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Transitive conformal holonomy groups. (English) Zbl 1278.53045

For a manifold \(M\) endowed with a conformal class \([g]\) of pseudo-Riemannian metrics, the 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:
(i)
\(\mathrm{SO}_0(p+1,q+1)\) for any \(p\), \(q\)
(ii)
\(\mathrm{SU}(n+1,m+1)\) for \(p=2n+1\), \(q=2m+1\)
(iii)
\(\mathrm{Sp}(1)\mathrm{Sp}(n+1,m+1)\) for \(p=4n+3\), \(q=4m+3\)
(iv)
\(\mathrm{Sp}_0(n+1,m+1)\) for \(p=4n+3\), \(q=4m+3\)
(v)
\(\mathrm{Spin}_0(1,8)\) for \(p=q=7\)
(vi)
\(\mathrm{Spin}_0(3,4)\) for \(p=q=3\)
(vii)
\(\mathrm{G}_{2,2}\) for \(p=3\), \(q=2\)
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 A. J. Di Scala and C. Olmos [Math. Z. 237, No. 1, 199–209 (2001; Zbl 0997.53051)] and A. J. Di Scala and 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 \((p+1,q+1)\).

MSC:

53C29 Issues of holonomy in differential geometry
53A30 Conformal differential geometry (MSC2010)
53C30 Differential geometry of homogeneous manifolds
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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References:

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