Theorem Of The Ball And Horseshoe Inequality

Sphere theorems have always been a central theme in global differential geometry, it give rise to a large number of questions. This has produced an intense activity which extends up to the present and constitutes one of the most vigorous branches of Differential Geometry." A sphere theorem with a pinching constant below 1/4 ", which appeared in Journal of Differential Geometry, by U. Abresch and W. T. Meyer. In their article, the main results as follows:Theorem A There exists a constant δ_odd ∈ (0, 1/4) such that any odd dimensional, compact, simply connected Riemannian manifold Mⁿ with δ_odd -pinched sectional curvature is homeomorphic to the sphere Sⁿ.Theorem B There exists a constant δ_ev ∈ (0, 1/4) such that for any even dimensional, compact, simply connected Riemannian manifold Mⁿ with δ_ev -pinched sectional curvature the cohomology rings H^*(Mⁿ;R) with coeffcients R ∈ {Q, Z₂} are isomorphic to the corresponding cohomology rings of one of the compact, rank one ,symmetric spaces Sⁿ,CP^n/2 ,HP^n/4 ; or the rings H^*(Mⁿ;R) are truncated polynomial rings generated by an element of degree 8.The center for the proofs of both theorems is to establish the horse shoe con-jicture of Bcrger, which had remained open until recently.Horse Shoe Inequality There exists a constant δ ∈ (0, 1/4) such that for any complete Riemannian manifold Mⁿ withδ ≤ K_M ≤ 1 and π ≤ inj Mⁿ ≤ diam Mⁿ ≤ π/2δ^1/2,the following holds: for any p₀ ∈ Mⁿ and any v ∈ S^n-1 T_P0 M, the distance between the antipodal points exp_P0(-πv) and exp_P0(πv) is bounded by π:...