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The prescribed Ricci curvature problem on three‐dimensional unimodular Lie groups
Author(s) -
Buttsworth Timothy
Publication year - 2019
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201800052
Subject(s) - unimodular matrix , mathematics , ricci curvature , lie group , uniqueness , pure mathematics , invariant (physics) , riemann curvature tensor , mathematical analysis , curvature , combinatorics , mathematical physics , geometry
Let G be a three‐dimensional unimodular Lie group, and let T be a left‐invariant symmetric (0,2)‐tensor field on G . We provide the necessary and sufficient conditions on T for the existence of a pair ( g , c ) consisting of a left‐invariant Riemannian metric g and a positive constant c such that R i c ( g ) = c T , where R i c ( g ) is the Ricci curvature of g . We also discuss the uniqueness of such pairs and show that, in most cases, there exists at most one positive constant c such that R i c ( g ) = c T is solvable for some left‐invariant Riemannian metric g .