论文标题
当地的保质黑森和统计歧管
Locally conformally Hessian and statistical manifolds
论文作者
论文摘要
统计歧管$ \ left（m，d，g \ right）$是带有无扭转连接$ d $和riemannian公制$ g $的歧管$ m $，因此张量$ d g $完全对称。如果$ d $是平坦的，则$ \ left（m，g，d \右）$是hessian歧管。当地的保质性黑森州（L.C.H）歧管是Hessian歧管$（C，\ nabla，g）$的商，因此单型组集团在Hessian Hysotheties上作用于$ C $，即此动作保留了一个组的$ \ nabla $和组性的$ G $ G $ $ G $。 L.C.H.等级是该字符的图像的等级，被视为单型组到实数的函数。 L.C.H.如果Lee Vector Field $ξ$杀死并满足$ \ nablaξ=λ\ id $，则称为辐射。我们证明了Radiant L.C.H. L.C.H.的指标等级1在所有Radiant L.C.H.的集合中都是密度的。指标。我们证明了紧凑型辐射L.C.H.的结构定理L.C.H.的歧管等级1。每一个这样的歧管$ c $都是在一个圆圈上纤维，纤维是恒定曲率的统计流形，纤维化是局部琐碎的，并且$ c $是从纤维上的统计结构和该纤维引起的单型自动形态的重构。
A statistical manifold $\left(M,D,g\right)$ is a manifold $M$ endowed with a torsion-free connection $D$ and a Riemannian metric $g$ such that the tensor $D g$ is totally symmetric. If $D$ is flat then $\left(M,g,D\right)$ is a Hessian manifold. A locally conformally Hessian (l.c.H) manifold is a quotient of a Hessian manifold $(C,\nabla,g)$ such that the monodromy group acts on $C$ by Hessian homotheties, i.e. this action preserves $\nabla$ and multiplies $g$ by a group character. The l.c.H. rank is the rank of the image of this character considered as a function from the monodromy group to real numbers. A l.c.H. manifold is called radiant if the Lee vector field $ξ$ is Killing and satisfies $\nabla ξ=λ\Id$. We prove that the set of radiant l.c.H. metrics of l.c.H. rank 1 is dense in the set of all radiant l.c.H. metrics. We prove a structure theorem for compact radiant l.c.H. manifold of l.c.H. rank 1. Every such manifold $C$ is fibered over a circle, the fibers are statistical manifolds of constant curvature, the fibration is locally trivial, and $C$ is reconstructed from the statistical structure on the fibers and the monodromy automorphism induced by this fibration.
