Differential Geometry I (course 436)

University of Maryland

Department of Mathematics

Autumn 2013






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.

    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 3-space: overview of team projects.

    HW4.

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

    HW3 solutions.

  • Lecture 24-25
    Project 2 work.

  • Lectures 26-28
    Project 2 presentations.