Differential Geometry : Math 561 (Spring 2012)

Lectures

Textbooks

Graders

Examinations and Homework

• HW #1: Due Thursday, May 10. There will be no late homework accepted!! Absolutely no exceptions! Note: the starred problems are required only for honors students. However, all students can get extra credit by attempting these.
  • Ch 2.2: 12, 13.
  • Ch 2.3: 1, 3, 4.
  • Ch 2.4: 1, 3, 12, 13.
  • Ch 2.5: 1, 3, *14.
  • Ch 3.2: 8.
  • Ch 3.3: 1, 5, *7, *21.
  • Ch 3.5: *14.

Brief Lecture notes

• Lecture 1: Tuesday, January 24
  
  1. Introduction and Outline.
• Lecture 2: Thursday, January 26
  
  1. Curves in R^n.
  2. 1-5: Curvature and torsion of curves in R^3.
  3. 1-5: Fundamental theorem of the local theory of curves.
• Lecture 3: Tuesday, January 31
  
  1. 1-6: Local canonical form.
  2. Wirtinger's Inequality.
• Lecture 4: Thursday, February 2
  
  1. 1-7: Isoperimetric Inequality.
• Lecture 5: Tuesday, February 7
  
  1. Chapter 2, Appendix A: Continuity in R^n.
• Lecture 6: Thursday, February 9
  
  1. Continuity: Inverse image of open sets are open.
  2. Connectedness.
  3. Connected subsets of real line are intervals.
  4. Intermediate Value Theorem.
• Lecture 7: Tuesday, February 14
  
  1. Appendix: Point-Set topology of Euclidean spaces.
  2. Closed sets and limit points.
  3. Continuity: Inverse image of closed sets are closed.
• Lecture 8: Thursday, February 16
  
  1. Closure of connected set is closed.
  2. Path Connectedness.
  3. Topologist's sine curve.
• Lecture 9: Tuesday, February 21
  
  1. Compactness: every finite cover has a finite subcover.
  2. Closed subspace of compact space is compact.
  3. Compact subset of R^n is closed.
  4. Image of compact space under continuous map is compact.
  5. Heine-Borel Theorem (a closed interval [a,b] is compact).
• Lecture 10: Thursday, February 23
  
  1. Subset A in R is compact <-> A is closed and bounded.
  2. Extreme Value Theorem.
  3. A closed cube in R^n is compact.
  4. Subset A in R^n is compact <-> A is closed and bounded.
  5. Product of compact subsets of R^n is compact.
  6. Let A be compact, if f : A -> R^n is bijective and continuous, then f is a homeomorphism.
  7. Examples of f: S^1 -> R^2, f: [a,b] -> R^2, f: [a,b) -> R^2.
• Lecture 11: Tuesday, February 28
  
  1. Chapter 2, Appendix B: Differentiability in R^n.
  2. The differential of a map in higher dimensions.
  3. Jacobian matrix.
  4. Chain Rule.
• Lecture 12: Thursday, March 1
  
  1. Basic rules of differentiation (constant maps, linear maps, sums, products, etc.).
• Lecture 13: Tuesday, March 6
  
  1. Partial derivatives.
  2. Example of function whose partial derivatives exist, but not differentiable (partials are not continuous).
• Lecture 14: Thursday, March 8
  
  1. Mean value theorem from calculus.
  2. Theorem: if partial derivatives exists, and are continuous, then differentiable.
  3. Maximum principle: differentiable and p and maximum achieved, then Df(p) =0.
  4. Df = 0 everywhere in U implies constant on connected components.
• Lecture 15: Tuesday, March 13
  
  1. Inverse function theorem.
• Lecture 16: Thursday, March 15
  
  1. Chapter 2: 2-dimensional surfaces in R^3.
• Lecture 17: Tuesday, March 20
  
  1. Exam Review.
• Thursday, March 22
  
  1. Exam I.
• Lecture 18: Tuesday, March 27
  
  1. Section 2.2: Intro to Regular surfaces.
  2. Critical and regular values.
  3. Proposition 2: Inverse image of a regular value is a regular surface.
• Lecture 19: Thursday, March 29
  
  1. Section 2.2, Example 6: Coordinate systems on a torus.
  2. Minimal number of coordinate systems possible for a torus: 3 (not in book).
• Tuesday, April 3
  
  1. Spring Break.
• Thursday, April 5
  
  1. Spring Break
• Lecture 20: Tuesday, April 10
  
  1. Section 2.3: Change of Parameters and Differentiable functions on surfaces.
  2. Overlap maps are diffeomorphisms.
  3. Example 4: Surfaces of revolution.
• Lecture 21: Thursday, April 12
  
  1. Section 2.4: Tangent Plane and differential of a map between surfaces..
• Lecture 22: Tuesday, April 17
  
  1. Section 2.5: First Fundamental Form and Area.
  2. Using the first fundamental form to find length
  3. Using the first fundamental form to find angle between coordinate curves
  4. Examples of these on a sphere
• Lecture 23: Thursday, April 19
  
  1. Section 2.5: The first fundamental form and area.
  2. Torus example.
  3. Section 2.6: Orientation.
  4. Theorem: S orientable if and only if there exists differentiable field of unit normal vectors.
  5. Sphere example, Mobius strip example.
• Lecture 24: Tuesday, April 24
  
  1. Chapter 3: Appendix on self-adjoint maps
  2. Section 3.2: Gauss map.
  3. Differential of Gauss map.
  4. Sphere example and intuitive torus example.
• Lecture 25: Thursday, April 26
  
  1. Section 3.2: Differential of Gauss map is self-adjoint
  2. Second fundamental form and normal curvature
  3. Sphere example, rotation of z=y^4 example, curved plane example
  4. Principal curvature and principal directions
  5. Normal curvature for any tangent v in terms of principal curvatures
  6. Definition 6: Determinant and trace of differential of Gauss map
• Lecture 26: Tuesday, May 1
  
  1. Definitions of I, II, E, F, G, e, f, g, K, and H.
• Lecture 27: Thursday, May 3
  
  1. Notes click here.
• Lecture 28: Tuesday, May 8
  
  1. Review of last few weeks.
  2. First fundamental Form.
  3. Gauss map.
  4. Second fundamental form.
  5. Gaussian and mean curvatures.
  6. How to compute all the above in coordinates.
• Lecture 29: Thursday, May 10
  
  1. TBA.