Quaternion-Kähler geometry (Q2771292)

This paper is a survey on the geometry of quaternion-Kähler (QK) manifolds. The author, S. M. Salamon, is an outstanding specialist in this field. He realises a complete picture of the most important results from the beginning up to today in the QK-geometry. His professional comments are very important both for the uninitiating and for the initiating geometers. The paper begins with an introduction that consists in a motivating exposure and in a lapidary description of every section of the work. NEWLINENEWLINENEWLINEThe first section entitled ``Almost-Complex Structures and the Canonical 4-Form'' is dealing with various possible definitions of QK-manifolds and their equivalence in different dimensions. Actually, a QK manifold is a Riemannian manifold of dimension \(4n\) whose holonomy group is contained in the group \(Sp(n)Sp(1)\). The frame bundle of a QK manifold reduces to a principal bundle with structure group \(Sp(n)Sp(1)\). The holonomy reduction of a QK manifold is characterized by the existence of a 4-form which is (i) in the same \(\text{GL}(4n,\mathbb{R})\)-orbit as \(\Omega =\omega _{1}\wedge \omega _{1}+\omega _{2}\wedge \omega _{2}+\omega _{3}\wedge \omega _{3}\) (with \(\omega _{i}(X,Y)=g(I_{i}X,Y)\), \(i=1,2,3\)) at each point and (ii) parallel. Any QK manifold of dimension \(4n\geq 8\) is Einstein, and its scalar curvature \(s\) vanishes if and only if it is locally hyper-Kähler, i.e. its restricted holonomy group is a subgroup of Sp\((n)\). Since Sp\((n)\text{Sp}(1)\) is not a subgroup of \(U(2n),\) a QK manifold is not in general a Kähler manifold. The well-known result that there is no almost complex structure on \(HP^{1}\cong S^{4}\) is improved by D. V. Alekseevsky, S. Marchiafava and M. Pontecorvo as follows: No positive QK manifold admits a compatible almost complex structure. NEWLINENEWLINENEWLINEIn the second section the classification of the so-called Wolf spaces is presented. These are QK-symmetric spaces with \(s>0\). Actually, there is a Wolf space corresponding to each simple compact Lie algebra. The topological properties of the Wolf spaces reflect general results on QK manifolds. It is conjectured that any positive QK manifold is a Wolf space. The following result is known on this direction. A positive QK manifold \(M\) of dimension \(4n\) is a symmetric space if one of the following is true: (i) \(n\leq 2,\) (ii) \(n\leq 2\) and \(b_{4}=1,\) (iii) the isometry group of \(M\) has rank at least \(n+1\). If \(M\) is a QK manifold with the isometry group \(G\) and \(g_{x}\) denotes the isotropy subgroup at \(x\in M\), then \(g_{x}\subseteq \text{sp}(n)+ \text{sp}(1).\) In case \(g_{x}\) contains sp(1) for all \(x\), \(M\) is locally symmetric. NEWLINENEWLINENEWLINESection 3 is dealing with the representation of the structure group \(\text{Sp}(n)\text{Sp}(1)\) on the complexified tangent space \((T_{x})_{c}\) of an arbitrary QK manifold. Moreover, the representation of this group on space of forms and applications is analyzed. In particular, the elliptic complexes on QK manifolds and the properties of the Dirac operator are discussed. A decomposition of the exterior forms on a QK manifold is presented. In particular, a decomposition of the curvature tensor of a QK manifold under the form \(R=R_{Q}+s\rho \), where \(R_{Q}\) takes values in \(S^{4}E\) and \(\rho \) is \(\text{Sp}(n)\text{Sp}(1)\)-invariant is given. NEWLINENEWLINENEWLINEIn Section 4, the quaternionic manifolds are defined as being the smooth manifolds of dimension \(4n\geq 8\) admitting a \(G\)-structure and a torsion-free \(G\)-connection, where \(G\) is the subgroup \(\text{GL}(n,H) \text{GL}(1,H)\) of \(\text{GL}(4n,\mathbb{R})\); in case \(n=1\) the definition must assure the existence of a self-dual structure on it. Next, the twistor space for a quaternionic is defined. Over a quaternionic manifold \(M,\) the total space \(Z\) admits a complex structure with the property that its fibres are rational curves with normal bundle \(O(1)\otimes C^{2n}\) and a local section \(s(M')\) (with \(M'\)- an open subset of \(M\)) is a complex submanifold if and only if it is an integrable complex structure. In case \(M\) is a hypercomplex manifold the twistor space \(Z\) is a trivial smooth bundle and the projection \(Z\rightarrow\mathbb{C} P^{1}\) is holomorphic. A generalization of Yang-Mills equations to connections on bundles over quaternionic manifolds is given.NEWLINENEWLINENEWLINEIn the next section, entitled ``Fano twistor space'' some remarkable properties of the twistor space of a QK manifold with positive scalar curvature are reviewed. If \(M\) is a QK manifold with \(s>0\) then its twistor space \(Z\) has a Kähler-Einstein metric. LeBrun has proved that \(Z\) is the twistor space of some QK manifold if \(Z\) is a compact Kähler-Einstein manifold with a holomorphic contact structure. It is remarked that, up to a homothety, there are only finitely many positive QK manifolds of dimension \(4n\). NEWLINENEWLINENEWLINEIn Section 6, following the model of hyper-Kähler manifolds, the twistor functions are defined on a QK manifold. In case \(M\) is a QK manifold of dimension \(4n\geq 8\) with nonzero scalar curvature, the mapping \(\varsigma \rightarrow X_{\varsigma }\) establishes a bijective correspondence between the space of twistor functions and the space of Killing vector fields. A necessary condition assuring that a QK manifold is a Wolf space is given in terms of its (compact Fano) twistor space.NEWLINENEWLINENEWLINEThe next section is entitled ``Divisors and Quotients''. If \(M\) is a QK manifold of dimension \(4n\geq 4,\) \(X\) is a Killing vector field and \(\varsigma \) is its corresponding twistor function, then the set \(M_{0}=\{x\in M\mid \varsigma (x)=0\}\) is considered. Battaglia had proved the following classifying result: if \(S^{1}\) acts on a positive QK manifold \(M\) of dimension \(4n\geq 4\) and the action is free on \(M_{0}\), then \(M\) is isometric to \(HP^{n}\). Some results on the isometry group of a QK manifold are presented.NEWLINENEWLINENEWLINEThe last section is devoted to the characteristic classes and Betti numbers of a QK manifold.NEWLINENEWLINENEWLINEThe paper ends with a Bibliography having 119 quotations.NEWLINENEWLINEFor the entire collection see [Zbl 0961.00021].