# Article

 Title: On sprays and connections (English) Author: Kozma, László Language: English Journal: Proceedings of the Winter School "Geometry and Physics" Volume: Issue: 1989 Year: Pages: [113]-116 . Category: math . Summary: [For the entire collection see Zbl 0699.00032.] \par 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 $G\sp i(x,y)=y\sp j\Gamma\sp i\sb j(x,y),$ where $G\sp i$ and $\Gamma\sp i\sb j$ express the semi-spray S and the connection map H, resp. In the linear case of H its geodesic spray S is quadratic: $G\sp i(x,y)=\Gamma\sp i\sb{jk}(k)y\sp jy\sp k.$ On the contrary, the homogeneity condition of S induces the relation for the compatible connection H, $y\sp j(\Gamma\sp i\sb j\circ \mu\sb t)=ty\sp j\Gamma\sp i\sb j,$ 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 {\it H. B. Levine} [Phys. Fluids 3, 225-245 (1960; Zbl 0106.209)], and {\it M. Crampin} [J. Lond. Math. Soc., II. Ser. 3, 178-182 (1971; Zbl 0215.510)]. (English) MSC: 53C05 MSC: 53C30 MSC: 58A30 MSC: 58F17 idZBL: Zbl 0707.53025 idMR: MR1061793 . Date available: 2009-07-13T21:24:46Z Last updated: 2012-09-18 Stable URL: http://hdl.handle.net/10338.dmlcz/701466 .

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