Curvature and symmetry of Milnor spheres

By K. Grove, W. Ziller. 2000

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Since Milnor’s discovery of exotic spheres [Mi], one of the most intriguing problems in Riemannian geometry has been whether there are exotic spheres with positive curvature. It is well known that there are exotic spheres that do not even admit metrics with positive scalar curvature [Hi]. On the other hand, there are many examples of exotic spheres with positive Ricci curvature (cf. [Ch1], [He], [Po], and [Na]) and this work recently culminated in [Wr] where it is shown that every exotic sphere that bounds a parallelizable manifold has a metric of positive Ricci curvature. This includes all exotic spheres in dimension 7. So far, however, no example of an exotic sphere with positive sectional curvature has been found. In fact, until now, only one example of an exotic sphere with nonnegative sectional curvature was known, the so-called Gromoll-Meyer sphere [GM] in dimension 7. As one of our main results we prove:

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/397931fcf81faef4d6f0818ac413a9e117572d25

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Nonnegatively and Positively curved manifolds (2007, background): “9 (Grove and Ziller). Any principal SO(n)-bundle over S(4) admits a cohomogeneity one action of S(3) × SO(n) with singular orbits of codimension 2. The proof uses the classification of bundles over S(4) in terms of characteristic classes. Grove and Ziller endow S(4) with the unique cohomogeneity one”
Homeomorphism classification of positively curved manifolds with almost maximal symmetry rank (2003, background): “the metric product of the unit spheres, S 3 × S 3 and S 3 × S 4 . By [GZ], each S 3 -bundle over S 4 admits infinitely many non-negatively curved metrics.”
Cheeger Manifolds and the Classification of Biquotients (2002, background): “…the distance between biquotients and general manifolds of nonnegative sectional curvature, since Grove and Ziller have constructed nonnegatively curved metrics on all S3-bundles over S4, giving infinitely many homotopy types of 2-connected 7-manifolds with nonnegative sectional curvature [16].”
Positively curved cohomogeneity one manifolds and 3-Sasakian geometry (2005, background): “In section 12, we will see that the above construction, as well as the principal bundle construction for cohomogeneity one manifolds in [ GZ ], naturally carries over to a large class within the more general context of orbifolds.”
Examples of Riemannian Manifolds with non-negative sectional curvature (2007, background): “We have the following existence theorem [42]:”
Cohomogeneity one manifolds with positive Ricci curvature (2002, background): “In [ GZ ] we conjectured that any cohomogeneity one manifold admits a metric with nonnegative sectional curvature (conrmed in the setting of almost nonnegative curvature in [ST]).”
Positively Curved Manifolds with Almost Maximal Symmetry Rank (2002, background): “[AW], [Be], [Ber], [Wa], compare [GS2], [ GZ ]) and the diffeomorphic classification of positively curved manifolds with the maximal symmetry rank, cf. [GS1].”
Metrics of positive Ricci curvature on quotient spaces (2003, background): “For m = 2, the first class was already considered in ([ GZ1 ], Corollary 3.9) where it was shown that the manifolds N 6 d are S 2 -bundles over S4 and admit metrics of even nonnegative sectional curvature.”