Holomorphic Injectivity and the Hopf Map
نویسندگان
چکیده
We give sharp conditions on a local biholomorphism F : X → C n which ensure global injectivity. For n ≥ 2, such a map is injective if for each complex line l ⊂ C n , the pre-image F −1 (l) embeds holomorphically as a connected domain into CP 1 , the embedding being unique up to Möbius transformation. In particular, F is injective if the pre-image of every complex line is connected and conformal to C. The proof uses the topological fact that the natural map RP 2n−1 → CP n−1 associated to the Hopf map admits no continuous sections and the classical Bieberbach-Gronwall estimates from complex analysis.
منابع مشابه
A Schneider Type Theorem for Hopf Algebroids
Comodule algebras of a Hopf algebroid H with a bijective antipode, i.e. algebra extensions B ⊆ A by H, are studied. Assuming that a lifted canonical map is a split epimorphism of modules of the (non-commutative) base algebra of H, relative injectivity of the H-comodule algebra A is related to the Galois property of the extension B ⊆ A and also to the equivalence of the category of relative Hopf...
متن کاملLocally homogeneous structures on Hopf surfaces
We study holomorphic locally homogeneous geometric structures modelled on line bundles over the projective line. We classify these structures on primary Hopf surfaces. We write out the developing map and holonomy morphism of each of these structures explicitly on each primary Hopf surface.
متن کاملThe Hopf Invariant and Simplex Straightening
Let M be a closed oriented 3-manifold which can be triangulated with N simplices. We prove that any map from M to a genus 2 surface has Hopf invariant at most C . Let X be a closed oriented hyperbolic 3-manifold with injectivity radius less than ǫ at one point. If there is a degree non-zero map from M to X, then we prove that ǫ is at least C . In the 1970’s, Thurston invented simplex straighten...
متن کاملPeriodic Points of
In this paper we study the set of periods of holomorphic maps on compact manifolds, using the periodic Lefschetz numbers introduced in D] and Ll], which can be computed from the homology class of the map. We show that these numbers contain information about the existence of periodic points of a given period and, if we assume the map to be transversal then they give us the exact number of such p...
متن کاملInjectivity Criteria for Holomorphic Curves in C
Combining the definition of Schwarzian derivative for conformal mappings between Riemannian manifolds given by Osgood and Stowe with that for parametrized curves in Euclidean space given by Ahlfors, we establish injectivity criteria for holomorphic curves φ : D → C. The result can be considered a generalization of a classical condition for univalence of Nehari.
متن کامل