Transitive conformal holonomy groups
For a manifold \(M\) endowed with a conformal class \([g]\) of pseudo-Riemannian metrics, the \textit{conformal holonomy group} \(\mathrm{Hol}(M,[g])\) is defined as follows: A conformal manifold \((M,[g])\) is the base of a canonically defined \(P\)-principal bundle \(\mathcal{G}\) with fibre \(G/P\), where \(G=\text{PO}(p+1,q+1)\) and \(P\) is the image in \(G\) of a parabolic subgroup fixing a certain null line \(\ell\). Associated to this is a \(G\)-principal bundle \(\mathcal{G}\times_P G\). Now \(\mathrm{Hol}(M,[g])\) is defined to be the holonomy group of the associated Cartan connection \(\omega\) of this \(G\)-principal bundle. The author studies the holonomy groups \(H\) of conformal manifolds \((M,[g])\) with the additional assumption that \(H\) acts transitively on the conformal Möbius sphere \(S^{p,q}\). The main result is the classification in case \(H\) acts irreducibly. Then \(H\) is one of the following: {\parindent=0.9cm \begin{itemize}\item[(i)] \(\mathrm{SO}_0(p+1,q+1)\) for any \(p\), \(q\) \item [(ii)] \(\mathrm{SU}(n+1,m+1)\) for \(p=2n+1\), \(q=2m+1\) \item [(iii)] \(\mathrm{Sp}(1)\mathrm{Sp}(n+1,m+1)\) for \(p=4n+3\), \(q=4m+3\) \item [(iv)] \(\mathrm{Sp}_0(n+1,m+1)\) for \(p=4n+3\), \(q=4m+3\) \item [(v)] \(\mathrm{Spin}_0(1,8)\) for \(p=q=7\) \item [(vi)] \(\mathrm{Spin}_0(3,4)\) for \(p=q=3\) \item [(vii)] \(\mathrm{G}_{2,2}\) for \(p=3\), \(q=2\) \end{itemize}} If \(H\) does not act irreducibly, then there exists a \(g_0\in[g]\) such that \((M,g_0)\) is locally a special Einstein product. The first step in the proof is to show that the connected semisimple Lie groups acting irreducibly on \(\mathbb R^{p+1,q+1}\) and transitively on \(S^{p,q}\) are precisely the groups (i) to (vii), drawing on previous work from \textit{A. J. Di Scala} and \textit{C. Olmos} [Math. Z. 237, No. 1, 199--209 (2001; Zbl 0997.53051)] and \textit{A. J. Di Scala} and \textit{T. Leistner} [Isr. J. Math. 182, 103--121 (2011; Zbl 1222.22008)]. Additionally, the maximal compact subgroup \(K\) of \(H\) must act transitively on \(S^p\times S^q\). The candidates for \(K\) can then be identified by a list of Kamerich. This leads to the possible candidates for the group \(H\), and among these the author then identifies those which act irreducibly and preserve a metric of signature \((p+1,q+1)\).
scientific article