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[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) |
