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On sprays and connections. (English) Zbl 0707.53025
Geometry and physics, Proc. 9th Winter Sch., Srní/Czech. 1989, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 22, 113-116 (1990).
[For the entire collection see Zbl 0699.00032.]
A connection structure (M,H) and a path structure (M,S) on the manifold M are called compatible, if $$S(v)=H(v,v),\forall v\in TM,$$ locally $$G^ i(x,y)=y^ j\Gamma^ i_ j(x,y),$$ where $$G^ i$$ and $$\Gamma^ i_ j$$ express the semi-spray S and the connection map H, resp. In the linear case of H its geodesic spray S is quadratic: $$G^ i(x,y)=\Gamma^ i_{jk}(k)y^ jy^ k.$$ On the contrary, the homogeneity condition of S induces the relation for the compatible connection H, $$y^ j(\Gamma^ i_ j\circ \mu_ t)=ty^ j\Gamma^ i_ j,$$ whence it follows not that H is linear, i.e. if a connection structure is compatible with a spray, then the connection is not necessarily homogeneous. This fact supplements the investigations of H. B. Levine [Phys. Fluids 3, 225-245 (1960; Zbl 0106.209)], and M. Crampin [J. Lond. Math. Soc., II. Ser. 3, 178-182 (1971; Zbl 0215.510)].
Reviewer: M.Rahula
MSC:
 53C05 Connections (general theory) 58A30 Vector distributions (subbundles of the tangent bundles)
Keywords:
connection structure; path structure