Conformally flat Lie groups
The aim of the present paper is on the one hand to prove that in dimension \(n \geq 3\) there is a one-to-one correspondence between left-invariant Möbius structures and left-invariant conformally flat structures on Lie groups and on the other hand to describe the complete list of Möbius and conformally flat Lie groups, respectively. Moreover, it is shown that any such Lie group is isometric, as a Riemannian manifold, to one of \(E^n\), \(H^n\), \(S^3\), \(E^1 \times S^3\), \(E^1 \times H^{n-1}\), or \(S^3 \times H^{n-3}\) endowed with their standard metrics. Let \(M(n)\), \(n \geq 2\), be the group of diffeomorphisms of the sphere \(S^n\) which map round \((n-1)\)-spheres to round \((n-1)\)-spheres. A manifold \(M\) is said to be modelled on \((S^n, M(n))\) if \(M\) has a maximal atlas of charts \((U, \phi)\), where \(\phi: U \rightarrow S^n\) is a homeomorphism on its image such that \(\phi_1 \phi_2^{-1}\) is a restriction of an element in \(M(n)\). Such a structure is called a Möbius structure and can also be interpreted as a conformally flat pseudogroup structure on \(M\), the pseudogroup being given by the restrictions of elements in \(M (n)\) to open subsets of the sphere [\textit{N. H. Kuiper}, Ann. Math., II. Ser. 50, No. 4, 916-924 (1949; Zbl 0041.09303)]. A Möbius structure on a Lie group \(G\) is called left invariant if the left action of \(G\) on itself is via Möbius transformations, i.e., diffeomorphisms of \(M\) which preserve the Möbius structure. In general a conformally flat structure on a manifold is a class of conformally equivalent metrics such that locally there exist conformal diffeomorphisms onto open subsets of Euclidean space. In dimension \(n \geq 3\), conformally flat structures induce Möbius structures and this sets up a one-to-one correspondence between Möbius structure and flat conformal structure [\textit{P. Kulkarni} and \textit{U. Pinkall}, Differential geometry, Lect. Notes Math. 1209, 190-209 (1986; Zbl 0612.57017)]. Then it is possible to define a left-invariant conformally flat structure on a Lie group \(G\) of dimension \(n \geq 3\) as a set \({\mathfrak c}\) of left-invariant metrics all of which are conformally flat, such that given \(g_1, g_2 \in {\mathfrak c}\) there is some \(\lambda \in R^+\) with \(g_1 = \lambda g_2\). The pair \((G, {\mathfrak c})\) is called a conformally flat Lie group.
scientific article; zbMATH DE number 1185897