Splitting, parallel gradient and Bakry-Emery Ricci curvature

In this paper we obtain a splitting theorem for the symmetric diffusion operator ${\mathrm{\Delta }}_{\varphi }=\mathrm{\Delta }-<\mathrm{\nabla }\varphi ,\mathrm{\nabla }>$ and a non-constant $C^3$ function $f$ in a complete Riemannian manifold $M$, under the assumptions that the Ricci curvature associated with ${\mathrm{\Delta }}_{\varphi }$ satisfies ${\mathrm{R}\mathrm{i}\mathrm{c}}_{\varphi }(\mathrm{\nabla }f,\mathrm{\nabla }f)\ge 0$, that $\mathrm{\mid }\mathrm{\nabla }f\mathrm{\mid }$ attains a maximum at $M$ and that ${\mathrm{\Delta }}_{\varphi }$ is non-decreasing along the orbits of $\mathrm{\nabla }f$. The proof uses the general fact that a complete manifold $M$ with a non-constant smooth function $f$ with parallel gradient vector field must be a Riemannian product $M=N\times \mathbb{R}$, where $N$ is any level set of $f$.