Suppose $S$ is a smooth compact oriented surface without boundary. Let $g_0$ and $g_1$ be two smooth Riemannian metrics on $S$. Consider the interpolating path of metrics $g_t=g_1t+g_0(1-t)$. Recall that a Riemannian metric $g$ on an oriented surface defines a unique (integrable) almost complex structure $J$ satisfying $J(g)=g$, $J^2=-1$. So we get a path $\gamma: [0,1]\to M$ in the moduli space of Riemann surfaces.

**Question.** Is it true that $\gamma$ is a real analytic path in $M$? If so, how can I convince myself in this? (the statement strikes me as counter-intuitive...) (We recall that the moduli space of Riemann surfaces (of fixed genus) has a natural real analytic structure (for example given by Fenchel–Nielsen coordinates))

**PS.** The comment of abx below suddenly makes this statement much more plausible for me. Indeed if we look at the path of $J_t$, then at each point $x\in S$, ${J_t}_x$ depends analytically on $t$ (by obvious linear algebra). But still, how to go from here to saying that the path in the moduli space is real analytic?

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