The answer is yes — the change of variables between is necessarily analytic.

The harmonic functions have to be analytic in both of your coordinates. So you can pass from one of your coordinates to some harmonic coordinates and then pass from the harmonic coordinates to the other of your coordinates.

The answer is yes — the change of variables between is necessarily analytic.

The harmonic functions have to be analytic in both of your coordinates. So you can pass from one of your coordinates to some harmonic coordinates and then pass from the harmonic coordinates to the other of your coordinates.

P.S. This idea is due to Shefel, and it was rediscovered independently by DeTurck and Kazdan. See correspondingly

• Шефель, С. З. "Конформное соответствие метрик и гладкость изометрических погружений." Сибирский математический журнал 20.2 (1979): 397—401.
• Dennis DeTurck, Jerry Kazdan Some regularity theorems in Riemannian geometry. Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 3, 249–260.

The naswer is yes --- the change of variables between is necessarily analytic.

The harmonic functions have to be analytic in both of your coordinates. So you can pass from one of your coordinates to some harmonic coordinates and then pass from the harmonic coordinates to the other of your coordinates.

The answer is yes — the change of variables between is necessarily analytic.

The harmonic functions have to be analytic in both of your coordinates. So you can pass from one of your coordinates to some harmonic coordinates and then pass from the harmonic coordinates to the other of your coordinates.