Let $M$ be a closed smooth manifold, and let $g_0 \in \Gamma(\operatorname{Sym}^2 T^*M)$ be a smooth Riemannian metric on $M$. Then there exists $\varepsilon > 0$ and a smooth one-parameter family of Riemannian metrics
are smooth Ricci flows on $M$ with $g_1(0)=g_2(0)=g_0$, then there exists $\delta>0$ such that $g_1(t)=g_2(t)$ for every $t \in [0,\delta)$. Equivalently, the short-time solution is unique among smooth Ricci flows on the closed manifold $M$ with initial metric $g_0$.