On Alexandrov spaces of curvature bounded from above (Q2713882)

The author considers a branched covering \(\pi\: M' \to M\) of a Riemannian manifold \(M\) with metric \(d\) by a manifold \(M'\) and proves that the metric \(d'\) obtained by lifting the metric \(d\) is the unique metric on \(M'\) which makes \(\pi\) a submetry (this means that \(\pi\) maps \(d'\)-balls onto \(d\)-balls). In the case when \(M\) and \(M'\) are two-surfaces, the author shows that if \(M\) has sectional curvature bounded from above by a constant \(C\) then \(M'\) has curvature bounded from above by the same constant \(C\) in the sense of Alexandrov (this fact is actually an immediate consequence of the following observation: if \(p=\pi(p')\) is a branch point and a point \(q=\pi(q')\) moves around \(p\) at the angle \(2\pi\) then \(q'\) moves around \(p'\) at the angle \(2 k \pi\), where \(k\) is an integer). NEWLINENEWLINENEWLINEThe author studies the following particular cases in more detail: 1) \(M\) and \(M'\) are Riemann spheres and \(\pi\) is a rational function; 2) \(M\) is a Riemann sphere, \(M'\) is a two-torus, and \(\pi\) is an even elliptic function; 3) \(M\) is a Riemann sphere, \(M'\) is a two-surface of genus \(g >1\), and \(\pi\) is some particular map.NEWLINENEWLINEFor the entire collection see [Zbl 0935.00013].