Lectures on Closed Geodesics
The question of existence of c10sed geodesics on a Riemannian manifold and the properties of the corresponding periodic orbits in the geodesic flow has been the object of intensive investigations since the beginning of global differential geo metry durin
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Editors
S. S. Chem J. L. Doob J. Douglas, jr. A. Grothendieck E. Heinz F. Hirzebruch E. Hopf S. Mac Lane W. Magnus M. M. Postnikov W. Schmidt D. S. Scott K. Stein J. Tits B.L. van der Waerden Managing Editors
B. Eckmann
J.K. Moser
Wilhelm Klingenberg
Lectures on Closed Geodesics
Springer-Verlag Berlin Heidelberg New York 1978
Wilhelm Klingenberg Mathematisches Institut der Universitiit Bonn, D-5300 Bonn
AMS Subject Classifications (1970): Primary: 49 C 05, 49 F 15, 53 C 20, 55 D 35, 58 B 20, 58 E 05, 5H E 10, 58 F 05,58 F 20 Secondary: 34 C 25, 49 B 05,53 B 20, 55 C 30, 55 E 05,57 A 20, 58 D 15, 58 F 15
ISBN-13: 978-3-642-61883-3 DOI: 10.1007/978-3-642-61881-9
e-ISBN-13: 978-3-642-61881-9
Library of Congress Cataloging in Publication Data. Klingenberg, Wilhelm, 1924-. Lectures on closed geodesics (Grundlehren der mathematischen Wissenschaften; 230). Bihliography: p. IncJudes index. 1. Riemannian manifolds. 2. Curves an surfaces. 1. Tîtle.Il. SerÎes: Die Grundlehren der mathematischen Wissenschaften m Einzeldarstellungen: 230. QA649.K54. 516'.362.77-13147. This work is subject to copyright. All rights are reserved, whcthcr thc whole Of part of the material 15 concerned. ~pecifically those of translation, reprinting, re-use of iIIustrations, hroadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private llse. a fee is payable to the publisher, the amount of the fee tu be determÎned by agreement with the publisher. ,
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1
(x)
Let K be a connection on 'l:: E-+M. For ~EEp=n-l(p) we define T~vE=ker
(Tn :
T~E-+TpM),
and
T~hE=ker (K~ : T~E-+Ep)
to be the vertical and the horizontal subspace of
T~E
respectively.
1.1.1 Proposition. A connection K on n : E-+M defines a splitting
of the tangent bundle with
More precisely, under the canonical identificat ion of can write this decomposition as
T~vE
with Ep,p=n(O, we
Proof By looking at the local representation K", of K we see that, if we identify with Ep, i. e. if we identify
T~vE
{(x,~, O, Y/)}E{(X,
with
{(x, y/)}E{X} X lli,
O} xIM xlli
Chapter 1. The Hilbert Manifold of Closed Curves
4
The local representation of TShE is {(x,~, y, -r(x)(y,
O}E{(X, O} xlM xlE.
Given a connection K on n : E--+M we define the covariant derivative of a differentiable section ~ : M --+ E by
Note that
V'~
is a section in the bundle
L(r; n): L(TM; E)--+M.
V'~
Using the local representation (ci>, if;, U) of E we see that the principal part of is represented by
where (p: if;(U)--+lEis the principal part of the local representation of
~.
Note. Let S(M) and Se(M) denote the space of sections of the bundles and n. Then a covariant derivative defines a map
'M
S(M) x SE (M)--+SE(M)
(v, O .... W. v
which has a local representation given as above. Whereas for Euc1idean vector bundles over Euc1idean manifolds such a map V' always defines a connection K, in our more general situation this need not always be true; see [FK] for further details.
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