A co-dimensional 3 sub-Riemannian structure on Gromoll-Meyer exotic sphere

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

We recognize the Gromoll-Meyer sphere Sigma^7 as the geodesic join of a simple closed geodesic and a minimal subsphere Sigma^5, which can be equivariantly identified with the Brieskorn sphere W^5_3. As applications we in particular determine the full isometry group of Sigma^7, classify all closed subgroups that act freely, determine the homotopy type of the corresponding orbit spaces, identify the Hirsch-Milnor involution in dimension 5 with the Calabi involution of W^5_3, and obtain explicit formulas for diffeomorphisms between the Brieskorn spheres W^5_3 and W^13_3 with standard Euclidean sp…

Extrinsic geometry of the Gromoll-Meyer sphere

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 quotiente $\widetilde{N}^{11}=(Sp(2)\times S^4)/S^3$, we construct a homogeneous isoparametric foliation with isoparametric hypersurfaces diffeomorphic to Sp(2). Furthermore, on the quotiente $\widetilde{N}^{11}/S^3$, 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…

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。

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 …

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.

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.

Sub-Riemannian geodesics on nested principal bundles

We study the interplay between geodesics on two non-holono\-mic systems that are related by the action of a Lie group on them. After some geometric preliminaries, we use the Hamiltonian formalism to write the parametric form of geodesics. We present several geometric examples, including a non-holonomic structure on the Gromoll-Meyer exotic sphere and twistor space.

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) $RP^7$.

一个外来7维球面的曲率

Curvature of an exotic 7-sphere

本文利用 Kaluza-Klein 构造研究 Gromoll-Meyer 外来7球面，结合 $k=1,2$ 的 $SU(2)$ 瞬子得到最大对称度规并计算 Ricci 张量，给出正曲率半径界及截面曲率分析基础。

通过$\star$-图对球面T-对偶性的几何实现

A Geometric Realization of Spherical T-Duality via $\star$-Diagrams

研究$\mathrm{S}^4$上$\mathrm{S}^3$丛的球面T-对偶性，将其与$\star$-图及对数变换推广联系，证明T-对偶对应空间是双自由$\star$-图，给出相同欧拉类丛的T-对偶情况，引入保积广义对数变换实现同伦7-球面间T-对偶性。