Global and Local Volume Bounds and the Shortest Geodesic Loops
Communications in analysis and geometry
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  the isoembolic theorem Vol(M ) ≥ C n inj(M ) n for all complete Riemannian manifolds
... iemannian manifolds with a sharp positive constant C n . A local version was then established by C. Croke in  VolB(x 0 , R) ≥ C n R n for all R ≤ 1 2 inj(M ) 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 . In the same way, an extension of C. Croke's local version was then established by R. Greene and P. Petersen in . 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 , 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 that an n-dimensional manifold M is by definition k-essential if there exists a continuous map ψ : M −→ K into a K(π, k) space such Extension process and short geodesic loops. In this part, we construct a map extension process and prove Theorem A.