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.
General
Imprint: |
Springer-Verlag
|
Country of origin: |
Germany |
Series: |
Grundlehren der mathematischen Wissenschaften, 230 |
Release date: |
October 2011 |
First published: |
1978 |
Authors: |
W. Klingenberg
|
Dimensions: |
235 x 155 x 13mm (L x W x T) |
Format: |
Paperback
|
Pages: |
230 |
Edition: |
Softcover reprint of the original 1st ed. 1978 |
ISBN-13: |
978-3-642-61883-3 |
Categories: |
Books >
Science & Mathematics >
Mathematics >
Geometry >
Differential & Riemannian geometry
|
LSN: |
3-642-61883-9 |
Barcode: |
9783642618833 |
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