论文标题
多场宇宙学模型的黑森歧管和黑森对称性
Hesse manifolds and Hessian symmetries of multifield cosmological models
论文作者
论文摘要
我简要概述了多场宇宙学模型的Noethy对称性的数学理论，这些模型将自然分解为可见和黑森（又称“隐藏”）对称性。虽然可见的对称性对应于标量场图的riemannian目标空间的无限异构体，即保留标量势，但黑森对称性具有更深的理论。后者对应于Hesse函数，定义为目标空间所谓的黑森方程的解。从定义上讲，黑森歧管是一种riemannian歧管，它承认非平凡的黑森函数 - 不要与Hessian歧管混淆（后者是Riemannian歧管，其指标是函数的本地Hessian）。所有Hesse $ n $ -manifolds $ {\ cal m} $都是不合理的，其特征是其索引，定义为Hesse函数空间的维度，它带有天然的对称双线性配对。黑森指数从上面限制为$ n+1 $，并且，当指标完成后，如果$ {\ cal m} $获得该界限，则可以通过$ \ mathbb {r}^{1，n} $ sable shemomorphism构造的weierss sap sab sabs sabs neff $ {\ cal m} $进行poincaréBall。更一般而言，任何基本双曲线空间形式都是完整的黑森歧管，而局部黑森指数最大的任何黑森歧管都是双曲线。特别是，完整的黑森表面类别与基本双曲线表面相吻合，因此，任何这样的表面都与庞加莱磁盘，双曲线刺穿盘或双曲线圆环的等速线相吻合。在完整的黑森歧管$（{\ cal m}，g）$上，任何黑森函数$λ$的值可以通过$ {\ cal m} $的特征子集的距离来表示。
I give a brief overview of the mathematical theory of Noether symmetries of multifield cosmological models, which decompose naturally into visible and Hessian (a.k.a. 'hidden') symmetries. While visible symmetries correspond to those infinitesimal isometries of the Riemannian target space of the scalar field map which preserve the scalar potential, Hessian symmetries have a much deeper theory. The latter correspond to Hesse functions, defined as solutions of the so-called Hesse equation of the target space. By definition, a Hesse manifold is a Riemannian manifold which admits nontrivial Hesse functions -- not to be confused with a Hessian manifold (the latter being a Riemannian manifold whose metric is locally the Hessian of a function). All Hesse $n$-manifolds ${\cal M}$ are non-compact and characterized by their index, defined as the dimension of the space of Hesse functions, which carries a natural symmetric bilinear pairing. The Hesse index is bounded from above by $n+1$ and, when the metric is complete, this bound is attained iff ${\cal M}$ is a Poincaré ball, in which case the space of Hesse functions identifies with $\mathbb{R}^{1,n}$ through an isomorphism constructed from the Weierstrass map. More generally, any elementary hyperbolic space form is a complete Hesse manifold and any Hesse manifold whose local Hesse index is maximal is hyperbolic. In particular, the class of complete Hesse surfaces coincides with that of elementary hyperbolic surfaces and hence any such surface is isometric with the Poincaré disk, the hyperbolic punctured disk or a hyperbolic annulus. On a complete Hesse manifold $({\cal M},G)$, the value of any Hesse function $Λ$ can be expressed though the distance from a characteristic subset of ${\cal M}$ determined by $Λ$.
