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On the properties of certain metrics with constant
scalar curvature

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*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