Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The prescribed Ricci curvature problem on 5-dimensional nilpotent Lie groups

Marius Landry Foka, Romain Pefoukeu Nimpa, Salomon Joseph Mbatakou, Michel Bertrand Ngaha Djiadeu, Thomas Bouetou Bouetou

TL;DR

This work investigates the prescribed Ricci curvature problem on connected 5-dimensional nilpotent Lie groups by leveraging Milnor-type reductions of left-invariant metrics. It constructs explicit moduli-space representatives for all 5D nilpotent algebras, derives Milnor-type bases, and computes the Ricci curvature for the resulting left-invariant metrics. The authors then provide global existence criteria for metrics satisfying Ric$(g)=t^2T$ (with $t\neq0$) across all cases, translating solvability into concrete polynomial compatibility conditions on the tensor $T$’s components. The results give a complete, case-by-case solvability landscape, enabling direct verification of prescribed-Ricci-curvature feasibility on these groups and highlighting the interplay between algebraic structure and curvature in noncompact homogeneous spaces.

Abstract

In this paper, using the Milnor-type theorem technique, we provide on each nilpotent five dimensional Lie group, some global existence result of a pair (g, c) consisting of a left-invariant Riemannian metric g and a positive constant c such that Ric(g) =cT, where Ric(g) is the Ricci curvature of g and T a given left-invariant symmetric (0, 2)-tensor field.

The prescribed Ricci curvature problem on 5-dimensional nilpotent Lie groups

TL;DR

This work investigates the prescribed Ricci curvature problem on connected 5-dimensional nilpotent Lie groups by leveraging Milnor-type reductions of left-invariant metrics. It constructs explicit moduli-space representatives for all 5D nilpotent algebras, derives Milnor-type bases, and computes the Ricci curvature for the resulting left-invariant metrics. The authors then provide global existence criteria for metrics satisfying Ric (with ) across all cases, translating solvability into concrete polynomial compatibility conditions on the tensor ’s components. The results give a complete, case-by-case solvability landscape, enabling direct verification of prescribed-Ricci-curvature feasibility on these groups and highlighting the interplay between algebraic structure and curvature in noncompact homogeneous spaces.

Abstract

In this paper, using the Milnor-type theorem technique, we provide on each nilpotent five dimensional Lie group, some global existence result of a pair (g, c) consisting of a left-invariant Riemannian metric g and a positive constant c such that Ric(g) =cT, where Ric(g) is the Ricci curvature of g and T a given left-invariant symmetric (0, 2)-tensor field.
Paper Structure (44 sections, 42 theorems, 128 equations)

This paper contains 44 sections, 42 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Preliminary on Milnor-type theorems
  3. Set of representatives of $\mathfrak{PM}$ for $5$-dimensional nilpotent Lie algebras
  4. Case of $\mathfrak{g}=5A_{1}$
  5. Case of $\mathfrak{g}=A_{5,4}$
  6. Case of $\mathfrak{g}=A_{3,1}\oplus 2A_1$
  7. Case of $\mathfrak{g}=A_{4,1}\oplus A_{1}$
  8. Case of $\mathfrak{g}=A_{5,6}$
  9. Case of $\mathfrak{g}=A_{5,5}$
  10. Case of $\mathfrak{g}=A_{5,3}$
  11. Case of $\mathfrak{g}=A_{5,1}$
  12. Case of $\mathfrak{g}=A_{5,2}$
  13. Milnor-type theorems for left-invariant Riemannian metrics on $5$-dimensional nilpotent Lie groups
  14. Case of $\mathfrak{g}=5A_{1}$
  15. Case of $\mathfrak{g}=A_{5,4}$
  16. ...and 29 more sections

Key Result

Lemma 1

taka A subset $U\subset GL_n(\mathbb{R})$ is a set of representatives of $\mathfrak{P}\mathfrak{M}$ if and only if for every $g\in GL_n(\mathbb{R})$ there exist $h\in U$ such that $h\in[[g]]$.

Theorems & Definitions (47)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Theorem 1
  • Remark 1
  • Lemma 2
  • Lemma 3
  • Proposition 1
  • Lemma 4
  • Proposition 2
  • ...and 37 more