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A perspective on canonical Riemannian metrics. (English) Zbl 1478.53003

Progress in Mathematics 336. Cham: Birkhäuser (ISBN 978-3-030-57184-9/hbk; 978-3-030-57187-0/pbk; 978-3-030-57185-6/ebook). xix, 247 p. (2020).
The title of the book, “A perspective on canonical Riemannian metrics”, is intriguing. What is a canonical metric? The authors explain this in a very readable introduction, which encourages the reader to delve into the subject. Canonical metric means ‘the best metric’ one can define on a given manifold. This best metric in understood from two points of view. The first one is that of the properties of its curvature, “curvature conditions” in the terminology of the book. For instance, the constancy of the Riemannian curvature or that of some of their associated tensors. Space forms, Einstein manifolds, Yamabe metrics are examples of manifolds endowed with good metrics respect to the properties of the curvature. Besides, one can consider the vanishing of some differential operators acting on some of the curvature tensor fields. Thus, one obtains locally symmetric metrics, metrics with parallel Ricci curvature, and manifolds having harmonic curvature metric. The second point of view to obtain the best metric is that of Critical Metrics, which means being a critical solution of a suitable Riemannian functional.
The book is organized as follows. The first two chapters are devoted to show the basic concepts of Riemannian Geometry and technical properties of Riemaniann manifolds, such as commutation rules for covariant derivatives of tensors fields and variations of geometric objects, the book thus being self-contained. The third chapter is about the Weyl tensor, including the Weyl-Schouten and Aubin theorems and the case of four-dimensional manifolds. Although the book is not written as a text book, these first three chapters contain a lot of basic, and not so basic, information about Riemannian manifolds.
Chapter four is about curvature conditions whilst chapter five is devoted to critical metrics of Riemannian functionals. According to the aim of the book, these chapters are the core of the book. The germ of chapter four is the authors’ previous paper [Ann. Global Anal. Geom. 55, No. 4, 719–748 (2019; Zbl 1415.53019)], where they consider Riemannian manifolds endowed with a potential function. Such a function is a function satisfying a differential equation \(\mathfrak{F}[g,f]=0\), where \(\mathfrak{F}\) is a differential operator acting on the metric \(g\) and on the potential function \(f\). Then new classes of Riemannian structures appear, including a general framework of classical canonical structures such as Einstein, harmonic curvature and Yamabe metrics, and gradient Ricci solitons. The authors are able to classify some of these new structures, such as \(f\)-space forms, \(f\)-locally symmetric and \(f\)-locally symmetric Einstein metrics. Besides they show significant examples and present open problems.
In Chapter five they consider the other point of view: that of critical metrics for suitable functionals. The classical Einstein-Hilbert functional is recalled and a basis for quadratic curvature functionals is introduced, deriving the corresponding Euler-Lagrange equations, and proving some rigidity results for critical metrics. In this case, the seminal paper is that of the first author [Calc. Var. Partial Differential Equations, 54, No. 3, 2921–2937 (2015; Zbl 1327.53053)]. In this chapter, authors are able to prove that critical metrics for a class of quadratic curvature functionals with non-negative (resp. non-positive) sectional curvature are Einstein.
In the last chapters more specific, recent results of the authors are shown. In Chapter six, a general Bochner-Weitzenböck formula is introduced, and a characterization of anti-self-dual metrics is obtained. Besides, the case of four-dimensional manifolds is carefully studied. Chapter seven contains recent results concerning the classification of Ricci solitons. The last chapter is devoted to the study of the existence of minimizers in the conformal class for the quadratic functional given by the rescaled \(L^{2}\)-norm of the divergence of the Weyl tensor.
The book is carefully written. It is not an encyclopedia of Riemannian geometry, but a deep introduction to active research topics around the idea of searching good metrics on a manifold. Moreover, it offers a good motivation to these topics, showing the relationship among different classes of Riemannian manifolds.

MSC:

53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53C20 Global Riemannian geometry, including pinching
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58E11 Critical metrics
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