Extrinsic geometry of the Gromoll-Meyer sphere

Almost Positive Curvature on the Gromoll-meyer Sphere

Gromoll and Meyer have represented a certain exotic 7-sphere Σ 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 has strictly positive sectional curvature at all points outside four subvarieties of codimension ≥ 1 which we describe explicitly.

A Minimal Brieskorn 5-sphere in the Gromoll-meyer Sphere and Its Applications

We recognize the Gromoll-Meyer sphere Σ as the geodesic join of a simple closed geodesic and a minimal subsphere Σ ⊂ Σ, which can be equivariantly identified with the Brieskorn sphere W 5 3 . As applications we in particular determine the full isometry group of Σ, classify all closed subgroups which act freely, determine the homotopy types of the corresponding orbit spaces, identify the Hirsch-...

An Infinite Family of Gromoll-meyer Spheres

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.

Extrinsic sphere and totally umbilical submanifolds in Finsler spaces

‎Based on a definition for circle in Finsler space‎, ‎recently proposed by one of the present authors and Z‎. ‎Shen‎, ‎a natural definition of extrinsic sphere in Finsler geometry is given and it is shown that a connected submanifold of a Finsler manifold is totally umbilical and has non-zero parallel mean curvature vector field‎, ‎if and only if its circles coincide with circles of the ambient...

Extrinsic Differential Geometry