Einstein solvmanifolds as submanifolds of symmetric spaces

TL;DR

This paper demonstrates that Einstein solvmanifolds can be embedded as submanifolds in symmetric spaces, providing a geometric realization that bridges Lie group representations and submanifold theory.

Contribution

It establishes that all Einstein solvmanifolds can be realized as submanifolds within symmetric spaces, extending previous work and connecting geometric and algebraic structures.

Findings

All Einstein solvmanifolds can be embedded as submanifolds in symmetric spaces.

The work includes Riemannian 2-step nilpotent Lie groups and Ricci soliton solvmanifolds.

Provides a geometric framework linking Lie groups and symmetric spaces.

Abstract

Lying at the intersection of Ado's theorem and the Nash embedding theorem, we consider the problem of finding faithful representations of Lie groups which are simultaneously isometric embeddings. Such special maps are found for a certain class of solvable Lie groups which includes all Einstein and Ricci soliton solvmanifolds, as well as all Riemannian 2-step nilpotent Lie groups. As a consequence, we extend work of Tamaru by showing that all Einstein solvmanifolds can be realized as submanifolds (in the submanifold geometry) of a symmetric space.