inverse function theorem on manifolds

Question

suppose there are two 3-manifolds(consider them as orthogonal matrices $SL(2,\mathbb R)$), and there is $F:SL(2,\mathbb R)\to SL(2,\mathbb R)$, given by $F(A）=A^3$. Can we apply inverse function theorem on these two manifolds so we can say that there are two open sets $U,V$, for each $B\in V \subset SL(2,\mathbb R)$), there is $A\in U\subset SL(2,\mathbb R)$, such that $A^3=B$. Usually we can only use inverse function theorem on two open sets of $\mathbb R^k$, is there a generalized version on manifolds? If it exists, how can we justify it?