# 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…)