We provide necessary and sufficient conditions for some particular couples $(g,\nabla )$ of pseudo-Riemannian metrics and affine connections to be statistical structures if we have gradient almost Einstein, almost Ricci, almost Yamabe solitons, or a more general type of solitons on the manifold. In particular cases, we establish a formula for the volume of the manifold and give a lower and an upper bound for the norm of the Ricci curvature tensor field.

中文翻译：

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统计流形上的梯度孤子

我们为某些特定夫妇提供必要和充分的条件 $（G，\nabla ）$如果我们在流形上具有近似爱因斯坦，近似里奇，近似Yamabe孤子或更普通的孤子梯度，则将伪黎曼度量和仿射联系确定为统计结构。在特定情况下，我们为流形的体积建立一个公式，并给出Ricci曲率张量场范数的下限和上限。