Suppose that $\omega$ is an 1-form on a Reimannain Manifold $(M,g)$ and $s$ is a $(0,2)$ symmetric tensor which be considered as $(1,1)$ symmetric tensor whenever it is convenient. If for all $s$ the following integral vanishes,

$$\int_M <‎\big( ‎‎-\mathrm{Ric} ‎+‎‎\dfrac{1}{2}‎Rg+‎‎2\mathrm{Sym}‎‎(\nabla ‎\omega)‎ +\omega\otimes \omega -|\pi |^2 ‎g\big) ‎,s>dV_‎g\\ +2\int_M (<s,\nabla U>+<‎\vec{\nabla}\mathrm{tr}(s),‎U> )‎dV_‎g =0$$

Then what we can obtain from this equality?

Note that $U$ is is defined by $w=g(U,.)$.


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