Differential Geometry of Manifolds.

Differential Geometry of Manifolds, Second Edition presents the extension of differential geometry from curves and surfaces to manifolds in general. The book provides a broad introduction to the field of differentiable and Riemannian manifolds, tying together classical and modern formulations. It in...

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Bibliographic Details
Online Access: Full Text (via Taylor & Francis)
Main Author: Lovett, Stephen (Stephen T.)
Format: eBook
Language:English
Published: Milton : CRC Press LLC, 2019.
Edition:2nd ed.
Series:Textbooks in mathematics (Boca Raton, Fla.)
Subjects:
Table of Contents:
  • Cover
  • Half Title
  • Series Page
  • Title Page
  • Copyright Page
  • Table of Contents
  • Preface
  • Acknowledgments
  • 1: Analysis of Multivariable Functions
  • 1.1 Functions from ℝn to ℝm
  • 1.2 Continuity, Limits, and Differentiability
  • 1.3 Differentiation Rules
  • Functions of Class Cr
  • 1.4 Inverse and Implicit Function Theorems
  • 2: Variable Frames
  • 2.1 Frames Associated to Coordinate Systems
  • 2.2 Frames Associated to Trajectories
  • 2.3 Variable Frames and Matrix Functions
  • 3: Differentiable Manifolds
  • 3.1 Definitions and Examples.
  • 3.2 Differentiable Maps between Manifolds
  • 3.3 Tangent Spaces
  • 3.4 The Differential of a Differentiable Map
  • 3.5 Manifolds with Boundaries
  • 3.6 Immersions, Submersions, and Submanifolds
  • 3.7 Orientability
  • 4: Multilinear Algebra
  • 4.1 Hom Space and Dual
  • 4.2 Bilinear Forms and Inner Products
  • 4.3 Adjoint, Self-Adjoint, and Automorphisms
  • 4.4 Tensor Product
  • 4.5 Components of Tensors over V
  • 4.6 Symmetric and Alternating Products
  • 4.7 Algebra over a Field
  • 5: Analysis on Manifolds
  • 5.1 Vector Bundles on Manifolds
  • 5.2 Vector and Tensor Fields on Manifolds.
  • 5.3 Lie Bracket and Lie Derivative
  • 5.4 Differential Forms
  • 5.5 Pull-Backs of Covariant Tensor Fields
  • 5.6 Lie Derivative of Tensor Fields
  • 5.7 Integration on Manifolds
  • Definition
  • 5.8 Integration on Manifolds
  • Applications
  • 5.9 Stokes' Theorem
  • 6: Introduction to Riemannian Geometry
  • 6.1 Riemannian Metrics
  • 6.2 Connections and Covariant Differentiation
  • 6.3 Vector Fields along Curves
  • Geodesics
  • 6.4 Curvature Tensor
  • 6.5 Ricci Curvature and Einstein Tensor
  • 7: Applications of Manifolds to Physics
  • 7.1 Hamiltonian Mechanics
  • 7.2 Special Relativity.
  • Pseudo-Riemannian Manifolds
  • 7.3 Electromagnetism
  • 7.4 Geometric Concepts in String Theory
  • 7.5 Brief Introduction to General Relativity
  • A: Point Set Topology
  • A.1 Metric Spaces
  • A.2 Topological Spaces
  • B: Calculus of Variations
  • B.1 Formulation of Several Problems
  • B.2 Euler-Lagrange Equation
  • B.3 Several Dependent Variables
  • B.4 Isoperimetric Problems and Lagrange Multipliers
  • C: Further Topics in Multilinear Algebra
  • C.1 Binet-Cauchy and k-Volume of Parallelepipeds
  • C.2 Volume Form Revisited
  • C.3 Hodge Star Operator
  • Bibliography
  • Index.