Spring Semester 2014

**Instructor:** Sergei Chmutov

- Office: MA-118
- Telephone: 292-0429
- email: chmutov@math.ohio-state.edu
- https://people.math.osu.edu/chmutov.1/13-14/m5540H/math5540H.htm
- Office Hours: Monday 1:00 - 2:15 pm.

**Classes:** Monday 2:15--4:05 pm. Wednesday, Friday 2:20--3:40 pm.
at Caldwell Lab 0171.

**Carmen:**
https://carmen.osu.edu

**TEXTBOOKS:**

Prentice-Hall, 1976, ISBN:9780132125895.

Publish or perish, 1999. ISBN Vol.

**Additional books:**

Revised Second Edition, Academic Press (Elsevier), March 2006.

Second Edition, AMS Student Mathematical Library Series Vol.

**Grading:**
There will be weekly homework assignments (25%), two midterms (25% each), and a final exam
(25%).

**GRADING SCALE:**

A | A- | B+ | B | B- | C+ | C | C- | D+ | D |

90 | 87 | 83 | 80 | 77 | 73 | 70 | 67 | 63 | 60 |

**TOPICS:
1. Curvature of curves.**

Plane curves. Curves in 3D. Frenet equations.

Curves on surfaces, Euler theorem, Meusnier theorem.

Curvature of surfaces. Gauss Map. The first and second fundamental forms.

Gaussian curvature. Egregium theorem.

Riemannian metrics. Gauss-Bonnet theorem.

Parallel transport. Geodesics. Christoffel symbols.

Variations of arc length. Euler-Lagrange equation.

& Jacobi fields and conjugate points.

The additional topics which we may disscuss during the course.

** Vector fields and bundles. Tensors.
Differential forms. General relativity theory.
**

Handout #1: Curvature of plane curves.

Handout #2: Global properties of plane curves.

Handout #3: Curves in * R ^{n}*.

Handout #4: Multilinear algebra.

Handout #5: Tensors.

Handout #6: Tensor fields.

Handout #7: Calculus of differential forms.

Handout #8: Vector fields and differential operators.

Handout #9: Christoffel symbols and Riemannian metric.

Assignment #1 (due Monday, January 13, 2014)

Assignment #2 (due Wednesday, January 22, 2014)

Assignment #3 (due Monday, January 27, 2014)

Assignment #4 (due Monday, February 3, 2014)

Assignment #5 (due Monday, February 17, 2014)

Assignment #6 (due Monday, February 24, 2014)

Assignment #7 (due Monday, March 3, 2014)

Assignment #8 (due Monday, March 17, 2014)

Assignment #9 (due Monday, March 24, 2014)

Assignment #10 (due Monday, March 31, 2014)

Assignment #11 (due Monday, April 14, 2014)

Assignment #12 (due Monday, April 21, 2014)