On the properties of certain metrics with constant scalar curvature

A. Raouf Chouikha

E-mail: chouikha@math.univ-paris13.fr

Abstract. In a preceding paper [Ch1], we showed some curvature properties of the Delaunay type metrics. These metrics are locally conformally flat of constant scalar curvature on the Riemannian product \ $(S^1 \times S^{n-1},dt^2+d{\xi}^2)$, a circle of length $T$ crossed with the standard (n-1)-sphere. In particular we have determined the number of these metrics in a conformal class $[g_0]_{_{T}}$. These metrics have a harmonic Riemannian curvature and a non parallel Ricci tensor, except for the cylindric one. We also remark a natural link between these metrics and the Derdzinski metrics, which are warped products: $ dt^2 + {f^2}(t) d{\xi}^2 $, and classify a family of Riemannian manifolds. Furthermore, we interest in the singularities of these Delaunay solutions for certain dimensions.

AMSclassification. 53C21, 53C25, 58G30

Keywords. Conformal class, Singular Yamabe metrics, Harmonic curvature