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Generalized Ricci Flow

Mario Garcia-Fernandez and Jeffrey Streets
Publication Date: 
Number of Pages: 
University Lecture Series
[Reviewed by
Andrew D. Hwang
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The book Generalized Ricci Flow by Mario Garcia-Fernandez and Jeffrey Streets is an invitation and literature guide to “generalized geometry” and its connections to curvature flow and the existence of metrics with special curvature.  The intended reader is a working mathematician or advanced graduate student conversant with smooth real manifolds, and familiar with Kähler metrics, symplectic structures, and parabolic PDEs.
“Classical geometry” connotes here the mathematical realm of smooth manifolds, encompassing exterior calculus on the cotangent bundle, and Lie derivatives, Riemannian metrics, complex structures, symplectic forms, connections, and curvatures on the tangent bundle. Existence of special metrics, such as Einstein or Einstein-Kähler, and the curvature flow of Hamilton and Perelman, are classical in this sense.
The setting of “generalized geometry” is an “exact Courant algebroid” over a manifold: a vector bundle fitting into a short exact sequence with the cotangent bundle as subbundle and the tangent bundle as quotient, and equipped with bracket and metric structures satisfying axioms analogous to the classical Lie bracket and a Riemannian metric. The complex setting includes, in addition, one or more vector bundle automorphisms acting as square root of minus the identity. An exact Courant algebroid has “non-diffeomorphism” symmetries arising from a differential 2-form on the base manifold. These “b-field symmetries” have physical meaning related to actions in field theories, and account for the interest in generalized geometry among string theorists.
Garcia-Fernandez and Streets develop the theory of exact Courant algebroids along lines analogous to the usual development of smooth manifolds and tensors, Riemannian metrics, holomorphic and symplectic structures. A dominant theme is the geometry and analysis of connections. Unlike the situation in classical Riemannian geometry, a generalized Riemannian structure on an exact Courant algebroid does not determine a unique torsion-free connection, nor are torsion-free connections necessarily those best-adapted to particular geometric situations.  Instead, there is rich, subtle interplay shaped by torsion of connections and decompositions of connections determined by subbundles.
The book is divided roughly into thirds. The first portion develops exact Courant algebroids in a manner comparable to first graduate courses in smooth manifolds and Riemannian geometry. The second portion concerns curvature flow, giving appropriate attention to analytic prerequisites for readers more familiar with geometry than analysis. The third portion develops the theory of complex Courant algebroids and generalizations of classical curvature flow for Kähler metrics. A final chapter introduces the concept of T-duality in physics and its relationship with generalized geometry. Under T-duality, known solutions of generalized curvature flow give rise to additional examples.
The authors provide textbook recommendations for background material on manifolds and PDEs, and optional references (which may be read concurrently) for classical Ricci flow. Otherwise, the book is self-contained, and strikes a comfortable balance between thoroughness and manageable size. There is no index, so a reader may be well-advised to keep notes on first appearances of terms and notations. There are no formal exercises, but routine omitted steps of “generalized” calculations are clearly marked as exercises for the reader.  Despite the book’s advanced prerequisites and sometimes-formidable computations, the tone is pleasantly informal. The authors draw parallels between “classical” and “generalized” concepts, examples, and phenomena, and supply useful philosophical observations and idioms. The extensive bibliography should serve as a valuable guide to the literature.
Generalized geometry is still a young field, and even expository accounts are generally found in papers. In the authors’ words, “The primary purpose of this book is to provide an introduction to the fundamental geometric, algebraic, topological, and analytic aspects of the generalized Ricci flow equation.”  Many results about generalized geometry and generalized Ricci flow are currently open. The authors note, “The secondary purpose of this book is to formulate questions and conjectures about the generalized Ricci flow as an invitation to the reader.”
The book seems likely to succeed on both counts for a new generation of researchers.
Andrew D. Hwang is an associate professor of mathematics at the College of the Holy Cross.