Geometry (1)

Instructor: Tsung-Ju Lee（李宗儒）

Term: Fall（秋季）

Location: 數學系館

Lecture Time: Tuesdays 9:00AM - 10:00AM and Thursdays 10:10AM - 12:00PM

TA Recitation: Tuesdays 8:00AM - 8:50AM

Course Overview

This course provides a comprehensive introduction to differential geometry.

Prerequisites

  • Calculus (1)(2), Advanced Calculus (1)(2), Linear Algebra (1)(2).

Textbooks

  • [dC] Differential geometry of curves and surfaces, Manfredo P. do Carmo.
  • Modern Geometry - Methods and Applications. Part I. The Geometry of Surfaces, Transformation Groups, and Fields, B.A. Dubrovin, A.T. Fomenko, and S.P. Novikov.

Grading

  • Midterm: 35%
  • Final: 35%
  • Weekly HW assignments: 30%

Schedule

Week Date Topic Materials
1 Sep 8, Sep 10 Curves in 3-dimensional Euclidean space (I)

Parameterized curves; arc-length parameterization.
  • [dC] 1.1-1.5
2 Sep 15, Sep 17 Curves in 3-dimensional Euclidean space (II)

The local theory and local canonical form for space curves.
  • [dC] 1.5-1.6
3 Sep 22, Sep 24 Regular surfaces (I)

Regular surfaces; change of parameters; differentiable functions on surfaces.
  • [dC] 2.1-2.3
4 Sep 29, Oct 1 Regular surfaces (II)

Tangent planes; the differential of a map.
  • [dC] 2.4
5 Oct 6, Oct 8 Regular surfaces (III)

The first fundamental form; areas; orientation of surfaces.
  • [dC] 2.5-2.6
6 Oct 13, Oct 15 The geometry of the Gauss map (I)

The definition of the Gauss map and its fundamental properties.
  • [dC] 3.1-3.2
7 Oct 20, Oct 22 The geometry of the Gauss map (II)

The Gauss map in local coordinates (the 2nd fundamental form).
  • [dC] 3.3
8 Oct 27, Oct 29 Midterm / The geometry of the Gauss map (III)

Midterm. Vector fields and the stronger form of the ODE theorem.
  • [dC] 3.4
9 Nov 3, Nov 5 The geometry of the Gauss map (IV)

Vector fields and the stronger form of the ODE theorem.
  • [dC] 3.4
10 Nov 10, Nov 12 The intrinsic geometry of surfaces (I)

Conformal maps and isothermal coordinates.
  • [dC] 4.1
11 Nov 17, Nov 19 The intrinsic geometry of surfaces (II)

The Gauss theorem and the equations of compatibility.
  • [dC] 4.2
12 Nov 24, Nov 26 The intrinsic geometry of surfaces (III)

Parallel transport; geodesics.
  • [dC] 4.3
13 Dec 1, Dec 3 The intrinsic geometry of surfaces (IV)

The Gauss–Bonnet theorem and its applications.
  • [dC] 4.4
14 Dec 8, Dec 10 The intrinsic geometry of surfaces (V)

The exponential map; geodesics polar coordinates.
  • [dC] 4.5
15 Dec 15, Dec 17 Global differential geometry (I)

First and second variation of arc-length; Bonnet’s theorem.
  • [dC] 5.4
16 Dec 22, Dec 24 Global differential geometry (II)

Complete surfaces; Hopf–Rinow theorem.
  • [dC] 5.3
17 Dec 29, Dec 31 Final Exam

Final Exam.