Related papers: An exotic sphere with positive sectional 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…

Gromoll and Meyer have represented a certain exotic 7-sphere $\Sigma^7$ 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 $\Sigma^7$ has strictly…

We study the geometry of the Gromoll-Meyer sphere, one of Milnor's exotic $7$-spheres. We focus on a Kaluza-Klein Ansatz, with a round $S^4$ as base space, unit $S^3$ as fibre, and $k=1,2$ $SU(2)$ instantons as gauge fields, where all…

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…

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

We examine homogeneous metrics on spheres and determine which ones have positive sectional curvature. The answer is subtle and surprisingly difficult to prove. In some cases we also determine their pinching constants. This completes the…

We prove that there exists a metric of positive curvature in a three-sphere which admits a given torus knot as a closed geodesic.We also sketch a construction of a metric in a four sphere, very likely of positive curvature, which admits a…

Since the advent of new pairwise non-diffeomorphic structures on smooth manifolds, it has been questioned whether two topologically identical manifolds could admit different geometries. Not surprisingly, physicists have wondered whether a…

We consider the blowup of a point of a compact K\"ahler manifold and a metric of the form $\mu^*h + t b$ on it, where $h$ is a K\"ahler metric on the original manifold and $b$ is Hermitian form that looks like the Fubini--Study metric near…

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…

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…

A simple proof is given of the necessary and sufficient condition on a triple of positive numbers A,B,C for the existence of a conformal metric of constant positive curvature on the sphere, with three conic singularities of total angles…

The main result of this note essentially is that if the base and fibers of a compact fibration carry Hermitian metrics of positive holomorphic sectional curvature, then so does the total space of the fibration. The proof is based on the use…

A necessary and sufficient condition for the existence and uniqueness of a conformal metric on 2-sphere of constant curvature 1 and with three conical singularities of prescribed order is given.

We show that any space with a positive upper curvature bound has in a small neighborhood of any point a closely related metric with a negative upper curvature bound.

This is a survey on cohomogeneity one manifolds with positive curvature. We discuss the known examples of this type and their geometry and the functions that describe the metric. We also describe the classification of cohomogeneity one…

In this paper, we give a new generalization of positive sectional curvature called positive weighted sectional curvature. It depends on a choice of Riemannian metric and a smooth vector field. We give several simple examples of Riemannian…

In this paper two metric properties on geodesic length spaces are introduced by means of the metric projection, studying their validity on Alexandrov and Busemann NPC spaces. In particular, we prove that both properties characterize the…

We obtain a conceptually new differential geometric proof of P.F. Klembeck's result that the holomorphic sectional curvature of a strictly pseudoconvex domain approaches (in the boundary limit) the constant sectional curvature of the…

This thesis discusses exotic 7-spheres, i.e. manifolds that are homeomorphic but not diffeomorphic to the ordinary 7-sphere, using a set of analytical and computational tools from theoretical physics. The theory of fibre bundles and…