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Closed geodesics in simply connected Riemannian spaces of negative curvature. (English. Russian original) Zbl 0960.53025

Sib. Math. J. 41, No. 5, 880-883 (2000); translation from Sib. Mat. Zh. 41, No. 5, 1076-1080 (2000).
It is known that if a Riemannian variety of negative sectional curvature is homeomorphic to the 2-dimensional plane then there is no closed geodesic on this variety. Examples are known of Riemannian varieties of negative sectional curvature which are homeomorphic to \(\mathbb R^n\), \(n>2\), and have closed geodesics. In the article under review, the author obtains Riemannian varieties with sectional curvature \(\leq -1\) which are diffeomorphic to \(\mathbb R^n\) and have closed geodesics.

MSC:

53C22 Geodesics in global differential geometry
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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References:

[1] Pogorelov A. V., Differential Geometry [in Russian], Nauka, Moscow (1974).
[2] Ionin V. K., ”Isoperimetric inequalities in simply connected Riemannian spaces of nonpositive curvature,” Dokl. Akad. Nauk SSSR,203, No. 2, 282–284 (1972). · Zbl 0258.52011
[3] Einsenhart L. P., Riemannian Geometry [Russian translation], Izdat. Inostr. Lit., Moscow (1948).
[4] Bishop R. L. andO’Neill B., ”Manifolds of negative curvature,” Trans. Amer. Math. Soc.,145, 1–49 (1969). · Zbl 0191.52002 · doi:10.1090/S0002-9947-1969-0251664-4
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