Differential Geometry : Math 561 (Spring 2015)

Lectures

Textbooks

Graders

Examinations and Homework

• Note: the starred problems are required only for honors students.
• HW #1, due Tuesday, March 3 at the beginning of class: Do Carmo, Section 1.2: 1,3,5. Section 1.3: 6, 10. Section 1.4: 1, 4, 9. Section 1.5: 1, 7(a), 8, 9, 11, 13.
• HW #2, due Thursday, April 16: Do Carmo, Section 2.2: 7, 12. Section 2.3: 1, 3, 4. Section 2.4: 1, 3, 13, *17, 18, 21, *24. Section 2.5: 1, 5, *11, *14.
• HW #3, due Thursday, April 23: Do Carmo, Section 3.2: 8. Section 3.3: 1, 2, 5, 6, *7, 16, *18, 20, *21, *22.

Brief Lecture notes

• Lecture 1: Tuesday, January 20
  
  1. Introduction and Outline.
• Lecture 2: Thursday, January 22
  
  1. Differentiation of curves in R^n.
  2. The derivative is a linear map.
• Lecture 3: Tuesday, January 27
  
  1. Examples of curves.
  2. Tangent vectors.
  3. Arclength.
  4. Mathematica demonstration.
• Lecture 4: Thursday, January 29
  
  1. Dot product, norm and orthogonal group.
  2. Regular curves and arclength.
  3. Rectifiable curves.
  4. Equality of length (in term of sup definition) with integral definition, for C^1 curves.
  5. Mathematica demonstration.
• Lecture 5: Tuesday, February 3
  
  1. Law of cosines
  2. More details about rectifiable curves.
  3. Some non-rectifiable curves.
  4. Mathematica demonstration.
• Lecture 6: Thursday, February 5
  
  1. Arclength is independent of parametrization.
  2. Arclength of a parabola.
  3. Arclength of an ellipse.
  4. Arclenth of logarithmic spiral.
• Lecture 7: Tuesday, February 10
  
  1. Orientations.
  2. Cross product.
• Lecture 8: Thursday, February 12
  
  1. Derivative with respect to arclength.
  2. Unit tangent vector
  3. Unit normal vector and curvature.
  4. Unit binormal.
  5. Mathematica demonstration.
• Lecture 9: Tuesday, February 17
  
  1. Frenet formulas.
  2. Curvature is zero if and only if curve is part of a line.
  3. If curvature is nonzero then torsion vanishes if and only if curve lies in a plane.
  4. Curvature and torsion of helix.
• Lecture 10: Thursday, February 19
  
  1. Curve is part of a circle if and only if curvature is constant and positive, and torsion vanishes.
  2. Frenet formulas for general curve (not necessarily parametrized by arclength)
  3. Formulas for cuvature and torsion of a general curve.
  4. Mathematica demonstration.
• Lecture 11: Tuesday, February 24
  
  1. Fundamental theorem for space curves.
  2. Uniqueness proof
• Lecture 12: Thursday, February 26
  
  1. Existence part of fundamental theorem.
  2. Method of successive approximation.
• Lecture 13: Tuesday, March 3
  
  1. Local canonical form.
  2. Isoperimetric inequality.
• Lecture 14: Thursday, March 5
  
  1. Definition of surface.
  2. Parametrized surfaces (local charts).
  3. Global definition of surfaces (overlap maps are diffeomorphisms).
  4. Graph of differentiable function is a surface.
  5. Sphere is a surface.
• Lecture 15: Tuesday, March 10
  
  1. Stereographic projection.
• Lecture 16: Thursday, March 12
  
  1. Proof that overlap maps are diffeomorphisms.
  2. Inverse image of regular value is a surfaces.
  3. Exam review.
• Thursday, March 19
  
  1. Exam I: in class.
• Lecture 17: Tuesday, March 24
  
  1. Differential of a map betwen surfaces.
• Lecture 18: Thursday, March 26
  
  1. First fundamental form.
  2. Surfaces of revolution.
  3. Area of surfaces.
• Lecture 19: Tuesday, April 7
  
  1. Orientable surfaces.
  2. Orientation.
  3. Normal vector fields.
• Lecture 20: Thursday, April 9
  
  1. Gauss map
  2. Differential of the Gauss map
  3. Self-adjoint linear mappings, symmetric bilinear forms, and quadratic forms.
  4. Diagonalization theorem.
  5. Second fundamental form and principal curvatures.
• Lecture 21: Tuesday, April 14
  
  1. Normal curvature of a curve.
  2. Meusnier's Theorem.
  3. Line of curvature.
  4. Gaussian curvature and mean curvature.
  5. Elliptic, hyperbolic, parabolic, and planar points.
  6. Hyperbolic paraboloid.
• Lecture 22: Thursday, April 16
  
  1. Dupin Indicatrix and intersection of surface with tangent plane translated in normal direction.
  2. Second fundamental form and principal curvatures, Gaussian curvature, and mean curvature in local coordinates.
  3. Gauss curvature of the torus.