We consider a family of $\alpha$-connections defined by a pair of generalized dual quasi-statistical connections $(\hat{\nabla },{\hat{\nabla }}^{\ast })$ on the generalized tangent bundle $(TM\oplus T^{\ast }M,\check{h})$ and determine their curvature, Ricci curvature and scalar curvature. Moreover, we provide the necessary and sufficient condition for ${\hat{\nabla }}^{\ast }$ to be an equiaffine connection and we prove that if $h$ is symmetric and $\nabla h=0$, then $(TM\oplus T^{\ast }M,\check{h},{\hat{\nabla }}^{\left(\alpha \right)},{\hat{\nabla }}^{(-\alpha )})$ is a conjugate Ricci-symmetric manifold. Also, we characterize the integrability of a generalized almost product, of a generalized almost complex and of a generalized metallic structure w.r.t. the bracket defined by the $\alpha$-connection. Finally we study $\alpha$-connections defined by the twin metric of a pseudo-Riemannian manifold, $(M,g)$, with a non-degenerate $g$-symmetric $(1,1)$-tensor field $J$ such that $d^{\nabla }J=0$, where $\nabla$ is the Levi-Civita connection of $g$.

我们考虑一个家庭 $\alpha$-由一对广义双重准统计连接定义的连接 $（\hat{\nabla }，{\hat{\nabla }}^{\ast }）$ 在广义切线束上 $（Ť\mathrm{中号}\oplus {Ť}^{\ast }\mathrm{中号}，\check{H}）$并确定它们的曲率，里氏曲率和标量曲率。此外，我们提供了必要的充分条件${\hat{\nabla }}^{\ast }$ 成为等式连接，我们证明如果 $H$ 是对称的 $\nabla H=0$， 然后 $（Ť\mathrm{中号}\oplus {Ť}^{\ast }\mathrm{中号}，\check{H}，{\hat{\nabla }}^{（\alpha ）}，{\hat{\nabla }}^{（-\alpha ）}）$是共轭Ricci对称流形。同样，我们用括号定义了广义概乘积，广义概复形和广义金属结构的可集成性。$\alpha$-联系。最后我们学习$\alpha$-由伪黎曼流形的孪生度量定义的连接， $（\mathrm{中号}，G）$，具有非简并的 $G$不对称 $（\mathrm{1个}，\mathrm{1个}）$张量场 $Ĵ$ 这样 $d^{\nabla }Ĵ=0$， 在哪里 $\nabla$ 是Levi-Civita的连接 $G$。