so the concept of set of measure zero makes sense, which is independent of the choice of smooth measure.

0.3cm Let $M, N$ be two $n$-dimensional smooth manifolds, and let $\varphi:M\rightarrow N$ be a smooth map. If $dy$ is a smooth (or quasi-smooth) measure on $N$ which can be expressed locally as $dy=f(y)dy_1\cdots dy_n$, we can define the *pull bake* $\varphi^*(dy)$ of $dy$ locally as $$\label{E:pullback}
\varphi^* (dy)=f(\varphi(x))\left|\det\left(\frac{\partial
y}{\partial x}\right)\right|dx_1\cdots dx_n.$$ It is easily to check that the definition is compatible when we choose different coordinate charts, and $\varphi^*(dy)$ is a quasi-smooth measure on $M$. We can not expect $\varphi^*(dy)$ is smooth in general, even if $dy$ is smooth, since $\varphi$ may have critical points. But if $\varphi$ is a local diffeomorphism and $dy$ is smooth, then $\varphi^*(dy)$ is smooth.

0.3cm If $M, N$ are Riemannian manifolds and $dx, dy$ are the associated Riemannian measures, then we can express the pull back measure $\varphi^*(dy)$ globally. To do this, first we need some comments on the “determinant" of a linear map between two different inner product vector spaces of the same dimension. Suppose $V,W$ are two $n$-dimensional vector spaces with inner products. For $n$ vectors $v_1,\cdots,v_n\in V$, let $a_{ij}=\langle v_i,v_j\rangle$ for 