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New Special Einstein Pseudo-Riemannian Metrics on Solvable Lie Algebras

Federico A. Rossi

TL;DR

This work develops a concrete approach to constructing Einstein pseudo-Riemannian metrics on solvable Lie algebras that are simultaneously pseudo-Kähler or para-Kähler with nonzero scalar curvature. The method links pseudo-Riemannian nilsolitons to Einstein solvmanifolds via rank-one pseudo-Iwasawa extensions and then seeks a parallel symplectic form and an integrable endomorphism to realize a complex or para-complex structure compatible with the metric. It yields explicit classifications in low dimensions for rank-one pseudo-Iwasawa extensions of certain nilsolitons and provides numerous new examples based on generalized Heisenberg algebras and diagonal nilsolitons. The results broaden the catalog of invariant Einstein pseudo-Riemannian metrics and demonstrate a practical strategy for generating further examples and exploring connections between nilsoliton geometry and special pseudo-Riemannian Einstein structures.

Abstract

We exhibit a concrete procedure to construct Einstein pseudo-Kähler and para-Kähler metrics on solvable Lie algebras. We apply this method to classify all the rank-one pseudo-Iwasawa extensions of type-(Nil4) nilsoliton in low dimension. We prove that such metrics exists on the rank-one pseudo-Iwasawa extension of the generalized Heisenberg Lie algebra. Further ideas and suggestions to produce more special Einstein pseudo-Riemannian metrics are exposed.

New Special Einstein Pseudo-Riemannian Metrics on Solvable Lie Algebras

TL;DR

This work develops a concrete approach to constructing Einstein pseudo-Riemannian metrics on solvable Lie algebras that are simultaneously pseudo-Kähler or para-Kähler with nonzero scalar curvature. The method links pseudo-Riemannian nilsolitons to Einstein solvmanifolds via rank-one pseudo-Iwasawa extensions and then seeks a parallel symplectic form and an integrable endomorphism to realize a complex or para-complex structure compatible with the metric. It yields explicit classifications in low dimensions for rank-one pseudo-Iwasawa extensions of certain nilsolitons and provides numerous new examples based on generalized Heisenberg algebras and diagonal nilsolitons. The results broaden the catalog of invariant Einstein pseudo-Riemannian metrics and demonstrate a practical strategy for generating further examples and exploring connections between nilsoliton geometry and special pseudo-Riemannian Einstein structures.

Abstract

We exhibit a concrete procedure to construct Einstein pseudo-Kähler and para-Kähler metrics on solvable Lie algebras. We apply this method to classify all the rank-one pseudo-Iwasawa extensions of type-(Nil4) nilsoliton in low dimension. We prove that such metrics exists on the rank-one pseudo-Iwasawa extension of the generalized Heisenberg Lie algebra. Further ideas and suggestions to produce more special Einstein pseudo-Riemannian metrics are exposed.
Paper Structure (7 sections, 15 theorems, 52 equations, 1 figure, 3 tables)

This paper contains 7 sections, 15 theorems, 52 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Invariant Structures on Lie Groups
  3. Pseudo-Riemannian Einstein Special Metrics
  4. Einstein pseudo-Riemannian nilpotent Lie algebras
  5. Pseudo-Riemannian Nilsolitons and Einstein Solvmanifolds
  6. A Strategy and New Examples
  7. More Examples, Ideas and Remarks

Key Result

Theorem 4.1

Let ${{\mathfrak{g}}}$ be a nilpotent not Abelian Lie algebra, then it has no Einstein Riemannian invariant metric.

Figures (1)

  • Figure 1: A nice diagram $\Delta$.

Theorems & Definitions (34)

  • Theorem 4.1: Milnor:curvatures
  • Theorem 4.2: Dotti:RicciCurvature
  • Theorem 4.3: ContiRossi:EinsteinNilpotent
  • Theorem 4.4: ContiRossi:EinsteinNilpotent
  • Example 4.5: ContiRossi:EinsteinNilpotent
  • Example 4.6
  • Lemma 4.7: FPS:skt6d
  • Proposition 4.8: ContiRossi:RicciTensor
  • Example 4.9
  • Definition 5.1
  • ...and 24 more