Isoparametric functions on exotic spheres

Qian, Chao; Tang, Zizhou

doi:10.1016/j.aim.2014.12.020

Cited by 44 publications

(41 citation statements)

References 28 publications

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“…At first, observe that the function f : E → R, (p, v) → v, v is a transnormal function(cf. Theorem 2.2 in [QT15]). Next, we need to compute the principal curvatures of S r (ξ) ⊂ E for any r > 0.…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

Ricci curvature of double manifolds via isoparametric foliations

Peng

Qian

2017

Advances in Mathematics

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Given a closed manifold M and a vector bundle ξ of rank n over M, by gluing two copies of the disc bundle of ξ, we can obtain a closed manifold D(ξ, M), the so-called double manifold.In this paper, we firstly prove that each sphere bundle S r (ξ) of radius r > 0 is an isoparametric hypersurface in the total space of ξ equipped with a connection metric, and for r > 0 small enough, the induced metric of S r (ξ) has positive Ricci curvature under the additional assumptions that M has a metric with positive Ricci curvature and n ≥ 3.As an application, if M admits a metric with positive Ricci curvature and n ≥ 2, then we construct a metric with positive Ricci curvature on D(ξ, M). Moreover, under the same metric, D(ξ, M) admits a natural isoparametric foliation.For a compact minimal isoparametric hypersurface Y n in S n+1 (1), which separates S n+1 (1) into S n+1 + and S n+1 − , one can get double manifolds D(S n+1 + ) and D(S n+1 − ). Inspired by Tang, Xie and Yan's work on scalar curvature of such manifolds with isoparametric foliations(cf.[TXY12]), we study Ricci curvature of them with isoparametric foliations in the last part.2010 Mathematics Subject Classification. 53C12, 53C40, 53C07.

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Section: Proof Of Theorem 12mentioning

confidence: 99%

Ricci curvature of double manifolds via isoparametric foliations

Peng

Qian

2017

Advances in Mathematics

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“…[20]). Moreover, if E ± are of rank greater than 1 and M ± are closed, (E ϕ , F ϕ ) can become isoparametric by a more careful choice of the one-parameter family of metrics on ∂E + as shown in [20].…”

Section: Disk Bundle Decomposition By Singular Riemannian Foliationmentioning

confidence: 99%

Isoparametric foliations, diffeomorphism groups and exotic smooth structures

2016

Advances in Mathematics

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Abstract. In this paper, we are concerned with interactions between isoparametric theory and differential topology. Two foliations are called equivalent if there exists a diffeomorphism between the foliated manifolds mapping leaves to leaves. Using differential topology, we obtain several results towards the classification problem of isoparametric foliations up to equivalence. In particular, we show that each homotopy n-sphere has the "same" isoparametric foliations as the standard sphere S n has except for n = 4, reducing the classification problem on homotopy spheres to that on the standard sphere. Moreover, we prove the uniqueness up to equivalence of isoparametric foliations with two points as the focal submanifolds on each sphere S n except for n = 5. Besides, we show that the uniqueness holds on S 5 if and only if π0(Diff(S 4 )) ≃ Z2, i.e., pseudo-isotopy implies isotopy for diffeomorphisms on S 4 . At last, some ideas behind the proofs enable us to discover new exotic smooth structures on certain manifolds.

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“…Noticing that F is a Morse-Bott function on N 8 with critical set S 4 ⊔ S 4 , we remark that according to the fundamental construction in Theorem 1.1 of [QT15], there exists a metric on N 8 such that F is an isoparametric function and the the focal submanifolds are still S 4 and totally geodesic. However, one cannot know more about the intrinsic geometric properties of N 8 and the isoparametric hypersurfaces.…”

Section: Introductionmentioning

confidence: 96%

Extrinsic geometry of the Gromoll-Meyer sphere

Qian

Tang

Yan

2020

Differential Geometry and its Applications

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Among a family of 2-parameter left invariant metrics on Sp(2), we determine which have nonnegative sectional curvatures and which are Einstein. On the quotient N 11 = (Sp(2) × S 4 )/S 3 , we construct a homogeneous isoparametric foliation with isoparametric hypersurfaces diffeomorphic to Sp(2). Furthermore, on the quotient N 11 /S 3 , we construct a transnormal system with transnormal hypersurfaces diffeomorphic to the Gromoll-Meyer sphere Σ 7 . Moreover, the induced metric on each hypersurface has positive Ricci curvature and quasi-positive sectional curvature simultaneously.2010 Mathematics Subject Classification. 53C12, 53C20, 53C40.

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Isoparametric functions on exotic spheres

Cited by 44 publications

References 28 publications

Ricci curvature of double manifolds via isoparametric foliations

Ricci curvature of double manifolds via isoparametric foliations

Isoparametric foliations, diffeomorphism groups and exotic smooth structures

Extrinsic geometry of the Gromoll-Meyer sphere

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