The authors obtain the following integral geometric inequality on a closed Riemannian manifold with \(\infty\)-Bakry-Émery Ricci curvature bounded from below: If \((M,g)\) is a closed Riemannian manifold of dimension \(n\geq3\) and \(f\) is a \(C^2(M)\) function, then \[ (*)\quad \int_M(N_f-\overline{N_f})^2e^{-f}dv_g\leq 4\Big(1+\frac{K}{\eta_1}\Big)\int_M|\mathrm{Ric}_f-\lambda g|^2e^{-f}dv_g, \] where \(\eta_1\) denotes the first non-zero eigenvalue for the \(f\)-Laplacian \(\Delta_f\) on \(M\), \(K\) is a non-negative constant such that the \(\infty\)-Bakry-Émery Ricci curvature \(\mathrm{Ric}_f\) satisfies \(\mathrm{Ric}_f\geq-K\), and \(\lambda\geq-K\) is a real constant. Here \[ N_f=R+2\Delta f-|\nabla f|^2+2\lambda f,\quad \overline{N_f}=\int_MN_fe^{-f}dv_g\Big/\int_Me^{-f}dv_g. \] Moreover, equality holds in \((*)\) if and only if \(\mathrm{Ric}_f=\lambda g\).
A similar inequality is obtained for Riemannian manifolds with totally geodesic boundary. Suppose \((M,g)\) is a compact Riemannian manifold of dimension \(n\geq 3\) with totally geodesic boundary \(\partial M\), and \(f\) is a \(C^2(M)\) function. If \(f\) satisfies the Dirichlet boundary condition or the Neumann boundary condition, then \[ (**)\quad\int_M(N_f-\overline{N_f})^2e^{-f}dv_g\leq 4\Big(1+\frac{K}{\xi_1}\Big)\int_M|\mathrm{Ric}_f-\lambda g|^2e^{-f}dv_g, \] where \(\xi_1\) denotes the first non-zero Neumann eigenvalue of \(\Delta_f\) on \(M\), \(K\) is a non-negative constant such that \(\mathrm{Ric}_f\geq-K\), and \(\lambda\geq-K\) is a real constant. Moreover, equality holds in \((**)\) if and only if \(\mathrm{Ric}_f=\lambda g\).
These results generalize previous results of J.-Y. Wu [Geom. Dedicata 169, 273–281 (2014; Zbl 1293.53050)].