## Differential Geometry I (course 436)

### 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 10-11, Fridays 3-4 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 pull-back on the other side.

• Lecture 12
Volume, area, and volume forms.

• Lecture 13
Project 1 work.

• Lecture 14
Volume, area, and volume forms - continued.

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.

• 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 3-space: overview of team projects.

HW4.

• Lecture 23
Geodesics - variational and ODE formulations. Completeness.

• Lecture 24-25
Project 2 work.

• Lectures 26-28
Project 2 presentations.