MATH-UA 377: Differential Geometry
Calendar (Updated December 21, 2020)
Week Date Reference Topics
1 Monday
2/1/2021
Notes: B.1-B.4
  • Abstract linear algebra
    • Abstract vector space
    • Linear independence
    • Basis and Dimension
Wednesday
2/3/2021
Notes: Chapter 3
  • Basis and dimension
  • Affine space
  • Affine geometry
2 Monday
2/8/2021
  • Concrete versus Abstract Linear Algebra
  • Linear Maps
  • Dot Product on \(\mathbb{R}^m\)
Wednesday
2/10/2021
Notes, Chapter 6
  • Euclidean geometry
    • Dot product
    • Cauchy-Schwarz inequality
    • Triangle inequality
    • Euclidean vector space
    • Length of vectors
    • Angle between vectors
3 Monday
2/15/2021
No class. President's Day
Wednesday
2/17/2021
Notes, Chapter 6
  • Euclidean geometry
    • Abstract inner product
    • Orthonormal basis
    • Orthonormal transformation
    • Orthogonal matrices
Thursday
2/18/2021
Notes, Chapter 6
  • Euclidean geometry
    • Euclidean space
    • Distance between two points
    • Rigid motions
    • Isometries
4 Monday
2/22/2021
Notes, Chapter 7
  • Parameterized curves in affine space
    • Velocity, acceleration
  • Geometry of curves in Euclidean \(2\)-space
    • Velocity, speed, acceleration
    • Arclength
    • Constant speed parameterization
    • Curvature
    • Frenet-Serret frame and equations
    • Curvature implies uniqueness up to rigid motion
Wednesday
2/24/2021
Notes, Chapter 7
  • Geometry of curves in Euclidean \(2\)-space
    • Orientation of space and curves
    • Frenet-Serret frame and equations of unit speed curve
5 Monday
3/1/2021
Notes, Chapter 7
  • Geometry of curves in Euclidean \(3\)-space
    • Frenet-Serret frame and equations of parameterized curve
Wednesday
3/3/2021
Notes, Chapter 8
  • Surfaces in \(\mathbb{R}^3\)
6 Monday
3/8/2021
Notes, Chapters 8 and 9
  • Submanifolds of \(\mathbb{R}^n\)
  • Surfaces in \(\mathbb{A}^3\)
  • Tangent Space of a Surface
Wednesday
3/10/2021
  • Differential calculus on a parameterized surface
    • Directional derivative
    • Differential of a function
  • Dual vector space
    • Linear functions of a vector space
    • Dual basis
7 Monday
3/15/2021
Notes, B.9, C.1-8
  • Multilinear algebra
    • Tensors
    • Symmetric and antisymmetric tensors
    • Tensor and wedge products
    • Exterior \(2\)-tensors
Wednesday
3/17/2021
  • Oriented area and volume
    • Oriented area of parallelogram
    • Oriented volume of parallelopiped
    • \(m\)-form as measure of oriented \(m\)-dimensional volume
8 Monday
3/22/2021
  • Vector fields on \(\mathbb{R}^m\)
  • Differential forms on \(\mathbb{R}^m\)
    • Wedge product
    • Differential of Function
Wednesday
3/24/2021
  • Midterm
9 Monday
3/29/2021
  • Differential forms
    • Exterior derivative of functions and \(1\)-forms
    • Pullback of \(1)-form on parameterized curve
    • Integral of \(1\)-form on oriented curve
    • Fundamental Theorem of Line Integrals
Wednesday
3/31/2021
  • Pushforward of a vector field
  • Pullback of a differential form
10 Monday
4/5/2021
    • Integration on a rectangular region in \(\mathbb{R}^m\)
    • Integral of differential form on oriented rectangular region
    • Fundamental theorem of calculus on a rectangular region
    • Integral of $2$-form over a parameterized domain
Wednesday
4/7/2021
  • Integration on a surface
    • Orientation of a surface in \(\mathbb{R}^3\)
    • Orientation of the boundary of a surface
    • Integration on an oriented surface
    • Green's theorem
11 Monday
4/12/2021
  • Geometry of curve in a surface in \(\mathbb{E}^3\)
    • Adapted frame along curve
    • Intrinsic and extrinsic curvature
    • Holonomy
Wednesday
4/14/2021
  • Geometry of surface in \(\mathbb{E}^3\)
    • First fundamental form
    • Second fundamental form
    • Gauss map
12 Monday
4/19/2021
No class, Spring Break
Wednesday
4/21/2021
  • Geometry of surface in \(\mathbb{E}^3\)
    • Covariant derivative of vector fields
    • Geometric invariants as tensors
13 Monday
4/26/2021
  • Geometry of surface in \(\mathbb{E}^3\)
    • Adapted frame and coframe
    • Connection forms
    • Structure equations
Wednesday
4/28/2021
  • Gauss-Bonnet Theorem for a rectangular surface
14 Monday
5/3/2021
  • Gauss-Bonnet Theorem for a closed surface
Wednesday
5/5/2021
  • Riemannian \(2\)-manifolds
  • Curvature on a Riemannian \(2\)-manifold
  • Levi-Civita connection
15 Monday
5/10/2021
  • Gauss-Bonnet theorem for closed \(2\)-manifold