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Global and Local Volume Bounds and the Shortest Geodesic Loops

2004
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Communications in analysis and geometry
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Introduction. The relations between the volume of complete Riemannian manifolds and the length of their shortest nontrivial geodesic loops under no curvature assumption are studied in this paper. We present and compare two lower bounds on the global volume, one of which admits a local version. Previous curvature-free estimates have been first obtained with the injectivity radius. Namely, M. Berger proved in [3] the isoembolic theorem Vol(M ) ≥ C n inj(M ) n for all complete Riemannian manifolds

doi:10.4310/cag.2004.v12.n5.a3
fatcat:oxw2hn2ckbfkrgn7vdgz633fsy