Modern Geometry (2013 Fall)

• Room: ºî¤T631
• Time: W789

Course Description

• 1. Smooth manifolds
• 2. Tangent spaces and cotangent spaces
• 3. Vector Bundles
• 4. Differential forms
• 5. Integration on manifolds
• 6. De Rham cohomology
• 7. Lie groups
• 8. Principal bundles, connections and curvatures (possible)

References

• L.Tu, An Introduction to Manifolds, UTX.
(easy to read)
• M. do Carmo, Differential forms and applications, UTX.
(easy to read)
• D. Barden and C. Thomas, An Introduction to Differential Manifolds.
(easy to read)
• S. Morita, Geometry of Differential forms.
(easy to read)
• S. Kobayashi, Fundations of Differential Geometry I & II.
• Frank Warner, Foundations of Differentiable Manifolds and Lie Groups, GTM.
• W.Boothby, An Introduction to Differential Manifolds and Riemannian Geometry.
• V.Guillemin and A.Pollack, Differential Topology.
• I.M.Singer and J.A.Thorpe, Lecture notes on Elementary Topology and Geometry, UTM.
• R.Bott and L.Tu, Differential Forms in Algebraic Topology, GTM.
• M.Spivak, A Comprehensive Introduction to Differential Geometry I.

Syllabus

• 9/18: Review topology, manifold (section 5)
• 9/25: Maps on and between manifolds (section 6, 9, 11, and part of 8)
• 10/2: Tangent vectors as derivations, tangent space, vector bundles (section 12, and part of section 8 and 6) (required: section 2)
• 10/9: Tangent bundle, partitions of unity, vector fields (section 13, 14)
• 10/16: Vector fields and their properties
• 10/23: Differential 1-forms (section 17) (required section 3, 4)
• 10/30: Tensor, Differential k-forms (section 18) (required section 3, 4)
• 11/6: Move to 1/22,
• 11/13: Midterm I
• 11/20: University Holiday
• 11/27: Operators on differential forms (section 19, 20)
• 12/4: Orientations, manifolds with boundary (section 21, 22)
• 12/11: Stokes Theorem, de Rham cohomology (section 23, 24)
• 12/18: Poincare Lemma, Homotopy operator (section 27)(Poincare Lemma can be found in do Carmo or Morita's books)
• 12/25: Hodge Theory, Poincare duality (Jost's Riemannian geometry and geometric analysis)
• 1/1: National Holiday
• 1/8: Exact sequence, Mayer-Vietoris Sequence
• 1/22: Final presentation, 10 am - 4 pm

Evaluation

• Midterm 50%, Final presentation 50%.