probability – maps a random variable to a Gaussian

If I have a random variable $ X in mathbb {R} ^ n $, under what conditions there is a $ C ^ 1 $ function $ varphi: mathbb {R} ^ n rightarrow mathbb {R} ^ n $ such that $ varphi (X) sim mathcal {N} (0, I_n) $ (vector $ n $ independent normal variables)?

$ X $ It probably has to have a density to begin with … But I'm a bit stuck there. I know that if $ n = 1 $, the inverse transformation sampling can be used to solve this problem, but I have trouble finding a generalization to $ R ^ n $.