PUBLICACIONES de la Real Sociedad Matem´atica Espa˜nola Volumen 9 Proceedings of XIII Fall Workshop on Geometry and Physics Murcia, Spain September 20–22, 2004 Edited by: Luis J. Al´ıas Linares Angel Ferr´andez Izquierdo Mar´ıa Angeles Hern´andez Cifre Pascual Lucas Saor´ın Jos´e Antonio Pastor Gonz´alez Organizing Committee Luis J. Al´ıas Linares ([email protected]) Pablo Miguel Chac´on Mart´ın ([email protected]) Angel Ferr´andez Izquierdo ([email protected]) Mar´ıa Angeles Hern´andez Cifre ([email protected]) Pascual Lucas Saor´ın ([email protected]) Pablo Mira Carrillo ([email protected]) Jos´e Antonio Pastor Gonz´alez ([email protected]) All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, whithout written permission of the Editors. L.J. Al´ıas, A. Ferr´andez, M.A. Hern´andez, P. Lucas, J.A. Pastor, editors Proceedings of XIII Fall Workshop on Geometry and Physics ISBN: 84-933610-6-2 Dep´osito legal: MU-1816-2005 Impreso en Espa˜na - Printed in Spain List of contributions Mini-Course J. Fernando Barbero G. Introductiontogeneralrelativity...................... 3 Invited lectures AnaBelaCruzeiro Some probabilistic estimates of heat kernel derivatives and applications 45 Eric Loubeau An aper¸cuofharmonicmapsandmorphisms............... 55 Juan Carlos Marrero and Diana Sosa Tulczyjew’s triple associated with a Lie affgebroid and a Hamiltonian section.................................. 66 Elena Medina and Mar´ıa Jos´eMar´ın New solutions of the Nizhnik-Novikov-Veselov model through symmetry transformations............................. 80 Stefano Montaldo and Irene I. Onnis Invariant surfaces in H2 × R with constant (mean or Gauss) curvature . 91 Jeanette E. Nelson and Roger F. Picken Classical and quantum geometry of moduli spaces in three-dimensional gravity.................................. 104 Estelita G.L.R. Vaz Onwarpedanddoublewarpedspace-times................ 115 Jean-Claude Zambrini Brownianmotionandsymplecticgeometry................ 126 Communications Antonio Ca˜nete Least-perimeterpartitionsofthedisk................... 141 Isabel Fern´andez and Francisco J. L´opez Periodic maximal graphs in the Lorentz-Minkowki space L3 ....... 149 4 Benito Hern´andez–Bermejo Hamiltonization and the study of Jacobi equations: some perspectives andresults................................ 155 Josep Llosa and Daniel Soler SemiRiemannian metrics as deformations of a constant curvature metric 161 Antonio L´opez Almorox Cohomology of Lie groups associated to compatible linear cocycles and Lie group structures on T ∗G ...................... 167 Jos´e M.M. Senovilla TrappedsubmanifoldsinLorentziangeometry.............. 175 Posters Rui Albuquerque Riemannian geometry of the twistor space of a symplectic manifold . 183 Angel Ballesteros, Francisco J. Herranz and Orlando Ragnisco q-Poisson coalgebras and integrable dynamics on spaces with variable curvature................................. 189 J. Fernando Barbero, Guillermo A. Mena Marug´an and Eduardo J.S. Villase˜nor Perturbative and non-perturbative quantum Einstein-Rosen waves . 195 Manuel Calixto Generalized W∞ higher-spinalgebrasandquantization.......... 202 Fernando Etayo, Rafael Santamar´ıa and Uju´eR.Tr´ıas Triplestructuresonamanifold....................... 208 Xavier Gr`acia and Rub´en Mart´ın Vector hulls and jet bundles . .................... 215 Julio Guerrero and Jos´eManuelP´erez Quantization of the regularized Kepler and related problems . 221 David Iglesias, Juan Carlos Marrero, Edith Padr´on and Diana Sosa Compatibility between Lie affgebroid and affine Jacobi structures on an affine bundle . .............................. 227 Manuel de Le´on, Jes´us Mar´ın-Solano, Juan Carlos Marrero, Miguel C. Mu˜noz-Lecanda and Narciso Rom´an-Roy Constraintalgorithmforsingularfieldtheories.............. 233 Eva Miranda A normal form theorem for integrable systems on contact manifolds . 240 C´esar Rosales Isoperimetric problems inside Euclidean convex domains . 247 Preface The professional dialogue between Mathematicians and Physicists is some- thing as natural as enriching in the daily life of a secondary school teacher. It seems that it is in the following educational step, the university, where the distance begins to be significant, situation that becomes usual as the specializ- ation level grows. Identical situation occurs in the realm of own Physicists or Mathematicians of the different areas, a classical characteristic of the Spanish people that we already see almost natural that a topologist and a statistician nothing must say themselves mathematically speaking. Perhaps it is exagger- ated, but it is what we daily observe. Luckily, who we define ourselves as geometricians or physicists had been thirteen years already trying to be the exception to so displeased rule and in that time giant steps have occurred. We have no doubt that we continue undergoing wide gaps of understanding, but every year a shortening of that ancestral distance is clearly observed. We think that the convergence is going to be achieved, essentially, through the Differential Geometry tools, mainly with theoretical physicists. It turns also out to be true that any other area of Mathematics can contribute in the same way to the progress, not only of Physics, but of any other science. The participants in these series of workshops have bet to foster a better understanding between Mathematicians and Physicists in order to undertake joint projects of greater scope, such as the society and the current research are demanding from them. We think that this is the right way to promote the Spanish science and these Proceedings will contribute to that aim. Further- more, they will be the best test of our commitment and support to Spanish quality publications. In Oviedo, where Murcia was nominated to organize this Workshop, and later in Coimbra, they left the strip in the highest, but we accepted the chal- 5 6 lenge to surpass it and we spared no effort to get it. We have done one’s bit to shorten the distance between geometry and physics; and what is better, to increase the quality, all over the world recognized, of those working in these areas. We take one’s chance to reiterate our public gratefulness, by its commend- able perseverance and example, to those who were very deserving of a ham, the colleagues Eduardo Mart´ınez Fern´andez, Miguel Carlos Mu˜noz Lecanda and Narciso Rom´an Roy. The University of Murcia, the S´eneca Foundation, the Spanish Ministry of Education and Science, the Cajamurcia Foundation, the Caja de Ahorros del Mediterr´aneo and the Real Spanish Mathematical Society made the Workshop and these Proceedings possible, so that they deserve our more sincere grate- fulness. As well as to all those that participated either learning, or teaching or, simply, engaging in a dialogue. Murcia, September 2005 The Editors Luis J. Al´ıas Linares Angel Ferr´andez Izquierdo Mar´ıa Angeles Hern´andez Cifre Pascual Lucas Saor´ın Jos´e Antonio Pastor Gonz´alez List of participants Name (Institution) e-mail address Alma-Luisa ALBUJER BROTONS (Univ. Murcia) [email protected] Rui Pedro ALBUQUERQUE (Univ. Evora)´ [email protected] Luis Jos´eAL´IAS LINARES (Univ. Murcia) [email protected] Luis Carlos de ANDRES´ DOMINGO (Univ. Pa´ıs Vasco) [email protected] Josu ARROYO OLEA (Univ. Pa´ıs Vasco) [email protected] Mar´ıa BARBERO LIN˜AN´ (Univ. Polit`ecnica de Catalunya) Manuel BARROS D´IAZ (Univ. Granada) [email protected] Roberto BENITO PRIETO (CSIC) rbenito@imaff.cfmac.csic.es Michele BENYOUNES (CNRS, Univ. Brest) [email protected] Miguel BROZOS VAZQUEZ´ (Univ. Santiago de Compostela) [email protected] Mar´ıa Teresa de BUSTOS MUNOZ˜ (Univ. Salamanca) Jos´e Luis CABRERIZO JARA´IZ (Univ. Sevilla) [email protected] Manuel CALIXTO MOLINA (Univ. Polit´ecnica de Cartagena) [email protected] Antonio Jes´us CANETE˜ MART´IN (Univ. Granada) [email protected] Jos´e Fernando CARINENA˜ MARZO (Univ. Zaragoza) [email protected] Alexandre CORTES´ AYASO (Univ. Santiago de Compostela) Ana Bela CRUZEIRO (Inst. Sup. T´ecnico, Lisboa) [email protected] Marina DELGADO TELLEZ´ DE CEPEDA (Univ. Carlos III) [email protected] Pablo Miguel CHACON´ MART´IN (Univ. Salamanca) [email protected] Alberto ENCISO CARRASCO (Univ. Complutense de Madrid) aenciso@fis.ucm.es Laura ESTEVE CORBALAN´ (Univ. Valencia) Fernando ETAYO GORDEJUELA (Univ. Cantabria) [email protected] Ra´ul Manuel FALCON´ GANFORNINA (Univ. Sevilla) [email protected] Rui Loja FERNANDES (Inst. Sup. T´ecnico, Lisboa) [email protected] Manuel FERNANDEZ´ ANDRES´ (Univ. Sevilla) [email protected] Isabel FERNANDEZ´ DELGADO (Univ. Granada) [email protected] Antonio FERNANDEZ´ MART´INEZ (Univ. Salamanca) [email protected] Jos´eFERNANDEZ´ NU´NEZ˜ (Univ. Oviedo) [email protected] Angel FERRANDEZ´ IZQUIERDO (Univ. Murcia) [email protected] Jos´e Antonio FERRY RUIZ [email protected] Carlos Alberto FUERTES PASCUAL (Univ. Salamanca) Pedro Luis GARC´IA PEREZ´ (Univ. Salamanca) [email protected] Xavier GRACIA` SABATE´ (Univ. Polit`ecnica de Catalunya) [email protected] Julio GUERRERO GARC´IA (Univ. Murcia) [email protected] Benito HERNANDEZ´ BERMEJO (Univ. Rey Juan Carlos) [email protected] Mar´ıa Angeles´ HERNANDEZ´ CIFRE (Univ. Murcia) [email protected] Pedro Jose HERRERO PINEYRO˜ (Univ. Murcia) [email protected] 8 Ana Mar´ıa HURTADO CORTEGANA (Univ. Valencia) [email protected] Miguel Angel´ JAVALOYES VICTORIA (Univ. Sao Paulo) [email protected]