On complex Berwald metrics which are not conformal changes of complex Minkowski metrics
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Hongchuan Xia
Abstract
We investigate a class of complex Finsler metrics on a domain D ⊂ ℂn. Necessary and sufficient conditions for these metrics to be strongly pseudoconvex complex Finsler metrics, or complex Berwald metrics, are given. The complex Berwald metrics constructed in this paper are neither trivial Hermitian metrics nor conformal changes of complex Minkowski metrics. We give a characterization of complex Berwald metrics which are of isotropic holomorphic curvatures, and also give characterizations of complex Finsler metrics of this class to be Kähler Finsler or weakly Kähler Finsler metrics. Moreover, in the strongly convex case, we give characterizations of complex Finsler metrics of this class to be projectively flat Finsler metrics or dually flat Finsler metrics.
Funding: This work is supported by the National Natural Science Foundation of China (Grant Nos. 11671330, 11701494, 11571288, 11771357), the Nanhu Scholars Program for Young Scholars of XYNU, and the Scientific Research Fund Program for Young Scholars of XYNU (No. 2017-QN-029).
References
[1] M. Abate, T. Aikou, G. Patrizio, Preface for “Complex Finsler geometry”. In: Finsler geometry (Seattle, WA, 1995), volume 196 of Contemp. Math., 97–100, Amer. Math. Soc. 1996. MR1403581 Zbl 0853.5304910.1090/conm/196/02435Search in Google Scholar
[2] M. Abate, G. Patrizio, Finsler metrics—a global approach. Springer 1994. MR1323428 Zbl 0837.5300110.1007/BFb0073980Search in Google Scholar
[3] M. Abate, G. Patrizio, Finsler metrics of constant curvature and the characterization of tube domains. In: Finsler geometry (Seattle, WA, 1995), volume 196 of Contemp. Math., 101–107, Amer. Math. Soc. 1996. MR1403582 Zbl 0861.5307110.1090/conm/196/02436Search in Google Scholar
[4] M. Abate, G. Patrizio, Holomorphic curvature of Finsler metrics and complex geodesics. J. Geom. Anal. 6 (1996), 341–363 (1997). MR1471896 Zbl 0896.3201310.1007/BF02921655Search in Google Scholar
[5] T. Aikou, On complex Finsler manifolds. Rep. Fac. Sci. Kagoshima Univ. Math. Phys. Chem. 24 (1991), 9–25. MR1172107 Zbl 0783.53019Search in Google Scholar
[6] T. Aikou, Some remarks on locally conformal complex Berwald spaces. In: Finsler geometry (Seattle, WA, 1995), volume 196 of Contemp. Math., 109–120, Amer. Math. Soc. 1996. MR1403583 Zbl 0857.5301510.1090/conm/196/02437Search in Google Scholar
[7] D. Bao, S.-S. Chern, Z. Shen, An introduction to Riemann-Finsler geometry. Springer 2000. MR1747675 Zbl 0954.5300110.1007/978-1-4612-1268-3Search in Google Scholar
[8] L. Huang, X. Mo, On some explicit constructions of dually flat Finsler metrics. J. Math. Anal. Appl. 405 (2013), 565–573. MR3061034 Zbl 1306.5306310.1016/j.jmaa.2013.04.028Search in Google Scholar
[9] L. Huang, X. Mo, On some dually flat Finsler metrics with orthogonal invariance. Nonlinear Anal. 108 (2014), 214–222. MR3238303 Zbl 1301.5301810.1016/j.na.2014.05.017Search in Google Scholar
[10] L. Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule. Bull. Soc. Math. France109 (1981), 427–474. MR660145 Zbl 0492.3202510.24033/bsmf.1948Search in Google Scholar
[11] L. Lempert, Holomorphic retracts and intrinsic metrics in convex domains. Anal. Math. 8 (1982), 257–261. MR690838 Zbl 0509.3201510.1007/BF02201775Search in Google Scholar
[12] G. Munteanu, Complex spaces in Finsler, Lagrange and Hamilton geometries, volume 141 of Fundamental Theories of Physics. Kluwer, Dordrecht 2004. MR2102340 Zbl 1064.5304710.1007/978-1-4020-2206-7Search in Google Scholar
[13] M.-Y. Pang, Finsler metrics with properties of the Kobayashi metric on convex domains. Publ. Mat. 36 (1992), 131–155. MR1179607 Zbl 0754.5305410.5565/PUBLMAT_36192_10Search in Google Scholar
[14] Z. Shen, Riemann-Finsler geometry with applications to information geometry. Chinese Ann. Math. Ser. B27 (2006), 73–94. MR2209953 Zbl 1107.5301310.1007/s11401-005-0333-3Search in Google Scholar
[15] L. Sun, C. Zhong, Characterizations of complex Finsler connections and weakly complex Berwald metrics. Differential Geom. Appl. 31 (2013), 648–671. MR3093496 Zbl 1319.5308310.1016/j.difgeo.2013.07.003Search in Google Scholar
[16] C. Zhong, On real and complex Berwald connections associated to strongly convex weakly Kähler–Finsler metric. Differential Geom. Appl. 29 (2011), 388–408. MR2795846 Zbl 1219.5302810.1016/j.difgeo.2011.03.006Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Graphs and metric 2-step nilpotent Lie algebras
- Stein manifolds of nonnegative curvature
- Homogeneous spin Riemannian manifolds with the simplest Dirac operator
- On the quantum periods of del Pezzo surfaces with ⅓ (1, 1) singularities
- Harmonicity of vector fields on a class of Lorentzian solvable Lie groups
- Uniform cover inequalities for the volume of coordinate sections and projections of convex bodies
- Criteria for Hattori–Masuda multi-polytopes via Duistermaat–Heckman functions and winding numbers
- On complex Berwald metrics which are not conformal changes of complex Minkowski metrics
- Continuous space-time transformations
Articles in the same Issue
- Frontmatter
- Graphs and metric 2-step nilpotent Lie algebras
- Stein manifolds of nonnegative curvature
- Homogeneous spin Riemannian manifolds with the simplest Dirac operator
- On the quantum periods of del Pezzo surfaces with ⅓ (1, 1) singularities
- Harmonicity of vector fields on a class of Lorentzian solvable Lie groups
- Uniform cover inequalities for the volume of coordinate sections and projections of convex bodies
- Criteria for Hattori–Masuda multi-polytopes via Duistermaat–Heckman functions and winding numbers
- On complex Berwald metrics which are not conformal changes of complex Minkowski metrics
- Continuous space-time transformations
