The Schwarzian derivative and conformal transformation on Finsler manifolds
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Thurston, in 1986, discovered that the Schwarzian derivative has mysterious properties similar to the curvature on a manifold. After his work, there are several approaches to develop this notion on Riemannian manifolds. Here, a tensor field is identified in the study of global conformal diffeomorphisms on Finsler manifolds as a natural generalization of the Schwarzian derivative. Then, a natural definition of a Mobius mapping on Finsler manifolds is given and its properties are studied. In particular, it is shown that Mobius mappings are mappings that preserve circles and vice versa. Therefore, if a forward geodesically complete Finsler manifold admits a Mobius mapping, then the indicatrix is conformally diffeomorphic to the Euclidean sphere $ S^{n-1}$ in $ \mathbb{R}^n $. In addition, if a forward geodesically complete absolutely homogeneous Finsler manifold of scalar flag curvature admits a non-trivial change of Mobius mapping, then it is a Riemannian manifold of constant sectional curvature.
Abstract
Thurston, in 1986, discovered that the Schwarzian derivative has mysterious properties similar to the curvature on a manifold. After his work, there are several approaches to develop this notion on Riemannian manifolds. Here, a tensor field is identified in the study of global conformal diffeomorphisms on Finsler manifolds as a natural generalization of the Schwarzian derivative. Then, a natural definition of a Mobius mapping on Finsler manifolds is given and its properties are studied. In particular, it is shown that Mobius mappings are mappings that preserve circles and vice versa. Therefore, if a forward geodesically complete Finsler manifold admits a Mobius mapping, then the indicatrix is conformally diffeomorphic to the Euclidean sphere $ S^{n-1}$ in $ \mathbb{R}^n $. In addition, if a forward geodesically complete absolutely homogeneous Finsler manifold of scalar flag curvature admits a non-trivial change of Mobius mapping, then it is a Riemannian manifold of constant sectional curvature.
참고문헌 (18)
- H. Akbar-Zadeh, "Sur les espaces de Finsler a courbures sectionnelles constantes", Acad. Roy. Belg. Bull. Cl. Sci., 74(10), pp.281~322 (1988)
- H. Akbar-Zadeh, "Generalized Einstein manifolds", J. Geom. Phys, 17(4), pp.342~380 (1995)[DOI]
- D. Bao, "Graduate Texts in Mathematics, 200" (2000), Springer-Verlag
- D. Bao, "Ricci and ag curvatures in Finsler geometry, in A sampler of Riemann-Finsler geometry", Math. Sci. Res. Inst. Publ., 50, pp.197~259 (2004), Cambridge Univ. Press
- B. Bidabad, "A classication of complete Finsler manifolds through the conformal theory of curves", Dierential Geom. Appl, 35, pp.350~360 (2014)[DOI]
- B. Bidabad, "On Sobolev spaces and density theorems on Finsler manifolds", AUT J. Math. Com, 1(1), pp.37~45 (2020)
- Behroz Bidabad, "Deformation of Cartan curvature on Finsler manifolds", 대한수학회보, 54(6), pp.2119~2139 (2017), 대한수학회
- B. Bidabad, "On a projectively invariant pseudo-distance in Finsler geometry", Int. J. Geom. Methods Mod. Phys, 12(4) (2015)[DOI]
- B. Bidabad, "Circle-preserving transformations in Finsler spaces", Publ. Math. Debrecen, 81(3-4), pp.435~445 (2012)[DOI]
- K. Carne, "The Schwarzian derivative for conformal maps", J. Reine Angew. Math, 408, pp.10~33 (1990)[DOI]
- X. Cheng, "Finsler Geometry" (2012), Science Press Beijing
- M. Hashiguchi, "On conformal transformations of Finsler metrics", J. Math. Kyoto Univ, 16(1), pp.25~50 (1976)[DOI]
- B. Osgood, "The Schwarzian derivative and conformal mapping of Rie-mannian manifolds", Duke Math. J, 67(1), pp.57~99 (1992)[DOI]
- M. K. Sedaghat, "On a class of complete Finsler manifolds", Internat. J. Math, 26(11), pp.9~ (2015)[DOI]
- M. Sepasi, "On a projectively invariant distance on Finsler spaces of constant negative Ricci scalar", C. R. Math. Acad. Sci. Paris, 352(12), pp.999~1003 (2014)[DOI]
- Z. Shen, "Lectures on Finsler geometry" (2001), World Scientic Publishing Co.
- W. P. Thurston, "Zippers and univalent functions, in The Bieberbach conjecture", Math. Surveys Monogr., 21, pp.185~197 (1986), Amer. Math. Soc[DOI]
- B. Y. Wu, "Comparison theorems in Finsler geometry and their applica-tions", Math. Ann, 337(1), pp.177~196 (2007)[DOI]