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Example of compact Riemannian manifold with only one geodesic.



The 2019 Stack Overflow Developer Survey Results Are In
Unicorn Meta Zoo #1: Why another podcast?
Announcing the arrival of Valued Associate #679: Cesar ManaraWhy are we interested in closed geodesics?Existence of geodesic on a compact Riemannian manifoldCompleteness of a Riemannian manifold with boundaryTotally geodesic hypersurface in compact hyperbolic manifoldTriangle equality in a Riemannian manifold implies “geodesic colinearity”?Example for conjugate points with only one connecting geodesicExample for infinitely many points with more than one minimizing geodesic to a point?Examples of compact negatively curved constant curvature manifoldCompact totally geodesic submanifolds in manifold with positive sectional curvatureClosed geodesic on a non-simply connected Riemannian manifold










2












$begingroup$


The Lyusternik-Fet theorem states that every compact Riemannian manifold has at least one closed geodesic.



Are there any easy-to-construct1 examples of compact Riemannian manifolds for which it is easy to see they only have one closed geodesic?2



If there aren't any such examples, are there any easy-to-construct examples that only have one closed geodesic but where proving this might be difficult?



And if there aren't any examples of this, are there any examples at all of compact manifolds with only one closed geodesic?




1 Of course, the $1$-sphere $S^1$ contains just one closed geodesic, but I'm interested in examples besides this one.



2 By the theorem of the three geodesics, this example cannot be a topological sphere.










share|cite|improve this question











$endgroup$
















    2












    $begingroup$


    The Lyusternik-Fet theorem states that every compact Riemannian manifold has at least one closed geodesic.



    Are there any easy-to-construct1 examples of compact Riemannian manifolds for which it is easy to see they only have one closed geodesic?2



    If there aren't any such examples, are there any easy-to-construct examples that only have one closed geodesic but where proving this might be difficult?



    And if there aren't any examples of this, are there any examples at all of compact manifolds with only one closed geodesic?




    1 Of course, the $1$-sphere $S^1$ contains just one closed geodesic, but I'm interested in examples besides this one.



    2 By the theorem of the three geodesics, this example cannot be a topological sphere.










    share|cite|improve this question











    $endgroup$














      2












      2








      2





      $begingroup$


      The Lyusternik-Fet theorem states that every compact Riemannian manifold has at least one closed geodesic.



      Are there any easy-to-construct1 examples of compact Riemannian manifolds for which it is easy to see they only have one closed geodesic?2



      If there aren't any such examples, are there any easy-to-construct examples that only have one closed geodesic but where proving this might be difficult?



      And if there aren't any examples of this, are there any examples at all of compact manifolds with only one closed geodesic?




      1 Of course, the $1$-sphere $S^1$ contains just one closed geodesic, but I'm interested in examples besides this one.



      2 By the theorem of the three geodesics, this example cannot be a topological sphere.










      share|cite|improve this question











      $endgroup$




      The Lyusternik-Fet theorem states that every compact Riemannian manifold has at least one closed geodesic.



      Are there any easy-to-construct1 examples of compact Riemannian manifolds for which it is easy to see they only have one closed geodesic?2



      If there aren't any such examples, are there any easy-to-construct examples that only have one closed geodesic but where proving this might be difficult?



      And if there aren't any examples of this, are there any examples at all of compact manifolds with only one closed geodesic?




      1 Of course, the $1$-sphere $S^1$ contains just one closed geodesic, but I'm interested in examples besides this one.



      2 By the theorem of the three geodesics, this example cannot be a topological sphere.







      differential-geometry examples-counterexamples geodesic






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 53 mins ago







      Peter Kagey

















      asked 1 hour ago









      Peter KageyPeter Kagey

      1,57072053




      1,57072053




















          2 Answers
          2






          active

          oldest

          votes


















          4












          $begingroup$

          First of all, you have to exclude constant maps $S^1to M$ from consideration: They are all closed geodesics. Secondly, you have to talk about geometrically distinct closed geodesics: Geodesics which have the same image are regarded as "the same". Then, it is a notorious conjecture/open problem:



          Conjecture. Every compact Riemannian manifold of dimension $n >1$ contains infinitely many geometrically distinct nonconstant geodesics.



          See for instance this survey article by Burns and Matveev.



          This is even unknown if $M$ is diffeomorphic to the sphere $S^n$, $nge 3$.






          share|cite|improve this answer











          $endgroup$




















            2












            $begingroup$

            If you analyze the geodesics using Clairaut's relation, you'll find that the only closed geodesic on a hyperboloid of one sheet is the central circle. Indeed, the same holds for a concave surface of revolution of the same "shape" as the hyperboloid of one sheet.



            EDIT: Apologies for missing the crucial compactness hypothesis.






            share|cite|improve this answer











            $endgroup$












            • $begingroup$
              Lovely example, but a hyperboloid isn't compact, right?
              $endgroup$
              – Peter Kagey
              50 mins ago










            • $begingroup$
              Oops. Sloppy reading. I'll delete.
              $endgroup$
              – Ted Shifrin
              49 mins ago










            • $begingroup$
              It's a nice example; you should leave it.
              $endgroup$
              – Peter Kagey
              49 mins ago











            Your Answer








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            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

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            active

            oldest

            votes






            active

            oldest

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            4












            $begingroup$

            First of all, you have to exclude constant maps $S^1to M$ from consideration: They are all closed geodesics. Secondly, you have to talk about geometrically distinct closed geodesics: Geodesics which have the same image are regarded as "the same". Then, it is a notorious conjecture/open problem:



            Conjecture. Every compact Riemannian manifold of dimension $n >1$ contains infinitely many geometrically distinct nonconstant geodesics.



            See for instance this survey article by Burns and Matveev.



            This is even unknown if $M$ is diffeomorphic to the sphere $S^n$, $nge 3$.






            share|cite|improve this answer











            $endgroup$

















              4












              $begingroup$

              First of all, you have to exclude constant maps $S^1to M$ from consideration: They are all closed geodesics. Secondly, you have to talk about geometrically distinct closed geodesics: Geodesics which have the same image are regarded as "the same". Then, it is a notorious conjecture/open problem:



              Conjecture. Every compact Riemannian manifold of dimension $n >1$ contains infinitely many geometrically distinct nonconstant geodesics.



              See for instance this survey article by Burns and Matveev.



              This is even unknown if $M$ is diffeomorphic to the sphere $S^n$, $nge 3$.






              share|cite|improve this answer











              $endgroup$















                4












                4








                4





                $begingroup$

                First of all, you have to exclude constant maps $S^1to M$ from consideration: They are all closed geodesics. Secondly, you have to talk about geometrically distinct closed geodesics: Geodesics which have the same image are regarded as "the same". Then, it is a notorious conjecture/open problem:



                Conjecture. Every compact Riemannian manifold of dimension $n >1$ contains infinitely many geometrically distinct nonconstant geodesics.



                See for instance this survey article by Burns and Matveev.



                This is even unknown if $M$ is diffeomorphic to the sphere $S^n$, $nge 3$.






                share|cite|improve this answer











                $endgroup$



                First of all, you have to exclude constant maps $S^1to M$ from consideration: They are all closed geodesics. Secondly, you have to talk about geometrically distinct closed geodesics: Geodesics which have the same image are regarded as "the same". Then, it is a notorious conjecture/open problem:



                Conjecture. Every compact Riemannian manifold of dimension $n >1$ contains infinitely many geometrically distinct nonconstant geodesics.



                See for instance this survey article by Burns and Matveev.



                This is even unknown if $M$ is diffeomorphic to the sphere $S^n$, $nge 3$.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited 26 mins ago

























                answered 45 mins ago









                Moishe KohanMoishe Kohan

                48.6k344110




                48.6k344110





















                    2












                    $begingroup$

                    If you analyze the geodesics using Clairaut's relation, you'll find that the only closed geodesic on a hyperboloid of one sheet is the central circle. Indeed, the same holds for a concave surface of revolution of the same "shape" as the hyperboloid of one sheet.



                    EDIT: Apologies for missing the crucial compactness hypothesis.






                    share|cite|improve this answer











                    $endgroup$












                    • $begingroup$
                      Lovely example, but a hyperboloid isn't compact, right?
                      $endgroup$
                      – Peter Kagey
                      50 mins ago










                    • $begingroup$
                      Oops. Sloppy reading. I'll delete.
                      $endgroup$
                      – Ted Shifrin
                      49 mins ago










                    • $begingroup$
                      It's a nice example; you should leave it.
                      $endgroup$
                      – Peter Kagey
                      49 mins ago















                    2












                    $begingroup$

                    If you analyze the geodesics using Clairaut's relation, you'll find that the only closed geodesic on a hyperboloid of one sheet is the central circle. Indeed, the same holds for a concave surface of revolution of the same "shape" as the hyperboloid of one sheet.



                    EDIT: Apologies for missing the crucial compactness hypothesis.






                    share|cite|improve this answer











                    $endgroup$












                    • $begingroup$
                      Lovely example, but a hyperboloid isn't compact, right?
                      $endgroup$
                      – Peter Kagey
                      50 mins ago










                    • $begingroup$
                      Oops. Sloppy reading. I'll delete.
                      $endgroup$
                      – Ted Shifrin
                      49 mins ago










                    • $begingroup$
                      It's a nice example; you should leave it.
                      $endgroup$
                      – Peter Kagey
                      49 mins ago













                    2












                    2








                    2





                    $begingroup$

                    If you analyze the geodesics using Clairaut's relation, you'll find that the only closed geodesic on a hyperboloid of one sheet is the central circle. Indeed, the same holds for a concave surface of revolution of the same "shape" as the hyperboloid of one sheet.



                    EDIT: Apologies for missing the crucial compactness hypothesis.






                    share|cite|improve this answer











                    $endgroup$



                    If you analyze the geodesics using Clairaut's relation, you'll find that the only closed geodesic on a hyperboloid of one sheet is the central circle. Indeed, the same holds for a concave surface of revolution of the same "shape" as the hyperboloid of one sheet.



                    EDIT: Apologies for missing the crucial compactness hypothesis.







                    share|cite|improve this answer














                    share|cite|improve this answer



                    share|cite|improve this answer








                    edited 49 mins ago

























                    answered 52 mins ago









                    Ted ShifrinTed Shifrin

                    65k44792




                    65k44792











                    • $begingroup$
                      Lovely example, but a hyperboloid isn't compact, right?
                      $endgroup$
                      – Peter Kagey
                      50 mins ago










                    • $begingroup$
                      Oops. Sloppy reading. I'll delete.
                      $endgroup$
                      – Ted Shifrin
                      49 mins ago










                    • $begingroup$
                      It's a nice example; you should leave it.
                      $endgroup$
                      – Peter Kagey
                      49 mins ago
















                    • $begingroup$
                      Lovely example, but a hyperboloid isn't compact, right?
                      $endgroup$
                      – Peter Kagey
                      50 mins ago










                    • $begingroup$
                      Oops. Sloppy reading. I'll delete.
                      $endgroup$
                      – Ted Shifrin
                      49 mins ago










                    • $begingroup$
                      It's a nice example; you should leave it.
                      $endgroup$
                      – Peter Kagey
                      49 mins ago















                    $begingroup$
                    Lovely example, but a hyperboloid isn't compact, right?
                    $endgroup$
                    – Peter Kagey
                    50 mins ago




                    $begingroup$
                    Lovely example, but a hyperboloid isn't compact, right?
                    $endgroup$
                    – Peter Kagey
                    50 mins ago












                    $begingroup$
                    Oops. Sloppy reading. I'll delete.
                    $endgroup$
                    – Ted Shifrin
                    49 mins ago




                    $begingroup$
                    Oops. Sloppy reading. I'll delete.
                    $endgroup$
                    – Ted Shifrin
                    49 mins ago












                    $begingroup$
                    It's a nice example; you should leave it.
                    $endgroup$
                    – Peter Kagey
                    49 mins ago




                    $begingroup$
                    It's a nice example; you should leave it.
                    $endgroup$
                    – Peter Kagey
                    49 mins ago

















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