Suppose that $(M,g)$ is a Riemannain manifold and $\nabla$ denotes its Levi-Civita connection. Define a $(0,2)$ tensor field $\pi$ by $$\pi (X,Y)=(\nabla_X \omega)(Y)-\omega (X)\omega (Y)+\dfrac{1}{2}\omega (U)g(X,Y)$$ where $\omega$ is an 1-form on $(M,g)$ and $U$ is its equivalent vector field, in fact for all $X\in \mathcal{X}(M)$ $$\omega (X)=g(X,U).$$ Suppose $g(t)=g+ts$ and $\omega (t)=\omega +t\delta$, I want to linearize $|\pi |^2$. By means of a local coordinate system on $M$ with local frame $\partial_i$, set $\pi (\partial_i,\partial_j)=\pi_{ij}$. consequently, $$ \pi_{ij}=\nabla_i \omega_j -\omega_i \omega_j +\dfrac{1}{2}g_{ij} |\omega|^2 $$ where, $\omega(\partial_i)=\omega_i$. We have $$|\pi |^2=g^{ij}\pi_{ij}=\mathrm{div}(\omega) +(\dfrac{n}{2}-1)|\omega |^2 .$$ Consequently, $$ (|\pi (t) |^2 )'(0)=<d \omega +(\dfrac{n}{2}-1)\omega\otimes \omega,-s>+<g ,d\delta +(n-2)\omega\otimes \delta>. $$
Is my result correct?