\(M\)-hyperbolic real subsets of complex spaces (Q1919888)

Let \(M\) be a complex space and \(V\) a subset of \(M\). The authors introduce a pseudodistance \(k_{V,M}\) on \(V\) which is an analog of the Kobayashi pseudodistance using holomorphic maps from the unit disk into \(M\) sending the open interval \((-1,1)\) into \(V\). They show that \(k_{V,M}\) decreases under a certain class of maps. This class includes all holomorphic maps between complex spaces but it may also contain non-holomorphic mappings. The authors give several examples of genuine distances \(k_{V,M}\) and examples of identically vanishing \(k_{V,M}\). In the case when \(M\) is a complex manifold and \(V\) is a connected closed real-analytic submanifold of \(M\) they prove that if \(k_{V,M}\) is a genuine distance, then it induces the topology of \(V\). An analog of the Kobayashi-Royden infinitesimal pseudometric for real-analytic submanifolds of complex manifolds is introduced and it is shown that \(k_{V,M}\) is the integrated form of this pseudometric. The authors establish also some properties of complex Lie groups of holomorphic transformations of \(M\) leaving \(V\) invariant. Finally, they study the ``geodesics'' of the pseudodistances \(k_{V,M}\).