Lower curvature bounds and cohomogeneity one manifolds.
\textit{K. Grove} and \textit{W. Ziller} [Ann. Math. (2) 152, 331--367 (2000; Zbl 0991.53016)] discovered a class of new examples of closed manifolds with nonnegative sectional curvature. All these manifolds admit a cohomogeneity one action, i.e. a smooth action of a compact Lie group whose principal orbit has codimension one. They conjectured that any cohomogeneity one manifold admits an invariant metric with nonnegative sectional curvature. Moreover, \textit{W. Tuschmann} and \textit{L. J. Schwachhöfer} [Curvature and cohomogeneity one'', preprint 62, Max Planck Inst., 2001, available at \texttt{http://www.mis.mpg.de/preprints/2001/prepr6201-abstr.html}] showed that any cohomogeneity one manifold admits invariant metrics of almost nonnegative sectional curvature. In the paper under review the author considers examples which give positive answer to the above conjecture. In particular he discusses metrics on homogeneous vector bundels, generalizing some ideas of J. Cheeger, and discusses when such bundles admit invariant metrics with normal homogeneous collar. Moreover, examples of manifolds with nonnegative or almost nonegative sectional curvature are discussed. Some survey of known obstructions for the existance of metrics of almost nonnegative curvature is also given.
scientific article; zbMATH DE number 1959901
scientific article