Codimension three nonnegatively curved submanifolds with infinite fundamental group
The authors investigate conditions on nonnegatively curved submanifolds of codimension three that imply the nonnegativity of the curvature operator. Their main result (Theorem 1.1) states that, if \(M^n \to {\mathbb{R}}^{n+3}\) is an isometric immersion of a complete Riemannian manifold \(M\) with nonnegative sectional curvature, then {\parindent6.5mm \begin{itemize}\item[(i)] if \(M\) has a line and \(f\) is noncylindrical (i.e., it does not split as an extrinsic product \(M_1\times {\mathbb{R}}^m\)), then \(M\) has nonnegative curvature operator; \item[(ii)] if \(M\) contains no lines and is covered by \(\widetilde M \times {\mathbb{R}}\), then the curvature operator of \(\widetilde M\) (and of \(M\)) is nonnegative. \end{itemize}} A result of \textit{J. Cheeger} and \textit{D. Gromoll} [Ann. Math. (2) 96, 413--443 (1972; Zbl 0246.53049)] states that if the fundamental group of a compact manifold of nonnegative curvature is infinite then its universal covering splits isometrically as \(\widetilde M \times {\mathbb{R}}^k\), for \(k > 0\). Using this, the authors classify codimension-three isometric immersions of non-flat compact manifolds with nonnegative sectional curvatures and infinite fundamental group up to diffeomorphism.
scientific article; zbMATH DE number 5859321
scientific article