On locally conformally flat critical metrics for quadratic functionals

Abstract

The aim of this note is to present some results about critical metrics for quadratic functional ${\mathcal {F}}_{t,s}$ defined on ${\mathcal {M}}_{1}=\{g\in {\mathcal {M}} | \mathrm{Vol}(g)=1\}$, where ${\mathcal {M}}$ is the space of smooth Riemannian metrics on a closed smooth Riemannian manifold $M^n$. We know that space form metrics are critical point for ${\mathcal {F}}_{t,s}$. We investigate when the converse is true. In particular, we show that locally conformally flat critical metrics with some additional conditions are space form metrics.