Session on Geometric Analysis in Mathematical Physics -- WCNA 2004 --
4th World Congress of Nonlinear Analysts,
Orlando, Florida,
June 30 - July 7, 2004

Organizers: Krishan Duggal (University of Windsor), Paul Ehrlich (University of Florida)



The WCNA 2004 web site for general conference information may be accessed at

WNCA - 2004 Web Site


The Second Announcement has been posted on the web site; we should especially call your attention to two new developments -- the article length is now limited to between 8 - 10 pages and also the manuscripts should be submitted to Professors Duggal and Ehrlich PRIOR to the WCNA - 2004 Conference, by March 31, 2004, according to the WCNA organizers.

LIST OF PARTICIPANTS



  • Jaedong Choi
    Department of Mathematics
    Korea Air Force Academy
    P.O. Box 335-2
    Cheongwon, Chungbuk 363-849
    South Korea
    jdong@afa.ac.kr


    Warped product space-times and Reissner-Nordstrom -AdS black holes

    Abstract -- We study a multiply warped product manifold associated with the Reissner-Nordstrom - AdS metric to investigate the physical properties inside the black hole event horizons.

    Our results include various limiting geometries, of the RN, Schwarzschild -- AdS and the Schwarzschild space-times, through the successive truncation procedure of parameters in the original curved space.



  • Andrzej Derdzinski
    Department of Mathematics
    Ohio State University
    Columbus, Ohio 43210
    andrzej@math.ohio-state.edu

    Quasi-Einstein metrics and Ricci solitons

    Abstract -- A survey of known facts about quasi-Einstein metrics and Ricci solitons is given, including their role in the Ricci flow and the case of Kahler-Ricci solitons.



  • G. N. Galanis
    Section of Mathematics
    Naval Academy of Greece
    Xatzikyriakion, Piraeus 185 39, Greece
    ggalanis@snd.eud.gr

    Bundles of accelerations on Banach manifolds (with C.T.J. Dodson)

    Abstract -- We consider an infinite dimensional manifold M modelled on a Banach space |E and we construct smooth fiber bundle structures on the tangent bundle of order two, T2M, which consists of all smooth curves of M that agree up to their acceleration, as well as on the corresponding second order frame bundle L2M. These bundles prove to be associated with respect to the identical representation of the general linear group GL(|E) that serves as the structural group of both of them. Moreover, a bijective correspondence between linear connections on T2M and connection forms L2M is revealed.



  • J. C. Diaz-Ramos
    Department of Geometry and Topology
    Faculty of Mathematics
    University of Santiago de Compostela
    15782 Santiago de Compostela
    Spain
    xtjosec@usc.es

    Comparison theorems for volumes of geodesic celestial spheres in Lorentzian geometry, (joint with Eduardo Garia-Rio)

    Abstract -- In this lecture we will discuss some volume comparison results for different objects in Lorentzian geometry, with special attention to geodesic celestial spheres, and the corresponding objects in Lorentzian space forms. Also, some rigidity results are shown which allow one to detect locally isotropic Lorentzian manifolds by some instrinsic properties of their celestial geodesic spheres.

  • Claus Gerhardt
    Institut fur Angewandte Mathematik
    Ruprecht - Karls - Universitat Heidelberg
    Im Neuenheimer Feld 298
    69120 Heidelberg
    Germany
    gerhardt@math.uni-heidelberg.de

    The inverse mean curvature in cosmological spacetimes -- transition from big crunch to big bang

    Professor Gerhardt is unable to participate personally at this conference, but wishes me to announce that research articles covering this previously announced topic may be downloaded from his web site.




  • Paul Ehrlich
    Department of Mathematics
    University of Florida
    Gainesvile, FL 32611-8105
    ehrlich@math.ufl.edu

    The index form of a warped product(with S.-B. Kim)

    Abstract -- We construct the index form along timelike geodesics on a Lorentzian warped product manifold and apply this index form to generalized Robertson-Walker (GRW) space-times.



  • Hans-Peter Kunzle
    Dept. of Mathematics and Statistical Sciences
    University of Alberta
    Edmonton, Canada T6G2G1
    hp.kunzle@ualberta.ca


    Spherically symmetric Einstein-Yang-Mills-Higgs equations for general compact gauge groups



  • David Metzler
    Department of Mathematics
    P.O. Box 118105
    University of Florida
    Gainesville, FL 32611-8105
    metzler@math.ufl.edu

    Fine structure of orbifolds

    Abstract -- Orbifolds have usually been viewed as mildly singular spaces, very similar to manifolds. I will present a couple of ways to see orbifolds as spaces with additional structure: namely, as stacks and as noncommutative spaces. If time allows, I will mention how these structures enter into orbifold string theory.

    Phillip E. Parker
    Math. Dept. # 33
    Witchita St. Univ.
    Wichita, KS 67260-0033
    phil@math.twsu.edu

    Geometry of Nonlinear Connections

    Abstract -- We show that locally diffeomorphic exponential maps can be defined for any second-order differential equation (2ODE) over a manifold, and give a (possibly nonlinear) covariant derivative for any (possibly nonlinear) connection. In the process, we introduce vertically homogeneous  connections, which allow us to include Finsler spaces among the applications.

    We provide significant support for the prospect of studying nonlinear connections via  2ODEs. One of the most important pieces is a generalized Ambrose-Palais-Singer correspondence, a major part of our motivation for regarding 2ODEs as generalized sprays.



  • Volker Perlick
    Institute fur Theoretische Physik
    Universitat zu Koln
    50923 Koln
    Germany
    perlick@thp.Uni-Koeln.DE


    On totally umbilic timelike submanifolds in Lorentzian manifolds

    Abstract -- I consider, in Lorentzian manifolds, timelike submanifolds that are totally umbilic, i.e., for which the second fundamental form is a multiple of the first fundamental form. I discuss their physical meaning, based on general relativity, and I give various criteria for their existence or non-existence in Lorentzian manifolds with certain properties.


    Ehrlich's editorial comment -- a fascinating discussion of photon surfaces from the viewpoint of theoretical physics which is related to the announced topic may be found in an article
    T. Foertsch, W. Hasse, V. Perlick, Inertial forces and photon surfaces in arbitrary spacetimes, Classical and Quantum Grav. 20 (2003), 4635- 4652.




  • Miguel Sanchez
    Departamento de Geometria y Topologia
    Facultad de Ciencias
    Avda. Fuentenueva s/n
    Universidad de Granada
    E-18071 Granada
    Spain
    sanchezm@ugr.es

    On the Geometry of Static Space-times

    Abstract -- Static spacetimes (i.e., a Lorentzian manifold endowed with a timelike irrotational Killing vector field K) are one of the simplest classes of Lorentzian manifolds, which includes classical Scharzschild spacetime. Thus, many of their geometric properties have been studied from different viewpoints and , recently, there has been renewed progress made.

    Our purpose is to review some of the properties and techniques, paying special attention to the following two:

    1. Variational Methods are applicable to obtain definitive results on geodesic connectedness as, for example: a static spacetime (M,g) is geodesically connected if g is geodesically complete and |g(K,K)| grows at most quadratically with the Riemannian distance on K per .

    2. Causality Theory and different geometrical tools apply for the problem of closed geodesics and connectedness by causal geodesics: if M is compact, then it admits a closed timelike geodesic and each two points are also connectable by a timelike geodesic.



  • Krishan Duggal
    Department of Mathematics
    University of Windsor
    Windsor, Ontario Canada
    yq8@uwindsor.ca

    Conformal Killing Vector Fields on Spacetime Solutions of Einstein's Equations and Initial Data ( with Ramesh Sharma)

    Abstract -- This talk presents spacetime solutions of Einstein's equations with a conformal Killing vector field and initial data. We follow the 3 + 1 split formalism due to Arnowitt, Deser, Misner (the so-called ADM - formalism) for the evolution of the spacetime and analyze the structure of the initial data (the metric and intrinsic curvature) of a spacelike hypersurface of the 1-parameter family of spacelike hypersurfaces folitating the spacetime.