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Isoparametric functions on exotic spheres. (English) Zbl 1312.53059

This paper is an extension of a previous work belonging to the second author and to J. Ge and Z. Tang [J. Reine Angew. Math. 683, 161–180 (2013; Zbl 1279.53057)], work which was widely reviewed in Zbl Math. The authors establish a lot of results about existence and non-existence of isoparametric functions on exotic spheres and Eells-Kuiper projective planes. To enunciate the main results from this category recall that a smooth function \(f\) on a smooth manifold \(N\) is called a Morse-Bott function or a generalized Morse function if :(i) the critical set defined by the equation \(df=0\) consists of a family of smooth submanifolds, the so-called critical submanifolds; (ii) the Hessian \(H_f\) is a non-degenerate quadratic form in the normal direction of each critical submanifold to \(N\).
{Theorem 1.1} (A fundamental construction). Let \(N\) be a closed connected smooth manifold and \(f\) a Morse-Bott function on \(N\) with critical set \(C(f)=M_{+}\cup M_{-}\), where \(M_{+}\) and \(M_{-}\) are both closed connected submanifolds of codimensions more than 1. Then there exists a metric \(g\) on \(N\) such that \(f\) is an isoparametric function. Moreover, the metric can be chosen so that the critical submanifolds \(M_{+}\) and \(M_{-}\) are both totally geodesic.
{Corollary 1.1}. Every homotopy \(n\)-sphere with \(n>4\) admits a metric and an isoparametric function with 2 points as the focal set.
The authors remark that the result of Corollary 1.1 is in strong contrast to the classification of cohomogenity-one actions on homotopy spheres [E. Straume, Mem. Am. Math. Soc. 569, 93 p. (1996; Zbl 0854.57033)].
{Proposition 1.1}. Let \(\Sigma ^n\) be a homotopy sphere which admits a metric \(g\) and an isoparametric function \(f\) with 2 points as the focal set. Suppose that the level hypersurfaces are all totally umbilical. Then \(\Sigma^n\) is diffeomorphic to \(S^n\).
{Theorem 1.2.} Every odd-dimensional exotic sphere admits no totally isoparametric functions with 2 points as the focal set.
{Proposition 3.2.} For \(m=4\) or 8, each Eells-Kuiper projective plane \(M^{2m}\) admits a metric and an isoparametric function such that one component of the focal set is a single point and the other is diffeomorphic to \(S^m\).
As an application, the authors give the following existence theorem which improves a beautiful result of L. Bérard Bergery [Ann. Inst. Fourier 27, No. 1, 231–249 (1977; Zbl 0309.53038)].
{Theorem 5.1.} On any homotopy sphere \(\Sigma^{2n}\) in \(2\Theta_{2n}\), \(n\geq 3\), there exists a Riemannian metric so that it possesses the \(\mathrm{SC}^p\)-property at two points, say \(m_{+} \) and \(m_{-}\). Furthermore under the same metric, there exists an isoparametric function \(f\) on \(\Sigma^{2n}\) with focal set \(C(f)=\{m_{+},m_{-}\}\). The later property means that \(\Sigma^{2n}\) is locally harmonic at both points \(m_{+}\) and \(m_{-}\).
Here, \(\Theta_l\) denotes the group of oriented homotopy \(l\)-spheres up to relation \(h\)-cobordant, and a Riemannian manifold \((M,g)\) is said to have the \(\mathrm{SC}^p\)-property at a point \(p\in M\) if all geodesics emanating from \(p\) are simply closed geodesics with the same length.
In connection with Proposition 3.2 the authors make the following remark: “Very recently, Z. Tang and W. Zhang [Adv. Math. 254, 41–48 (2014; Zbl 1290.51005)] solved a problem of Berard-Bergery and Besse. That is, they showed that every Eells-Kuiper quaternionic projective plane carries a Riemannian metric with the \(\mathrm{SC}^p\) property for A certain point \(p\). We do not know how to improve our Proposition 3.2. For instance, we do not know whether there is a metric on every Eells-Kuiper quaternionic projective plane with not only the property in Proposition 3.2, but also the \(\mathrm{SC}^p\)-property.”
Reviewer: Ioan Pop (Iaşi)

MSC:

53C20 Global Riemannian geometry, including pinching
57R60 Homotopy spheres, Poincaré conjecture
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References:

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