In the present paper we describe one family of closed Riemannian manifolds with positive sectional curvature. Now the list of known examples is not large (for instance, all known manifolds with dimension > 24 are diffeomorphic to compact rank one symmetric spaces) (we restrict ourselves only by pointing out simply connected manifolds) : 1) Berger described all normally homogeneous closed positi...

Using the Green's function and some comparison theorems, we obtain a lower bound on the rst Dirichlet eigenvalue for a domain D on a complete manifold with curvature bounded from above. And the lower bound is given explicitly in terms of the diameter of D and the dimension of D. This result can be considered as an analogue for nonpositively curved manifolds of Li-Schoen L-Sc] and Li-Yau's L-Ya]...

Article history: Received 5 April 2016 Accepted 20 July 2016 Available online 22 July 2016 Editor: M. Trodden Background bouncing cosmologies, driven by a single scalar field, having a quasi-matter domination period during the contracting phase, i.e., depicting the so-called Matter Bounce Scenario, are reconstructed for cosmologies with spatial positive curvature. These cosmologies lead to a ne...

We derive and study necessary and sufficient conditions for an S-bundle to admit an invariant metric of positive or nonnegative sectional curvature. In case the total space has an invariant metric of nonnegative curvature and the base space is odd dimensional, we prove that the total space contains a flat totally geodesic immersed cylinder. We provide several examples, including a connection me...

• Coxeter groups with respect to the standard generating set [5, p. 45, Exercise 26], the argument being that of Solomon [18]; • virtually abelian groups with respect to any generating set [4]; • two-step nilpotent groups with infinite cyclic commutator subgroup with respect to some generating set [19]. (On the other hand, it is shown in [19] that if G has Heisenberg rank at least 2, then G has...

We discuss a characterization of positively curved surfaces M with the property that at each point the tangent plane to M is not a support plane for the entire surface. A one parameter family of examples which have special relevance with respect to the characterization is also given. Each member of this family is a smooth embedded surface in R 3 that is topologically a disk, has everywhere posi...

We consider nonnegatively curved 4-manifolds that admit effective isometric actions by nite groups and from that draw topological conclusions about the manifold. Our rst theorem shows that if the man-ifolds admits an isometric Zp Zp for p large enough that the manifold has Euler characteristic less than or equal to ve. Our second theorem requires no hypothesis on the structure of the group othe...

An (n − 2)–dimensional smooth manifold in Gr(n, 2) is said to be curved if its restriction to each m–space of R is no more than (m − 2)– dimensional. A K 2 -set is a set of zero Lebesgue measure containing a translate of every plane in a curved manifold. We show that this is a natural class of sets with respect to the Kakeya problem and prove that dimH(E) ≥ 7/2 for all K 2 -sets E. When the und...

In this work, we continue with the classification for positively curved homogeneous Finsler spaces (G/H,F ). With the assumption that the homogeneous space G/H is odd dimensional and the positively curved metric F is reversible, we only need to consider the most difficult case left, i.e. when the isotropy group H is regular in G. Applying the fixed point set technique and the homogeneous flag c...

For non-reversible Finsler metrics of positive flag curvature on spheres and projective spaces we present results about the number and the length of closed geodesics and about their stability properties. 2000 MSC classification: 53C22; 53C60; 58E10