Differential Geometry in the Large

Differential Geometry in the Large is the proceedings of the “Australian–German Workshop on Differential Geometry in the Large,” held in February 2019 at a campus of the University of Melbourne in Creswick, Australia. The book is a collection of expository surveys and research articles covering recent progress in several active areas of research in differential geometry. The book is organized into three parts corresponding to broad topics, each containing five or six independent chapters. The latter two parts contain a mix of research articles and expository surveys with some new proofs and results. Each of the parts includes a well-chosen mix of chapters indicating the wide range of current work in the area as well as the major questions and themes arising in these investigations.

 

Part I: Geometric Evolution Equations and Curvature Flow contains expository chapters introducing and summarizing recent progress on questions related to the Ricci, mean curvature, and related flows including some new results and new or simplified proofs of previous results. An exception is the first chapter, Real Geometric Invariant Theory by Christoph Böhm and Ramiro A. Lafuente, which gives a self-contained introduction to the geometric invariant theory of real reductive Lie groups, including proofs of the Kempf–Ness and Kirwan–Ness stratification theorems, using geometric techniques and hence accessible to those without a significant background in algebraic geometry.

 

Part II: Structures on Manifolds and Mathematical Physics includes chapters on a diverse range of topics including the existence of metrics with specific properties, e.g. Sasaki metrics with constant scalar curvature, metrics with prescribed Ricci curvature, and Einstein metrics with non-negative self-dual Weyl tensor; the rigorous construction of the supersymmetric path integral; and identifying minimal representatives of weak

equivalence classes of Poincaré differential graded commutative algebras.

 

Part III: Recent Developments on Non-Negative Sectional Curvature covers progress on identifying new examples of manifolds with non-negative sectional curvature; the structure and properties of limits of manifolds and Alexandrov spaces with sectional curvature bounded below; a Gauss–Bonnnet–Chern theorem for a generalization of simplicial complexes with metrics along with a related simplicial sectional curvature and its connection to the Hopf conjecture; and the homogeneity conjecture for manifolds with positive sectional curvature.

 

Though some of the research articles are technical and the scope of topics covered is diverse, most of the book is accessible to anyone with a background in differential geometry or geometric analysis. The expository survey chapters give overviews of active areas and open questions, including concise accounts of the background and related work, and hence function well as introductions to current activity in differential geometry for graduate students or non-specialists. 

 

The volume includes important additions to the literature including new results, new proofs of previous results, and simplified expositions, and also an excellent collection of surveys on recent activity. It is well written and offers a generous overview and invitation to a variety of modern, active topics in differential geometry.

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Chris Seaton is Professor of Mathematics at Rhodes College.