Examples of Riemannian manifolds with positive curvature almost everywhere. (English) Zbl 0963.53020
Let \(M\) be a manifold with quasi-positive sectional curvature (that is, with sectional curvature nonnegative everywhere and positive at a point). Can \(M\) be deformed to one with positive curvature almost everywhere? And if \(M\) is of positive sectional curvature almost everywhere, can it be deformed to one with positive curvature? The answers are not known. In this paper, the authors prove several theorems that enlighten these problems. The main result can be stated as follows:
“The unit tangent bundle of \(S^4\) admits a metric with positive sectional curvature at almost every point with the following properties.
(1) The connected component of the identity of the isometry group is isomorphic to \(SO(4)\) and contains a free \(S^3\)-subaction.
(2) The set of points where there are 0 sectional curvatures contains totally geodesic flat 2-tori and is the union of two copies of \(S^3\times S^3\) that intersect along \(S^2\times S^3\).”
In particular, by taking a circle subgroup of the free \(S^3\)-action in (1), the following corollary can be obtained:
“There is a manifold \(M^6\) with the homology of \(\mathbb{C} P^3\) but not the cohomology ring of \(\mathbb{C} P^3\), that admits a metric with positive sectional curvature almost everywhere”.
“The unit tangent bundle of \(S^4\) admits a metric with positive sectional curvature at almost every point with the following properties.
(1) The connected component of the identity of the isometry group is isomorphic to \(SO(4)\) and contains a free \(S^3\)-subaction.
(2) The set of points where there are 0 sectional curvatures contains totally geodesic flat 2-tori and is the union of two copies of \(S^3\times S^3\) that intersect along \(S^2\times S^3\).”
In particular, by taking a circle subgroup of the free \(S^3\)-action in (1), the following corollary can be obtained:
“There is a manifold \(M^6\) with the homology of \(\mathbb{C} P^3\) but not the cohomology ring of \(\mathbb{C} P^3\), that admits a metric with positive sectional curvature almost everywhere”.
Reviewer: Pascual Lucas Saorín (Murcia)
MSC:
| 53C20 | Global Riemannian geometry, including pinching |
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