Dimension estimations for the space of invariant affine connections which are compatible with the adjoint representation of the semisimple Lie group (Q1896738)

If \(G\) is an \(n\)-dimensional real Lie group then the dimension of the vector space \(\Lambda\) of all left-invariant torsion-free affine connections on \(G\) is equal to \({1\over 2} n^2 (n+1)\). The author establishes the following main result: let \(G\) be an \(n\)- dimensional real semisimple Lie group having \(r\) simply connected invariant subgroups; then the subspace of \(\Lambda\) consisting of the connections compatible with the adjoint representation of \(G\) has codimension \({1\over 2} n^2 (n - 1) + r\).