Ricci curvature in the neighborhood of rank-one symmetric spaces.
In his previous paper [J. Math. Pure. Appl. (9) 78, 389--430 (1999; Zbl 0970.53014)], the first author showed that every field of bilinear forms, close enough to the appropriate multiple of the standard metric, may be seen as the Ricci curvature of a close metric (in a weighted functional spaces topology) with the same structure at infinity as the model space. In the present paper these results are put in a general setting and some global aspects of the Ricci curvature of a Riemannian manifold, seen as a differential operator acting on the space of all metrics on the manifold and especially in a neighborhood of the standard metric of a rank-one noncompact symmetric space, are studied by using the techniques recently developed by \textit{O. Biquard} [Métriques d'Einstein asymptotiquement symmétriques, Astérisque, 265, Soc. Math. France (2000; Zbl 0967.53030)]. More precisely, the Ricci curvature can be prescribed in a neighborhood of the standard metric of any of the spaces \(R\mathbb{H}^n\) \((n\geq 10)\), \(c\mathbb{H}^n\) \((m\geq 5)\), \(H\mathbb{H}^p\) \((p\geq 2)\) and the Cayley hyperbolic plane \(O\mathbb{H}^2\) in topologies given by the weighted functional spaces. So, any symmetric bilinear field close enough to the standard may be realized as the Ricci curvature of a close metric if its decay rate at infinity (its weight) belongs to some precisely known interval. The case if the decay rate is too small or too large is studied.
scientific article; zbMATH DE number 1777182
scientific article