Let M be a closed simply connected manifold and 0 < δ ≤ 1. Klingenberg and Sakai conjectured that there exists a constant i0 = i0(M, δ) > 0 such that the injectivity radius of any Riemannian metric g on M with δ ≤ Kg ≤ 1 can be estimated from below by i0. We study this question by collapsing and Alexandrov space techniques. In particular we establish a bounded version of the KlingenbergSakai co...

Only two examples of extremally Ricci pinched G2-structures can be found in the literature and they are both homogeneous. We study this paper existence structure such very special closed on Lie groups. Strong structural conditions algebra proved to hold. As an application, we obtain three new that all necessarily steady Laplacian solitons. The deformation rigidity structures also studied.

We prove that a quasi-isometric map, and more generally coarse embedding, between pinched Hadamard manifolds is within bounded distance from unique harmonic map.

QCD finite energy sum rules, together with the latest updated ALEPH data on hadronic decays of the tau-lepton are used in order to determine the vacuum condensates of dimension d = 2 and d = 4. These data are also used to check the validity of the Weinberg sum rules, and to determine the chiral condensates of dimension d = 6 and d = 8, as well as the chiral correlator at zero momentum, proporti...