Related papers: Almost positive curvature on the Gromoll-Meyer sph…

In this paper, firstly we study some left invariant Riemannian metrics on para-hypercomplex 4-dimensional Lie groups. In each Lie group, the Levi-Civita connection and sectional curvature have been given explicitly. We also show these…

we show that the space of metrics of positive scalar curvature on a manifold is, when nonempty, homotopy equivalent to a space of metrics of positive scalar curvature that restrict to a fixed metric near a given submanifold of codimension…

A consequence of the surgery theorem of Gromov and Lawson is that every closed, simply-connected 6-manifold admits a Riemannian metric of positive scalar curvature. For metrics of positive Ricci curvature it is widely open whether a similar…

This note is about the interplay between the almost-hermitian and Riemannian geometries of a manifold. These geometries can be seen to interact through curvature. The main result is an obstruction equation to the integrability of…

We show that there exists a metric with positive scalar curvature on S2xS1 and a sequence of embedded minimal cylinders that converges to a minimal lamination that, in a neighborhood of a strictly stable 2-sphere, is smooth except at two…

Let L\subset V=\bR^{k,l} be a maximally isotropic subspace. It is shown that any simply connected Lie group with a bi-invariant flat pseudo-Riemannian metric of signature (k,l) is 2-step nilpotent and is defined by an element \eta \in…

A spherical conical metric $g$ on a surface $\Sigma$ is a metric of constant curvature $1$ with finitely many isolated conical singularities. The uniformization problem for such metrics remains largely open when at least one of the cone…

In this paper we develop new methods of study of generalized normal homogeneous Riemannian manifolds. In particular, we obtain a complete classification of generalized normal homogeneous Riemannian metrics on spheres. We prove that for any…

A theorem of Grove and Searle directly establishes that positive curvature 2d manifolds M with effective circular symmetry group of dimension 8 or less have positive Euler characteristic X(M): the fixed point set N consists of even…

In this article, we investigate the geometry of a central extension $\widehat{\mathcal D}_{\mu}(S^{2})$ of the group of volume-preserving diffeomorphisms of the 2-sphere equipped with the $L^{2}$-metric, whose geodesics correspond solutions…

We prove nonuniqueness results for constant sixth order $Q$-metrics on complete locally conformally flat $n$-dimensional Riemannian manifolds with $n\geqslant 7$. More precisely, assuming a positive Green function exists for the sixth order…

We give sharp sectional curvature estimates for complete immersed cylindrically bounded $m$-submanifolds $\phi:M\to N\times\mathbb{R}^{\ell}$, $n+\ell\leq 2m-1$ provided that either $\phi$ is proper with the second fundamental form with…

Positive curvature and bosons Compact positive curvature Riemannian manifolds M with symmetry group G allow Conner-Kobayashi reductions M to N, where N is the fixed point set of the symmetry G. The set N is a union of smaller-dimensional…

In this work we construct a sequence of Riemannian metrics on the three-sphere with scalar curvature greater than or equal to $6$ and arbitrarily large widths. Our procedure is based on the connected sum construction of positive scalar…

I classify the Finsler structures on the 2-sphere that have constant Finsler-Gauss curvature and whose geodesics are the great circles. Modulo diffeomorphism, there is a 2-parameter family of such Finsler structures, only one of which is…

Natural metric structures on tangent bundles and tangent sphere bundles enclose many important problems, from the topology of the base to the determination of their holonomy. We make here a brief study of the topic. We find the…

We show there are precisely 15 inhomogeneous biquotients of the form $Sp(3)// Sp(1)^2$ and show that at least 8 of them admit metrics of quasi-positive curvature.

We derive a curvature-variation formula for a path of left-invariant metrics on a compact Lie group, beginning at a bi-invariant metric. We prove rigidity theorems for paths which remain nonnegatively curved, and we make progress towards a…

A conformal metric $g$ with constant curvature one and finite conical singularities on a compact Riemann surface $\Sigma$ can be thought of as the pullback of the standard metric on the 2-sphere by a multi-valued locally univalent…

Let $M$ be a smooth closed spin (resp. oriented and totally non-spin) manifold of dimension $n\geq 5$ with fundamental group $\pi$. It is stated, e.g. in [RS95], that $M$ admits a metric of positive scalar curvature (pscm) if its…