dg.differential geometry – Paths $tg_1+(1-t)g_0$ in the moduli space of Riemann surfaces

Suppose $S$ is a smooth compact orientable surface without boundary. Let $g_0$ and $g_1$ be two smooth Riemannin metrics on $S$. Consider the interpolating path of metrics $g_t=g_1t+g_0(1-t)$. This gives us a path $gamma: (0,1)to M$ in the moduli space of Riemann surfaces (since each $g_t$ defines an integrable complex structure on $S$).

Question. Is is 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…)