L. Biliotti, F. Mercuri, P. Piccione
Following the lines of the celebrated Riemannian result of Gromoll and Meyer, we use infinite dimensional equivariant Morse theory to establish the existence of infinitely many geometrically distinct closed geodesics in a class of globally hyperbolic stationary Lorentzian manifolds.
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Gromoll and Meyer have represented a certain exotic 7-sphere $Σ^7$ as a biquotient of the Lie group $G = Sp(2)$. We show for a 2-parameter family of left invariant metrics on $G$ that the induced metric on $Σ^7$ has strictly positive sectional curvature at all points outside four subvarieties of codimension $\geq 1$ which we describe explicitly.
We construct a co-dimension $3$ completely non-holonomic sub-bundle on the Gromoll-Meyer exotic $7$ sphere based on its realization as a base space of a Sp(2)-principal bundle with the structure group Sp(1). The same method is valid for constructing a co-dimension 3 completely non-holonomic sub-bundle on the standard 7 sphere (or more general on a $4n+3$ dimensional standard sphere). In the latter case such a construction based on the Hopf bundle is well-known. Our method provides an alternated simple proof for the standard sphere $\mathbb{S}^7$.
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。
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。
We construct a new infinite family of models of exotic 7-spheres. These models are direct generalizations of the Gromoll-Meyer sphere. From their symmetries, geodesics and submanifolds half of them are closer to the standard 7-sphere than any other known model for an exotic 7-sphere.
Here we generalize the Gromoll-Meyer construction of an exotic 7-sphere by producing geometric models of exotic 8, 10 and Kervaire spheres as quotients of sphere bundles over spheres by free isometric actions. We give a geometric application at the end.
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…
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'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。
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 …
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.
We show that there is a metric on the Gromoll-Meyer sphere with positive sectional curvature.
We describe a few abstract principles that are used in the deformation of the Gromoll-Meyer metric to positive curvature.
In this paper we prove several multiplicity results of $t$-periodic light rays in conformally stationary spacetimes using the Fermat metric and the extensions of the classical theorems of Gromoll-Meyer and Bangert-Hingston to Finsler manifolds. Moreover, we exhibit some stationary spacetimes with a finite number of $t$-periodic light rays and compute a lower bound for the period of the light rays when the flag curvature of the Fermat metric is $η$-pinched.
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。
We prove the shifting theorems of the critical groups of critical points and critical orbits for the energy functionals of Finsler metrics on Hilbert manifolds of $H^1$-curves, and two splitting lemmas for the functionals on Banach manifolds of $C^1$-curves. Two results on critical groups of iterated closed geodesics are also proved; their corresponding versions on Riemannian manifolds are based on the usual splitting lemma by Gromoll and Meyer (1969). Our approach consists in deforming the square of the Finsler metric in a Lagrangian which is smooth also on the zero section and then in using …
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