Time: Tuesdays, Thursdays at 2pm.
Room: B0421 Mathematics Building.
Teacher: Y.A.
Rubinstein. Office hours: Tuesday, Thursdays at 1:10pm.
TA:
James Murphy. Office hours: Thursdays 1011, Fridays 34
in MATH 2121. Email: jmurphy4@math.umd.edu
Course plan:
The goal will be to give an introduction to modern
differential geometry that will prepare students to
either MATH 734 or MATH 742.
Requirement: Homeworks (%40), three team projects (%60). The
second and third projects should be typeset in TeX.
Some references:
J.J. Callahan, The geometry of spacetime, Springer, 2000.
M. Spivak, A Comprehensive Introduction To Differential Geometry,
Vol. I, 3rd Ed, 1999.
Additional references will be given as we go along.
Schedule:

Lecture 1
Overview. Basic definitions of Riemannian geometry: a metric
and a manifold from an intuitive viewpoint. Rigorizing
these definitions: coordinate charts, transformations between
charts, measuring lengths in different charts.

Lecture 2
Towards the notion of a tensor. The notion of a vector, metric
tensor. Covariance and contravariance.

Lecture 3
Definitions of manifold, (co)tangent bundle, bundles, sections,
Riemannian metric, induced metric from an Euclidean embedding.
HW1.

Lecture 4
An example: The 1D circle (using ambient Cartesian coordinates).

Lecture 5
The 1D circle continued (using intrinsic spherical coordinate).

Lecture 6
The 1D circle continued (using intrinsic spherical coordinate).

Lecture 7
The 1D circle continued (using intrinsic spherical coordinate).

Lecture 8
The 1D circle example wrapped up.

Lecture 9
Review of all previously defined notions.
Inducing metrics from an embedding.
HW2.

Lecture 10
Project 1 work.

Lecture 11
Split classroom:
Reconciling the physics notation with the mathematics notation
on one side, and the notion of pullback on the other side.

Lecture 12
Volume, area, and volume forms.

Lecture 13
Project 1 work.

Lecture 14
Volume, area, and volume forms  continued.
HW1 solutions.
HW3.

Lecture 15
Presentations of Project 1 (groups 1,2,6).

Lecture 16
Presentations of Project 1 (groups 3,4,5).

Lecture 17
Review of concepts and outlook towards holonomy.
HW2 solutions.

Lecture 18
The Gaussian curvature.

Lecture 19
Gaussian curvature  continued. Examples of computation of Gaussian
curvature.

Lecture 20
A negatively curved surface.
Gauss' Theorem  Teorema Egregium.

Lecture 21
Gauss' Theorem  Teorema Egregium  continued.

Lecture 22
Extrinsic vs. intrinsic. Review of Gauss' Theorem.
Problems concerning isometric embedding of surfaces
in 3space: overview of team projects.
HW4.

Lecture 23
Geodesics  variational and ODE formulations. Completeness.
HW3 solutions.

Lecture 2425
Project 2 work.

Lectures 2628
Project 2 presentations.
