We study the Riemannian geometry of 3D axisymmetric ideal fluids. We prove that the $L^2$ exponential map on the group of volume-preserving diffeomorphisms of a $3$-manifold is Fredholm along axisymmetric flows with sufficiently small swirl. Along the way, we define the notions of axisymmetric and swirl-free diffeomorphisms of any manifold with suitable symmetries and show that such diffeomorphisms form a totally geodesic submanifold of infinite $L^2$ diameter inside the space of volume-preserving diffeomorphisms whose diameter is known to be finite. As examples we derive the axisymmetric Euler equations on $3$-manifolds equipped with each of Thurston's eight model geometries.

中文翻译：

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黎曼三流形上的轴对称微分同胚和理想流体

我们研究 3D 轴对称理想流体的黎曼几何。我们证明了 $3$-流形的体积保持微分同胚群上的 $L^2$ 指数映射是沿轴对称流的 Fredholm，涡旋足够小。在此过程中，我们定义了具有适当对称性的任何流形的轴对称和无涡旋微分同胚的概念，并表明这种微分同胚在体积保持微分同胚的空间内形成了一个无限 $L^2$ 直径的完全测地子流形，其直径为已知是有限的。作为示例，我们在配备 Thurston 的八种模型几何形状中的每一种的 $3$-流形上推导出轴对称欧拉方程。