Sur de nouvelles variétés riemanniennes d’Einstein. (French) Zbl 0544.53038

Bérard Bergery, Lionel

Inst. Elie Cartan, Univ. Nancy I 6, 1-60 (1983).

D. Page constructed a non-homogeneous Riemannian Einstein metric on \(\overline{CP}^ 2\#\overline{CP}^ 2\) with positive scalar curvature [A compact rotating gravitational instanton (Preprint)]. This metric has a 4-dimensional isometry group which admits a 3-dimensional principal orbit. In this paper, the author considers another representation of Page’s metric which may permit a characterization of this metric and a generalization for higher dimensions. He obtains the following results. Let (M,g) be a compact, connected and simply connected 4-dimensional Einstein manifold whose isometry group is of dimension \(\geq\) 4. Then, (M,g) is symmetric or it is \(CP^ 2\#\overline{CP}^ 2\) with Page’s metric (up to a factor). Let be a compact Kählerian Einstein manifold with positive scalar curvature, and suppose that the first Chern class \(c_ 1(N)\) is \(p\alpha\), where p is an integer \(>1\), and \(\alpha\) is an indivisible class with integer coefficients. For any integer q, let \(E_ q\to N\) be the complex line bundle of class \(q\alpha\) and \(M_ q\to N\) be the associated bundle of \(E_ q\) with fibre \(S^ 2=CP^ 1\). Then, if \(0<q<p\), \(M_ q\) admits a Riemannian Einstein metric being Hermitian and conformal to Kählerian (but not Kählerian). Moreover, for any integer q, the complex line bundle \(E_ q\) admits a one parameter family of complete Hermitian Einstein metrics with negative scalar curvature. If \(q>p\), there is a Kählerian metric, if 1\(\leq q\leq p\), there is a metric (non-Kählerian) with zero Ricci curvature, and if \(q=p\), \(E_ q\) admits a Kählerian metric with zero Ricci curvature.

Reviewer: S.Takizawa

MSC:

53C25	Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C55	Global differential geometry of Hermitian and Kählerian manifolds

Keywords:

Einstein metric; positive scalar curvature; Page’s metric; Kählerian Einstein manifold; complex line bundle; Hermitian Einstein metrics; Ricci curvature