Math 761 Differentiable Manifolds (Fall 2006)

Lectures

Textbooks

Graders

Examinations and Homework

• HW #1: Due Thursday, September 21. Warner Chapter 1: 9, 10, 12.
• HW #2: Due Thursday, October 5. Warner Chapter 1: 13, 15, 18, 21, 22, 24.
• HW #3: Due Thursday, October 26. Warner Chapter 2: 2, 8, 9, 10, 12, 14, 15, 16.
• HW #4: Due Thursday, November 9. Warner Chapter 4: 2, 3, 4, 5.
• HW #5: Due Thursday, November 16. Warner Chapter 4: 12, 13, 14; Spivak Chapter 8: 2, 3, 4, 7, 8, 14.
• HW # 6: Due end of semester. Spivak Chapter 8: 17, 21, 22-26, 31, 32. Chapter 11: 1, 2, 3, 6.

Brief lecture outline

• Lecture 1: Tuesday, September 5
  
  1. Introduction and outline.
  2. Differentiable manifolds and differentiable structures.
  3. Smooth functions.
  4. Group actions.
  5. The Poincare Conjecture.
• Lecture 2: Thursday, September 7
  
  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 12
  
  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 14
  
  1. Vector bundles.
  2. Dual bundle.
  3. Immersions, submanifolds, and imbeddings.
  4. Inverse function theorem.
  5. Local structure of immersions.
• Lecture 5: Tuesday, September 19
  
  1. Implicit function theorem.
  2. Conditions when inverse images of submanifolds are submanifolds.
  3. Vector fields.
  4. The Lie bracket.
• Lecture 6: Thursday, September 21
  
  1. Distributions
  2. Some basic Lie groups.
• Lecture 7: Tuesday, September 25
  
  1. Distributions
  2. Lie derivatives.
  3. Outline of proof of Frobenius Theorem.
• Lecture 8: Thursday, September 28
  
  1. Details of proof of Frobenius Theorem.
• Lecture 9: Tuesday, October 3
  
  1. Tensor products.
  2. Exterior algebra
  3. Identification of exterior algebra with alternating multilinear maps.
  4. Wedge products.
  5. Left hook.
• Lecture 10: Thursday, October 5
  
  1. Tensor bundles
  2. Tensors = maps linear over C^{\infty} functions.
  3. Exterior derivative operator d: existence and uniqueness.
• Lecture 11: Tuesday, October 10
  
  1. Pull-back of forms.
  2. Pull-back commutes with d.
  3. Properties of Lie derivatives on k-forms.
• Lecture 12: Thursday, October 12
  
  1. More formulas with Lie derivatives.
  2. Invariant Expression for d using Lie derivatives.
  3. Restatement of Frobenius Theorem using differential ideals.
  4. Riemannian metrics.
  5. The musical isomorphisms, sharp and flat.
• Lecture 13: Tuesday, October 17
  
  1. More index raising and lowering.
  2. Trace of certian tensors.
  3. Induced inner product on exterior algebra.
  4. Orientations.
• Lecture 14: Tuesday, October 24
  
  1. Hodge star operator.
  2. Self-dual and anti-self-dual 2-forms in dimension 4.
• Lecture 15: Thursday, October 26
  
  1. Orientable manifolds.
• Lecture 16: Tuesday, October 31
  
  1. Integration on manifolds.
  2. Stokes' Theorem for chains.
  3. Manifolds with boundary.
• Lecture 17: Thursday, November 2
  
  1. Manifolds with boundary.
  2. Stokes' Theorem on manifolds with boundary.
  3. Adjoint of d.
  4. Hodge Laplacian.
  5. Integration by parts.
• Lecture 18: Tuesday, November 7
  
  1. Chains and homology.
  2. De Rham cohomology.
  3. De Rham's Theorem.
• Lecture 19: Thursday, November 9
  
  1. Cohomology with compact support.
  2. Poincare Lemma.
  3. Homotopic maps induce same map on cohomology.
• Lecture 20: Tuesday, November 14
  
  1. If M^n is connected and orientable, then H^n_{cpt}(M) = R.
• Lecture 21: Thursday, November 16
  
  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.
• Lecture 22: Tuesday, November 21
  
  1. Sard's Theorem.
  2. Degree of a map between connected orientable n-manifolds.
  3. S^n has a nowhere-vanishing vector field if and only if n is odd.
• Lecture 23: Tuesday, November 28
  
  1. Co-chain complexes.
  2. Long exact sequence in cohomology.
  3. Mayer-Vietoris sequence for cohomology.
• Lecture 24: Thursday, November 30
  
  1. Euler characteristic and triangulations.
• Lecture 25: Tuesday, December 5
  
  1. Mayer-Vietoris for cohomology with compact supports.
  2. Exact sequence of a pair, for N = submanifold of M.
  3. Smooth Jordan Curve Theorem.
• Lecture 26: Thursday, December 7
  
  1. Poincare Duality.
  2. Generalized smooth Jordan Curve Theorem.