Global and Local Volume Bounds and the Shortest Geodesic Loops

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 for all complete Riemannian manifolds with a sharp positive constant C_n. A local version was then established by C. Croke in [4] for all complete Riemannian manifolds but with a nonsharp constant. Replacing the notion of injectivity radius with one of local geometric contractibility, M. Gromov extended M. Berger's global volume estimate in [7]. In the same way, an extension of C. Croke's local version was then established by R. Greene and P. Petersen in [6]. Other results have been obtained by M. Gromov and C. Croke, who compared the volume with the length of the shortest nontrivial closed geodesic, noted scg(M). In [7], M. Gromov proved that every 1-essential closed Riemannian manifold M satisfies Vol(M) = C_n scg(M)ⁿ for some positive constant C_n depending only on the dimension n of M. Recall the condition for an n-dimensional manifold M to be k-essential. The homology coefficients are in Z, if M is orientable, and in Z2, otherwise. For the two-dimensional sphere, the previous inequality still holds as it was proved by C. Croke in [5]. These two statements are the only results known in this direction and no counter-example exist so far. Throughout this paper, we will consider geodesic loops rather than closed geodesics. By definition, a geodesic loop (based at a point P) is a geodesic arc with endpoints P. In particular, a closed geodesic is a geodesic loop whose tangent vectors at its endpoints agree. The length of the shortest nontrivial geodesic loop of a Riemannian manifold M is denoted by sgl(M). The first theorem we will prove provides lower bounds on the diameter, on the volume of the whole manifold and on the volume of sufficiently small balls.

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Published 1 January 2004