On sprays and connections

Kozma, László

Proceedings of the Winter School "Geometry and Physics", GDML_Books, (1990), p. [113]-116 / Harvested from

Résumé

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

MR MR1061793 | Zbl 0707.53025

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/