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Combining several results on related (or conjugate) connections, defined on banachable fibre bundles, we set up a machinery, which permits to study various transformations of linear connections. Global and local methods are applied throughout. As an application, we get an extension of the classical affine transformations to the context of infinite-dimensional vector bundles. Another application shows that, realising the ordinary linear differential equations (in Banach spaces) as connections, we get the usual transformations of (equivalent) equations. Thus, some classical results on differential equations, such as the Theorem of Floquet, can have a “geometric” interpretation.
Abstract
Pseudoconnections (or quasi connections) were defined as a generalization of linear connections by Y.-C. Wong in [14], and were developed mainly by Italian and Rumanian mathematicians. The purpose of this paper is to study some properties of a special type of pseudoconnections: the so-called associated pseudoconnections oginirating from linear connections in a very simple manner. In 1 we give a necessary and sufficient condition for a pseudoconnection to be associated, the in 2 we study the geodesics of an associated pseudoconnection. This notion has an immediate application in Finsler geometry, this is the theme of 3. Some questions connecting the curvature of associated pseudoconnections were studied by the author in [7].
The purpose of this paper is to study the principal fibre bundle (P, M, G, π p ) with Lie group G, where M admits Lorentzian almost paracontact structure (Ø, ξ p , η p , g) satisfying certain condtions on (1, 1) tensor field J, indeed possesses an almost product structure on the principal fibre bundle. In the later sections, we have defined trilinear frame bundle and have proved that the trilinear frame bundle is the principal bundle and have proved in Theorem 5.1 that the Jacobian map π * is the isomorphism.
. 1975 On semi-symmetric and quarter-symmetric linear connections Tensor (N.S.) 29 249 – 254 . [6] Hayden , H. A
Trans. Amer. Math. Soc. 99 325 – 341 10.1090/S0002-9947-1961-0121751-2 . [18] Wong , Y.-C. 1962 Linear connexions with zero
., 1966 . [32] Ch.-L . Terng . Natural vector bundles and natural differential operators . Amer. J. Math ., 100 : 775 – 828 , 1978 . [33] K . Yano and A. J . Ledger . Linear connections on tangent bundles . J. London Math. Soc ., 39 : 495 – 500
