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On conformal equivalence of Riemann-Finsler metrics. (English) Zbl 0907.53044
In the first part of this paper, the authors give a brief, very clear and self contained setting for Finsler geometry, focused on its conformal theory. The following list of keywords from the text points to the richness of the topics: Frölicher-Nijenhuis theory, homogeneity, semispray, spray, complete lift, horizontal endomorphism, semibasic trace, Riemann-Finsler metric, the fundamental lemma of Finsler geometry, Barthel endomorpism, Cartan tensor, gradient operator. The consistent computational method is motivated basically by J. Grifone’s article [Ann. Inst. Fourier, Grenoble 22, No. 1 and No. 3, 287-334 and 291-338 (1972; Zbl 0237.53028 and Zbl 0242.53011)].
The second part is devoted to the conformal equivalence of Riemann-Finsler metrics. The authors derive how the Barthel endomorphism, and the curvature of the Barthel endomorphism, the basic connections change. They give some applications and reformulate and prove in a new way some celebrated results: Knebelman’s theorem (stating that the scale function must be a vertical lift) and the famous result which states that in the case of a simultaneous conformal and projective change, the scale function is constant.

MSC:
53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
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