The form sum and the Friedrichs extension of Schrödinger-type operators on Riemannian manifolds

We consider $H_V=\Delta_M+V$, where $(M,g)$ is a Riemannian manifold (not necessarily complete), and $\Delta_M$ is the scalar Laplacian on $M$. We assume that $V=V_0+V_1$, where $V_0\in L_{\operatorname{loc}}^2(M)$and $-C\leq V_1\in L_{\operatorname{loc}}^1(M)$ ($C$ is a constant) are real-valued, and $\Delta_M+V_0$ is semibounded below on $C_{c}^{\infty}(M)$. Let $T_0$ be the Friedrichs extension of $(\Delta_M+V_0)\vert _{C_{c}^{\infty}(M)}$. We prove that the form sum $T_0\tilde{+} V_1$ coincides with the self-adjoint operator $T_F$ associated to the closure of the restriction to $C_{c}^{\infty}(M)\times C_{c}^{\infty}(M)$ of the sum of two closed quadratic forms of $T_0$ and $V_1$. This is an extension of a result of Cycon. The proof adopts the scheme of Cycon, but requires the use of a more general version of Kato's inequality for operators on Riemannian manifolds.