Riemannian manifolds of conullity two.

*(English)*Zbl 0904.53006
Singapore: World Scientific. xvii, 300 p. (1996).

This book deals with a topic from Riemannian geometry which has attracted many geometers since the original work of Élie Cartan and which has produced a great variety of beautiful and interesting results. One of the fundamental objects associated to a Riemannian manifold is its Riemannian curvature tensor \(R\). This tensor satisfies certain algebraic relations, and in general a tensor on some real vector space satisfying these algebraic relations is called an algebraic curvature tensor. Élie Cartan proved that an algebraic curvature tensor \(K\) on some vector space \(V\) is that of a Riemannian symmetric space if and only if \(K(X,Y) \cdot K = 0\) holds for all \(X,Y \in V\), where the dot indicates that \(K(X,Y)\) acts on \(K\) as a derivation. This result leads naturally to the question: what are the Riemannian manifolds satisfying \(R(X,Y) \cdot R = 0\)? Or equivalently, what are the Riemannian manifolds for which at each point the Riemannian curvature tensor is that of a Riemannian symmetric space? These Riemannian manifolds are known nowadays as semi-symmetric spaces.

In 1968, K. Nomizu conjectured that in all dimensions \(\geq 3\) every semi-symmetric space is locally symmetric. Already four years later, H. Takagi constructed a 3-dimensional semi-symmetric space which is not locally symmetric, and then K. Sekigawa constructed counterexamples in arbitrary dimensions. In 1982, Z. Szabó derived a full local classification of semi-symmetric spaces. Locally, each semi-symmetric space belongs to one of the following three classes: (1) the trivial class consisting of all locally symmetric spaces and all 2-dimensional Riemannian manifolds; (2) the exceptional class of all elliptic, hyperbolic, Euclidean and Kählerian cones; (3) the typical class of all Riemannian manifolds foliated by Euclidean leaves of codimension two. The trivial class is well-understood, and the exceptional class has been studied thoroughly by Z. Szabó. In this book the authors present a detailed study of the typical class of semi-symmetric spaces.

The book is divided into 12 chapters, and their contents is as follows. In Chapter 1, the authors give a brief introduction to semi-symmetric spaces, discuss an extrinsic analogue in submanifold theory, and present details concerning the early development of this topic. In Chapter 2, Szabó’s local classification result and the main steps in order to derive it are discussed. Szabó also studied foliated semi-symmetric spaces, that is, the semi-symmetric spaces in the typical class. He obtained an implicit classification in terms of a system of partial differential equations. In Chapter 3, the authors use another approach to derive a system of partial differential equations whose solutions give locally the metrics of semi-symmetric spaces. In Chapter 4, this PDE system is used to derive an explicit local classification of all curvature homogeneous semi-symmetric spaces, that is, all semi-symmetric spaces which have the curvature tensor of the same symmetric space at each point.

The new geometric concepts of asymptotic foliation and algebraic rank are introduced in Chapter 5. These concepts are used in Chapters 6-8 to derive explicit local classifications of semi-symmetric spaces. In Chapter 9, the authors turn their attention to the global situation and study complete semi-symmetric spaces. Such spaces have already been treated by Z. Szabó. The authors review his work on this topic, and then proceed by constructing new explicit examples.

In Chapter 10, the authors apply their results for the construction of hypersurfaces in 4-dimensional Euclidean space with type number two. A result of particular interest here is the explicit construction of such a hypersurface which admits exactly one non-trivial isometric deformation. In Chapter 11, the authors discuss a particular generalization of semi-symmetric spaces. Eventually, in Chapter 12 a survey is given about curvature homogeneous Riemannian manifolds of non-symmetric type. These are Riemannian manifolds which have the same algebraic curvature tensor \(K\) at each point, and \(K\) is not a curvature tensor of a symmetric space.

The many references in the book and an extensive bibliography at the end are very helpful for everyone who wishes to start with research in this field.

In 1968, K. Nomizu conjectured that in all dimensions \(\geq 3\) every semi-symmetric space is locally symmetric. Already four years later, H. Takagi constructed a 3-dimensional semi-symmetric space which is not locally symmetric, and then K. Sekigawa constructed counterexamples in arbitrary dimensions. In 1982, Z. Szabó derived a full local classification of semi-symmetric spaces. Locally, each semi-symmetric space belongs to one of the following three classes: (1) the trivial class consisting of all locally symmetric spaces and all 2-dimensional Riemannian manifolds; (2) the exceptional class of all elliptic, hyperbolic, Euclidean and Kählerian cones; (3) the typical class of all Riemannian manifolds foliated by Euclidean leaves of codimension two. The trivial class is well-understood, and the exceptional class has been studied thoroughly by Z. Szabó. In this book the authors present a detailed study of the typical class of semi-symmetric spaces.

The book is divided into 12 chapters, and their contents is as follows. In Chapter 1, the authors give a brief introduction to semi-symmetric spaces, discuss an extrinsic analogue in submanifold theory, and present details concerning the early development of this topic. In Chapter 2, Szabó’s local classification result and the main steps in order to derive it are discussed. Szabó also studied foliated semi-symmetric spaces, that is, the semi-symmetric spaces in the typical class. He obtained an implicit classification in terms of a system of partial differential equations. In Chapter 3, the authors use another approach to derive a system of partial differential equations whose solutions give locally the metrics of semi-symmetric spaces. In Chapter 4, this PDE system is used to derive an explicit local classification of all curvature homogeneous semi-symmetric spaces, that is, all semi-symmetric spaces which have the curvature tensor of the same symmetric space at each point.

The new geometric concepts of asymptotic foliation and algebraic rank are introduced in Chapter 5. These concepts are used in Chapters 6-8 to derive explicit local classifications of semi-symmetric spaces. In Chapter 9, the authors turn their attention to the global situation and study complete semi-symmetric spaces. Such spaces have already been treated by Z. Szabó. The authors review his work on this topic, and then proceed by constructing new explicit examples.

In Chapter 10, the authors apply their results for the construction of hypersurfaces in 4-dimensional Euclidean space with type number two. A result of particular interest here is the explicit construction of such a hypersurface which admits exactly one non-trivial isometric deformation. In Chapter 11, the authors discuss a particular generalization of semi-symmetric spaces. Eventually, in Chapter 12 a survey is given about curvature homogeneous Riemannian manifolds of non-symmetric type. These are Riemannian manifolds which have the same algebraic curvature tensor \(K\) at each point, and \(K\) is not a curvature tensor of a symmetric space.

The many references in the book and an extensive bibliography at the end are very helpful for everyone who wishes to start with research in this field.

Reviewer: Jürgen Berndt (Hull)

##### MSC:

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

53B20 | Local Riemannian geometry |

53C12 | Foliations (differential geometric aspects) |

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