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 1 4

Course Information

• Syllabus
• Information
• Ceiba
• Lectures: Wednesday, 16:30 ~ 17:20 and Friday 10:20 ~ 12:10 at Astro-Math 102
• Office hours: Tuesday 11:00 ~ 11:45 and 16:00 ~ 17:00, or by appointment, at Astro-Math 458
       Office hour during the exam week: 16:00 ~ 17:00 on January 13th and 14th (Tuesday and Wednesday).
• Grading scheme:
  • Homework 30%
         You have three jokers: the lowest three grades will be discarded.
  • Midterm 30%
  • Final Exam 35%
         The final exam will be held on January 16th (Friday), from 9:30 to 12:10.
  • Course participation 5%
• Course Assistant: Yu-Ching Sun, whose office is Astro-Math 445.
  • CA office hour: Thursday, 16:00 ~ 17:00.
• References:
  • [T] Clifford Taubes, Differential geometry. Bundles, connections, metrics and curvature. MR
  • [dC] Manfredo do Carmo, Riemannian Geometry. MR
  • [CE] Jeff Cheeger and David Ebin, Comparison theorems in Riemannian Geometry. MR
  • [W] Frank Warner, Foundations of differentiable manifolds and Lie groups. MR
  • [BT] Raoul Bott and Loring Tu, Differential forms in algebraic topology. MR

Lecture summaries and references

• (9/17)  topological manifolds and smooth manifolds.   Reference: [T, §1]
• (9/19)  submanifolds, immersion and embedding, projective space.   Reference: [T, §1]
• (9/24)  partition of unity and exhaustion function.   Reference: [T, §1] and note0924
• (9/26)  embedding into Euclidean spaces, Lie group, matrix groups.   Reference: [T, §2]
• (10/01)  vector bundles, tautological bundle.   Reference: [T, §3]
• (10/03)  tangnet bundle and cotangent bundle.   Reference: [T, §3]
• (10/08)  vector field and derivation, maps between vector bundles.   Reference: [T, §3]
• (10/15)  subbundle and quotient bundle.   Reference: [T, §4] and note1015
• (10/17)  pull-back bundle, symmetric square and exterior powers of cotangent bundle, push-forward and pull-back, exterior derivative.   Reference: [T, §4, §5 and §12.1]
• (10/22)  exterior derivative, orientation and integration, de Rham cohomologies.   Reference: [T, §12.1 and §12.2] and [BT, §3]
• (10/24)  manifold with boundary, Stokes theorem, the notion of left/right invariant for Lie groups, exponential map for matrix groups.   Reference: [BT, §3] and [T, §5.4 and §5.5]
• (10/29)  exponential map (continued), almost complex structure.   Reference: [T, §2.4 and §6]
• (10/31)  complex vector bundle, orientation for vector bundle, metric on vector bundle.   Reference: [T, §6 and §7]
• (11/05)  Riemannian manifold, geodesics.   Reference: [T, §8]
• (11/07)  examples of geodesics: hypersurface, special orthogonal group.   Reference: [T, §8]
• (11/12)  (geodesic) exponential map.   Reference: [T, §9]
• (11/14)  exponential map (continued), Gaussian coordinate and Gauss lemma, properties of geodesics.   Reference: [T, §9] and [CE, §1.2 ~ §1.3]
• (11/19)  properties of geodesics (continued), spherical geometry and hyperbolic geometry.   Reference: [T, §9]
• (11/21)  Midterm
• (11/24 ~ 11/28)  fall break
• (12/03)  principal G-bundle.   Reference: [T, §10]
• (12/05)  group action, associated vector bundles.   Reference: [T, §10]
• (12/10)  covariant derivative.   Reference: [T, §11]
• (12/12)  covariant derivative (continued), connection on principal G-bundle, corresponding covariant derivative on associated bundle.   Reference: [T, §11]
• (12/17)  local expression of connection on principal G-bundle.   Reference: [T, §11]
• (12/19)  Lie derivative, Cartan formula, curvature of covariant derivative.   Reference: [T, §12]
• (12/24)  curvature of connection on principal G-bundle, curvature and commuting derivatives, Frobenius theorem.   Reference: [T, §12~13]
• (12/26)  Frobenius theorem and flat connection, flat connection over the circle, holonomy map.   Reference: [T, §13]
• (12/31)  holonomy and curvature, construction of first Chern class from curvature.   Reference: [T, §13~14]
• (1/07)  example for first Chern class, total Chern class.   Reference: [T, §14]
• (1/09)  example for second Chern class, Pontryagin class.   Reference: [T, §14]

Homework

• Homework 01: due September 24.
• Homework 02: due October 1.
• Homework 03: due October 8.
• Homework 04: due October 15.
• Homework 05: due October 22.
• Homework 06: due October 29.
• Homework 07: due November 5.
• Homework 08: due November 12.
• Homework 09: due November 19.
• Homework 10: due December 3.
• Homework 11: due December 10.
• Homework 12: due December 17.
• Homework 13: due December 24.
• Homework 14: due December 31.

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