Lectures on Closed Geodesics: 230 (Grundlehren der mathematischen Wissenschaften) - Softcover

Buch 2 von 125: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/A Series of Modern Surveys in Mathematics

Klingenberg, W.

 
9783642618833: Lectures on Closed Geodesics: 230 (Grundlehren der mathematischen Wissenschaften)
Softcover
ISBN 10: 3642618839 ISBN 13: 9783642618833
Verlag: Springer, 2011

Inhaltsangabe

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.

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Reseña del editor

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.

„Über diesen Titel“ kann sich auf eine andere Ausgabe dieses Titels beziehen.

Verlag
Springer
Erscheinungsdatum
2011
Sprache
Englisch
ISBN 10 
3642618839
ISBN 13 
9783642618833
Einband
Tapa blanda
Anzahl der Seiten
248
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9783540083931: Lectures on Closed Geodesics: 230 (Grundlehren der mathematischen Wissenschaften, 230)

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Hardcover