A COURSE IN DIFFERENTIAL GEOMETRY THIERRY AUBIN PDF

A Course in Differential Geometry Share this page Thierry Aubin This textbook for second-year graduate students is intended as an introduction to differential geometry with principal emphasis on Riemannian geometry. Chapter I explains basic definitions and gives the proofs of the important theorems of Whitney and Sard. Chapter II deals with vector fields and differential forms. Chapter IV develops the notion of connection on a Riemannian manifold considered as a means to define parallel transport on the manifold. The author also discusses related notions of torsion and curvature, and gives a working knowledge of the covariant derivative.

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A Course in Differential Geometry Share this page Thierry Aubin This textbook for second-year graduate students is intended as an introduction to differential geometry with principal emphasis on Riemannian geometry. Chapter I explains basic definitions and gives the proofs of the important theorems of Whitney and Sard. Chapter II deals with vector fields and differential forms. Chapter IV develops the notion of connection on a Riemannian manifold considered as a means to define parallel transport on the manifold.

The author also discusses related notions of torsion and curvature, and gives a working knowledge of the covariant derivative. Chapter V specializes on Riemannian manifolds by deducing global properties from local properties of curvature, the final goal being to determine the manifold completely.

The author is well known for his significant contributions to the field of geometry and PDEs—particularly for his work on the Yamabe problem—and for his expository accounts on the subject.

The text contains many problems and solutions, permitting the reader to apply the theorems and to see concrete developments of the abstract theory. Readership Graduate students, research mathematicians, and mathematics educators interested in differential geometry. The presentation is very successful, and I can strongly recommend the book to anybody willing to learn differential geometry, as well as to teachers of the subject.

This makes it a much more approachable text than many other traditional sources … an excellent textbook for a first course on basic differential geometry, very helpful to both the instructors and their students.

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Thierry Aubin

Aubin submitted his thesis and graduated with a doctorate in but, in the previous year, he had already been appointed to the University of Lille. He writes in [ 6 ] about this outstanding graduate course:- His graduate course at the Pierre et Marie Curie University in was designed to present several theorems published by him in ; it had a remarkable intensity. I remember Aubin only presented what was strictly necessary, seldom consulting his notes He included some results outside those which were directly relevant, but gave little explanation or commentary, allowing each student to work alone on the meaning and scope of the course. Hidehiko Yamabe published Conformal deformation of Riemannian structures on compact manifolds in which he proved the following: given a compact Riemannian manifold without boundary of dimension greater than two with a given metric, there exists a metric with constant scalar curvature which is conformal to the given metric.

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Thierry Émilien Flavien Aubin

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