MATH-UA 377: Differential Geometry
Calendar (Last update on January 31, 2020)
Week Date Reference Topics
1 Monday
1/27/2020
Notes: Chapters 1, 2 Euclidean geometry
Cartesian coordinates
Abstract vector spaces
Wednesday
1/29/2020
Axler: Chapters 1, 2 Notes: Chapter 3 Abstract vector spaces
Linear indepdendence
Linear basis
Linear maps
2 Monday
2/3/2020
Notes: Chapter 4 Affine spaces
Affine independence
Affine basis
Affine maps
Wednesday
2/5/2020
Notes: Chapter 6
O'Neill: Chapter 3
Affine and vector space versions of \(\mathbb{R}^m\)
Inner product on a vector space
Euclidean space
3 Monday
2/10/2020
Notes: Chapter 7
O'Neill: Sections 2.1-2.3
Curves in Euclidean space
Velocity, speed, acceleration
Curvature of a curve in Euclidean 2-space
Frenet-Serret frame and equations in Euclidean 2-space
Wednesday
2/12/2020
Notes: Chapter 7
O'Neill: Sections 2.3-2.4
Frenet-Serret frame and equations in Euclidean 2-space
4 Monday
2/17/2020
No class, Presidents' Day
Wednesday
2/19/2020
Notes: Section 7.2
O'Neill, Chapter 4
Curvature, torsion of a curve in Euclidean 3-space
Frenet-Serret frame and equations in Euclidean 3-space
Curvature and torsion imply uniqueness up to rigid motion
5 Monday
2/24/2020
Notes: Chapter 8 and 9
O'Neill: Sections 4.1-4.2
Definitions of a surface in \(\widetilde{\mathbb{R}}^3\)
Wednesday
2/26/2020
Notes: Chapter 9
O'Neill: Section 4.3
Coordinate charts
Sphere
Torus
Tangent space
6 Monday
3/2/2020
Notes: Chapter 10
O'Neill: Chapter 1
Tangent space
First fundamental form
Wednesday
3/4/2020
Notes: Chapter 10
O'Neill, Chapter 1
First fundamental form
7 Monday
3/9/2020
Geometry of curve in surface in \(\mathbb{E}^3\)
Second fundamental form
Gauss map
Wednesday
3/11/2020
Notes: Chapter 10 Dual vector space
Dual basis
Natural isomorphism \(V^{**}\simeq V\)
Directional derivative
Differential of function
  3/16/2020, 3/18/2020 No class, Spring Recess
8 Monday
3/23/2020
O'Neill, Chapter 4
  • 4.1: Surfaces in \(\mathbb{R}^3\)
  • 4.2: Patch computations
  • 4.3: Differentiable functions and tangent vectors
Wednesday
3/25/2020
O'Neill, Chapter 4
  • 4.4: Differential forms on a surface
9 Monday
3/30/2020
O'Neill, Chapter 4
  • 4.4: Differential forms on a surface
Wednesday
4/1/2020
O'Neill, Chapters 4
  • 4.4: Differential forms on a surface
10 Monday
4/6/2020
O'Neill: Chapters 2 and 6
  • 2.6: Frame fields
  • 2.7: Connection forms
  • 2.8: Structural equations
Wednesday
4/8/2020
O'Neill: Chapter 2
  • 2.6: Frame fields
  • 2.7: Connection forms
  • 2.8: Structural equations
11 Monday
4/13/2020
O'Neill: Chapter 6
  • 6.1: The fundamental equations
  • 6.2: Form computations
Wednesday
4/15/2020
O'Neill: Chapter 6
  • 6.5: Intrinsic geometry of surfaces in \(\mathbb{R}^3\)
12 Monday
4/20/2020
O'Neill: Section 3.3
  • Area of parallelogram
  • Volume of parallelopiped
  • Volume as a tensor
  • Orientation
Wednesday
4/22/2020
O'Neill: Chapter 4
  • Integral of \(1\)-form on an oriented curve
  • Fundamental theorem of line integrals
  • Integration on a rectangular region
  • Integration of a differential form on an rectangular region
13 Monday
4/27/2020
O'Neill: Chapter 4
  • Orientation of domains in \(\mathbb{R}^m\)
  • Orientation and integration on a curve
  • Orientation and integration on a surface
Wednesday
4/29/2020
O'Neill: Chapters 4 and 6
  • Orientation of domains in \(\mathbb{R}^m\)
  • Orientation and integration on a curve
  • Orientation and integration on a surface
14 Monday
5/4/2020
O'Neill: Chapters 4 and 7
  • Integration along oriented surface with boundary
  • Stokes's Theorem
  • Gauss-Bonnet on a surface with boundary
Wednesday
5/6/2020
15 Monday
5/11/2020