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http://dx.doi.org/10.4134/JKMS.j160619

VANISHING OF PROJECTIVE VECTOR FIELDS ON COMPACT FINSLER MANIFOLDS  

Shen, Bin (School of Mathematics Southeast University)
Publication Information
Journal of the Korean Mathematical Society / v.55, no.1, 2018 , pp. 1-16 More about this Journal
Abstract
In this paper, we give characteristic differential equations of a kind of projective vector fields on Finsler manifolds. Using these equations, we prove the vanishing theorem of projective vector fields on any compact Finsler manifold with the negative mean Ricci curvature, which is defined in [10]. This result involves the vanishing theorem of Killing vector fields in the Riemannian case and the work of [1, 14].
Keywords
Finsler metric; projective vector fields; mean Ricci curvature; Bochner formula;
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1 H. Akbar-Zadeh, Initiation to Global Finslerian Geometry, North-Holland Mathematical Library, 68, Elsevier Science B.V., Amsterdam, 2006.
2 D. Bao, S.-S. Chern, and Z. Shen, An Introduction to Riemann-Finsler Geometry, Graduate Texts in Mathematics, 200, Springer-Verlag, New York, 2000.
3 G. S. Hall, Symmetries and Curvature Structure in General Relativity, World Scientific Lecture Notes in Physics, 46, World Scientific Publishing Co., Inc., River Edge, NJ, 2004.
4 B. Li, On the classification of projectively flat Finsler metrics with constant flag curvature, Adv. Math. 257 (2014), 266-284.   DOI
5 R. L. Lovas, Affne and projective vector fields on spray manifolds, Period. Math. Hungar. 48 (2004), no. 1-2, 165-179.   DOI
6 V. S. Matveev, H. Rademacher, M. Troyanov, and A. Zeghib, Finsler conformal Lichnerowicz-Obata conjecture, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 3, 937-949.   DOI
7 D. D. Monticelli, Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 3, 611-654.
8 P. Petersen, Riemannian Geometry, Second edition, Graduate Texts in Mathematics, 171, Springer, New York, 2006.
9 M. Schottenloher, A mathematical Introduction to Conformal Field Theory, translated from the German, Lecture Notes in Physics. New Series m: Monographs, 43, Springer-Verlag, Berlin, 1997.
10 B. Shen, Vanishing of Killing vector fields on Finsler manifolds, Kodai Math. J., to be appeared.
11 Z. Shen, Lectures on Finsler Geometry, World Scientific Publishing Co., Singapore, 2001.
12 Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, Dordrecht, 2001.
13 Z. Shen, Projectively flat Finsler metrics of constant flag curvature, Trans. Amer. Math. Soc. 355 (2003), no. 4, 1713-1728.   DOI
14 H. Tian, Projective vector fields on Finsler manifolds, Appl. Math. J. Chinese Univ. Ser. B 29 (2014), no. 2, 217-229.   DOI
15 H. H. Wu, The Bochner technique in differential geometry, Math. Rep. 3 (1988), no. 2, i-xii and 289-538.
16 K. Yano, Integral Formulas in Riemannian Geometry, Pure and Applied Mathematics, No. 1, Marcel Dekker, Inc., New York, 1970.