Math 225B: Differential Geometry
Winter 2018
Instructor: Sucharit Sarkar.
Class: MWF 12-12:50, MS 7608.
Office hours: By appointment.
Textbook:
- Spivak (S), A comprehensive introduction to differential geometry,
volume 1, chapters 1-8, 11.
Additional reading:
- Milnor (M), Topology from a differentiable viewpoint.
- Warner (W), Foundations of differentiable manifolds and Lie groups.
- Guillemin and Pollack (GP), Differential topology.
- Bott and Tu (BT), Differential forms in algebraic topology.
- Petersen (P), Course notes.
Useful
links: MyUCLA,
Department
page,
old
Qualifying exams (QE).
Topics: 225B is a course in differential geometry. The students
are required to know the material of 131AB (real analysis), 120A
(basic differential geometry), 121 (point set topology), and 225A
(differential topology). This is intended to be a course for the first
year graduate students to help them prepare for the Topology
Qualifiers, but advanced Math majors with sufficient background are
allowed to take the course as well. The course covers an assortment of
topics such as Lie derivatives, integrable distributions and Frobenius
theorem, differential forms, integration and Stokes theorem, de Rham
cohomology, including Mayer-Vietoris sequence, Poincare duality, Thom
classes, degree theory and Euler characteristic revisited from
viewpoint of de Rham cohomology, Riemannian metrics, gradients, volume
forms, and interpretation of classical integral theorems as aspects of
Stokes theorem for differential forms.
Homework: Homeworks need not be submitted and will not be graded. However, you are encouraged to work through the following homework problems, which improves your understanding of the course material. (The homeworks are numbered according to week.)
HW1,
HW2,
HW3,
HW4,
HW5,
HW6,
HW7,
HW8,
HW9.
Exams: There is a single 3 hour final exam which is non-collaborative
and closed-book. You are not allowed to use books, notes, or any
electronic devices (such as calculators, phones, computers) during the
exam. The exam problems will be based on the homework problems.
Location, Date, Time: TBD.
There will be no make-up exams. Attending the final exam is
mandatory. In particular, note that university policy requires that a
student who misses the finals be automatically given F, unless the
absence is due to extreme and documented circumstances, in which case,
if the performance in the course is otherwise satisfactory, the grade
might be I.
Grading: The final letter grade will be based on your
performance in the Final. You may view your score in MyUCLA.
Tentative schedule:
| Week |
Topic |
| 1 |
S 2-3, smooth manifolds, tangent bundles, vector fields |
| 2 |
S 3-4, vector bundles, orientations, duals, tensor products |
| 3 |
S 5, flows along vector fields, Lie derivatives |
| 4 |
S 6, Frobenius integrability theorem, local and global |
| 5 |
S 7, differential forms, deRahm cochain complex |
| 6 |
S 7, Cartan's magic formula, Poincare lemma |
| 7 |
S 11, Mayer Vietoris sequence, cohomology of S^n |
| 8 |
S 8, Stokes theorem, cohomology with compact support |
| 9 |
S 11, Poincare duality, degree, deRahm theorem |
| 10 |
S 11, Thom class, Euler class, index of vector fields |