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A non-Standard Indefinite Einstein Solvmanifold

Federico A. Rossi

TL;DR

This paper constructs a concrete nonstandard example of an indefinite Einstein solvmanifold. It presents a 5-dimensional solvable Lie algebra with a left-invariant Einstein metric whose nilradical coincides with the derived algebra and whose orthogonal complement is nonabelian, failing standard and pseudo-Iwasawa decompositions. The metric is explicitly computed to satisfy $\widetilde{Ric}=\frac{4096}{175}\mathrm{Id}$ with $\lambda=\frac{4096}{175}$. The work demonstrates that existing techniques for the Riemannian case do not generalize to indefinite metrics and highlights new obstacles in pseudo-Riemannian geometry.

Abstract

We describe an example of an indefinite invariant Einstein metric on a solvmanifold which is not standard, and whose restriction on the nilradical is nondegenerate.

A non-Standard Indefinite Einstein Solvmanifold

TL;DR

This paper constructs a concrete nonstandard example of an indefinite Einstein solvmanifold. It presents a 5-dimensional solvable Lie algebra with a left-invariant Einstein metric whose nilradical coincides with the derived algebra and whose orthogonal complement is nonabelian, failing standard and pseudo-Iwasawa decompositions. The metric is explicitly computed to satisfy with . The work demonstrates that existing techniques for the Riemannian case do not generalize to indefinite metrics and highlights new obstacles in pseudo-Riemannian geometry.

Abstract

We describe an example of an indefinite invariant Einstein metric on a solvmanifold which is not standard, and whose restriction on the nilradical is nondegenerate.
Paper Structure (4 sections, 9 theorems, 26 equations)

This paper contains 4 sections, 9 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Invariant Structures on Lie Groups
  3. Definite vs. Indefinite Einstein Solvmanifolds
  4. A non-Standard Indefinite Einstein Solvmanifold

Key Result

Lemma 2.1

The Ricci tensor $\mathop{\mathrm{ric}}\nolimits$ of a left-invariant pseudo-Riemannian metric $g$ on a Lie algebra ${{\mathfrak{g}}}$ is given by: where $B(v,w)=\mathop{\mathrm{tr}}\nolimits(\mathop{\mathrm{ad}}\nolimits v\circ\mathop{\mathrm{ad}}\nolimits w)$ is the Killing form.

Theorems & Definitions (22)

  • Lemma 2.1: ContiRossi:EinsteinNilpotent
  • Definition 2.2
  • Proposition 2.3: ContiSegnanRossi:PseudoSasakiEinsteinSolv
  • Remark 2.4
  • Definition 3.1
  • Definition 3.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 12 more