Hardcover ISBN:  9780821839294 
Product Code:  GSM/71 
List Price:  $92.00 
MAA Member Price:  $82.80 
AMS Member Price:  $73.60 
Electronic ISBN:  9781470421106 
Product Code:  GSM/71.E 
List Price:  $86.00 
MAA Member Price:  $77.40 
AMS Member Price:  $68.80 

Book DetailsGraduate Studies in MathematicsVolume: 71; 2006; 633 ppMSC: Primary 53; Secondary 57;
The book presents the basics of Riemannian geometry in its modern form as geometry of differentiable manifolds and the most important structures on them. The authors' approach is that the source of all constructions in Riemannian geometry is a manifold that allows one to compute scalar products of tangent vectors. With this approach, the authors show that Riemannian geometry has a great influence to several fundamental areas of modern mathematics and its applications. In particular,
 Geometry is a bridge between pure mathematics and natural sciences, first of all physics. Fundamental laws of nature are formulated as relations between geometric fields describing various physical quantities.
 The study of global properties of geometric objects leads to the farreaching development of topology, including topology and geometry of fiber bundles.
 Geometric theory of Hamiltonian systems, which describe many physical phenomena, led to the development of symplectic and Poisson geometry. Field theory and the multidimensional calculus of variations, presented in the book, unify mathematics with theoretical physics.
 Geometry of complex and algebraic manifolds unifies Riemannian geometry with modern complex analysis, as well as with algebra and number theory.
Prerequisites for using the book include several basic undergraduate courses, such as advanced calculus, linear algebra, ordinary differential equations, and elements of topology.ReadershipGraduate students and research mathematicians interested in modern geometry and its applications.

Table of Contents

Chapters

Chapter 1. Cartesian spaces and Euclidean geometry

Chapter 2. Symplectic and pseudoEuclidean spaces

Chapter 3. Geometry of twodimensional manifolds

Chapter 4. Complex analysis in the theory of surfaces

Chapter 5. Smooth manifolds

Chapter 6. Groups of motions

Chapter 7. Tensor algebra

Chapter 8. Tensor fields in analysis

Chapter 9. Analysis of differential forms

Chapter 10. Connections and curvature

Chapter 11. Conformal and complex geometries

Chapter 12. Morse theory and Hamiltonian formalism

Chapter 13. Poisson and Lagrange manifolds

Chapter 14. Multidimensional variational problems

Chapter 15. Geometric fields in physics


Additional Material

Reviews

The book is designed for students in mathematics and theoretical physics but it will be very useful for teachers as well. ...has a much wider scope than the usual textbook on differential geometry.
European Mathematical Society Newsletter 
The textbook offers an abundance of general theories, concrete examples, and algebraic computations. It is a readable introduction to a wide number of areas of geometrical and algebraic themes in its interrelation with physics, appropriate for students of mathematics and theoretical physics.
Hubert Gollek for Zentralblatt MATH 
This excellent textbook offers a modern treatment of most differential geometrical notions and tools used today, in pure mathematics as well as in theoretical physics. The approach used by the authors is most remarkable...the reviewer thinks that this is an outstanding book, highly recommended to mathematicians and mathematical physicists, from beginners up to advanced researchers.
CharlesMichel Marle for Mathematical Reviews


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The book presents the basics of Riemannian geometry in its modern form as geometry of differentiable manifolds and the most important structures on them. The authors' approach is that the source of all constructions in Riemannian geometry is a manifold that allows one to compute scalar products of tangent vectors. With this approach, the authors show that Riemannian geometry has a great influence to several fundamental areas of modern mathematics and its applications. In particular,
 Geometry is a bridge between pure mathematics and natural sciences, first of all physics. Fundamental laws of nature are formulated as relations between geometric fields describing various physical quantities.
 The study of global properties of geometric objects leads to the farreaching development of topology, including topology and geometry of fiber bundles.
 Geometric theory of Hamiltonian systems, which describe many physical phenomena, led to the development of symplectic and Poisson geometry. Field theory and the multidimensional calculus of variations, presented in the book, unify mathematics with theoretical physics.
 Geometry of complex and algebraic manifolds unifies Riemannian geometry with modern complex analysis, as well as with algebra and number theory.
Prerequisites for using the book include several basic undergraduate courses, such as advanced calculus, linear algebra, ordinary differential equations, and elements of topology.
Graduate students and research mathematicians interested in modern geometry and its applications.

Chapters

Chapter 1. Cartesian spaces and Euclidean geometry

Chapter 2. Symplectic and pseudoEuclidean spaces

Chapter 3. Geometry of twodimensional manifolds

Chapter 4. Complex analysis in the theory of surfaces

Chapter 5. Smooth manifolds

Chapter 6. Groups of motions

Chapter 7. Tensor algebra

Chapter 8. Tensor fields in analysis

Chapter 9. Analysis of differential forms

Chapter 10. Connections and curvature

Chapter 11. Conformal and complex geometries

Chapter 12. Morse theory and Hamiltonian formalism

Chapter 13. Poisson and Lagrange manifolds

Chapter 14. Multidimensional variational problems

Chapter 15. Geometric fields in physics

The book is designed for students in mathematics and theoretical physics but it will be very useful for teachers as well. ...has a much wider scope than the usual textbook on differential geometry.
European Mathematical Society Newsletter 
The textbook offers an abundance of general theories, concrete examples, and algebraic computations. It is a readable introduction to a wide number of areas of geometrical and algebraic themes in its interrelation with physics, appropriate for students of mathematics and theoretical physics.
Hubert Gollek for Zentralblatt MATH 
This excellent textbook offers a modern treatment of most differential geometrical notions and tools used today, in pure mathematics as well as in theoretical physics. The approach used by the authors is most remarkable...the reviewer thinks that this is an outstanding book, highly recommended to mathematicians and mathematical physicists, from beginners up to advanced researchers.
CharlesMichel Marle for Mathematical Reviews