Summary: In this paper, we prove the existence of normalized ground state solutions for the following Schrödinger equation \[\begin{cases} -\Delta u-a(x)f(u)=\lambda u, \quad x\in \mathbb{R}^N;\\ u\in H^1(\mathbb{R}^N), \end{cases}\] and give a better representation of its geometrical structure, where \(N\geq 1\), \(\lambda \in \mathbb{R}\), \(a\in \mathcal{C}(\mathbb{R}^N, [0,\infty))\) with \(0<a_{\infty}:=\lim_{|y|\rightarrow \infty}a(y)\leq a(x)\) and \(f\in \mathcal{C}(\mathbb{R},\mathbb{R})\) satisfies general assumptions. In particular, we propose a new approach to recover the compactness for a minimizing sequence on a suitable manifold, and overcome the essential difficulties due to the nonconstant potential \(a\).