Copies of the classnotes are on the internet in PDF format as given below. The notes for the sections may contain hyperlinks to posted webpages; the links appear in red fonts. The "Proofs of Theorems" files were prepared in Beamer and they contain proofs of the results from the class notes. The "Printout of Proofs" are printable PDF files of the Beamer slides without the pauses. These notes and supplements have not been classroom tested (and so may have some typographical errors).
- Chapter 0. Background Material.
- Chapter 1. Differentiable Manifolds.
- Chapter 2. Tangent Space.
- Chapter 3. Integration of Vector Fields and Differential Forms.
- Chapter 4. Linear Connections.
- Chapter 5. Riemannian Manifolds.
Chapter 0. Background Material.
Chapter 1. Differentiable Manifolds.
Chapter 2. Tangent Space.
- Section 2.1. Tangent Vector.
- Section 2.2. Linear Tangent Mapping.
- Section 2.3. Vector Bundles.
- Section 2.4. The Bracket [X, Y].
- Section 2.5. Exterior Differential.
- Section 2.6. Orientable Manifolds.
- Section 2.7. Manifolds with Boundary.
- Study Guide 2.
Chapter 3. Integration of Vector Fields and Differential Forms.
- Section 3.1. Integration of Vetor Fields.
- Section 3.2. Lie Derivative.
- Section 3.3. The Frobenius Theorem.
- Section 3.4. Integrability Criteria.
- Study Guide 3.
Chapter 4. Linear Connections.
- Section 4.1. First Definitions.
- Section 4.2. Christoffel Symbols.
- Section 4.3. Torsion and Curvature.
- Section 4.4. Parallel Transport. Geodesics.
- Section 4.5. Covariant Derivative.
- Study Guide 4.
Chapter 5. Riemannian Manifolds.
- Section 5.1. Some Definitions.
- Section 5.2. Riemannian Connection.
- Section 5.3. Exponential Mapping.
- Section 5.4. Some Operators on Differential Forms.
- Section 5.5. Spectrum of a Manifold.
- Study Guide 5.
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