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On a class of complete Finsler manifolds

Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Hafez Ave., 15914 Tehran, Iran

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B. Bidabad

Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Hafez Ave., 15914 Tehran, Iran

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https://doi.org/10.1142/S0129167X15500913Cited by:0 (Source: Crossref)

Abstract

Here, it is shown that if a forward geodesically complete Finsler manifold admits a circle preserving change of metric then its indicatrix is conformally diffeomorphic to the Euclidean sphere S^n-1. Moreover, if the Finsler manifold is absolutely homogeneous and of scalar flag curvature then it is a Riemannian manifold of constant sectional curvature. These results provide a geometric interpretation for existence of solutions to the certain ODE on the Riemannian tangent space.

Keywords:

AMSC: 53C60, 58B20

References

H. Akbar-Zadeh , Acad. Roy. Belg. Bull. Cl. Sci. (5) 74 , 281 ( 1988 ) . Google Scholar
H. Akbar-Zadeh , Initiation to Global Finslerian Geometry 68 ( Elsevier Science , 2006 ) . Google Scholar
A. Asanjarani and B. Bidabad , Differential Geom. Appl. 26 , 434 ( 2008 ) . Crossref, Web of Science, Google Scholar
D. Bao , S. Chern and Z. Shen , An Introduction to Riemann–Finsler Geometry , Graduate Texts in Mathematics 200 ( Springer , 2000 ) . Crossref, Google Scholar
B. Bidabad , C. R. Acad. Sci. Paris Ser. I 349 , 1191 ( 2011 ) . Crossref, Web of Science, Google Scholar
B. Bidabad , Differential Geom. Appl. 35 , 350 ( 2014 ) . Crossref, Web of Science, Google Scholar
B. Bidabad and P. Joharinad , Differential Geom. Appl. 33 , 75 ( 2014 ) . Crossref, Web of Science, Google Scholar
B. Bidabad and Z. Shen , Publ. Math. Debrecen 81 , 435 ( 2012 ) . Crossref, Web of Science, Google Scholar
S. Kobayashi and K. Nomizu , Foundations of Differential Geometry 1 ( Wiley-Interscience , New York , 1963 ) . Google Scholar
S. Kobayashi and K. Nomizu , Foundations of Differential Geometry 2 ( Wiley-Interscience , New York , 1969 ) . Google Scholar
K. Nomizu and K. Yano, Math. Ann. 210(2), 163 (1974). Crossref, Web of Science, Google Scholar
Z. Shen , Lectures on Finsler Geometry ( World Scientific , 2001 ) . Link, Google Scholar
Y. Tashiro , Trans. Amer. Math. Soc. 117 , 251 ( 1965 ) . Crossref, Web of Science, Google Scholar
K. Yano , Proc. Imp. Acad. Tokyo 16 , 505 ( 1940 ) . Crossref, Google Scholar
V. Zadnik , Rend. Circ. Mat. Palermo (2) Suppl. 75 , 347 ( 2005 ) . Google Scholar