Suppose $\Omega_s \subset \mathbb{R}^n$ is a compact subset for each $s \in [0,T]$. I have a linear operator $p_t^s:H^1(\Omega_t) \to H^1(\Omega_s)$ which maps functions on $\Omega_t$ to functions on $\Omega_s$, and $p_t^s$ is continuous with continuous inverse $p_s^t$. It does this via a diffeomorphism $P_t^s:\Omega_s \to \Omega_t$ with $p_t^s f = f \circ P_t^s$.
Question We can define (the Levi-Civita connection) a modified gradient on $\Omega_t$ by $\nabla_{\Omega_t} f := (\nabla f)^T := \nabla f - (\nabla f \cdot N)N$ where the superscript $T$ denotes projection onto the tangent space of $\Omega_t$ and $N$ is the normal vector on $\Omega_t$. My question is for functions $g \in C^1(\Omega_s)$, is it true that $\nabla_{\Omega_t} (fg) =g\nabla_{\Omega_t} (f)?$ So can I take it out as a constant?
Essentially, I want to use this condition in an integral over $\Omega_t$. I tried writing everything out in coordinates (eg. $x$ on $\Omega_t$ and $y$ on $\Omega_s$) and used the diffeomorphism but I am very confused.