Manifolds with positive sectional curvature almost everywhere.
The following deformation conjecture is mentioned in a paper of \textit{D. Gromoll} and \textit{W. Meyer} [Ann. Math. 100, 401-406 (1974; Zbl 0293.53015)]: Let \((M,g)\) be a complete Riemannian manifold with nonnegative sectional curvature and suppose that there is one point \(p\) in \(M\) such that all planes based at \(p\) have positive curvature. Then \(M\) admits a metric with positive sectional curvature. In the paper under review, the author mentions previous work related to the deformation conjecture, due to several people. He also mentions the relation with another conjecture, known as the \textit{Hopf conjecture}, which states that the product of spheres \(S^n\times S^m\) does not admit a Riemannian metric with positive sectional curvature. The main result of this paper is a series of examples of compact manifolds which admit a Riemannian metric with positive sectional curvature on an open dense set of points. These examples provide counterexamples to the deformation conjecture. In particular, the author shows that this conjecture is wrong in dimensions \(4n+1\) for all \(n\geq 1\), as well as in dimensions 6, 12 and 24. He proves that each of the manifolds \(\mathbb{R} P^3 \times \mathbb{R} P^2\) and \(\mathbb{R} P^7\times \mathbb{R} P^6\) as well as their universal covers \(S^3 \times S^2\) and \(S^7 \times S^6\) admit a metric with positive sectional curvature on dense open subsets. Another result in this paper is that there are infinitely many different homotopy types of simply connected \((4n-1)-\)dimensional manifolds with positive sectional curvature on open dense sets, for all \(n\geq 3\). The paper also contains a list of old and new open problems concerning the subject.
scientific article; zbMATH DE number 1777240