有旋轴对称理想流体流动的几何
The Geometry of Axisymmetric Ideal Fluid Flows with Swirl
摘要：
黎曼流形$M$的保体积同态群的截面曲率可以给出关于$M$上无粘不可压缩流体流动的稳定性的信息。本文证明了由轴对称旋转流体流产生的固体平面环面的体态群的子流形（记为$\mathcal{D}_{μ,E}(M)$）在包含场$X = u(r)\partial_θ$的每个截面上具有正截面曲率，当且仅当$\partial_r(ru^2)>0$ .这是在形成鲜明对比的情况下，$\mathcal{D}_μ(M)$，其中只有Killing场$X$具有非负截面曲率在所有部分包含它。我们还表明，该标准保证了存在的共轭点上$\mathcal{D}_{μ,E}(M)$沿测地线定义的$X$。
英文摘要：
The sectional curvature of the volume preserving diffeomorphism group of a Riemannian manifold $M$ can give information about the stability of inviscid, incompressible fluid flows on $M$. We demonstrate that the submanifold of the volumorphism group of the solid flat torus generated by axisymmetric fluid flows with swirl, denoted by $\mathcal{D}_{μ,E}(M)$, has positive sectional curvature in every section containing the field $X = u(r)\partial_θ$ iff $\partial_r(ru^2)>0$. This is in sharp contrast to the situation on $\mathcal{D}_μ(M)$, where only Killing fields $X$ have nonnegative sectional curvature in all sections containing it. We also show that this criterion guarantees the existence of conjugate points on $\mathcal{D}_{μ,E}(M)$ along the geodesic defined by $X$.
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