method for computing the tension field in some particular situations. Such a method is shown to be adequate for many tensor fields defined on manifolds M equipped with a G-structure compatible with . This leads to the construction of several new examples of differential forms which are harmonic sections or determine a harmonic map from into its sphere bundle.     

The structure of singular spaces of dimension 2

In this paper, we study the structure of locally compact metric spaces of Hausdorff dimension 2. If such a space has non-positive curvautre and a local cone structure, then every simple closed curve bounds a conformal disk. On a surface (a topological manifold of dimension 2), a distance function with non-positive curvature and whose metric topology is equivalent to the surface topology gives a structure of a Riemann surface. The construction of conformal disks in these spaces uses minimal surface theory; in particular, the solution of the Plateau Problem in metric spaces of non-positive curvature. Received: 18 November 1997/ Revised versions: 15 January and 7 June 1999     

Affine and projective transformations of Berwald connections

We discuss the infinitesimal affine transformations of the Berwald connection of a spray, and the relation between the projective transformations of a spray and the affine transformations of its Berwald-Thomas-Whitehead connection.     

The energy of Hopf vector fields

The 3-dimensional Hopf vector field is shown to be a stable harmonic section of the unit tangent bundle. In contrast, higher dimensional Hopf vector fields are unstable harmonic sections; indeed, there is a natural variation through smooth unit vector fields which is locally energy-decreasing, and whose asymptotic limit is a singular vector field of finite energy. This energy is explicitly calculated, and conjectured to be the infimum of the energy functional over all smooth unit vector fields. Received: 17 March 1999     

Torsion of SU(2)-structures and Ricci curvature in dimension 5

Following the approach of Bryant [R. Bryant, Some remarks on G₂-structures, in: S. Akbulut, T. Önder, R.J. Stern (Eds.), Proceeding of Gökova Geometry-Topology Conference 2005, International Press, 2006], we study the intrinsic torsion of an SU(2)-structure on a 5-dimensional manifold deriving an explicit expression for the Ricci and the scalar curvature in terms of torsion forms and its derivative. As a consequence of this formula we prove that the α-Einstein condition forces some special SU(2)-structures to be Sasaki-Einstein.     

Classification of homogeneous almost cosymplectic three-manifolds

The purpose of this paper is to classify all simply connected homogeneous almost cosymplectic three-manifolds. We show that each such three-manifold is either a Lie group G equipped with a left invariant almost cosymplectic structure or a Riemannian product of type R×N, where N is a Kähler surface of constant curvature. Moreover, we find that the Reeb vector field of any homogeneous almost cosymplectic three-manifold, except one case, defines a harmonic map.     

Global inversion and covering maps on length spaces

In order to obtain global inversion theorems for mappings between length metric spaces, we investigate sufficient conditions for a local homeomorphism to be a covering map in this context. We also provide an estimate of the domain of invertibility of a local homeomorphism around a point, in terms of a kind of lower scalar derivative. As a consequence, we obtain an invertibility result using an analog of the Hadamard integral condition in the frame of length spaces. Some applications are given to the case of local diffeomorphisms between Banach-Finsler manifolds. Finally, we derive a global inversion theorem for mappings between stratified groups.     

Some results on geodesic mappings of Riemannian manifolds satisfying the conditionR.R.=Q(S,R)

In this paper geodesically corresponding metricsg and on a manifoldM, dim 5, under the assumption that the tensorsR andS of the metricg satisfyR.R=Q(S, R), are considered. It is stated that the corresponding tensors and of not necessarily must satisfy . Certain relations between the curvatures ofg and are obtained.Supported by a post-doctoral fellowship of the researchcouncil of the KU Leuven; Bitnet FGBDA3O at BLEKUL11     

Biharmonic maps, conformal deformations and the Hopf maps

We consider harmonic semi-conformal maps between two Riemannian manifolds. By deforming conformally the codomain metric, we construct new examples of non-harmonic biharmonic maps.     

Minimal two spheres in Kähler-Einstein Fano manifolds

In this paper we show the existence of stable symplectic non-holomorphic two-spheres in Kähler manifolds of positive constant scalar curvature of real dimension four and in Kähler-Einstein Fano manifolds of real dimension six. Some of the techniques used involve deformation theory of algebraic cycles.     

Generic equi-centro-affine differential geometry of plane curves

We study the equi-centro-affine invariants of plane curves from the view point of the singularity theory of smooth functions. We define the notion of the equi-centro-affine pre-evolute and pre-curve and establish the relationship between singularities of these objects and geometric invariants of plane curves.