## Math 225B: Differential Geometry

### Winter 2018

**Instructor:** Sucharit Sarkar.

**Class:** MWF 1-1:50pm GEOLOGY 6704.

**Office hours:** M 2-4pm MS 6909.

**TA:** Sangjin
Lee.

**TA review session:** Th 1-1:50pm GEOLOGY 6704.

**TA office hours:** Tu 4-6pm MS 3919.

**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.

**Useful
links:** University
page,

Department
page,

MyUCLA,

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 are due at the beginning of lecture on Wednesdays.
Do not submit homework by e-mail. No late homework will be accepted.

You are encouraged to work in groups on your homework; this is
generally beneficial to your understanding and helps you learn how to
communicate clearly about mathematics. However, you must write up all
solutions yourself. Moreover, since crediting your collaborators is an
important element of academic ethics, you should write down with whom
you worked at the top of each assignment. You should also cite any
sources (other than lectures and the textbook) that you use.

**Exams:** There is a single 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.

**Location, Date, Time:** Tu 3/20 11:30am-2:30pm GEOLOGY 6704.

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:** Numerical grades will be recorded in the MyUCLA
gradebook and the composite numerical grade will be computed as 70% HW
+ 30% Final, and the final letter grade will be assigned based on
that.

If you believe a problem on a homework or an exam has been graded
incorrectly, or that your score was not correctly recorded in the
MyUCLA gradebook, you must bring this to the attention of the
instructor within 10 calendar days of the due date of the assignment
in question, or the date of the exam, and before the end of the
quarter (3/23). Grading complaints not initiated within this period
of time will not be considered. Please verify in a timely manner that
your scores are correctly recorded on MyUCLA.

**Tentative schedule:**
Week |
Lectures |
Dates |
Topic |

1 |
3 |
1/8-1/12 |
S 2-3, smooth manifolds, tangent bundles, vector fields |

2 |
2 |
1/17-1/19 |
S 3-4, vector bundles, orientations, duals, tensor products |

3 |
3 |
1/22-1/26 |
S 5, flows along vector fields, Lie derivatives |

4 |
3 |
1/29-2/2 |
S 6, Frobenius integrability theorem, local and global |

5 |
3 |
2/5-2/9 |
S 7, differential forms, deRahm cochain complex |

6 |
3 |
2/12-2/16 |
S 7, Cartan's magic formula, Poincare lemma |

7 |
2 |
2/21-2/23 |
S 11, Mayer Vietoris sequence, cohomology of S^n |

8 |
3 |
2/26-3/2 |
S 8, Stokes theorem, cohomology with compact support |

9 |
3 |
3/5-3/9 |
S 11, Poincare duality, degree, deRahm theorem |

10 |
3 |
3/12-3/16 |
S 11, Thom class, Euler class, index of vector fields |