D I F F E R E N T I A L    G E O M E T R Y    I ,    F A L L   2 0 2 2

Course Information

• NTU COOL
• Lectures: Wednesday, 16:30 ~ 17:20 and Friday 10:20 ~ 12:10 at Astro-Math 101
• Office hours: 15:00 ~ 16:00 every Wednesday
• Grading scheme:
  • Homework 30%
         You have two jokers: the lowest two grades will be discarded.
  • Midterm 35%
  • Final Exam 35%
• Grader: 李宸寬
• Textbook:
  • [W] Frank Warner, Foundations of differentiable manifolds and Lie groups. MR
• Other references:
  • [dC] Manfredo do Carmo, Riemannian Geometry. MR
  • [T] Clifford Taubes, Differential geometry. Bundles, connections, metrics and curvature. MR
  • [W'] Chin-Lung Wang, Differential Geometry. in link

Lecture summaries and references

• (week 1)  smooth manifold.  Reference: [W] §1.2, §1.3.
• (week 2)  derivation, tangent bundle, differential, immersion, submersion, embedding.  Reference: [W] §1.4, §1.5.
• (week 3)  weak Whitney embedding, tangent vector field, flow, Lie bracket.  Reference: [W] §1. and [W'] §1.5, §1.6.
• (week 4)  Lie derivative, Frobenius theorem.  Reference: [W] §1.7, §1.8.
• (week 5)  tensor, differential form, exterior derivative.  Reference: [W] §2.1, §2.2, §2.3.
• (week 6)  Lie derivative of differential form, manifold with boundary, orientation, integration, Stokes theorem.  Reference: [W] §2.3, §4.1, §4.2.
• (week 7)  Frobenius theorem in terms of differential ideals, Poincaré lemma, Mayer--Vietoris sequence.  Reference: [W] §2.4, §4.3.
• (week 8)  Midterm
• (week 9)  Riemannian metric, Levi-Civita connection, Riemann curvature tensor.  Reference: [W'] §3.1, §3.4.
• (week 10)  first variational formula, geodesic, exponential map, Gauss map.  Reference: [W'] §3.1, §3.2.
• (week 11)  Hopf-Rinow theorem, Jacobi field, geodesic normal coordinate.  Reference: [W'] §3.2, §3.3.
• (week 12)  the method of moving frame, Hodge theory.  Reference: [W] §6.1, §6.2.
• (week 13)  Hodge theory (continued).  Reference: [W] §6.1, §6.2.
• (week 14)  Bochner formula, Sobolev spaces and its properties.  Reference: [W] §6.3.
• (week 15)  elliptic operator, elliptic estimates.  Reference: [W] §6.3, §6.4, §6.6.

Homework

• Homework 1: due September 16.
• Homework 2: due September 23.
• Homework 3: due September 30.
• Homework 4: due October 7.
• Homework 5: due October 14.
• Homework 6: due October 21.
• Homework 7: due Novmeber 11.
• Homework 8: due Novmeber 18.
• Homework 9: due Novmeber 25.
• Homework 10: due December 9.
• Homework 11: due December 16.

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