Tensors and Riemannian Geometry

Nail H. Ibragimov, Higher Education Press

• Éditeur: De Gruyter
• Année: 2015
• ISBN:
  9783110379648
• Édition: 1
• Sujet: Mathematics
• Sous-sujet: Geometry

This book is based on the experience of teaching the subject by the author in Russia, France, South Africa and Sweden. The author provides students and teachers with an easy to follow textbook spanning a variety of topics on tensors, Riemannian geometry and geometric approach to partial differential equations. Application of approximate transformation groups to the equations of general relativity in the de Sitter space simplifies the subject significantly.

Part I
Tensors and Riemannian spaces

Tensor calculus has been invented by G. Ricci. He called the new branch of mathematics an absolute differential calculus and developed it during the ten years of 1887—1896. The tensor calculus provides an elegant language, e.g. for presenting the special and general relativity.

The concept of tensors was motivated by development of Riemannian geometry of general manifolds (Riemann, 1854) and by E. B. Christoffel’s transformation theory of quadratic differential forms (Christoffel, 1869). Subsequently, the tensor notation has been generally accepted in differential geometry, continuum mechanics and theory of relativity (see (5)).

The tensor calculus and Riemannian spaces furnish a profound mathematical background for theoretical physics and differential equations of mathematical physics.

Chapter 1 contains a collection of selected formulae from the classical vector calculus and an easy to follow introduction to the index notation used in the present book.

Chapter 2 includes a variety of topics on conservation laws from the basic concepts and examples through to modern developments in this field.

Since the present book is designed for graduate courses in differential equations and mathematical modelling, I provide in Chapter 3 a simple introduction to tensors and Riemannian spaces with emphasis on calculations in local coordinates rather than on the global geometric language.

The concepts of isometric, conformal and generalized motions in Riemannian spaces, given in Chapter 4, are useful in various applications in physics and theory of differential equations.

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