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Overview

This notes uses Einstein's summation convention.

Topics

• Frenet–Serret formulas

• Intro to Surfaces

  • Rectifiable Surfaces

• Exterior Algebra of Vector Spaces

  • Exterior Algebra of Multicovectors
    • The Dual Functor and the Multicovector Functor
    • Alternating Tensor Algebra

• Topological Manifolds

  • Smooth Manifolds
    • Smooth Functions on Smooth Manifolds
    • Partial Derivatives on Manifolds
    • Spheres in Rn
    • Real Projective Space
      • Grassmannian Spaces
      • Complex Projective Space

• Smooth Partitions of Unity for Manifolds

• Tangent Space for Manifolds

• Submersions, Immersions and Local Diffeomorphism of Smooth Manifolds

  • Embeddings for Smooth Manifolds
    • Embedded Smooth Submanifolds
    • Immersed Smooth Submanifolds
      • Smooth Maps on and Between Submanifolds
      • Tangent Spaces and Vector Fields on Submanifolds
      • Smooth Submanifold of Manifolds with a Boundary
  • Rank and Local Normal Forms of Smooth Manifolds
  • Smooth Covering Maps

• Sets of Measure Zero in Smooth Manifolds and Sard's Theorem

  • The Whitney Embedding Theorem
  • The Whitney Approximation Theorem
  • Transversailty Theorems for Smooth Manifolds

• The Tangent Bundle

  • Vector Fields on Smooth Manifolds
    • Tangent Spaces and Vector Fields on Submanifolds

• Integral Curves, Flows and Flowouts on Smooth Manifolds

  • Lie Derivative
  • Time-Dependent Vector Fields and Flows
  • Flows and Flowouts on Manifold with Boundary
  • Lie Derivative
  • Solution of Linear and Quasilinear First Order PDEs using Vector Flows

• Vector Bundles on Smooth Manifolds

  • Local and Global Sections of Vector Bundles
  • Vector Subbundles on Smooth Manifolds
  • Bundle Homomorphisms for Vector Bundles
    • Vector Subbundles on Smooth Manifolds
  • Fibre Bundles on Smooth Manifolds

• The Cotangent Bundle

  • Covector Fields on Smooth Manifolds
    • Line Integrals of Differential 1-forms
  • Differential Forms on Smooth Manifolds
    • The Exterior Derivative on Smooth Manifolds
    • Symplectic Forms on Smooth Manifolds

• The Tensor Bundles

• Lie Groups

  • Lie Subgroups
  • Lie Algebra of a Lie Group
  • Lie Group Actions
    • Homogeneous Spaces in the Case of Lie groups

• Riemannian Metrics on Smooth Manifolds

  • The Riemannian Distance Function
  • The Tangent-Cotangent Bundle Isomorphism for Riemannian Manifolds
  • Psuedo-Riemannian Metrics on Smooth Manifolds

• Orientations of Smooth Manifolds

  • Riemannian Volume Form on Smooth Manifolds
  • Orientations and Covering Maps for Smooth Manifolds

• Integration of Differential Forms on Smooth Manifolds

  • Stokes's Theorem and Smooth Manifolds with Corners
  • Integration on Riemannian Manifolds
    • Laplacian Operator on Riemannian Manifolds
      • Laplace-Beltrami Operator on Riemannian Manifolds
  • Densities on Smooth Manifolds

• The de Rham Cohomology Groups

  • Mayer-Vietoris Theorem for de Rham Cohomology
    • Compactly Supported de Rham Cohomology
  • Smooth Singular Homology
    • The de Rham Theorem and Stokes's Theorem on Chains

• Tangent Distributions and Involutivity on Smooth Manifolds

  • Foliations on Smooth Manifolds