Cartan geometries and dynamics (Q1573676)

The authors study Cartan connections in a \(J\)-principal bundle \(Q\) over a closed manifold \(M\). To such a connection associated is a Lie group \(L\), with Lie algebra \(l\), which admits \(J\) as a closed subgroup. The standard example of a Cartan connection arises if \(L\) admits a discrete subgroup \(\Gamma\) such that the double coset space \(\Gamma\setminus L/J\) is a closed manifold. In this case every subgroup \(G\) of the centralizer of \(J\) in \(L\) acts on \(\Gamma\setminus L\) as a group of principal bundle automorphisms preserving the Cartan connection. On the other hand, the existence of a sufficiently large group \(G\) of principal bundle automorphisms forces the Cartan connection \(\Theta\) to have constant curvature or to be standard. The authors derive several rigidity results of this kind. As an example, they show that \(\Theta\) is standard if \(G\) is a connected semisimple Lie group of rank \(\geq 2\) without compact factors, \(J\) is compact, the action of \(G\) is ergodic with respect to an invariant volume on the bundle \(Q\) and that there is a 1-parameter subgroup \(g_t\) of \(G\) which preserves the Cartan connection up to a diagonalizable 1-parameter subgroup of \(GL(l)\).