Einstein extensions of Riemannian manifolds

TL;DR

This paper investigates conditions under which Riemannian manifold extensions, constructed via symmetric endomorphisms, are Einstein, revealing finite eigenvalue types, explicit classifications, and connections to special geometric structures like Calabi-Yau manifolds.

Contribution

It characterizes Einstein extensions of Riemannian manifolds using symmetric endomorphisms, providing classifications for specific eigenvalue types and linking to known geometric structures.

Findings

Eigenvalues of D are constant and integer (up to scaling) if det D ≠ 0.

Finite eigenvalue types of D exist in each dimension due to arithmetic relations.

Complete classification of Einstein extensions when D has two eigenvalues, one multiplicity free.

Abstract

Given a Riemannian space $N$ of dimension $n$ and a field $D$ of symmetric endomorphisms on $N$, we define the extension $M$ of $N$ by $D$ to be the Riemannian manifold of dimension $n+1$ obtained from $N$ by a construction similar to extending a Lie group by a derivation of its Lie algebra. We find the conditions on $N$ and $D$ which imply that the extension $M$ is Einstein. In particular, we show that in this case, $D$ has constant eigenvalues; moreover, they are all integer (up to scaling) if $\det D \ne 0$. They must satisfy certain arithmetic relations which imply that there are only finitely many eigenvalue types of $D$ in every dimension (a similar result is known for Einstein solvmanifolds). We give the characterisation of Einstein extensions for particular eigenvalue types of $D$, including the complete classification for the case when $D$ 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 K\"{a}hler 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 $N$ is a Lie group with a left-invariant metric, under some additional assumptions.