Differential Geometry of Manifolds

1. Auflage 2010
ISBN: 978-1-56881-457-5
Verlag: Taylor & Francis Inc

Buch, Englisch, 440 Seiten, Format (B × H): 196 mm x 241 mm, Gewicht: 923 g

Reihe: Textbooks in Mathematics

From the coauthor of Differential Geometry of Curves and Surfaces, this companion book presents the extension of differential geometry from curves and surfaces to manifolds in general. It provides a broad introduction to the field of differentiable and Riemannian manifolds, tying together the classical and modern formulations. The three appendices provide background information on point set topology, calculus of variations, and multilinear algebra—topics that may not have been covered in the prerequisite courses of multivariable calculus and linear algebra.
Differential Geometry of Manifolds takes a practical approach, containing extensive exercises and focusing on applications of differential geometry in physics, including the Hamiltonian formulation of dynamics (with a view toward symplectic manifolds), the tensorial formulation of electromagnetism, some string theory, and some fundamental concepts in general relativity.

Autoren/Hrsg.

Lovett, Stephen T.

Fachgebiete

• Mathematik | Informatik Mathematik Geometrie Differentialgeometrie

Inhaltsverzeichnis

Analysis of Multivariable Functions Functions from Rn to Rm Continuity, Limits, and Differentiability Differentiation Rules: Functions of Class Cr Inverse and Implicit Function Theorems Coordinates, Frames, and Tensor Notation Curvilinear Coordinates Moving Frames in Physics Moving Frames and Matrix Functions Tensor Notation Differentiable Manifolds Definitions and Examples Differentiable Maps between Manifolds Tangent Spaces and Differentials Immersions, Submersions, and Submanifolds Chapter Summary Analysis on Manifolds Vector Bundles on Manifolds Vector Fields on Manifolds Differential Forms Integration on Manifolds Stokes’ Theorem Introduction to Riemannian Geometry Riemannian Metrics Connections and Covariant Differentiation Vector Fields Along Curves: Geodesics The Curvature Tensor Applications of Manifolds to Physics Hamiltonian Mechanics Electromagnetism Geometric Concepts in String Theory A Brief Introduction to General Relativity Point Set Topology Introduction Metric Spaces Topological Spaces Proof of the Regular Jordan Curve Theorem Simplicial Complexes and Triangulations Euler Characteristic Calculus of Variations Formulation of Several Problems The Euler-Lagrange Equation Several Dependent Variables Isoperimetric Problems and Lagrange Multipliers Multilinear Algebra Direct Sums Bilinear and Quadratic Forms The Hom Space and the Dual Space The Tensor Product Symmetric Product and Alternating Product The Wedge Product and Analytic Geometry

Über Autor(innen)

Stephen Lovett is an associate professor of mathematics at Wheaton College in Illinois. Lovett has also taught at Eastern Nazarene College and has taught introductory courses on differential geometry for many years. Lovett has traveled extensively and has given many talks over the past several years on differential and algebraic geometry, as well as cryptography.