Math 761 Differentiable Manifolds (Fall 2008)

Lectures

Textbooks

Graders

Examinations and Homework

• HW #1: Due Thursday, September 18. Warner Chapter 1: 9, 10, 12.
• HW #2: Due Thursday, October 2. Warner Chapter 1: 13, 15, 18, 21, 22, 24.
• HW #3: Due Friday, November 14. Warner Chapter 2: 2, 8, 9, 10, 12, 14, 15, 16.
• HW #4: Due Tuesday, November 25. Warner Chapter 4: 2, 3, 4, 5, 12, 13, 14. Spivak Chapter 8: 2, 3, 4, 7, 8, 14.
• HW # 5: Due end of semester. Spivak Chapter 8: 17, 21, 31, 32. Chapter 11: 1, 2, 3.

Brief lecture outline

• Lecture 1: Tuesday, September 2
  
  1. Introduction and outline.
  2. Differentiable manifolds and differentiable structures.
  3. Smooth functions.
  4. Group actions.
  5. The Poincare Conjecture.
• Lecture 2: Thursday, September 4
  
  1. Tangent vectors.
  2. Derivations, Taylor's Theorem.
  3. Equivalence classes of curves definition.
  4. Transformation formula for components of vectors.
  5. Equivalence of two definitions of tangent space.
• Lecture 3: Tuesday, September 9
  
  1. The differential of a map.
  2. Equivalence of two definitions of differential.
  3. Chain rule.
  4. Cotangent space.
  5. Coordinate differentials dx^i.
  6. Tangent bundle and cotangent bundle.
• Lecture 4: Thursday, September 11
  
  1. Vector bundles.
  2. Dual bundle.
  3. Immersions, submanifolds, and imbeddings.
  4. Inverse function theorem.
  5. Local structure of immersions.
• Lecture 5: Tuesday, September 16
  
  1. Implicit function theorem.
  2. Conditions when inverse images of submanifolds are submanifolds.
  3. The orthogonal group O(n) is a submanifold of GL(n,R).
  4. SO(3) = SU(2) = Sp(1) is diffeomorphic to RP^3.
• Lecture 6: Thursday, September 18
  
  1. Vector fields (Warner Proposition 1.43).
  2. One-parameter group of diffeomorphisms (Spivak Chapter 5, Theorem 5).
  3. Vector field with compact support is complete (Spivak Chapter 5, Theorem 6).
• Lecture 7: Tuesday, September 23
  
  1. The Lie bracket (Warner Proposition 1.45).
  2. Case of 1 vector field (Warner Proposition 1.53).
  3. Frobenius Theorem for 2-dimensional distribution of planes in R^3 (Spivak Chapter 6).
• Lecture 8: Thursday, September 25
  
  1. k-dimensional distributions on an n-manifold = subbundle of tangent bundle.
  2. Proof of Frobenius Theorem: Spivak Chapter 6, Theorem 5.
• Lecture 9: Tuesday, September 30
  
  1. \phi-related vector fields and the Lie bracket: Spivak Chapter 6, Proposition 3.
  2. More details used in the Frobenius proof: Spivak Chapter 5, pages 207-220.
  3. Example: contact distribution.
• Lecture 10: Thursday, October 2
  
  1. Tensor products.
  2. Exterior algebra
  3. Identification of exterior algebra with alternating multilinear maps.
• Lecture 11: Tuesday, October 7
  
  1. Computing wedge products.
  2. Left hook.
  3. Tensor bundles
  4. Tensors = maps linear over C^{\infty} functions.
• Lecture 12: Thursday, October 9
  
  1. Exterior derivative operator d: existence and uniqueness.
  2. Pull-back of forms.
  3. Pull-back commutes with d.
  4. Properties of Lie derivatives on k-forms.
• Lecture 13: Tuesday, October 21
  
  1. More formulas with Lie derivatives.
  2. Invariant Expression for d using Lie derivatives.
  3. Restatement of Frobenius Theorem using differential ideals.
• Lecture 14: Thursday, October 23
  
  1. Riemannian metrics.
  2. The musical isomorphisms, sharp and flat.
  3. Index raising and lowering.
  4. The dual inner product = inverse metric.
  5. Delta symbol is really a tensor.
  6. Traces of tensors.
• Lecture 15: Tuesday, October 28
  
  1. Induced inner product on exterior algebra.
  2. Orientations.
  3. Hodge star operator.
  4. Self-dual and anti-self-dual 2-forms in dimension 4.
• Lecture 16: Tuesday, November 11
  
  1. Orientable manifolds.
  2. Integration on manifolds.
  3. Stokes' Theorem for chains.
• Lecture 17: Thursday, November 13
  
  1. Manifolds with boundary.
  2. Stokes' Theorem on manifolds with boundary.
  3. Chains and homology.
  4. De Rham cohomology.
  5. De Rham's Theorem.
• Lecture 18: Tuesday, November 18
  
  1. Cohomology with compact support.
  2. Poincare Lemma.
  3. Homotopic maps induce same map on cohomology.
• Lecture 19: Thursday, November 20
  
  1. If M^n is connected and orientable, then H^n_{cpt}(M) = R.
• Lecture 20: Tuesday, November 25
  
  1. H^{n-1}(R^n \ {0} ) \neq 0.
  2. If M^n is connected and non-orientable, then H^n_{cpt}(M) = 0.
  3. If M^n is connected and non-compact, then H^n(M) = 0.
  4. Sard's Theorem.
  5. Degree of a map between connected orientable n-manifolds.
  6. S^n has a nowhere-vanishing vector field if and only if n is odd.
• Lecture 21: Tuesday, December 2
  
  1. Co-chain complexes.
  2. Long exact sequence in cohomology.
  3. Mayer-Vietoris sequence for cohomology.
• Lecture 22: Thursday, December 4
  
  1. Euler characteristic and triangulations.