Estimates of the first two buckling eigenvalues on spherical domains

In this paper, we study the first two eigenvalues of the buckling problem on spherical domains. We obtain an estimate of the second eigenvalue in terms of the first eigenvalue, which improves on a recent result obtained by Wang and Xia (2007) [1].     

Symmetries of parabolic contact structures

We generalize the concept of locally symmetric spaces to parabolic contact structures. We show that symmetric normal parabolic contact structures are torsion-free and some types of them have to be locally flat. We prove that each symmetry given at a point with non-zero harmonic curvature is involutive. Finally we give restrictions on the number of different symmetries which can exist at such a point.     

Almost complex structures on cotangent bundles and generalized geometry

We study a class of complex structures on the generalized tangent bundle of a smooth manifold M$M$ endowed with a torsion free linear connection, ∇$\nabla$. We introduce the concept of ∇$\nabla$-integrability and we study integrability conditions. In the case of the generalized complex structures introduced by Hitchin (2003) in [2], we compare the two concepts of integrability. Moreover, as an application, we describe almost complex structures on the cotangent bundle of M$M$ induced by complex structures on the generalized tangent bundle of M$M$.     

Extrinsic curvatures of distributions of arbitrary codimension

In this article, using the generalized Newton transformation, we define higher order mean curvatures of distributions of arbitrary codimension and we show that they agree with the ones from Brito and Naveira [F. Brito, A.M. Naveira, Total extrinsic curvature of certain distributions on closed spaces of constant curvature, Ann. Global Anal. Geom., 18 (2000) 371–383]. We also introduce higher order mean curvature vector fields and we compute their divergence for certain distributions and using this we obtain total extrinsic mean curvatures.     

On the Dirac spectrum of Riemannian foliations admitting a basic parallel 1-form

In this paper, in the special setting of a Riemannian foliation with basic, non-necessarily harmonic mean curvature we introduce a Weitzenböck-Lichnerowicz type formula which allows us to apply the classical Bochner-Lichnerowicz technique. We show that the lower bound for the eigenvalues of the basic Dirac operator can be calculated using only classical techniques. As another application, for general Riemannian foliations we calculate the above eigenvalue bound in the presence of a basic parallel 1-form, as an extension of a known result on a closed Riemannian manifold. Some results concerning the limiting case are obtained in the final part of the paper.     

Bounds on eigenvalues of the product and the Jordan product of two positive definite operators on a finite-dimensional Hilbert space

We show that R ²ⁿ with its standard symplectic structure is universal in that, subject to a mild topological restriction, essentially all symplectic manifolds can be obtained from it by reduction.Ford Foundation Fellow. Partially supported by NSF grant # DMS-8805699.Supported by the Netherlands Organization for Scientific Research (NWO).     

The Ricci tensor of SU(3)-manifolds

Following the approach of Bryant [R. Bryant, Some remarks on G₂-structures. e-print: math.DG/0305124] we study the intrinsic torsion of a SU(3)-manifold deriving a number of formulae for the Ricci and the scalar curvature in terms of torsion forms. As a consequence we prove that in some special cases the Einstein condition forces the vanishing of the intrinsic torsion.     

Generalized holomorphic structures

We investigate generalized holomorphic structures in generalized complex geometry. We find that a generalized holomorphic vector bundle carries a generalized complex structure on its total space if some additional conditions hold. We prove that generalized holomorphicity is equivalent to the integrability of a distribution on the total space, and a family of linear Dirac structures associated with this distribution is a generalized complex structure if a further condition holds. Under the same condition, we also prove that local generalized holomorphic frames exist around a regular point.     

A remark about static space times

We discuss a differential equation, whose unknowns are a function and a Riemannian metric. This equation occurs both in general relativity (static space times) and in the study of the space of Riemannian metrics on a manifold (singularities of the map from the space of metrics into the space of functions, which assigns to any metric its scalar curvature).     

Geometric realizations,curvature decompositions,and Weyl manifolds

We show that any Weyl curvature model can be geometrically realized by a Weyl manifold.     

Mathematical theory of nonrelativistic matter and radiation

We consider a system of finitely many nonrelativistic electrons bound in an atom or molecule which are coupled to the electromagnetic field via minimal coupling or the dipole approximation. Among a variety or results, we give sufficient conditions for the existence of a ground state (an eigenvalue at the bottom of the spectrum) and resonances (eigenvalues of a complex dilated Hamiltonian) of such a system. We give a brief outline of the proofs of these statements which will appear at full length in a later work.Dedicated to the memory of J. Schwinger, whose understanding of Quantum Electrodynamics was profound     

Six-dimensional nearly Kähler manifolds of cohomogeneity one

We consider six-dimensional strict nearly Kähler manifolds acted on by a compact, cohomogeneity one automorphism group G$G$. We classify the compact manifolds of this class up to G$G$-diffeomorphisms. We also prove that the manifold has constant sectional curvature whenever the group G$G$ is simple.     

Ruled CR-submanifolds of locally conformal Kähler manifolds

The purpose of this paper is to study the canonical totally real foliations of CR-submanifolds in a locally conformal Kähler manifold.     

On the conformal geometry of transverse Riemann–Lorentz manifolds

Physical reasons suggested in [J.B. Hartle, S.W. Hawking, Wave function of the universe, Phys. Rev. D41 (1990) 1815–1834] for the Quantum Gravity Problem lead us to study type-changing metrics on a manifold. The most interesting cases are Transverse Riemann–Lorentz Manifolds. Here we study the conformal geometry of such manifolds.     

Essential points of conformal vector fields

An essential point of a conformal vector field ξ$\xi$ on a conformal manifold (M,c)$(M,c)$ is a point around which the local flow of ξ$\xi$ preserves no metric in the conformal class c$c$. It is well-known that a conformal vector field vanishes at each essential point. In this note we show that essential points are isolated. This is a generalization to higher dimensions of the fact that the zeros of a holomorphic function are isolated. As an application, we show that every connected component of the zero set of a conformal vector field is totally umbilical.     

Some remarks on the converse of Weyl’s conformal theorem

One of Weyl’s classical theorems states that a certain tensor, the Weyl tensor, is unchanged when the metric from which it is constructed is replaced by another metric conformally related to it. This paper explores the converse of this theorem.     

On Nurowski’s conformal structure associated to a generic rank two distribution in dimension five

For a generic distribution of rank two on a manifold M$M$ of dimension five, we introduce the notion of a generalized contact form. To such a form we associate a generalized Reeb field and a partial connection. From these data, we explicitly construct a pseudo-Riemannian metric on M$M$ of split signature. We prove that a change of the generalized contact form only leads to a conformal rescaling of this metric, so the corresponding conformal class is intrinsic to the distribution.