Schouten like metrics on five dimensional nilpotents Lie groups
Marius Landry Foka, Michel Bertrand Ngaha Djiadeu, Thomas Bouetou Bouetou
TL;DR
This paper addresses the prescribed Ricci curvature problem on five-dimensional nilpotent Lie groups by introducing Schouten like metrics, which satisfy $\mathrm{Ric}_g=(s\lambda_0+c)g+g(D(\cdot),\cdot)$ with $D$ a symmetric derivation. It proves the equivalence between Schouten like metrics and algebraic Schouten solitons, and classifies such metrics using the Milnor-type classification of inner products on the five-dimensional nilpotent algebras. Through explicit analysis of each algebra in Table 2, the authors derive precise conditions on the structure constants that yield Schouten like metrics and identify corresponding nilsoliton instances. The results provide a detailed map of Schouten-like geometric structures in dimension five and contribute to the broader understanding of algebraic Schouten solitons on nilpotent Lie groups.
Abstract
The prescribed Ricci curvature problem involves finding a Riemannian metric g that satisfies the equation ric(g) = T, where T is a fixed symmetric (0, 2)-tensor field on a differential manifold M. In this paper, we introduce the concept of Schouten-like metrics as particular solutions to the prescribed Ricci curvature problem. We classify these metrics on five-dimensional nilpotent Lie groups by establishing a connection with algebraic Schouten solitons. This approach also enables us to classify five-dimensional nilsolitons, providing a comprehensive understanding of their geometric structures and properties.