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We provide a general B\"ochner type formula which enables us to prove somerigidity results for $V$-static spaces. In particular, we show that an$n$-dimensional positive static triple with connected boundary and positivescalar curvature must be isometric to the standard hemisphere, provided thatthe metric has zero radial Weyl curvature and satisfies a suitable pinchingcondition. Moreover, we classify $V$-static spaces with non-negative sectionalcurvature.

Remarks on critical metrics of the scalar curvature and volume functionals on compact manifolds with boundary