A minimal Brieskorn 5-sphere in the Gromoll-Meyer sphere and its applications

Gromoll-Meyer球面上正曲率的两阶段构造

A Two-Stage Construction of Positive Curvature on the Gromoll-Meyer Sphere

本文通过两阶段扰动构造了Gromoll-Meyer怪球面上收敛到Cheeger形变的显式单参数度量族，在足够小参数下实现严格正截面曲率，由AI代理发现。

Gromoll-Meyer球面上的一个正曲率度量

A positively curved metric on the Gromoll-Meyer sphere

该文在Sp(2)上构造双不变度量的显式形变，使Gromoll-Meyer exotic 7-球面在充分小正参数下获得正截面曲率，证明结合均匀估计与代数分析，并给出临界集上三次系数的正下界。

Gromoll--Meyer的作用与（奇异）时空的几何

Gromoll--Meyer's actions and the geometry of (exotic) spacetimes

本文提出一种计算方法，在经典和奇异球面上等变构造物理模型，首次在同胚但非微分同胚流形上实现具有相同物理性质的洛伦兹度量，并展望推广到更高维奇异球面。

Isoparametric functions and exotic spheres

The first part of the paper is to improve the fundamental theory of isoparametric functions on general Riemannian manifolds. Next we focus our attention on exotic spheres, especially on "exotic" 4-spheres (if exist) and the Gromoll-Meyer sphere. In particular, as one of main results we prove: there exists no properly transnormal function on any exotic 4-sphere if it exists. Furthermore, by projecting an $S^3$-invariant isoparametric function on $Sp(2)$, we construct a properly transnormal but not an isoparametric function on the Gromoll-Meyer sphere with two points as the focal varieties.

On the Stochastic Processes on $7$-Dimensional Spheres

We studied isometric stochastic flows of a Stratonovich stochastic differential equation on spheres, i.e. on the standard sphere and Gromoll-Meyer exotic sphere. The standard sphere $S^7_s$ can be constructed as the quotient manifold $\mathrm{Sp}(2, \mathbb{H})/S^3$ with the so-called ${\bullet}$-action of $S^3$, whereas the Gromoll-Meyer exotic sphere $Σ^7_{GM}$ as the quotient manifold $\mathrm{Sp}(2, \mathbb{H})/S^3$ with respect to the so-called ${\star}$-action of $S^3$. The Stratonovich stochastic differential equation which describes a continuous-time stochastic process on the standard …

An exotic sphere with positive sectional curvature

We show that there is a metric on the Gromoll-Meyer sphere with positive sectional curvature.

Some Principles for Deforming Nonnegative Curvature

We describe a few abstract principles that are used in the deformation of the Gromoll-Meyer metric to positive curvature.

一个物理学家对奇异球面的访问

A Physicist's Visit to Exotic Spheres

本文通过物理理论工具探讨奇异7球面的几何结构，推导了Gromoll-Meyer球面的Riemannian度量，并分析其曲率性质，最后提出基于机器学习的求解Einstein度量的算法。

An exotic sphere with positive curvature

A metric with positive sectional curvature on the Gromoll-Meyer exotic 7-sphere is constructed explicitly. The proof relies on a 2-parameter family of left invariant metrics on Sp(2) and a one-parameter family of conformal deformations via an isoparametric function F on it. One byproduct is a metric with positive sectional curvature on a homotopy (but not diffeomorphic) $RP^7$.

On the existence of infinitely many closed geodesics on non-compact manifolds

We prove that any complete (and possibly non-compact) Riemannian manifold $M$ possesses infinitely many closed geodesics provided its free loop space has unbounded Betti numbers in degrees larger than the dimension of $M$, and there are no close conjugate points at infinity. Our argument builds on an existence result due to Benci and Giannoni, and generalizes the celebrated theorem of Gromoll and Meyer for closed manifolds.

Morse theory, closed geodesics, and the homology of free loop spaces

This is a survey paper on Morse theory and the existence problem for closed geodesics. The free loop space plays a central role, since closed geodesics are critical points of the energy functional. As such, they can be analyzed through variational methods. The topics that we discuss include: Riemannian background, the Lyusternik-Fet theorem, the Lyusternik-Schnirelmann principle of subordinated classes, the Gromoll-Meyer theorem, Bott's iteration of the index formulas, homological computations using Morse theory, $SO(2)$- vs. $O(2)$-symmetries, Katok's examples and Finsler metrics, relations t…

Cheeger manifolds and the classification of biquotients

A closed manifold is called a biquotient if it is diffeomorphic to K\G/H for some compact Lie group G with closed subgroups K and H such that K acts freely on G/H. Biquotients are a major source of examples of Riemannian manifolds with nonnegative sectional curvature. We prove several classification results for biquotients: (1) We classify all simply connected rational homology spheres which are diffeomorphic to biquotients. For example, the Gromoll-Meyer exotic sphere is the only exotic sphere of any dimension which is a biquotient. (2) We determine exactly which Cheeger manifolds, the connec…