Nonlinear and direction connections

Vilms, Jaak

doi:10.1090/S0002-9939-1971-0279752-1

Nonlinear and direction connections
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Proc. Amer. Math. Soc. 28 (1971), 567-572 Request permission

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.

References

Pierre Dazord, Sur une généralisation de la notion de “spray”, C. R. Acad. Sci. Paris Sér. A-B 263 (1966), A543–A546 (French). MR 210054

Connexion de direction symétrique associée à un “spray” généralisé

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Joseph Grifone, Prolongement linéaire d’une connexion de directions, C. R. Acad. Sci. Paris Sér. A-B 269 (1969), A90–A93 (French). MR 244915
Makoto Matsumoto, A global foundation of Finsler geometry, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 33 (1960/61), 171–208. MR 124866, DOI 10.1215/kjm/1250776064
Jaak Vilms, Connections on tangent bundles, J. Differential Geometry 1 (1967), 235–243. MR 229168
Jaak Vilms, Curvature of nonlinear connections, Proc. Amer. Math. Soc. 19 (1968), 1125–1129. MR 238214, DOI 10.1090/S0002-9939-1968-0238214-8