Inhomogeneous Einstein metrics on low-dimensional spheres and other low-dimensional spaces. (English) Zbl 0965.53033

Böhm, Christoph

Invent. Math. 134, No. 1, 145-176 (1998).

The standard metric on the \(n\)-dimensional sphere \(S^{n}\) is Einstein. The first non-trivial examples of an Einstein metric on \(S^{n}\), \(n = 4m + 3\), was given by G. R. Jensen [J. Differ. Geom. 8, 599-614 (1973; Zbl 0284.53038)]. Five years later J. P. Bourguignon and H. Karcher described a further metric on \(S^{15}\) [Ann. Sci. Ec. Norm. Supér. 11, 71-92 (1978; Zbl 0386.53031)]. These metrics were the only known examples of Einstein metrics on spheres. All of them are homogeneous. It is also known there exists no further homogeneous metric on \(S^{n}\).
In the paper under review, infinitely many inhomogeneous Einstein metrics with positive scalar curvature on \(S^{n}\), \(n = 5, 6, 7, 8, 9\) as well as on other low-dimensional spaces, for instance \(S^{3} \times S^{2}\), \(S^{3} \times S^{3}\), \(S^{3} \times {\mathbb CP}^{2}\), \(S^{4} \times {\mathbb RP}^{2} \times {\mathbb RP}^{2}\), \(S^{3} \times S^{3} \times S^{3}\), \(S^{7} \times S^{2}\), are determined. A consequence of this construction are new examples of Riemannian manifolds where the Palais-Smale condition for the total scalar curvature functional is not fulfilled.

Reviewer: Ryszard Deszcz (Wroclaw)

MSC:

53C25	Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C30	Differential geometry of homogeneous manifolds

Keywords:

Einstein metric; homogeneous space; inhomogeneous space; total scalar curvature functional; warped product; Palais-Smale condition

Citations:

Zbl 0284.53038; Zbl 0386.53031