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Upper bounds on the length of the shortest closed geodesic on simply connected manifolds. (English) Zbl 0960.53026

In this paper the author proves three theorems relating the length of the shortest closed geodesic on a simply connected Riemannian manifold either to the diameter or to the volume of the manifold. The first estimate explicitly depends on the volume and an upper bound for the sectional curvature. More precisely, let \(M^n\) be a simply connected compact Riemannian manifold with a non-trivial second homology group, of sectional curvature \(K\leq 1\) and volume \(\leq V\). Then the length of the shortest closed geodesic \(\gamma(t)\) on the manifold \(M^n\) is bounded from above by \[ g(V,n)=(c_1(n)(V+1))^{c_2(n)(V+1)}, \] where \(c_1(n)=10^4(n!)^3\), \(c_2(n)=10^5(n!)^3\). The second estimate depends on the diameter, a positive lower bound for the volume and on a lower bound for the sectional curvature. The result is stated as follows. Let \(M^n\) be a simply connected compact Riemannian manifold with a non-trivial second homology group, of sectional curvature \(K\geq -1\), volume \(\geq v>0\) and diameter \(d\leq D\). Then the length of the shortest closed geodesic \(\gamma(t)\) on the manifold \(M^n\) is bounded from above by \[ f(n,D,v)=\exp({{e^{c_4(n)D}}\over{\min\{1,v\}^{c_5(n)}}}), \] where the constants \(c_4(n)\) and \(c_5(n)\) can be explicitly calculated. Finally, the third estimate depends on the diameter, on a lower bound for the sectional curvature and on a lower bound for the simply connectedness radius. More precisely, let \(M^n\) be a simply connected compact Riemannian manifold with a non-trivial second homology group, of sectional curvature \(K\geq -1\), and diameter \(d\leq D\). Assume that all metric balls of radius \(\leq c\) in \(M^n\) are simply connected. Then the length of the shortest closed geodesic \(\gamma(t)\) on the manifold \(M^n\) is bounded from above by a certain function \(h(n,D,c)\) of \(n\), \(D\) and \(c\). If we assume, in addition, that every closed curve \(\gamma\) in a metric ball of radius less than \(c\) can be contracted to a point inside this ball by a homotopy which contains only closed curves of length less than an explicitly given function of the length of \(\gamma\), then one is able to write down an explicit formula for \(h(n,D,c)\).

MSC:

53C22 Geodesics in global differential geometry
53C20 Global Riemannian geometry, including pinching
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