$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\TM}{T\mathcal{M}}$ $\newcommand{\Ric}{\operatorname{Ric}}$ $\newcommand{\Volg}{\operatorname{Vol}_g}$
I would like to find a reference for the following claim:
Let $(\M,g)$ be a closed oriented $d$-dimensional Riemannian manifold. Then $$ \int_{\M} \Ric(V,V) \Volg \le (d-1) \int_{\M} |\nabla V|^2 \Volg, $$ for every vector field $V \in \Gamma(\TM)$. ($\nabla$ is the Levi-civita connection).
For $d=2$ equality holds if and only if $V$ is a conformal vector field.
For $d \ge 3$ equality holds if and only if $V$ is a concircular* vector field.
*A vector field $V$ is called concircular if $\nabla V=h\text{Id}_{\TM}$ for some $h \in C^{\infty}(\M)$. Every such vector field is conformal, but not vice versa.
I am quite sure this should be classic, but I could not find this statement anywhere.
