Abstract
It is proved that if at every point of a closed, three-dimensional, Riemannian manifold with bounded sectional curvature the injectivity radius does not exceed a specific absolute constant, then the manifold is a special graph and its metric splits locally.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
T. Soma, "The Gromov invariant of links," Inven. Math.,64 445–454 (1981).
P. Scott, Geometry on Three-Dimensional Manifolds [Russian translation], Moscow (1986).
V. G. Borodin, Three-dimensional manifolds of nonpositive curvature with small injectivity radius [in Russian], Leningrad (1988). Manuscript deposited in VINITI, Dep. No. 5264-B88.
J. Chelger and M. Gromov, "Collapsing Riemannian manifolds while keeping their curvature constant," J. Diff. Geom.,23 No. 3, 309–346 (1986).
W. Ballmann, M. Gromov, and V. Schroeder, Manifolds of Nonpositive Curvature, Progress in Mathematics,61, Birkhäuser, Boston (1985).
D. B. A. Epstein, "Periodic flows on 3-manifolds," Ann. Math.,95 66–82 (1972).
Additional information
Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 10–19, 1991.
Rights and permissions
About this article
Cite this article
Borodin, V.G. Three-dimensional manifolds of nonpositive curvature with small injectivity radius. J Math Sci 69, 831–836 (1994). https://doi.org/10.1007/BF01250810
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01250810