This book focuses on a selection of special topics, with emphasis on past and present research of the authors on canonical Riemannian metrics on smooth manifolds.

On the backdrop of the fundamental contributions given by many experts in the field, the volume offers a self-contained view of the wide class of Curvature Conditions and Critical Metrics of suitable Riemannian functionals. The authors describe the classical examples and the relevant generalizations.

This monograph is the winner of the 2020 Ferran Sunyer i Balaguer Prize, a prestigious award for books of expository nature presenting the latest developments in an active area of research in mathematics.

Focusses on an active research field and contains many recent results Presents many open problems and possible research directions Offers a self-contained view of the wide class of Curvature Conditions and Critical Metrics of suitable Riemannian functionals

Zusammenfassung
"A very readable introduction, which encourages the reader to delve into the subject. ... 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." (Fernando Etayo Gordejuela, zbMATH 1478.53003, 2022)

Inhalt
Introduction.- Basic Concepts of Riemannian Geometry.- Commutations & Variations.- The Weyl Tensor.- Curvature Conditions.- Critical Metrics of Riemannian Functionals.- Bochner-Weitzenböck Formulas and Applications.- Ricci Solitons: Selected Results.- Existence Results of Canonical Metrics on Four Manifolds.- List of Symbols.- References.- Index.