The Morse index for manifolds with constant sectional curvature. (English) Zbl 1540.58015
Summary: We compute the Morse index of a critical submanifold of the energy functional on the loop space of a manifold with constant sectional curvature. The case of constant non-positive sectional curvature is a known result and the case of a sphere has been proved by W. Klingenberg [Q. J. Math., Oxf. II. Ser. 20, 11–31 (1969; Zbl 0177.26202); Lectures on closed geodesics. Springer, Cham (1978; Zbl 0397.58018)]. We adapt Klingenberg’s proof of the case of a sphere to the case of constant sectional curvature, to obtain the possible Morse indices of critical submanifolds of the energy functional.
MSC:
| 58J20 | Index theory and related fixed-point theorems on manifolds |
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
| 53C20 | Global Riemannian geometry, including pinching |
| 53C40 | Global submanifolds |
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