The structure of projective maps between real projective manifolds

In this paper we study the set of projective maps between compact properly convex real projective manifolds. We show that this set contains only finitely many distinct homotopy classes and each homotopy class has the structure of a real projective manifold. When domain is irreducible and the target is strictly convex, our results imply that each non-trivial homotopy class contains at most one projective map. These results are motivated by the theory of holomorphic maps between compact complex manifolds.     

The Schwarzian derivative on Finsler manifolds of constant curvature

Periodica Mathematica Hungarica - Lagrange introduced the notion of Schwarzian derivative and Thurston discovered its mysterious properties playing a role similar to that of curvature on Riemannian...     

Harmonic maps between complex projective spaces

Using the DoCarmo-Wallach theory, we classify (homogeneous) polynomial harmonic maps of complex projective spaces into spheres and complex projective spaces in terms of finite dimensional moduli spaces. We make use of representation theory of the (special) unitary group to give, for a spherical range, the exact dimension and, for complex projective spaces, a lower bound of the dimension of the moduli spaces.Supported in part by NSF Grant DMS 8603172.Research done in part while the third named author was on leave at University of California, Berkeley.     

The Schwarzian derivative in several complex variables IV

Using two different Lie groups, two different Schwarzian derivatives of holomorphic mappings on domains in Cⁿ are defined and discussed. The necessary and sufficient conditions for annihilation of these two different Schwarzian derivatives are given. Project supported by the National Natural Science Foundation of China.     

The Schwarzian derivative for harmonic mappings

Journal d'Analyse Mathématique -     

Schwarzian derivative in Kähler manifolds (II)

Two injectivity theorems are proved which are for objective manifoldN = ??ⁿ andN =B ⁿ (unit ball), both equipped with canonical metric. Thus, the injectivity theorems are established in three cases of objective manifolds, i.e. with different holomorphic sectional curvatures - 1, 0 and 1 respectively.     

The Schwarzian derivative in several complex variables (III)

In the point view of Lie group, the cross ratio and Schwarzian derivative in Cⁿ are defined and discussed, especially the Schwarzian derivative of holomorphic mappings on the domains in matrix space C^{m x n} is defined and discussed. It is proved that it is invariant up to similarity under the group of holomorphic automorphism of the Grassmann manifold CG(m, n). And it is also proved that the Schwarzian derivative equals zero if and only if the mapping is liaearly fractional. Project supported by the National Natural Science Foundation of China.     

Parallelizability of complex projective Stiefel manifolds

The question of parallelizability of the complex projective Stiefel manifolds is settled.

     

The degrees of maps between manifolds

We study the question: which integers k can be realized as the degree of a map between two given closed (n-1)-connected 2n-manifolds? Mathematics Subject Classification (2000): 57R19, 55M25 Received: 9 July 2001; in final form: 21 July 2002 //Published online: 24 February 2003     

Dual linear connections on fitted manifolds of a space with projective connection

This paper presents an investigation of dual linear connections (projective and affine), induced by different fittings of a space with a projective connection P_n,n, a regular hypersurface V_n-1P _n,n , and a regular hyperbandH _m P _n,n .Translated from Itogi Nauki i Tekhniki. Problemy Geometrii, Vol. 8, pp. 25–46, 1977.     

A Grothendieck-Riemann-Roch formula for maps of complex manifolds

Mathematische Annalen -     

The Schwarzian derivative and the Wiman-Valiron property

Let f be a transcendental meromorphic function in the plane such that f has finitely many critical values, the multiple points of f have bounded multiplicities, and the inverse function of f has finitely many transcendental singularities. Using the Wiman-Valiron method, it is shown that if the Schwarzian derivative S_f of f is transcendental, then f has infinitely many multiple points, the inverse function of S_f does not have a direct transcendental singularity over ∞, and ∞ is not a Borel exceptional value of S_f. The first of these conclusions was proved by Nevanlinna and Elfving via a fundamentally different method.     

Almost complex and complex structures on real projective Stiefel manifolds

We give a complete list of real projective Stiefel manifolds which admit almost complex structures and show that many of them are in fact complex manifolds. The first named author was supported in part by Grants 1/1486/94 and 2/1225/96 of VEGA (Slovakia) during the preparation of this work.