Consider the Navier-Stokes Equations (NSE) for viscous incompressible fluid flows on a non-compact, smooth, simply-connected and complete Einstein manifold \((\mathbf {M},g)\) with negative Ricci curvature tensor. We prove the existence and uniqueness of a time-periodic solution to NSE for vector fields on \((\mathbf {M},g)\). Our method is based on the dispersive and smoothing properties of the semigroup generated by the linearized Stokes equations to construct a bounded (in time) solution of the nonhomogeneous Stokes equation and on the ergodic method to obtain the periodic solution to Stokes equation. Then, using the fixed point arguments, we can pass from the Stokes equations to Navier-Stokes equations to obtain periodic solutions to NSE on the Einstein manifold \((\mathbf {M},g)\). We also prove the stability of the periodic solution.

考虑Navier-Stokes方程（NSE），用于粘性非压缩流体在非紧凑，光滑，简单连接且具有负Ricci曲率张量的完整爱因斯坦流形\（（\ mathbf {M}，g）\）上。我们证明了\（（\ mathbf {M}，g）\）上向量场的NSE时间周期解的存在性和唯一性。我们的方法基于线性化Stokes方程生成的半群的分散和平滑特性，以构造非齐次Stokes方程的有界（及时）解，并基于遍历方法获得Stokes方程的周期解。然后，使用不动点参数，我们可以将Stokes方程传递给Navier-Stokes方程，以获得关于Einstein流形上NSE的周期解\（（\ mathbf {M}，g）\）。我们还证明了周期解的稳定性。