Formula omitted]-structures on manifolds

Abstract

In this paper, we define and study two new structures on a differentiable manifold called by us an f(a,b)(3,2,1)-structure and a framed f(a,b)(3,2,1)-structure as a generalization of some geometric structures determined by polynomial structures, where a,b∈R and b≠0. At beginning, we present some examples regarding f(a,b)(3,2,1)-structures and establish their some fundamental properties. We also give a necessary condition for an f(a,b)(3,2,1)-structure to be an almost quadratic ϕ-structure. Later, it is shown that the existence of two semi-Riemannian metrics on differentiable manifolds admitting a framed f(a,b)(3,2,1)-structure, i.e., framed f(a,b)(3,2,1)-manifolds. In particular, a framed f(a,b)(3,2,1)-manifold endowed with the first semi-Riemannian metric mentioned above is called a framed metric f(a,b)(3,2,1)-manifold. Finally, we construct some examples to illustrate the existence of framed metric f(a,b)(3,2,1)-manifolds.