[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 $S\left(v\right)=H(v,v),\forall v\in TM,$ locally $G^i(x,y)=y^j{\Gamma }_j^i(x,y),$ where $G^i$ and ${\Gamma }_j^i$ express the semi-spray S and the connection map H, resp. In the linear case of H its geodesic spray S is quadratic: $G^i(x,y)={\Gamma }_{jk}^i\left(k\right)y^j{y}^k.$ On the contrary, the homogeneity condition of S induces the relation for the compatible connection H, $y^j({\Gamma }_j^i\circ {\mu }_t)=t{y}^j{\Gamma }_j^i,$ 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)].

EUDML-ID : urn:eudml:doc:221875
Mots clés:

@article{701466,
     title = {On sprays and connections},
     booktitle = {Proceedings of the Winter School "Geometry and Physics"},
     series = {GDML\_Books},
     publisher = {Circolo Matematico di Palermo},
     address = {Palermo},
     year = {1990},
     pages = {[113]-116},
     mrnumber = {MR1061793},
     zbl = {0707.53025},
     url = {http://dml.mathdoc.fr/item/701466}
}

Kozma, László. On sprays and connections, dans Proceedings of the Winter School "Geometry and Physics", GDML_Books,  (1990), pp. [113]-116. http://gdmltest.u-ga.fr/item/701466/