Consider $M$ a smooth compact connected manifold with $w$ a volum form. Take for example $M=\mathbb{S}^n$ with the Haar measure. For any smooth map $f:M\rightarrow M$ and its pullback measure $\nu = f^* w$ it is well known that we have $$\int_M \nu = \text{deg}(f)\int_M w $$with $\deg(f)\in \mathbb{Z}$. Is this condition sufficient? For any $\nu$ volum form on $M$ such that $\int_M \nu / \int_M w \in \mathbb{Z} $ does there exist $f:M\rightarrow M$ such that $\nu = f^*w$ ? Can we construct such an $f$ explicitly ?

Consider $M$ a smooth compact connected manifold with $w$ a volum form. Take for example $M=\mathbb{S}^n$ with the uniform measure. For any smooth map $f:M\rightarrow M$ and its pullback measure $\nu = f^* w$ it is well known that we have $$\int_M \nu = \text{deg}(f)\int_M w $$with $\deg(f)\in \mathbb{Z}$. Is this condition sufficient? For any $\nu$ volum form on $M$ such that $\int_M \nu / \int_M w \in \mathbb{Z} $ does there exist $f:M\rightarrow M$ such that $\nu = f^*w$ ? Can we construct such an $f$ explicitly ?