The Differential Geometry of Finsler Spaces
The present monograph is motivated by two distinct aims. Firstly, an endeavour has been made to furnish a reasonably comprehensive account of the theory of Finsler spaces based on the methods of classical differential geometry. Secondly, it is hoped that
- PDF / 24,943,317 Bytes
- 298 Pages / 439 x 666 pts Page_size
- 8 Downloads / 192 Views
MATHEMATISCHEN WISSENSCHAFTEN IN EINZELDARSTELLUNGEN MIT BESONDERER BERUCKSICHTIGUNG DERANWENDUNGSGEBIETE HERAUSGEGEBEN VON
R. GRAMMEL . F. HIRZEBRUCH . E. HOPF H. HOPF . W. MAAK . W. MAGNUS· F. K. SCHMIDT K. STEIN· B. L. VAN DER WAERDEN BAND 101
THE DIFFERENTIAL GEOMETRY OF FI NSLE R SPACE S BY
HANNO RUND
SPRINGER-VERLAG BERLIN· GOTTINGEN . HEIDELBERG 1959
THE DIFFERENTIAL GEOMETRY OF FINSLER SPACES BY
DR. HANNO RUND PROFESSOR OF APPLIED MATHEMATICS IN THE UNIVERSITY OF NATAL
S PRI N G E R -VE RLAG BERLIN· GOTTINGEN· HEIDELBERG 1959
ALLE RECHTE, INSBESONDERE DAS DER UBERSETZUNG IN FREMDE SPRACHEN, VORBEHALTEN OHNE AUSDRUCKLICHE GENEHMIGUNG DES VERLAGES 1ST ES AUCH NICHT GESTATTET, DIESES BUCH ODER TEILE DARAUS AUF PHOTOMECHANISCHEM WEGE (PHOTOKOPIE, MIKROKOPIE) ZU VERVIELFALTIGEN ISBN 978-3-642-51612-2 ISBN 978-3-642-51610-8 (eBook) DOI 10.1007/978-3-642-51610-8
© BY SPRINGER-VERLAG OHG., BERLIN· GOTTINGEN . HEIDELBERG 1959 Reprint of the original edition 1959
BRUHLSCHE UNIVERSITATSDRUCKEREI GIESSEN
Preface The present monograph is motivated by two distinct aims. Firstly, an endeavour has been made to furnish a reasonably comprehensive account of the theory of Finsler spaces based on the methods of classical differential geometry. Secondly, it is hoped that this monograph may serve also as an introduction to a branch of differential geometry which is closely related to various topics in theoretical physics, notably analytical dynamics and geometrical optics. With this second object in mind, an attempt has been made to describe the basic aspects of the theory in some detail - even at the expense of conciseness - while in the more specialised sections of the later chapters, which might be of interest chiefly to the specialist, a more succinct style has been adopted. The fact that there exist several fundamentally different points of view with regard to Finsler geometry has rendered the task of writing a coherent account a rather difficult one. This remark is relevant not only to the development of the subject on the basis of the tensor calculus, but is applicable in an even wider sense. The extensive work of H. BUSEMANN has opened up new avenues of approach to Finsler geometry which are independent of the methods of classical tensor analysis. In the latter sense, therefore, a full description of this approach does not fall within the scope of this treatise, although its fundamental significance cannot be doubted. Fortunately, a recent volume l covers this ground comprehensively and with far greater competence than could possibly be achieved in the present text. Consequently the theory of BUSEMANN is not included, and will be referred to only when it has a direct bearing upon a particular problem under discussion. We shall thus restrict our attention to the methods of classical differential geometry which have prevailed in the literature of the subject up to the present time. The application of tensor methods has been dominated by the initial impetus given to the theory of Finsler spaces by L. BERWALD a
