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Simple closed geodesics on convex surfaces. (English) Zbl 0768.53019

First, various estimates for lengths of simple closed geodesics (a geodesic is said to be simple if it has no self-intersections) in terms of the diameter \(D\), total area \(A\), and upper bound of curvature of a given smooth Riemannian surface \(M^ 2\) diffeomorphic to \(S^ 2\) are established. For example, if \(L\) denotes the length of the longest simple closed geodesic on such an \(M^ 2\) of class \(C^ 3\) and with curvature \(0 \leq K \leq 1\), then \(L \leq A/2\). This estimate is sharp and is achieved if and only if \(M^ 2\) is isometric to the unit sphere. On the other hand, the authors prove that the isoperimetric inequality Area\((M^ 2) \leq {8\over \pi}D^ 2\) is valid for any \(M^ 2\) with \(K \geq 0\). Second, it is shown that any nontrivial closed geodesic of the shortest length is simple on any \(C^ 3\)-smooth Riemannian 2-sphere of nonnegative curvature. The proof of this result introduces a completely different approach in comparison with the one given in order to prove the Lusternik-Schnirelmann theorem [see W. Ballmann, Bonn. Math. Schr. 102, 1-25 (1978; Zbl 0394.53027) or M. Grayson, Ann. Math., II. Ser. 129, No. 2, 71-111 (1989; Zbl 0686.53036)] or the one suggested by H. Poincaré and carried out correctly by C. Croke [J. Differ. Geom. 17, 595-634 (1980; Zbl 0501.53031)]. To illustrate the usefulness of these results, some applications are pointed out.

MSC:

53C22 Geodesics in global differential geometry
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