Differential Geometry of Manifolds (Hardcover, 2 ed)

1 Analysis of Multivariable Functions Legend 1.1 Functions from R to R cut 1.2 Continuity, Limits, and Differentiability new 1.3 Differentiation Rules: Functions of Class moved 1.4 Inverse and Implicit Function Theorems split 2 Coordinates, Frames, and Tensor Notation Variable Frames 2.1 Curvilinear Coordinates Frames Associated to Coordinate Systems 2.2 Moving Frames in Physics Frames Associate to Trajectories 2.3 Moving Variable Frames and Matrix Functions 2.4 Tensor Notation 3 Differentiable Manifolds 3.1 Definitions and Examples 3.2 Differentiable Maps between Manifolds 3.3 Tangent Spaces 3.4 Differentials of Maps between Manifolds 3.5 Manifolds with Boundary 3.6 Immersions, Submersions, and Submanifolds 3.7 Orientability 3.8 Chapter Summary 4 Multilinear Algebra (most of this chapter’s content comes from previous Appendix C) 4.1 Bilinear Forms, Quadratic Forms, and Inner Products 4.2 Adjoint, Self-Adjoint, Automorphisms (I will introduce/revisit Lorentz transforms here as automorphisms of a Lorentz metric) 4.3 The Hom Space and the Dual Space 4.4 The Tensor Product 4.5 Components of Tensors (some of the content will come from the deleted section on tensor notation in the previous section 2.4) 4.6 Symmetric and Alternating Products 5 Analysis on Manifolds 5.1 Vector Bundles on Manifolds 5.2 Vector Fields on Manifolds (I will be able to introduce the notions of Riemannian, Lorentzian, and symplectic manifolds here as way of comparison.) 5.3 The Lie Bracket 5.4 Differential Forms 5.5 Push-Forwards and Pull-Backs 5.6 Integration on Manifolds 5.7 Stokes’ Theorem 6 Introduction to Riemannian Geometry 6.1 Riemannian Metrics 6.2 Connections and Covariant Differentiation (Since this section is decoupled from the Levi-Civita connection, I may put this section in Chapter 5.) 6.3 The Levi-Civita Connection 6.4 Vector Fields Along Curves 6.5 Geodesics 6.6 The Curvature Tensor (I may split this section into two.) 7 Applications of Manifolds to Physics 7.1 Hamiltonian Mechanics 7.2 Electromagnetism 7.3 Metric Concepts in String Theory 7.4 A Brief Introduction to General Relativity (I will significantly rework this section.) Appendices A Point Set Topology A.1 Introduction (This currently is just 3 paragraphs to introduce the section) A.1 Metric Spaces A.2 Topological Spaces A.4 Proof of the Regular Jordan Curve Theorem A.5 Simplicial Complexes and Triangulations A.6 Euler Characteristic B Calculus of Variations B.1 Formulation of Several Problems (Same as former A.1) B.1 The Euler- Lagrange Equation B.2 Several Dependent Variables B.4 Isoperimetric Problems and Lagrange Multipliers C Some More Useful Multilinear Algebra C.1 Volume Forms and the Binet-Cauchy Theorem C.2 The Hodge Star Operator