具有平面场的三维流形的Godbillon-Vey型不变量
A Godbillon-Vey type invariant for a 3-dimensional manifold with a plane field
摘要：
我们考虑一个三维光滑流形$M$，它具有一个任意的、\textit{a priori}不可积的分布（平面场）${\cal D}$和一个横截于${\cal D}$的向量场$T$.利用1-形式$ω$使得${\cal D} = \ker\,ω$和$ω(T)=1$，我们构造了一个类似于定义叶理的Godbillon-Vey类的3-形式，并展示了这个形式如何依赖于$ω$和~ $T$ .对于$M$上相容的黎曼度量，我们用正规曲线的曲率和挠率以及${\cal D}$的非对称第二基本型来表示这个3-形式.我们推导了相应泛函的Euler-Lagrange方程：对于$M$上的变量$({\cal D},T)$，以及对于$(M,{\cal D})$上的变量Riemann或Randers度量.我们表明，对于测地线域$T$（例如，对于接触结构），这样的$({\cal D},T)$是临界的，当${\cal D}$是可积的时，刻画临界对，并证明这些临界对不是极值。
英文摘要：
We consider a 3-dimensional smooth manifold $M$ equipped with an arbitrary, \textit{a priori} non-integrable, distribution (plane field) ${\cal D}$ and a vector field $T$ transverse to ${\cal D}$. Using a 1-form $ω$ such that ${\cal D} = \ker\,ω$ and $ω(T)=1$ we construct a 3-form analogous to that defining the Godbillon-Vey class of a foliation, and show how does this form depend on $ω$ and~$T$. For a compatible Riemannian metric on $M$, we express this 3-form in terms of the curvature and torsion of normal curves and the non-symmetric second fundamental form of ${\cal D}$. We deduce Euler-Lagrange equations of associated functionals: for variable $({\cal D},T)$ on $M$, and for variable Riemannian or Randers metric on $(M,{\cal D})$. We show that for a geodesic field $T$ (e.g., for a contact structure) such $({\cal D},T)$ is critical, characterize critical pairs when ${\cal D}$ is integrable, and prove that these critical pairs are not extrema.
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