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We construct an analytic formula of Green's function for the Laplace-Beltramioperator on surfaces with a non-trivial Killing vector field, which contain allsurfaces of revolution. To represent the function, we explicitly build a chartwhich is defined on the whole space except for at most two points, where themetric is represented by one positive function and where the Killing vectorfield is represented as a constant vector field preserving this positivefunction. In this construction, we immediately obtain the formula for a givensurface with a Killing vector field by solving the geodesic equations. Asapplications, we show that the logarithm of the speed and the pressure of theKilling vector field as a steady solution of the Euler equations is equal tothe convolution of a hydrodynamic Green's function and the Gaussian curvatureand twice that up to a harmonic function. Moreover, we see that a similarrelation holds for a potential vector field on general curved surfaces. As anexample, we apply the formula to surfaces of revolution.

Green's function for the Laplace-Beltrami operator on surfaces with a non-trivial Killing vector field and its application to potential flows