RIEMANNIAN MANIFOLDS WITH POSITIVE SECTIONAL CURVATURE

Of special interest in the history of Riemannian geometry have been manifolds with positive sectional curvature. In these notes we want to give a survey of this subject and some recent developments. We start with some historical developments. 1. History and Obstructions It is fair to say that Riemannian geometry started with Gauss’s famous ”Disquisitiones generales ” from 1827 in which one finds a rigorous discussion of what we now call the Gauss curvature of a surface. Much has been written about the importance and influence of this paper, see in particular the article [Do] by P.Dombrowski for a careful discussion of its contents and influence during that time. Here we only make a few comments. Curvature of surfaces in 3-space had been studied previously by a number of authors and was defined as the product of the principal curvatures. But Gauss was the first to make the surprising discovery that this curvature only depends on the intrinsic metric and not on the embedding. Here one finds for example the formula for the metric in the form ds2 = dr2 + f(r, θ) 2dθ2. Gauss showed that every metric on a surface has this form in ”normal ” coordinates and that it has curvature K = −frr/f. In fact one can take it as the definition of the Gauss curvature and proves Gauss’s famous ”Theorema Egregium” that the curvature is an intrinsic invariant and does not depend on the embedding in R3. He also proved a local version of what we nowadays call the Gauss-Bonnet theorem (it is not clear what Bonnet’s contribution was to this result), which states that in a geodesic triangle ∆ with angles α, β, γ the Gauss curvature measures the angle ”defect”: Kdvol = α + β + γ − π Nowadays the Gauss Bonnet theorem also goes under its global formulation for a compact surface