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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Nonlinear and direction connections


Author: Jaak Vilms
Journal: Proc. Amer. Math. Soc. 28 (1971), 567-572
MSC: Primary 53.85
DOI: https://doi.org/10.1090/S0002-9939-1971-0279752-1
MathSciNet review: 0279752
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Abstract: Nonlinear connections and direction connections are two types of connections arising in Finsler geometry. In his work on generalized sprays, P. Dazord showed that there is a relationship between these two types (nonlinear connections were called sections by him). This relationship has also been used by J. Grifone in a work on prolongation of direction connections. In this paper we examine this relationship in a general setting. In particular, we show that E. Cartan's condition ``D'' is necessary and sufficient for a direction connection to define a nonlinear one. Also, we prove a nonuniqueness result for direction connections associated to a given nonlinear one.


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DOI: https://doi.org/10.1090/S0002-9939-1971-0279752-1
Keywords: Finsler geometry, nonlinear connection, direction connection, E. Cartan's condition ``D", spray
Article copyright: © Copyright 1971 American Mathematical Society