G E O M E T R Y ,    F A L L   2 0 2 5

Course Information

• NTU COOL
• Lectures: Wednesday, 10:20 ~ 12:10, and Friday 10:20 ~ 11:10; at Astro-Math 102
• Office hour: Monday, 14:00 ~ 15:00 at Astro-Math 458
• Course Assistant: 連焌凱
       Problem session: Friday 11:20 ~ 12:10; at Astro-Math 102
• Course prerequisite: linear algebra and mathematical analysis
       It will be helpful if you are familiar with general topology. (more precisely, ch.1 of Topology and Geometry by Glen Bredon mathscinet)
• Grading scheme:
  • Homework 30%
         You have two jokers: the lowest two grades will be discarded.
  • Midterm 35%   on October 22
  • Final Exam 35%   on December 17
• Main Topics:
  1. Theory of surfaces
  2. Differentiable manifolds
  3. Differential forms
• Textbooks:
  • Felix Schulze, Geometry of Curves and Surfaces. file
  • John M. Lee, Introduction to Smooth Manifolds. file (via NTU IP)
• Other references:
  • Manfredo do Carmo, Differential Geometry of Curves and Surfaces.
  • Theodore Shifrin, Differential Geometry: A First Course in Curves and Surfaces. file
  • Loring W. Tu, An Introduction to Manifolds. file (via NTU IP)

Lecture summaries and references

• (Week 1)  parametrized curves; rectifiable and arclength; Frenet frame for curves in R^3; Hopf Umlaufsatz; isoperimetric inequality.  ref: [Schulze] p.5 ~ 23
• (Week 2)  regular surface; applications of IFT; change of parametrization; smooth function and its differential; tangent plane; 1st fundamental form.  ref: [Schulze] p.26 ~ 39
• (Week 3)  smooth map between regular surface, and its differential; Weingarten map and 2nd fundamental form; mean curvature and Gauss curvature; compatibility: Gauss equation and Codazzi-Mainardi equation.  ref: [Schulze] p.32 ~ 34, 39 ~ 42, 49 ~ 52
• (Week 4)  Gauss Theorema Egregium; principal curvatures as the coefficients in Taylor series; totally umbilical surface; vector fields and covariant derivative; change of parametrization and covariant derivative; covariant derivative along a curve.  ref: [Schulze] p.53 ~ 62
• (Week 5)  parallel transport; geodesic and variation of energy; geodesic exponential map; geodesic curvature and normal curvature.  ref: [Schulze] p.60 ~ 70
• (Week 6)  Gauss-Bonnet theorem: local and global.  ref: [Schulze] p.70 ~ 77
• (Week 7)  mean curvature and variation of the area; hyperbolic geometry.  ref: [Shifrin] p.91 ~ 110
          smooth manifold.  ref: [Lee] ch.1
• (Week 8)  Midterm
• (Week 9)  derivation and tangent plane; smooth map between manifolds, and its differential; tangent bundle; immersion, submersion, embedding, submanifold.  ref: [Lee] ch.2, 3, 4
• (Week 10)  basic property of immersion and submersion; embedding into Euclidean spaces, and (weak) Whitney embedding theorem.  ref: [Lee] ch.6
• (Week 11)  vector fields; integral curves and flows; commuting derivatives and Lie brackets; Lie derivative; bracket and commutation of the flow.  ref: [Lee] ch.8, 9, 19
• (Week 12)  involutive distribution and Frobenius theorem; multi-linear algebra: tensor and alternating algebra; wedge product; differential forms.  ref: [Lee] ch.19, 14
• (Week 13)  exterior derivative; Lie derivative of differential form, Cartan magic formula, manifold with boundar; orientation.  ref: [Lee] ch.14, 1, 15
• (Week 14)  integration on manifolds; Stokes theorem; de Rham cohomology and homomorphism to R^1; homotopy invariance of de Rham cohomology.  ref: [Lee] ch.16, 17
• (Week 15)  Mayer-Vietoris sequence for de Rham cohomology; Frobenius theorem in terms of differential ideal.  ref: [Lee] ch.17, 19

Homework

• Homework 01: due September 12.
• Homework 02: due September 19.
• Homework 03: due September 26.
• Homework 04: due October 3.
• Homework 05: due October 8.
• Homework 06: due October 17.
• Homework 07: due October 31.
• Homework 08: due November 7.
• Homework 09: due November 14.
• Homework 10: due November 21.
• Homework 11: due November 28.
• Homework 12: due December 5.
• Homework 13: due December 12.

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