接触同构群的Riemannian geometry
Riemannian geometry of the contactomorphism group
摘要：
在接触同构群上定义了一个右不变的黎曼度量，并研究了它的Euler-Arnold方程。如果度量与接触形式相关联，则欧拉-阿诺德方程简化为$m_t + u(m) + (n+2) mE(f) = 0$，根据Reeb域$E$、流函数$f$、由$f$定义的接触向量场$u$和动量$m = f - Δf$。在这里，方程被认为是一个紧凑的流形$M$的尺寸$2n+1$。当$n=0$这减少到Camassa-Holm方程，我们强调与高阶方程的类比。利用Euler-Arnold方程的动量守恒定律，将测地线方程改写为Sobolev类$H^s$接触同构群上的光滑一阶方程，从而得到了光滑依赖于初值的解的局部存在性.此外，我们证明了一个整体存在的标准类似的Beale-Kato-Majda标准在流体力学，并显示如何自动满足这个标准的全测地子群的量子态。最后简要讨论了Euler-Arnold方程的奇异解和守恒律。
英文摘要：
We define a right-invariant Riemannian metric on the group of contactomorphisms and study its Euler-Arnold equation. If the metric is associated to the contact form, the Euler-Arnold equation reduces to $m_t + u(m) + (n+2) mE(f) = 0$, in terms of the Reeb field $E$, a stream function $f$, the contact vector field $u$ defined by $f$, and the momentum $m = f - Δf$. Here the equation is considered on a compact manifold $M$ of dimension $2n+1$. When $n=0$ this reduces to the Camassa-Holm equation, and we emphasize the analogy with the higher-order equation. We use the usual momentum conservation law for Euler-Arnold equations to rewrite the geodesic equation as a smooth first-order equation on the contactomorphism group of Sobolev class $H^s$, and thus obtain local existence in time of solutions which depend smoothly on initial data. In addition we prove a global existence criterion analogous to the Beale-Kato-Majda criterion in fluid mechanics, and show how this criterion is automatically satisfied on the totally geodesic subgroup of quantomorphisms. Finally we briefly discuss singular solutions and conservation laws of the Euler-Arnold equation.
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