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 $a,b\in \mathbb{R}$ and $b\ne 0$. 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.

在本文中，我们定义并研究了可微流形上的两个新结构，我们称之为 $F_{(\mathrm{一种},乙)}(3,2,1)$-结构和框架 $F_{(\mathrm{一种},乙)}(3,2,1)$-结构作为由多项式结构确定的某些几何结构的概括，其中 $\mathrm{一种},乙\in \mathbb{电阻}$ 和 $乙\ne 0$. 首先，我们提供一些关于$F_{(\mathrm{一种},乙)}(3,2,1)$-结构并建立它们的一些基本属性。我们还给出了一个必要条件$F_{(\mathrm{一种},乙)}(3,2,1)$-结构是一个几乎二次的ϕ -结构。后来，证明了在可微流形上存在两个半黎曼度量，允许一个框架$F_{(\mathrm{一种},乙)}(3,2,1)$-结构，即框架 $F_{(\mathrm{一种},乙)}(3,2,1)$-歧管。特别是，一个框架$F_{(\mathrm{一种},乙)}(3,2,1)$-流形赋予上述第一个半黎曼度量称为框架度量 $F_{(\mathrm{一种},乙)}(3,2,1)$-歧管。最后，我们构造了一些例子来说明框架度量的存在$F_{(\mathrm{一种},乙)}(3,2,1)$-歧管。