I am asking for either motivation on the requirement regarding $f_i^{-1}(0)$ in the following definition, or better yet a reference to a book dealing with this subject. The following is the definition of a compact Riemann manifold with piecewise smooth boundary from chapter 6 of Ergodic Theory by Cornfeld, Fomin, Sinai:
Suppose $Q_0$ is a closed Riemann manifold of class $C^\infty$, possibly noncompact. Suppose that $r$ functions $f_1,f_2,\dots,f_r$ of classs $C^\infty$ are given on $Q_0$. The set $Q = \{q \in Q_0\mid f_i(q) \geq 0, 1 \leq i \leq r\}$ is said to be a compact Riemann manifold with piecewise smooth boundary if 1.) $Q$ is compact, 2.) the set $f_i^{-1}(0)$ does not contain any critical points of the function $f_i$ and is therefore a $C^\infty$-submanifold of codimension 1, $1 \leq i \leq r$, 3.) the gradients $\nabla f_i, \nabla f_j$ are linearly independent at intersection points $q \in f_i^{-1}(0)\cap f_j^{-1}(0)$.
If I understood correctly, $f_1,\dots,f_r$ are just some smooth functions on $Q_0$. What is not clicking for me is what makes $0$ such a special value that we want to impose the said requirements on the inverse images of $f_1,\dots,f_r$?