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.     

On the connection theory of extensions of transitive Lie groupoids

Due to a result by Mackenzie, extensions of transitive Lie groupoids are equivalent to certain Lie groupoids which admit an action of a Lie group. This paper is a treatment of the equivariant connection theory and holonomy of such groupoids, and shows that such connections give rise to the transition data necessary for the classification of their respective Lie algebroids.     

Lift of the Finsler foliation to its normal bundle

E. Ghys in [E. Ghys, Appendix E: Riemannian foliations: Examples and problems, in: P. Molino (Ed.), Riemannian Foliations, Birkhäuser, Boston, 1988, pp. 297-314. [3]] has posed a question (still unsolved) if any Finslerian foliation is a Riemannian one? In this paper we prove that the natural lift of a Finslerian foliation to its normal bundle is a Riemannian foliation for some Riemannian transversal metric. The methods we used here are closely related to those used by M. Abate and G. Patrizio in [M. Abate, G. Patrizio, Finsler Metrics—A Global Approach, Springer-Verlag, Berlin, 1994].     

An index theory for paths that are solutions of a class of strongly indefinite variational problems

We generalize the Morse index theorem of [12,15] and we apply the new result to obtain lower estimates on the number of geodesics joining two fixed non conjugate points in certain classes of semi-Riemannian manifolds. More specifically, we consider semi-Riemannian manifolds admitting a smooth distribution spanned by commuting Killing vector fields and containing a maximal negative distribution for . In particular we obtain Morse relations for stationary semi-Riemannian manifolds (see [7]) and for the G?del-type manifolds (see [3]). Received: 4 April 2001 / Accepted: 27 September 2001 / Published online: 23 May 2002 The authors are partially sponsored by CNPq (Brazil) Proc. N. 301410/95 and N. 300254/01-6. Parts of this work were done during the visit of the two authors to the IMPA, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil, in January and February 2001. The authors wish to express their gratitude to all Faculty and Staff of the IMPA for their kind hospitality.     

Collapse of warped foliations

The aim of this note is to generalize the concept of warped product to a foliated manifold (M,F,g) as follows: If is a smooth function constant along the leaves of the foliation F then new metric structure gf on the manifold M is constructed as follows: gf(v,w)=f²g(v,w) if v,w are tangent to F and gf(v,w)=g(v,w) if v or w is perpendicular to F. A foliated manifold (M,F,gf) is called warped foliation while f is called warping function.Next, if is a sequence of warping functions on M, the question of the existence of the limit in Gromov-Hausdorff of a sequence ((M,F,gfn))_n∈N warped foliation is asked. A number of examples is considered such foliations with dense leaf or foliations consisting of finite number of Reeb components. Next, sufficient and necessary condition of converging in Gromov-Hausdorff sense of a Riemannian foliation with all leaves compact to the space of leaves with a metric defined by Hausdorff distance of leaves is developed. Finally some results on Hausdorff foliations with all leaves compact are shown.     

On twisted contact groupoids and on integration of twisted Jacobi manifolds

We introduce the concept of twisted contact groupoids, as an extension either of contact groupoids or of twisted symplectic ones, and we discuss the integration of twisted Jacobi manifolds by twisted contact groupoids. We also investigate the very close relationships which link homogeneous twisted Poisson manifolds with twisted Jacobi manifolds and homogeneous twisted symplectic groupoids with twisted contact ones. Some examples for each structure are presented.     

Classification of 1st order symplectic spinor operators over contact projective geometries

We give a classification of 1st order invariant differential operators acting between sections of certain bundles associated to Cartan geometries of the so-called metaplectic contact projective type. These bundles are associated via representations, which are derived from the so-called higher symplectic (sometimes also called harmonic or generalized Kostant) spinor modules. Higher symplectic spinor modules are arising from the Segal-Shale-Weil representation of the metaplectic group by tensoring it by finite dimensional modules. We show that for all pairs of the considered bundles, there is at most one 1st order invariant differential operator up to a complex multiple and give an equivalence condition for the existence of such an operator. Contact projective analogues of the well known Dirac, twistor and Rarita-Schwinger operators appearing in Riemannian geometry are special examples of these operators.     

Multiplicity and stability of closed geodesics on bumpy Finsler 3-spheres

We prove that for every Q-homological Finsler 3-sphere (M, F) with a bumpy and irreversible metric F, either there exist two non-hyperbolic prime closed geodesics, or there exist at least three prime closed geodesics. Huagui Duan: Partially supported by NNSF and RFDP of MOE of China. Yiming Long: Partially supported by the 973 Program of MOST, Yangzi River Professorship, NNSF, MCME, RFDP, LPMC of MOE of China, S. S. Chern Foundation, and Nankai University.     

Eigenvalue estimate of the basic Dirac operator on a Kähler foliation

Let F be a Kähler spin foliation of codimension q=2n on a compact Riemannian manifold M with the transversally holomorphic mean curvature form κ. It is well known [S.D. Jung, T.H. Kang, Lower bounds for the eigenvalue of the transversal Dirac operator on a Kähler foliation, J. Geom. Phys. 45 (2003) 75-90] that the eigenvalue λ of the basic Dirac operator Db satisfies the inequality , where σ∇ is the transversal scalar curvature of F. In this paper, we introduce the transversal Kählerian twistor operator and prove that the same inequality for the eigenvalue of the basic Dirac operator by using the transversal Kählerian twistor operator. We also study the limiting case. In fact, F is minimal and transversally Einsteinian of odd complex codimension n with nonnegative constant transversal scalar curvature.     

Weitzenböck formulas for Riemannian foliations

In this paper we study the interplay between adiabatic limits of a Riemannian foliation and the classical Weitzenböck formula. For the leafwise part, our study leads to a vanishing result for the first order term of differential spectral sequence associated with the foliation. For the transversal part we obtain a Weitzenböck type formula which is an extension of the previous formula for basic forms due to Ph. Tondeur, M. Min-Oo, and E. Ruh, and is also more general than a Weitzenböck formula for transverse fiber bundle due to Y. Kordyukov.     

Tannaka duality for proper Lie groupoids

By replacing the category of smooth vector bundles of finite rank over a manifold with the category of what we call smooth Euclidean fields, which is a proper enlargement of the former, and by considering smooth actions of Lie groupoids on smooth Euclidean fields, we are able to prove a Tannaka duality theorem for proper Lie groupoids. The notion of smooth Euclidean field we introduce here is the smooth, finite dimensional analogue of the usual notion of continuous Hilbert field.     

The eta invariant and metrics of positive scalar curvature

Research partially supported by the NSF     

Topological structure of non-contractible loop space and closed geodesics on real projective spaces with odd dimensions

In this paper, we use Chas–Sullivan theory on loop homology and Leray–Serre spectral sequence to investigate the topological structure of the non-contractible component of the free loop space on the real projective spaces with odd dimensions. Then we apply the result to get the resonance identity of non-contractible homologically visible prime closed geodesics on such spaces provided the total number of distinct prime closed geodesics is finite.     

Closed geodesics on positively curved Finsler spheres

In this paper, we prove that for every Finsler n-sphere (Sn,F) for n?3 with reversibility λ and flag curvature K satisfying , either there exist infinitely many prime closed geodesics or there exists one elliptic closed geodesic whose linearized Poincaré map has at least one eigenvalue which is of the form exp(πiμ) with an irrational μ. Furthermore, there always exist three prime closed geodesics on any (S³,F) satisfying the above pinching condition.     

On causality and closed geodesics of compact Lorentzian manifolds and static spacetimes

Some results related to the causality of compact Lorentzian manifolds are proven: (1) any compact Lorentzian manifold which admits a timelike conformal vector field is totally vicious, and (2) a compact Lorentzian manifold covered regularly by a globally hyperbolic spacetime admits a timelike closed geodesic, if some natural topological assumptions (fulfilled, for example, if one of the conjugacy classes of deck transformations containing a closed timelike curve is finite) hold. As a consequence, any compact Lorentzian manifold conformal to a static spacetime is geodesically connected by causal geodesics, and admits a timelike closed geodesic.     

Addendum to “Morse theory of causal geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric” [Ann. I. H. Poincaré – AN 27 (3) (2010) 857–876]

We give the details of the proof of equality (29) in Caponio et al. (2010) [3].     

Global hyperbolicity and Palais-Smale condition for action functionals in stationary spacetimes

In order to apply variational methods to the action functional for geodesics of a stationary spacetime, some hypotheses, useful to obtain classical Palais-Smale condition, are commonly used: pseudo-coercivity, bounds on certain coefficients of the metric, etc. We prove that these technical assumptions admit a natural interpretation for the conformal structure (causality) of the manifold. As a consequence, any stationary spacetime with a complete timelike Killing vector field and a complete Cauchy hypersurface (thus, globally hyperbolic), is proved to be geodesically connected.