tailieunhanh - Linear connections on light-like manifolds

In case of existence, there is an infinitude of connections with none distinguished. We propose a method to single out connections with the help of a special set of 1-forms by the condition that the 1-forms become parallel with respect to this connection. | Turk J Math 32 (2008) , 41 – 49. ¨ ITAK ˙ c TUB Linear Connections on Light-like Manifolds T. Dereli, S ¸ . Ko¸cak and M. Limoncu Abstract It is well-known that a torsion-free linear connection on a light-like manifold (M, g) compatible with the degenerate metric g exists if and only if Rad(T M ) is a Killing distribution. In case of existence, there is an infinitude of connections with none distinguished. We propose a method to single out connections with the help of a special set of 1-forms by the condition that the 1-forms become parallel with respect to this connection. Such sets of 1-forms could be regarded as an additional structure imposed upon the light-like manifold. We consider also connections with torsion and with non-metricity on light-like manifolds. 1. Introduction In the following we will adopt the terminology of the book Duggal-Bejancu [1]. A light-like manifold (M, g) is a smooth manifold M with a smooth symmetric tensor field of type (0,2) with constant index and nullity-degree (co-rank) on M . There are half a dozen other names for such manifolds, “degenerate manifolds” being maybe one of the most popular terms. There are some scattered (and only partly related and sometimes duplicated) works of mathematical and physical origin in the literature about connections on light-like manifolds (see [1] and references therein, and also [2], [3], [4],[5]). It is an important result (it could be called the fundamental theorem of connections on light-like manifolds) that a torsion-free linear connection ∇ on M compatible with g (∇g = 0) exists if and only if Rad(T M ) is a Killing distribution ([DB]). Here, Rad(T M ) denotes the radical of g, that is the sub-bundle of T M with (Rad(T M ))x = Rad(Tx M ) = AMS Mathematics Subject Classification: Primary 55S10, 55S05 41 ˙ KOC ˙ DERELI, ¸ AK, LIMONCU {ξ ∈ Tx M | g(ξ, v) = 0 ∀ v ∈ Tx M } for x ∈ M . Rad(T M ) is a distribution of rank equal to the constant nullity-degree (co-rank) of gx . A distribution