Lectures on Closed Geodesics:
Saved in:
| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Berlin, Heidelberg
Springer Berlin Heidelberg
1978
|
| Series: | Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics
230 |
| Subjects: | |
| Online Access: | Volltext |
| Item Description: | The question of existence of closed 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 geometry during the last century. The simplest case occurs for closed surfaces of negative curvature. Here, the fundamental group is very large and, as shown by Hadamard [Had] in 1898, every non-null homotopic closed curve can be deformed into a closed curve having minimal length in its free homotopy class. This minimal curve is, up to the parameterization, uniquely determined and represents a closed geodesic. The question of existence of a closed geodesic on a simply connected closed surface is much more difficult. As pointed out by Poincare [po 1] in 1905, this problem has much in common with the problem of the 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 closed geodesic of elliptic type, i. e., the corresponding periodic orbit in the geodesic flow is infinitesimally stable |
| Physical Description: | 1 Online-Ressource (XI, 230 p) |
| ISBN: | 9783642618819 9783642618833 |
| DOI: | 10.1007/978-3-642-61881-9 |
Staff View
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| 500 | |a The question of existence of closed 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 geometry during the last century. The simplest case occurs for closed surfaces of negative curvature. Here, the fundamental group is very large and, as shown by Hadamard [Had] in 1898, every non-null homotopic closed curve can be deformed into a closed curve having minimal length in its free homotopy class. This minimal curve is, up to the parameterization, uniquely determined and represents a closed geodesic. The question of existence of a closed geodesic on a simply connected closed surface is much more difficult. As pointed out by Poincare [po 1] in 1905, this problem has much in common with the problem of the 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 closed geodesic of elliptic type, i. e., the corresponding periodic orbit in the geodesic flow is infinitesimally stable | ||
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| institution | BVB |
| isbn | 9783642618819 9783642618833 |
| language | English |
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| publishDate | 1978 |
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| series2 | Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics |
| spelling | Klingenberg, Wilhelm Verfasser aut Lectures on Closed Geodesics by Wilhelm Klingenberg Berlin, Heidelberg Springer Berlin Heidelberg 1978 1 Online-Ressource (XI, 230 p) txt rdacontent c rdamedia cr rdacarrier Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics 230 The question of existence of closed 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 geometry during the last century. The simplest case occurs for closed surfaces of negative curvature. Here, the fundamental group is very large and, as shown by Hadamard [Had] in 1898, every non-null homotopic closed curve can be deformed into a closed curve having minimal length in its free homotopy class. This minimal curve is, up to the parameterization, uniquely determined and represents a closed geodesic. The question of existence of a closed geodesic on a simply connected closed surface is much more difficult. As pointed out by Poincare [po 1] in 1905, this problem has much in common with the problem of the 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 closed geodesic of elliptic type, i. e., the corresponding periodic orbit in the geodesic flow is infinitesimally stable Mathematics Global differential geometry Differential Geometry Mathematik Geschlossene geodätische Linie (DE-588)4157022-4 gnd rswk-swf Geodätische Linie (DE-588)4156669-5 gnd rswk-swf Riemannscher Raum (DE-588)4128295-4 gnd rswk-swf Geschlossene geodätische Linie (DE-588)4157022-4 s Riemannscher Raum (DE-588)4128295-4 s 1\p DE-604 Geodätische Linie (DE-588)4156669-5 s 2\p DE-604 Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics 230 (DE-604)BV049758308 230 https://doi.org/10.1007/978-3-642-61881-9 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Klingenberg, Wilhelm Lectures on Closed Geodesics Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics Mathematics Global differential geometry Differential Geometry Mathematik Geschlossene geodätische Linie (DE-588)4157022-4 gnd Geodätische Linie (DE-588)4156669-5 gnd Riemannscher Raum (DE-588)4128295-4 gnd |
| subject_GND | (DE-588)4157022-4 (DE-588)4156669-5 (DE-588)4128295-4 |
| title | Lectures on Closed Geodesics |
| title_auth | Lectures on Closed Geodesics |
| title_exact_search | Lectures on Closed Geodesics |
| title_full | Lectures on Closed Geodesics by Wilhelm Klingenberg |
| title_fullStr | Lectures on Closed Geodesics by Wilhelm Klingenberg |
| title_full_unstemmed | Lectures on Closed Geodesics by Wilhelm Klingenberg |
| title_short | Lectures on Closed Geodesics |
| title_sort | lectures on closed geodesics |
| topic | Mathematics Global differential geometry Differential Geometry Mathematik Geschlossene geodätische Linie (DE-588)4157022-4 gnd Geodätische Linie (DE-588)4156669-5 gnd Riemannscher Raum (DE-588)4128295-4 gnd |
| topic_facet | Mathematics Global differential geometry Differential Geometry Mathematik Geschlossene geodätische Linie Geodätische Linie Riemannscher Raum |
| url | https://doi.org/10.1007/978-3-642-61881-9 |
| volume_link | (DE-604)BV049758308 |
| work_keys_str_mv | AT klingenbergwilhelm lecturesonclosedgeodesics |