Abstract
Let \(F:=\alpha +|\beta |\) be a strong Randers metric on a complex manifold. We show that \(F\) is Kähler if and only if \(\beta \) is parallel with respect to \(\alpha \). Furthermore if \(\alpha \) has constant holomorphic sectional curvature, we show that the following assertions are equivalent: (i) \(F\) is Kähler; (ii) \(F=|v|^{2}+\langle c,\bar{v}\rangle \) is a Minkowskian metric unless \(F\) is usually Kählerian.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Abate, G. Patrizio, Finsler Metrics-A Global Approach with Applications to Geometric Function Theory, Lecture Notes in Mathematics, 1591 (Springer, Berlin, 1994), pp. x+180
M. Abate, G. Patrizio, Kähler Finsler manifolds of constant holomorphic curvature. Int. J. Math.8, 169–186 (1997)
N. Aldea, G. Munteanu, On the Geometry of Complex Randers Spaces, Proc. of the 14-th Nat. Sem. on Finsler, Lagrange and Hamilton Spaces (Brasov, 2006), pp. 1–8.
N. Aldea, G. Munteanu, On complex Finsler spaces with Randers metric. J. Korean Math. Soc.46, 949–966 (2009)
D. Bao, C. Robles, Z. Shen, Zermelo navigation on Riemannian manifolds. J. Differ. Geom. 66, 377–435 (2004)
J. Cao, P. Wong, Finsler geometry of projectivized vector bundles. J. Math. Kyoto Univ. 43, 369–410 (2003)
B. Chen, Y. Shen, Kähler Finsler metrics are actually strongly Kähler. Chin. Ann. Math. Ser. B 30, 173–178 (2009)
B. Chen, Y. Shen, On complex Randers metrics. Int. J. Math. 21, 971–986 (2010)
X. Cheng, Z. Shen, Finsler Geometry, An Appoach via Randers Spaces (Science Press, Beijing, 2012)
A. Futaki, An obstruction to the existence of Einstein Kähler metrics. Invent. Math. 73, 437–443 (1983)
S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vol. II. Reprint of the 1969 original. Wiley Classics Library. A Wiley-Interscience Publication (Wiley, New York, 1996), pp. xvi+468
X. Mo, L. Huang, On characterizations of Randers norms in a Minkowski space. Int. J. Math. 21, 523–535 (2010)
G. Tian, Kähler-Einstein metrics with positive scalar curvature. Invent. Math. 130, 1–37 (1997)
P. Wong, C. Zhong, On weighted complex Randers metrics. Osaka J. Math. 48, 589–612 (2011)
J. Xiao, T. Zhong, C. Qiu, Bochner technique on strong Kähler-Finsler manifolds. Acta Math. Sci. Ser. B Engl. Ed. 30, 89–106 (2010)
C. Zhong, A vanishing theorem on Kähler Finsler manifolds. Differ. Geom. Appl. 27, 551–565 (2009)
C. Zhong, On real and complex Berwald connections associated to strongly convex weakly Kähler-Finsler metric. Differ. Geom. Appl. 29, 388–408 (2011)
C. Zhong, T. Zhong, Hodge decomposition theorem on strongly Kähler Finsler manifolds. Sci. China Ser. A 49, 1696–1714 (2006)
Acknowledgments
This work is supported by the National Natural Science Foundation of China 11371032 and the Doctoral Program of Higher Education of China 20110001110069. Second author is supported by the doctoral scientific research foundation of Henan Normal University 01016500130.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mo, X., Zhu, H. Some results on strong Randers metrics. Period Math Hung 71, 24–34 (2015). https://doi.org/10.1007/s10998-014-0082-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10998-014-0082-8