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Julia E
Julia E
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Does there exist a compact Riemannan manifold $M^n$ and an $L > 0$ such that the number of homotopy classes of simple closed curves $\gamma$ on $M^n$ whose geodesicshortest representatives have length at most $L$ is infinite? For surfaces ($n=2$) with constant curvature metrics, this is impossible. Thanks!

Does there exist a compact Riemannan manifold $M^n$ and an $L > 0$ such that the number of homotopy classes of simple closed curves $\gamma$ on $M^n$ whose geodesic representatives have length at most $L$ is infinite? For surfaces ($n=2$) with constant curvature metrics, this is impossible. Thanks!

Does there exist a compact Riemannan manifold $M^n$ and an $L > 0$ such that the number of homotopy classes of simple closed curves $\gamma$ on $M^n$ whose shortest representatives have length at most $L$ is infinite? For surfaces ($n=2$) with constant curvature metrics, this is impossible. Thanks!

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Julia E
Julia E
  • 43
  • 3

Geodesics of bounded length on a Riemannian manifold

Does there exist a compact Riemannan manifold $M^n$ and an $L > 0$ such that the number of homotopy classes of simple closed curves $\gamma$ on $M^n$ whose geodesic representatives have length at most $L$ is infinite? For surfaces ($n=2$) with constant curvature metrics, this is impossible. Thanks!