Semiintegrable almost Grassmann structures

Abstract

In the present paper we study locally semiflat (we also call them semiintegrable) almost Grassmann structures. We establish necessary and sufficient conditions for an almost Grassmann structure to be α-or β-semiintegrable. These conditions are expressed in terms of the fundamental tensors of almost Grassmann structures. Since we are not able to prove the existence of locally semiflat almost Grassmann structures in the general case, we give many examples of α-and β-semiintegrable structures, mostly four-dimensional. For all examples we find systems of differential equations of the families of integral submanifolds Vα and V β of the distributions ∆α and ∆ β of plane elements associated with an almost Grassmann structure. For some examples we were able to integrate these systems and find closed form equations of submanifolds Vα and V β .

Curvature Properties of Some Four-Dimensional Manifolds

“…Every AG-metric fulfils the curvature condition In Section 2 we prove that the Bach tensor B ( [14]) of any 4-dimensional semi-Riemannian manifold (M,g), fulfilling (l)(ii), (l)(iii) and (2), satisfies B = • S. In Section 4 we state that every AG-metric fulfils We check that the AG-metrics g considered in the Examples 3.5-3.13 and 3.15 of [2] satisfy (4) u…”

“…In Section 2 we present a review of such conditions. In addition, in Section 3 we give some results on 4-dimensional warped products which we apply in Section 4 to prove that the AG-metric g, defined in Example 3.5 of [2], is a non-warped product metric. We prove also that this metric cannot be realized on a hypersurface in a 5-dimensional space of constant curvature.…”

“…The manifold M endowed with such structure is said to be an almost Grassmann manifold ( [1], Chapter 7; [2], Definition 1.1). Certain additional conditions lead to so-called semiintegrable almost Grassmann structures ( [2], Definition 1.2). The latter were studied in [2] and examples of such structures, mainly 4-dimensional, are presented there.…”

“…Certain additional conditions lead to so-called semiintegrable almost Grassmann structures ( [2], Definition 1.2). The latter were studied in [2] and examples of such structures, mainly 4-dimensional, are presented there. Certain semi-Riemannian metrics are related to these structures.…”

“…The problem arises to describe curvature properties and, in particular, curvature properties of pseudosymmetry type, of these metrics. In this paper we present curvature properties of 4-dimensional metrics g considered in the Examples 3.5 -3.16 of [2]. These metrics will be called Akivis-Goldberg, in short AG-metrics.…”