New Developments in Differential Geometry Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre/or Mathematics and Computer Science, Amsterdam, The Netherlands Volume 350 New Developments in Differential Geometry Proceedings of the Colloquium on Differential Geometry, Debrecen, Hungary, July 26-30, 1994 edited by L. Tamassy Institute of Mathematics and Informatics, Lajos Kossuth University, Debrecen, Hungary and 1. Szenthe Department of Geometry, Lordnd Eotvos University, Budapest, Hungary KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON A C.LP. Catalogue record for this book is available from the Library of Congress. Published by Kluwer Academic Publishers, P.O. Box 17,3300 AA Dordrecht, The Netherlands. Kluwer Academic Publishers incorporates the publishing programmes of D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, The Netherlands. ISBN-13: 978-94-010-6553-5 e-ISBN-13: 978-94-009-0149-0 DOl: 10.1007/978-94-009-0149-0 Printed on acid-free paper Softcover reprint of the hardcover 1st edition 1996 All Rights Reserved © 1996 Kluwer Academic Publishers No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. TABLE OF CONTENTS Preface .................................................. ix List of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. x Hypercomplex Structures on Quatemionic Manifolds D. V. Alekseevsky and S. Marchiafava ............................ . Time Inversion in Physics Tamas Antal ............................................. 21 Non Commutative Geometry of GLp-Bundles Akira Asada ............................................. 25 m Totally Umbilical Degenerate Monge Hypersurfaces of Aurel Bejancu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 51 The Left Exactness of the Smooth Left Puppe Sequence Paul Cherenack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 59 A Linear Connection Associated with Any Second Order Differential Equation Field M. Crampin ............................................. 77 Geometry of Geodesic Tubes on Sasakian Manifolds Mirjana Djoric ........................................... 87 Classification of Differential (n-l)-forms on an n-dimensional Manifold with Boundary Wojciech Domitrz ......................................... 103 Natural Relations between Connections in 2-fibred Manifolds Miroslav Doupovec and Alexandr Vondra . ........................ 113 Connections on Higher Order Frame Bundles Marek Elzanowski and Sergey Prishepionok ....................... 131 The Differential Geometry of Cosserat Media Marcelo Epstein and Manuel De Leon ........................... 143 vi TABLE OF CONTENTS The Parametric-Manifold Approach to Canonical Gravity Gyula Fodor and Zoltan Perjes ................................ 165 Admissible Operations and Product Preserving Functors Jacek Gancarzewicz, Wlodzimierz Mikulski and Zdzislaw Pogoda ........ 179 Curvature Properties of Para Kahler Manifolds E. Garda-RIo, L. Hervella and R. Vasquez-Lorenzo . ................. 193 Four Dimensional Osserman Lorentzian Manifolds Eduardo Garda-RIo and Demir N. Kupeli ........................ 201 The Eta Invariant and the Equivariant Spin Bordism of Spherical Space form 2 Groups Peter B. Gilkey and Boris Botvinnik ............................. 213 On Locally Conformal Kahler Structures Toyoko Kashiwada ......................................... 225 Torsion-Free Connections on Higher Order Frame Bundles Ivan Kolaf .............................................. 233 The Trace Decomposition of Tensors of Type (1,2) and (1,3) D. Krupka ............................................... 243 Higher-Order Constrained Systems on Fibred Manifolds: An Exterior Differential Systems Approach Olga Krupkova ........................................... 255 The Method of Separation of Variables for Laplace-Beltrami Equation in Semi -Riemannian Geometry Demir N. Kupeli .......................................... 279 A Geometrical Approach to Classical Field Theories: A Constraint Algorithm for Singular Theories Manuel de Leon, Jesus Marin-Solano and Juan C. Marrero ............ 291 Noether Type Theorems in Higher Order Analytical Mechanics R. Miron . ............................................... 313 The Electromagnetic Field in the Higher Order Relativistic Geometrical Optics Radu Miron and Tomoaki Kawaguchi ........................... 319 TABLE OF CONTENTS vii On a Riemannian Approach to the Order ex Relative Entropy M. Miyata, K. Kato, M. Yamada and T. Kawaguchi . ................. 325 A Differential Equation Related with Some General Connections Tominosuke Otsuki ......................................... 335 On Quasi Connections on Fibred Manifolds Paul Popescu ............................................ 343 Diffeomorphism Groups of a Manifold with Boundary Tomasz Rybicki ........................................... 353 Separability of Time-Dependent Second-Order Equations W. Sarlet ............................................... 363 The Field Equations of Generalized Conformally Flat Spaces of Metric gJlv(x,~, () = e20'(x'{,{)1]JlV P. C. Stavrinos, V. Balan and N. Prezas .......................... 373 Symmetries of Sprays and Admissible Lagrangians 1. Szenthe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 Curvature of Submanifolds in Points Finsler Spaces L. Tamassy .............................................. 391 Special Vector Fields on a Compact Riemannian Manifold Grigorios Tsagas .......................................... 399 Magnetic Dynamical Systems C. Udri§te, A. Udri§te, V. Balan and M. Postolache .................. 407 Totally Geodesic Orbits in Homogeneous Spaces Y. Villarroel ............................................. 415 Parallel and Transnormal Curves on Surfaces Bernd Wegner ............................................ 423 PREFACE In succession to our former meetings on differential geometry a Colloquium took place in Debrecen from July 26 to July 30, 1994. The Colloquium was organized by the University of Debrecen, the Debrecen Branch of the Hungarian Academy of Sciences and supported by the Janos Bolyai Mathematical Society. The Colloquium and especially this proceedings volume received an important financial contribution form OMFB in the framework of the ACCORD Programme no. H9112-0855. The Organizing Committee was the following: S. Bacso, P.T. Nagy, L. Kozma (secretary), Gy. Soos, J. Szenthe (chairman) and L. Tamassy (chairman). It was pleasant to meet both the returning participants of our former colloquia and the numerous new guests. The Colloquium had 68 participants from 22 foreign countries and 18 from Hungary. At the opening we commemorated the 25th anniversary of the death of Otto Varga, the late Professor of the Debrecen University, one of the founders of Finsler geometry, the master of many differential geometers of our country. The programme included 10 plenary lectures from: P.B. Gilkey, R. Miron, I. Kolar, B. Wegner, D. Lehmann, o. Kowalski, T. Otsuki, K.B. Marathe, M. Crampin, W. Sarlet and 68 short lectures in 3 sections. The meeting created an inspiring atmosphere for fruitful discussions between the participants. The historical sites of the town Debrecen and its famous surroundings offered ideal occasions to get to know Hungarian cultural traditions and for evening programmes. The present volume contains the written versions of the lectures presented at the Colloquium and also a list of the participants. The content of the papers covers a wide range of topics in differential geometry. The subjects receiving major emphasis were Riemannian geometry, Finsler geometry, the theory of submanifolds and applications of differential geometry in mathematical physics. It is our pleasant duty to thank our guests who contributed to the success of the Colloquium and especially to those who offered us their manuscript for publication. The Editors IX List of the participants M. ANASTASIEI (IiUii, Romania) K. MARATHE (Brooklyn, NY, USA) T. ANTAL (Budapest, Hungary) S. MARCHIAFAVA (Roma, Italy) A. ASADA (Matumoto, Japan) M. MARVAN (Opava, Czech Republic) G. ATANASIU (BriUiov, Romania) T. MELINTE (IiUii, Romania) S. BAcs6 (Debrecen, Hungary) J. MIKESH (Zlin, Czech Republic» V. BALAN (Bucharest, Romania) V. MIQUEL (Valencia, Spain) 1. BEJAN (IiUii, Romania) R. MIRON (IiUii, Romania) A. BEJANCU (IiUii, Romania) E. MOLNAR (Budapest, Hungary) T. BINH (Debrecen, Hungary) P. MULTARZYNSKI (Warszawa, Poland) N. BLAZIC (Belgrade, Hungary) P. NAGY (Szeged, Hungary) P. CHERENACK (Capetown, South Africa) S. NIKCEVIC (Belgrade, Yugoslavia) I. COMIC (Novi Sad, Yugoslavia) M. OKUMURA (Urawa, Japan) M. CRAMPIN (Milton Keynes, UK) T. OTSUKI (Yokohama, Japan) 1. DEL RIEGO (San L. Potosi, Mexico) B. PAAL (Budapest, Hungary) W. DOMITRZ (Warszawa, Poland) J. PARK (Kwang ju, Korea) M. DJORIC (Belgrade, Yugoslavia) M. PAUN (BriUiov, Romania) M. DOUPOVEC (Bmo, Czech Republic) O. PEKONEN (Jyvaskyla, Finland) M. ELZANOWSKI (Portland, USA) Z. PERJES (Budapest, Hungary) L. FRIEDLAND (Geneseo, NY, USA) M. POPESCU (Craiova, Romania) J. GANCARZEWICZ (Krakow, Poland) P. POPESCU (Craiova, Romania) E. GARCIA-RIO (Santiago, Spain) R. POPPER (Venezuala) P. GILKEY (Eugene, OR, USA) I. RADOMIR (BriUiov, Romania) V. GIRTU (Bacau, Romania) T. RAPCsAK (Budapest, Hungary) I. GOTTLIEB (IiUii, Romania) T. RYBICKI (Rzeszow, Poland) T. HAUSEL (Budapest, Hungary) W. SARLET (Gent, Belgium) 1. HERVELLA (Santiago, Spain) W. SASIN (Warszawa, Poland) R. IVANOVA (Sofia, Bulgaria) J. SLovAK (Bmo, Czech Republic) T. KASHlWADA (Tokyo, Japan) A. SZEMOK (Szeged, Hungary) T. KAWAGUCHI (Tsukuba, Japan) J. SZENTHE (Budapest, Hungary) A. KOBOTIS (Thessaloniki, Greece) J. SZILASI (Debrecen, Hungary) I. KOLAR (Bmo, Czech Republic) 1. TAMASSY (Debrecen, Hungary) Z. KovAcs (Nyiregyhcl.za, Hungary) G. TSAGAS (Thessaloniki, Greece) O. KOWALSKI (Praha, Czech Republic) A. UDRI~TE (Bucharest, Romania) J. KOZMA (Szeged, Hungary) C. UDRISTE (Bucharest, Romania) L. KOZMA (Debrecen, Hungary) S. VACARU (Chisianu, Moldavia) D. KRUPKA (Opava, Czech Republic) 1. VERHOCZKI (Budapest, Hungary) O. KRUPKOVA (Opava, Czech Republic) Y. VILLARROEL (Venezuala) A. KURUSA (Szeged, Hungary) W. VOGEL (Karlsruhe, Germany) D. LEHMANN (Montpellier, France) A. VONDRA (Bmo, Czech Republic) M. LEON (Madrid, Spain) B. WEGNER (Berlin, Germany) H. LOWE (Braunschweig, Germany) Z. ZEKANOWSKI (Warszawa, Poland) G. LAMER (Budapest, Hungary) N. ZHUKOVA (Nizhny Novgorod, Russia) V. ZOLLER (Budapest, Hungary) x Hypercomplex structures on quaternionic manifolds D.V. ALEKSEEVSKY and S. MARCHIAFAVA In memory of Franco Tricerri Abstract. Let (M, Q) be a quaternionic manifold. Conditions for existence of hypercom plex structures H subordinated to the quaternionic structure Q are determined, in particular for a quaternionic Kiihler manifold (M,g,Q). Some special systems of almost hypercom plex structures which are admissible for Q are also considered and their relationships with quaternionic transformations are indicated. 1. Introduction An almost hypercomplex structure H = (Jb J2, Js) on a manifold M is = a triple of anticommuting almost complex structures Ja, a 1,2,3, with Js = hh· H = (Ja") His called" Ha hypercomplex structure if there exist"sH a = = torsionless connection with Ja 0, a 1,2,3 (In such a case coincides with the Obata connection of H). An almost quaternionic structure Q on M is a 3-dimensional subbundle of the bundle of endomorph isms EndT M which is locally generated by almost hypercomplex structures H = (J1' h, Js). Q is quaternionic if there exists a torsionless connection " which preserves it. Any hypercomplex structure H on M determines in an obvious way a quater nionic structure Q =< H >, that is Qx = EaRJa Ix for any x E M: we will say that H is subordinated to Q. Note also that the quaternionic pro jective space Hpn, which carries a natural integrable quaternionic structure Work done under the program of G.N.S.A.G.A. of C.N.R. and partially financed by M.U.R.S.T. L. Tamtissy and J. Szenthe (eds.), New Developments in Differential Geometry, 1-19. © 1996 Kluwer Academic Publishers.