Archiv der Mathematik

, Volume 106, Issue 6, pp 573–580 | Cite as

Non-closed isometry-invariant geodesics



Let c be a non-closed and bounded geodesic in a complete Riemannian manifold M. Assume that c is invariant under an isometry A of M and that c is not contained in the set of fixed points of A. We prove that, for some \({k\ge 2}\), the geodesic flow line ċ corresponding to c is dense in a k-dimensional torus N embedded in TM. In particular, every geodesic with initial vector in N is A-invariant.

Mathematics Subject Classification

53C22 58E10 57S20 


Isometry-invariant geodesics Morse-Bott theory Actions of non-compact abelian Lie groups 


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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Abteilung Reine MathematikMathematisches Institut der Universität FreiburgFreiburgGermany

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