A bifurcation result for semi-Riemannian trajectories of the Lorentz force equation

We obtain a bifurcation result for solutions of the Lorentz equation in a semi-Riemannian manifold; such solutions are critical points of a certain strongly indefinite functionals defined in terms of the semi-Riemannian metric and the electromagnetic field. The flow of the Jacobi equation along each solution preserves the so-called electromagnetic symplectic form, and the corresponding curve in the symplectic group determines an integer valued homology class called the Maslov index of the solution.We study electromagnetic conjugate instants with symplectic techniques, and we prove at first, an analogous of the semi-Riemannian Morse Index Theorem (see (Calculus of Variations, Prentice-Hall, Englewood Cliffs, NJ, USA, 1963)). By using this result, together with recent results on the bifurcation for critical points of strongly indefinite functionals (see (J. Funct. Anal. 162(1) (1999) 52)), we are able to prove that each non-degenerate and non-null electromagnetic conjugate instant along a given solution of the semi-Riemannian Lorentz force equation is a bifurcation point.     

Some Classes of Almost Anti-Hermitian Structures on the Tangent Bundle

In [11] we have considered a family of natural almost anti-Hermitian structures (G, J) on the tangent bundle TM of a Riemannian manifold (M, g), where the semi-Riemannian metric G is a lift of natural type of g to TM, such that the vertical and horizontal distributions VTM, HTM are maximally isotropic and the almost complex structure J is a usual natural lift of g of diagonal type interchanging VTM, HTM (see [5], [15]). We have obtained the conditions under which this almost anti-Hermitian structure belongs to one of the eight classes of anti-Hermitian manifolds obtained in the classification given in [1]. In this paper we consider another semi-Riemannian metric G on TM such that the vertical and horizontal distributions are orthogonal to each other. We study the conditions under which the above almost complex structure J defines, together with G, an almost anti-Hermitian structure on TM. Next, we obtain the conditions under which this structure belongs to one of the eight classes of anti-Hermitian manifolds obtained in the classification in [1].Partially supported by the Grant 100/2003, MECT-CNCSIS, România.     

Morse Theory for Geodesics in Conical Manifolds

The aim of this paper is to extend the Morse theory for geodesics to the conical manifolds. In a previous paper [15] we defined these manifolds as submanifolds of with a finite number of conical singularities. To formulate a good Morse theory we use an appropriate definition of geodesic, introduced in the cited work. The main theorem of this paper (see Theorem 3.6, section 3) proofs that, although the energy is nonsmooth, we can find a continuous retraction of its sublevels in absence of critical points. So, we can give a good definition of index for isolated critical values and for isolated critical points. We prove that Morse relations hold and, at last, we give a definition of multiplicity of geodesics which is geometrical meaningful. In section 5 we compare our theory with the weak slope approach existing in literature. Some examples are also provided.     

Orthogonal trajectories on Riemannian manifolds and applications to Plane Wave type spacetimes

We present a result on trajectories of a Lagrangian system joining two given submanifolds of a Riemannian manifold, under the action of an unbounded potential. As an application, we consider geodesics in a class of semi-Riemannian manifolds, the Plane Wave type spacetimes.     

Prescribed scalar curvature on then-sphere

This paper considers the prescribed scalar curvature problem onS ⁿ forn>-3. We consider the limits of solutions of the regularization obtained by decreasing the critical exponent. We characterize those subcritical solutions which blow up at the least possible energy level, determining the points at which they can concentrate, and their Morse indices. We then show that forn=3 this is the only blow up which can occur for solutions. We use this in combination with the Morse inequalities for the subcritical problem to obtain a general existence theorem for the prescribed scalar curvature problem onS ³.This article was processed by the author using the style filepljourlm from Springer-Verlag.     

On the spectral flow in Lorentzian manifolds

In this paper we use functional analytical techniques to determine the differential equation satisfied by the eigenvalues of a smooth family of Fredholm operators, obtained from the index form along a Lorentzian geodesic. The formula is then applied to the study of the evolution of the index function, and, using a perturbation argument, we prove a version of the classical Morse index theorem for stationary Lorentzian manifolds. Received: January 31, 2000; in final form: March 13, 2002?Published online: February 20, 2003 The second author is partially sponsored by CNPq (Brazil), Grant 200615/01-7.     

Enveloppes inférieures de fonctions admissibles sur l'espace projectif complexe. Cas dissymétrique

This paper generalizes the first author's preceding works concerning admissible functions on certain Fano manifolds [A. Ben Abdesselem, Lower bound of admissible functions on sphere, Bull. Sci. Math. 126 (2002) 675-680 [2]; A. Ben Abdesselem, Enveloppes inférieures de fonctions admissibles sur l'espace projectif complexe. Cas symétrique, Bull. Sci. Math. 130 (2006) 341-353 [3]]. Here, we study a larger class of functions which can be less symmetric than the ones studied before. When the sup of these functions is null, we prove that they admit a lower bound, giving precisely Tian invariant [G. Tian, On Kähler-Einstein metrics on certain Kähler manifolds with C₁(M)>0, Invent. Math. 89 (1987) 225-246 [7]] (see also [T. Aubin, Réduction du cas positif de l'équation de Monge-Ampère sur les variétés Kählériennes à la démonstration d'une inégalité, J. Funct. Anal. 57 (1984) 143-153 [1]]) on these manifolds.     

Conjugate points and Maslov index in locally symmetric semi-Riemannian manifolds

We study the singularities of the exponential map in semi Riemannian locally symmetric manifolds. Conjugate points along geodesics depend only on real negative eigenvalues of the curvature tensor, and their contribution to the Maslov index of the geodesic is computed explicitly. We prove that degeneracy of conjugate points, which is a phenomenon that can only occur in semi-Riemannian geometry, is caused in the locally symmetric case by the lack of diagonalizability of the curvature tensor. The case of Lie groups endowed with a bi-invariant metric is studied in some detail, and conditions are given for the lack of local injectivity of the exponential map around its singularities.     

Lightlike submanifolds of semi-Riemannian manifolds

The purpose of this paper is to initiate a study of the differential geometry of lightlike (degenerate) submanifolds of semi-Riemannian manifolds. We construct the transversal vector bundle for an arbitrary lightlike submanifold and obtain results on the geometric structures induced on it.     

Transformations of linear connections,II

We elucidate [9] with two applications. In the first we view connections as differential systems. Specializing this to trivial bundles overS ¹ and applying the theory of Floquet, we obtain equivalent connections with constant Christoffel symbols. In the second application we prove that the canonical connections of parallelizable manifolds (in particular Lie groups) can be obtained from the canonical flat connection of appropriate trivial bundles. Thus, the formalisms of [1], [4], [5] and [6] fit in the general setting of [9].     

A Liouville-type result for quasi-linear elliptic equations on complete Riemannian manifolds

We extend a Liouville-type result of D. G. Aronson and H. F. Weinberger and E.N. Dancer and Y. Du concerning solutions to the equation Δ_pu=b(x)f(u) to the case of a class of singular elliptic operators on Riemannian manifolds, which include the ?-Laplacian and are the natural generalization to manifolds of the operators studied by J. Serrin and collaborators in Euclidean setting. In the process, we obtain an a priori lower bound for positive solutions of the equation in consideration, which complements an upper bound previously obtained by the authors in the same context.     

A curvature condition for a twisted product¶to be a warped product

It is shown that a mixed Ricci-flat twisted product semi-Riemannian manifold can be expressed as a warped product semi-Riemannian manifold. As a consequence, any Einstein twisted product semi-Riemannian manifold is in fact, a warped product semi-Riemannian manifold. Received: 8 March 2001 / Revised version: 25 July 2001     

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     

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.     

Asymptotique de la norme $L^2$ d'un cycle géodésique dans les revêtements de congruence d'une variété hyperbolique arithmétique

Let M be an arithmetic hyperbolic manifold and be a codimension 1 geodesic cycle. In this paper, we study the asymptotic growth of the -norm of the lifts of F in the congruence tower above M. We obtain an explicit value for the growth rate of this norm. In particular, we provide a new proof of a celebrated result of Millson [Mi] on the homology of the arithmetic hyperbolic manifolds. The method is quite general and gives a new way of getting non zero homology classes in certain locally symmetric spaces. Received: 20 April 2001; in final form: 26 September 2001 / Published online: 28 February 2002     

Kähler manifolds of quasi-constant holomorphic sectional curvature

In correspondence with the manifolds of quasi-constant sectional curvature defined (cf [5], [9]) in the Riemannian context, we introduce in the K?hlerian framework the geometric notion of quasi-constant holomorphic sectional curvature. Some characterizations and properties are given. We obtain necessary and sufficient conditions for these manifolds to be locally symmetric, Ricci or Bochner flat, K?hler η-Einstein or K?hler-Einstein, etc. The characteristic classes are studied at the end and some examples are provided throughout.      

Conformal holonomy of C-spaces, Ricci-flat, and Lorentzian manifolds

The main result of this paper is that a Lorentzian manifold is locally conformally equivalent to a manifold with recurrent lightlike vector field and totally isotropic Ricci tensor if and only if its conformal tractor holonomy admits a 2-dimensional totally isotropic invariant subspace. Furthermore, for semi-Riemannian manifolds of arbitrary signature we prove that the conformal holonomy algebra of a C-space is a Berger algebra. For Ricci-flat spaces we show how the conformal holonomy can be obtained by the holonomy of the ambient metric and get results for Riemannian manifolds and plane waves.     

An obstruction to the positivity of relative Yamabe invariants

For a supergroup , we describe an obstruction to the existence of positive scalar curvature metrics with minimal boundary condition on a compact n-dimensional -manifold W with nonempty boundary M, , in terms of the bordism class [M] in the Stolz obstruction group associated to [St2]. In par ticular, when W is a 5-dimensional spin manifold and the -invariant of a connected component of M is nonzero, we prove that W does not admit a positive scalar curvature metric with minimal boundary condition. Received: 4 July 2001; in final form: 5 February 2002 / Published online: 8 November 2002 RID="*" ID="*" Partially supported by the Grants-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science, No. 11640070.     

On weakly harmonic maps from Finsler to Riemannian manifolds

We prove global C0,α$C^{0,\alpha }$-estimates for harmonic maps from Finsler manifolds into regular balls of Riemannian target manifolds generalizing results of Giaquinta, Hildebrandt, and Hildebrandt, Jost and Widman from Riemannian to Finsler domains. As consequences we obtain a Liouville theorem for entire harmonic maps on simple Finsler manifolds, and an existence theorem for harmonic maps from Finsler manifolds into regular balls of a Riemannian target.     

On the Energy of Distributions, with Application to the Quaternionic Hopf Fibrations

 The energy of an oriented q-distribution ? in a compact oriented manifold M is defined to be the energy of the section of the Grassmannian manifold of oriented q-planes in M induced by ?. In the Grassmannian, the Sasaki metric is considered. We show here a condition for a distribution to be a critical point of the energy functional. In the spheres, we see that Hopf fibrations are critical points. Later, we prove the instability for these fibrations. (Received 30 December 2000; in revised form 11 April 2001)