Coverings and isometries for the Kobayashi infinitesimal metric (Q2731921)

Let \(M_1\), \(M_2\) be complex connected manifolds with \(\dim M_1=\dim M_2\). Fix a point \(a\in M_1\). Let \(f:M_1\rightarrow M_2\) be a holomorphic mapping such that \(F_{M_2}(f(a),f'(a)(v))=F_{M_1}(a,v)\), \(v\in T_aM_1\), where \(F_{M_j}\) denotes the Kobayashi-Royden pseudometric for \(M_j\), \(j=1,2\).NEWLINENEWLINENEWLINEThe main results of the paper are the following theorems:NEWLINENEWLINENEWLINE(1) If \(M_1\) is taut and the universal covering of \(M_2\) is biholomorphic to a bounded strictly convex domain, then \(f\) is a covering.NEWLINENEWLINENEWLINE(2) If the universal coverings of \(M_1\) and \(M_2\) are biholomorphic to bounded convex domains, then \(f\) is a covering.NEWLINENEWLINENEWLINE(3) Assume that \(M_1=M_2=D\subset\mathbb C\) is a bounded domain for which there exists a \(\delta>0\) such that for any piecewise \(\mathcal C^1\)-loop \(\gamma\subset D\), if \(\gamma\) is not homotopic to a point, then the length of \(\gamma\) is \(\geq\delta\) (e.g. \(D\subset\mathbb C\) is a non-degenerated annulus). Then \(f\) is biholomorphic.