Riemannian geometry and holonomy groups.

*(English)*Zbl 0685.53001
Pitman Research Notes in Mathematics Series, 201. Harlow: Longman Scientific & Technical; New York: John Wiley & Sons. 201 p. £17.00 (1989).

The present book is concerned with n-dimensional Riemannian manifolds (M,g) whose holonomy group H is a proper subgroup of SO(n) acting irreducibly on \({\mathbb{R}}^ n\). By results of M. Berger and J. Simons obtained in 1955 and 1962 the subgroups of SO(n) arising as holonomy groups of such manifolds are one of U(n/2), SU(n/2), Sp(n/4)Sp(1), Sp(n/4), \(G_ 2\), Spin(7) or Spin(9), provided M is oriented, simply connected, not symmetric and not a local product. Each of these groups defines its own class of Riemannian manifolds with its own geometry, for instance \(H=U(n/2)\) the class of Kähler- and \(H=Sp(n/4)Sp(1)\) the class of quaternionic Kähler manifolds. Recent results on geometric properties and classification problems are the main topic of this book. For several of these groups, especially \(G_ 2\), Spin(7) and Spin(9), it is rather hard to provide examples of Riemannian manifolds with this holonomy group.

The book is divided into 12 chapters of different character. Chapters 1 and 2 are dealing with concepts and basic properties of Riemannian manifolds, including a local version of the decomposition theorem of de Rham. Chapters 3 and 4 give a summary of certain aspects of complex- and Kähler geometry. Basic facts as for instance the interaction between symplectic and Kähler geometry as well as recent results of M. Gromov and D. MacDuff concerning symplectic manifolds admitting no Kähler structure are treated in short form.

In chapter 4 the so-called space \({\mathcal R}\) of curvature tensors of a Riemannian manifold is introduced. Its elements are the SO(n)-equivariant maps from the bundle P of orthonormal frames of M into the SO(n)-module \(\Lambda^ 2\otimes \Lambda^ 2\) (where \(\Lambda^ 2=\Lambda^ 2{\mathbb{R}}^{n*})\) taking values in the submodule defined by the usual symmetries of the curvature tensor R, which is an element of \({\mathcal R}\). The decomposition of \({\mathcal R}\) into irreducible components gives rise to the natural decomposition of R into the Weyl- and Ricci tensor and the scalar curvature. More generally, in the case of a smaller holonomy subgroup \(H\subset SO(n)\) the reduced space \({\mathcal R}^ H\) of curvature tensors consists of the restrictions of elements of \({\mathcal R}\) to the holonomy bundle Q(p) containing a fixed frame \(p\in P\). In this way, representation theory enters into the discussion, since the irreducible components of \({\mathcal R}^ H\) define new tensors, for instance the Bochner tensor in the case of \(H=U(n/2)\). This way of treating the curvature by means of representation theory is also used for a study of differential operators acting between associated bundles, so for instance the Laplacian.

Chapters 5 and 6 are devoted to symmetric spaces and representation theory respectively. Symmetric spaces are regarded from the point of view of curvature just explained, associating them with H-invariant elements R of \({\mathcal R}^ H\). Representation theory is included to provide a working tool for decomposing tensor products into irreducible representations.

The remaining second half of the book is concerned with explicit constructions of special Kähler, quaternionic manifolds and manifolds whose holonomy groups are subgroups of the exceptional group \(G_ 2\) or of Spin(7). At first, results on 4-dimensional Riemannian manifolds M are discussed. The decomposition \(\Lambda^ 2TM=\Lambda^ 2_+M\oplus \Lambda^ 2_-M\) into self dual and anti-self dual 2-forms is defined by means of a canonical double covering \(Sl(4,{\mathbb{R}})\to (SO_ 0(3,3).\) In contrast to Yang-Mills theory the total space of anti-self dual forms serves here for the construction of manifolds whose holonomy group is contained in \(G_ 2\). There exist several forms on \(\Lambda^ 2_-M\) satisfying certain identities, which are the essential ingredients of this construction carried out in chapter 11.

The twistor space ZM, defined as the 1-sphere bundle of \(\Lambda^ 2_- M\), is discussed in more detail. There exist for instance two almost complex structures \(J_ 1\) and \(J_ 2\) on ZM and \(J_ 1\) is complex if M is self dual. In this case and if M is Einstein, ZM also has a complex contact structure. Several topological properties and classification results are quoted or proved as consequences of the existence of these structures.

Chapter 8 is concerned with Riemannian manifolds whose holonomy group H is contained in SU(m) or Sp(k) (where \(m=2k)\) satisfying additional conditions such as being Einstein or Ricci-flat. For example, if M is Kähler-Einstein with non vanishing scalar curvature t then there exist Kähler metrics with vanishing Ricci tensor on certain open domains of the total space of the canonical bundle \(\lambda^{m,0}M\). The proof of the Calabi conjecture is used as a tool to provide other examples of Kähler manifolds with preassigned Ricci tensor.

A manifold is called hyperkähler if \(H\subset Sp(k)\). In this case there exist three complex structures \(I_ 1\), \(I_ 2\) and \(I_ 3\) on M which are covariantly constant and satisfy conditions similar to the multiplication rules of the quaternions. Examples of such manifolds M are for instance the total space of the cotangent bundle of the complex projective space and a compact one: Take a K3 surface, blow up the diagonal of \(K\times K\) to obtain a new manifold \(\tilde M\) and factor \(\tilde M\) by the involution induced by the natural involution of \(K\times K\) to obtain M. A special construction, called symplectic reduction of symplectic Kählerian or hypercomplex manifolds leads to new such manifolds. This construction is applied to the level sets N of the moment map of homogeneous symplectic manifolds.

Quaternionic manifolds M, treated in chapter 9, have holonomy group H contained in Sp(k)Sp(1). In general, they cannot be endowed with global smooth fields of complex structures as in the case of hyperkähler manifolds. Results on the structure of these manifolds, decompositions of the corresponding reduced spaces of curvature tensors and properties of an analogue ZM of the twistor space are quoted. Examples arise for instance from a result of J. A. Wolf stating that certain homogeneous spaces of simple compact Lie groups are the twistor spaces of certain quaternionic Kählerian symmetric spaces.

Chapter 10 contains a proof of the classification theorem of M. Berger mentioned at the beginning of this review. Finally, in chapters 11 and 12 the holonomy groups \(G_ 2\) and Spin(7) are discussed. Starting with several equivalent definitions of \(G_ 2\) and its relation to the standard representation of SO(4) on the space \(\mu =\Lambda^ 1\oplus \Lambda^ 2\) of dimension 7 it appears as the subgroup of Gl(7,\({\mathbb{R}})\) leaving fixed a certain 3-form on \(\mu\). General properties of reductions of the frame bundle of a 7-dimensional manifold to \(G_ 2\) and of Riemannian structures with holonomy group \(G_ 2\) are proved.

In a similar way, definitions and basic properties of the group Spin(7) and the related Clifford modules are provided. Riemannian manifolds with holonomy group Spin(7) are of dimension 8. A sporadic example of such manifolds is \(M={\mathbb{R}}^+\times SO(5)/SO(3).\) A more systematic construction starts with 4-dimensional self dual Einstein manifolds M which admit a spin structure \(\tilde P.\) The Riemannian metric with holonomy group H contained in Spin(7) then lives on the total space of the associated bundle \(\sigma_-M=P\times_{Spin(4)}\sigma_-,\) where \(\sigma_-\) is a basic representation of Sp(1).

The book is divided into 12 chapters of different character. Chapters 1 and 2 are dealing with concepts and basic properties of Riemannian manifolds, including a local version of the decomposition theorem of de Rham. Chapters 3 and 4 give a summary of certain aspects of complex- and Kähler geometry. Basic facts as for instance the interaction between symplectic and Kähler geometry as well as recent results of M. Gromov and D. MacDuff concerning symplectic manifolds admitting no Kähler structure are treated in short form.

In chapter 4 the so-called space \({\mathcal R}\) of curvature tensors of a Riemannian manifold is introduced. Its elements are the SO(n)-equivariant maps from the bundle P of orthonormal frames of M into the SO(n)-module \(\Lambda^ 2\otimes \Lambda^ 2\) (where \(\Lambda^ 2=\Lambda^ 2{\mathbb{R}}^{n*})\) taking values in the submodule defined by the usual symmetries of the curvature tensor R, which is an element of \({\mathcal R}\). The decomposition of \({\mathcal R}\) into irreducible components gives rise to the natural decomposition of R into the Weyl- and Ricci tensor and the scalar curvature. More generally, in the case of a smaller holonomy subgroup \(H\subset SO(n)\) the reduced space \({\mathcal R}^ H\) of curvature tensors consists of the restrictions of elements of \({\mathcal R}\) to the holonomy bundle Q(p) containing a fixed frame \(p\in P\). In this way, representation theory enters into the discussion, since the irreducible components of \({\mathcal R}^ H\) define new tensors, for instance the Bochner tensor in the case of \(H=U(n/2)\). This way of treating the curvature by means of representation theory is also used for a study of differential operators acting between associated bundles, so for instance the Laplacian.

Chapters 5 and 6 are devoted to symmetric spaces and representation theory respectively. Symmetric spaces are regarded from the point of view of curvature just explained, associating them with H-invariant elements R of \({\mathcal R}^ H\). Representation theory is included to provide a working tool for decomposing tensor products into irreducible representations.

The remaining second half of the book is concerned with explicit constructions of special Kähler, quaternionic manifolds and manifolds whose holonomy groups are subgroups of the exceptional group \(G_ 2\) or of Spin(7). At first, results on 4-dimensional Riemannian manifolds M are discussed. The decomposition \(\Lambda^ 2TM=\Lambda^ 2_+M\oplus \Lambda^ 2_-M\) into self dual and anti-self dual 2-forms is defined by means of a canonical double covering \(Sl(4,{\mathbb{R}})\to (SO_ 0(3,3).\) In contrast to Yang-Mills theory the total space of anti-self dual forms serves here for the construction of manifolds whose holonomy group is contained in \(G_ 2\). There exist several forms on \(\Lambda^ 2_-M\) satisfying certain identities, which are the essential ingredients of this construction carried out in chapter 11.

The twistor space ZM, defined as the 1-sphere bundle of \(\Lambda^ 2_- M\), is discussed in more detail. There exist for instance two almost complex structures \(J_ 1\) and \(J_ 2\) on ZM and \(J_ 1\) is complex if M is self dual. In this case and if M is Einstein, ZM also has a complex contact structure. Several topological properties and classification results are quoted or proved as consequences of the existence of these structures.

Chapter 8 is concerned with Riemannian manifolds whose holonomy group H is contained in SU(m) or Sp(k) (where \(m=2k)\) satisfying additional conditions such as being Einstein or Ricci-flat. For example, if M is Kähler-Einstein with non vanishing scalar curvature t then there exist Kähler metrics with vanishing Ricci tensor on certain open domains of the total space of the canonical bundle \(\lambda^{m,0}M\). The proof of the Calabi conjecture is used as a tool to provide other examples of Kähler manifolds with preassigned Ricci tensor.

A manifold is called hyperkähler if \(H\subset Sp(k)\). In this case there exist three complex structures \(I_ 1\), \(I_ 2\) and \(I_ 3\) on M which are covariantly constant and satisfy conditions similar to the multiplication rules of the quaternions. Examples of such manifolds M are for instance the total space of the cotangent bundle of the complex projective space and a compact one: Take a K3 surface, blow up the diagonal of \(K\times K\) to obtain a new manifold \(\tilde M\) and factor \(\tilde M\) by the involution induced by the natural involution of \(K\times K\) to obtain M. A special construction, called symplectic reduction of symplectic Kählerian or hypercomplex manifolds leads to new such manifolds. This construction is applied to the level sets N of the moment map of homogeneous symplectic manifolds.

Quaternionic manifolds M, treated in chapter 9, have holonomy group H contained in Sp(k)Sp(1). In general, they cannot be endowed with global smooth fields of complex structures as in the case of hyperkähler manifolds. Results on the structure of these manifolds, decompositions of the corresponding reduced spaces of curvature tensors and properties of an analogue ZM of the twistor space are quoted. Examples arise for instance from a result of J. A. Wolf stating that certain homogeneous spaces of simple compact Lie groups are the twistor spaces of certain quaternionic Kählerian symmetric spaces.

Chapter 10 contains a proof of the classification theorem of M. Berger mentioned at the beginning of this review. Finally, in chapters 11 and 12 the holonomy groups \(G_ 2\) and Spin(7) are discussed. Starting with several equivalent definitions of \(G_ 2\) and its relation to the standard representation of SO(4) on the space \(\mu =\Lambda^ 1\oplus \Lambda^ 2\) of dimension 7 it appears as the subgroup of Gl(7,\({\mathbb{R}})\) leaving fixed a certain 3-form on \(\mu\). General properties of reductions of the frame bundle of a 7-dimensional manifold to \(G_ 2\) and of Riemannian structures with holonomy group \(G_ 2\) are proved.

In a similar way, definitions and basic properties of the group Spin(7) and the related Clifford modules are provided. Riemannian manifolds with holonomy group Spin(7) are of dimension 8. A sporadic example of such manifolds is \(M={\mathbb{R}}^+\times SO(5)/SO(3).\) A more systematic construction starts with 4-dimensional self dual Einstein manifolds M which admit a spin structure \(\tilde P.\) The Riemannian metric with holonomy group H contained in Spin(7) then lives on the total space of the associated bundle \(\sigma_-M=P\times_{Spin(4)}\sigma_-,\) where \(\sigma_-\) is a basic representation of Sp(1).

Reviewer: H.Gollek

##### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53C10 | \(G\)-structures |

53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |

53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |

32L25 | Twistor theory, double fibrations (complex-analytic aspects) |