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

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.

### MSC:

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

### Citations:

Zbl 0284.53038; Zbl 0386.53031
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