Einstein extensions of Riemannian manifolds

Abstract

Given a Riemannian space N N of dimension n n and a field D D of symmetric endomorphisms on N N , we define the extension M M of N N by D D to be the Riemannian manifold of dimension n + 1 n+1 obtained from N N by a construction similar to extending a Lie group by a derivation of its Lie algebra. We find the conditions on N N and D D which imply that the extension M M is Einstein. In particular, we show that in this case, D D has constant eigenvalues; moreover, they are all integer (up to scaling) if det D ≠ 0 \det D \ne 0 . They must satisfy certain arithmetic relations which imply that there are only finitely many eigenvalue types of D D in every dimension (a similar result is known for Einstein solvmanifolds). We give the characterisation of Einstein extensions for particular eigenvalue types of D D , including the complete classification for the case when D D has two eigenvalues, one of which is multiplicity free. In the most interesting case, the extension is obtained, by an explicit procedure, from an almost Kähler Ricci flat manifold (in particular, from a Calabi-Yau manifold). We also show that all Einstein extensions of dimension four are Einstein solvmanifolds. A similar result holds valid in the case when N N is a Lie group with a left-invariant metric, under some additional assumptions.