Grundlehren der mathematischen Wissenschaften

Lectures on Closed Geodesics

Authors: Klingenberg, W.

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About this book

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 during the last century. The simplest case occurs for c10sed surfaces of negative curvature. Here, the fundamental group is very large and, as shown by Hadamard [Had] in 1898, every non-null homotopic c10sed curve can be deformed into a c10sed curve having minimallength in its free homotopy c1ass. This minimal curve is, up to the parameterization, uniquely determined and represents a c10sed geodesic. The question of existence of a c10sed geodesic on a simply connected c10sed surface is much more difficult. As pointed out by Poincare [po 1] in 1905, this problem has much in common with the problem ofthe existence of periodic orbits in the restricted three body problem. Poincare [l.c.] outlined a proof that on an analytic convex surface which does not differ too much from the standard sphere there always exists at least one c10sed geodesic of elliptic type, i. e., the corres­ ponding periodic orbit in the geodesic flow is infinitesimally stable.

Table of contents (5 chapters)

Table of contents (5 chapters)

Buy this book

eBook $84.99
price for USA in USD
  • ISBN 978-3-642-61881-9
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • Immediate eBook download after purchase and usable on all devices
  • Bulk discounts available
Hardcover $149.99
price for USA in USD
Softcover $109.99
price for USA in USD
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Bibliographic Information

Bibliographic Information
Book Title
Lectures on Closed Geodesics
Authors
Series Title
Grundlehren der mathematischen Wissenschaften
Series Volume
230
Copyright
1978
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Sringer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-61881-9
DOI
10.1007/978-3-642-61881-9
Hardcover ISBN
978-3-540-08393-1
Softcover ISBN
978-3-642-61883-3
Series ISSN
0072-7830
Edition Number
1
Number of Pages
XI, 230
Topics