Quaternionic contact Einstein structures and the quaternionic contact (Q2925664)

In the present paper the authors consider the Yamabe problem on the quaternionic Heisenberg group (three-dimensional center). This problem turns out to be equivalent to the quaternionic contact Yamabe problem on the unit \((4n+3)\)-dimensional sphere in the quaternionic space due to the quaternionic Cayley transform. The central notion is the quaternionic contact structure given by \textit{O. Biquard}, see [Métriques d'Einstein asymptotiquement symétriques. Paris: Société Mathématique de France (2000; Zbl 0967.53030)] and [in: Proceedings of the 2nd meeting on quaternionic structures in mathematics and physics, Roma, Italy, September 6--10, 1999. Rome: Dipartimento di Matematica ``Guido Castelnuovo'', Università di Roma ``La Sapienza''. 23--30 (2001; Zbl 0993.53017)]. The contents of this paper are the following:NEWLINENEWLINEIn Chapters 2 and 3 the authors describe in detail the notion of a quaternionic contact manifold and the Biquard connection.NEWLINENEWLINEIn Chapter 4 they write the explicit form of the Bianchi identities and derive a system of equations satisfied by the divergences of some tensors, then they show the following NEWLINENEWLINENEWLINE{Theorem 4.9} The qc-scalar curvature of a qc-Einstein manifolds (i.e., the qc-Ricci tensor is trace-free) of its dimension \(>7\) is a global constant. In addition, the vertical distribution \(V\) of a qc-Einstein structure is integrable. On a \(7\)-dimensional qc-Einstein manifold the constancy of the qc scalar curvature is equivalent to the integrability of the vertical distribution. NEWLINENEWLINENEWLINEThey show that the qc-Einstein condition is equivalent to the vanishing of the torsion of the Biquard connection, then they give one of their main results which is a local characterization of such spaces as \(3\)-Sasakian manifolds, i.e., NEWLINENEWLINENEWLINE {Theorem 1.3.} Let \((M^{4n+3},g,\mathbb{Q})\) be a qc manifold with positive qc scalar curvature \(\mathrm{Scal} >0\), assumed to be constant if \(n=1\). The following conditions are equivalent: {\parindent=6mm \begin{itemize}\item[a)] \((M^{4n+3},g,\mathbb{Q})\) is a qc-Einstein manifold. \item[b)] \(M\) is locally a \(3\)-Sasakian, i.e., locally there exists an \(\mathrm{SO}(3)\)-matrix \(\Psi\) with smooth entries, such that the local qc structure \((\frac{16n(n+2)}{Scal}\Psi\cdot\eta,\mathbb{Q})\) is \(3\)-Sasakian. \item[c)] The torsion of the Biquard connection is identically zero. NEWLINENEWLINE\end{itemize}} In particular, a qc-Einstein manifold of positive qc scalar curvature, assumed in addition to be constant if \(n=1\), is an Einstein manifold of positive Riemannian scalar curvature.NEWLINENEWLINEIn Chapter 5 the authors describe the conformal transformations preserving the qc-Einstein condition. They find all conformal transformations preserving the qc-Einstein condition on the quaternionic Heisenberg group. Then they prove the following result:NEWLINENEWLINE {Theorem 1.1} Let \(\Theta=\frac 1{2h}\tilde{\Theta}\) be a conformal transformation of the standard qc-structure \(\tilde\Theta\) on the quaternionic Heisenberg group \(G(\mathbb{H})\). If \(\Theta\) is also qc-Einstein, then up to a left translation the function \(h\) is given by NEWLINE\[NEWLINE h=c\left[(1+\nu|q|^2)^2+\nu^2(x^2+y^2+z^2)\right], NEWLINE\]NEWLINE where \(c\) and \(\nu\) are positive constants. All functions \(h\) of this form have this property. NEWLINENEWLINENEWLINEChapter 6 concerns a special class of functions defined respectively on the quaternionic space or on a quaternionic contact manifold, which play a role somewhat similar to that played by the CR functions, but the analogy is not complete. The real parts of such functions will be also of interest in connection with conformal transformations preserving the qc-Einstein tensor and should be thought of as generalizations of pluriharmonic functions. NEWLINENEWLINENEWLINEChapter 7 studies infinitesimal conformal automophisms of qc structures and shows that they depend on three functions satisfying some differential conditions. NEWLINENEWLINENEWLINEIn Chapter 8 the authors present a partial solution of the qc-Yamabe problem on the quaternionic sphere. The main results of this paper are NEWLINENEWLINENEWLINE{Proposition 8.2} Let \((M,\tilde\eta)\) be a compact qc-Einstein manifold of dimension \((4n+3)\). Let \(\tilde\eta=\frac 1{2h}\eta\) be a conformal transformation of the qc-structure \(\tilde\eta\) on \(M\). Suppose \(\eta\) has constant scalar curvature. {\parindent=6mm \begin{itemize}\item[a)] If \(n>1\), then any of the following two conditions implies that \(\eta\) is a qc-Einstein structure: i) the vertical space of \(\eta\) is integrable; ii) the qc structure \(\eta\) is qc-pseudo Einstein. \item[b)] If \(n=1\) and the vertical space of \(\eta\) is integrable then \(\eta\) is a qc-Einstein structure. NEWLINENEWLINE\end{itemize}} {Theorem 1.2} Let \(\eta=f\tilde\eta\) be a conformal transformation of the standard qc-structure \(\tilde\eta\) on the quaternionic sphere \(S^{4n+3}\). Suppose \(\eta\) has constant qc-scalar curvature. If the vertical space of \(\eta\) is integrable then up to a multiplicative constant \(\eta\) is obtained from \(\tilde\eta\) by a conformal quaternionic contact automorphism. In the case \(n>1\) the same condition holds when the function \(f\) is the real part of an anti-CRF function.