Notes on symmetric conformal geometries.

*(English)*Zbl 1374.53080This article summarizes the previous results of the authors on symmetric conformal geometries, some of which are specifications of the results for general parabolic geometries, and provides several new results for symmetric conformal geometries.

Let \((M,c)\) be a conformal manifold, i.e., \(c=[g]\) is the conformal class of a pseudo-Riemannian metric \(g\) of signature \((p,q)\), with \(p+q=n=\dim M>2\). This is an instance of parabolic geometry of the type \((\mathrm{PO}(p+1,q+1),P)\), where \(\mathrm{PO}\) is the pseudo-orthogonal group and \(\mathrm{P}\) is the subgroup of projectivized pseudo-orthogonal transformations fixing a null line. This setup, important in the proofs, is not essential for understanding the main results. Recall that conformal flatness is equivalent to the vanishing of the Weyl tensor for \(n\geq4\) and to the vanishing of the Cotton tensor \(C\) for \(n=3\).

A symmetry of the authors at a point \(x\in M\) is a conformal map \(S_x:M\to M\) fixing \(x\) and such that \(d_xS_x=-id\). The pair \((M,c)\) is conformally symmetric if there exists a conformal symmetry at each point \(x\) (no a priori assumption of smooth dependence on \(x\)).

The main result (summarized from two theorems) states that a conformally symmetric structure \(c\) on a connected manifold \(M\) is either locally flat or a nonflat conformally homogeneous space conformally covered by a pseudo-Riemannian symmetric space.

It is also demonstrated that not every locally flat conformal manifold is a conformally symmetric space. A relation to the notion of a conformally symmetric manifold by A. Derdzinski and W. Roter [Tohoku Math. J. (2) 59, No. 4, 565–602 (2007; Zbl 1146.53014)] is drawn.

Let \((M,c)\) be a conformal manifold, i.e., \(c=[g]\) is the conformal class of a pseudo-Riemannian metric \(g\) of signature \((p,q)\), with \(p+q=n=\dim M>2\). This is an instance of parabolic geometry of the type \((\mathrm{PO}(p+1,q+1),P)\), where \(\mathrm{PO}\) is the pseudo-orthogonal group and \(\mathrm{P}\) is the subgroup of projectivized pseudo-orthogonal transformations fixing a null line. This setup, important in the proofs, is not essential for understanding the main results. Recall that conformal flatness is equivalent to the vanishing of the Weyl tensor for \(n\geq4\) and to the vanishing of the Cotton tensor \(C\) for \(n=3\).

A symmetry of the authors at a point \(x\in M\) is a conformal map \(S_x:M\to M\) fixing \(x\) and such that \(d_xS_x=-id\). The pair \((M,c)\) is conformally symmetric if there exists a conformal symmetry at each point \(x\) (no a priori assumption of smooth dependence on \(x\)).

The main result (summarized from two theorems) states that a conformally symmetric structure \(c\) on a connected manifold \(M\) is either locally flat or a nonflat conformally homogeneous space conformally covered by a pseudo-Riemannian symmetric space.

It is also demonstrated that not every locally flat conformal manifold is a conformally symmetric space. A relation to the notion of a conformally symmetric manifold by A. Derdzinski and W. Roter [Tohoku Math. J. (2) 59, No. 4, 565–602 (2007; Zbl 1146.53014)] is drawn.

Reviewer: Boris S. Kruglikov (Tromsø)

##### MSC:

53C35 | Differential geometry of symmetric spaces |

53A30 | Conformal differential geometry (MSC2010) |