On sprays and connections

Kozma, László

  • Proceedings of the Winter School "Geometry and Physics", Publisher: Circolo Matematico di Palermo(Palermo), page [113]-116

Abstract

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[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 ) , ∀ v ∈ 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 ∘ 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)].

How to cite

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Kozma, László. "On sprays and connections." Proceedings of the Winter School "Geometry and Physics". Palermo: Circolo Matematico di Palermo, 1990. [113]-116. <http://eudml.org/doc/221875>.

@inProceedings{Kozma1990,
abstract = {[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)].},
author = {Kozma, László},
booktitle = {Proceedings of the Winter School "Geometry and Physics"},
keywords = {Geometry; Physics; Proceedings; Winter school; Srní (Czechoslovakia)},
location = {Palermo},
pages = {[113]-116},
publisher = {Circolo Matematico di Palermo},
title = {On sprays and connections},
url = {http://eudml.org/doc/221875},
year = {1990},
}

TY - CLSWK
AU - Kozma, László
TI - On sprays and connections
T2 - Proceedings of the Winter School "Geometry and Physics"
PY - 1990
CY - Palermo
PB - Circolo Matematico di Palermo
SP - [113]
EP - 116
AB - [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)].
KW - Geometry; Physics; Proceedings; Winter school; Srní (Czechoslovakia)
UR - http://eudml.org/doc/221875
ER -

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