Introduction to Differential Geometry
     110.439,  ThFr 2-3:15,  Room TBA
Professor:  Christina Sormani
                          Kriegar 220,  516-6637                Office Hours: Mondays?

Texts:  A first course in GeometricTopology and
                  Differential Geometry, by Ethan D. Bloch, Birkhauser
                  We will be covering Chapters 4-8 which provide a nice
                      introductory course to differential geometry and avoid
                      geometric topology.   The first three chapters are not
                      required but will be a good reference if you need to
                      study geometric topology in the future.    Section 4.2 is
                      difficult and deals with high dimensions, so we will begin
                      with 4.3 (curves) and return to 4.2 when it is needed
                      This book has good explanations of curves, surfaces
                      and curvature with plenty of diagrams.

                 Calculus on Manifolds, by M. Spivak, Addison Wesley
                      This small paperback is also required.  It explains higher
                       dimensional calculus including the inverse and implicit
                       function theorems as well as tensors.
                     (Flemming's Functions of Several Variables is more verbose)

Prerequisites:   Multivariable Calculus and Linear Algebra are
                required.   You should review multiple integrals, Inner products
                and cross products from Calculus.  You should review matrices
                eigenvalues and determinants from Linear Algebra.   It is also
                important to review abstract vector spaces (see Anton and
                Rorres, Chapter 5) in the beginning of the semester.

Grading Policy:

       Homework, 30%:    Not all problems will be graded.
              No late assignments will be accepted without a doctor's
              note.    Assignments are due at the beginning of class on
             Thursdays.  You may consult with classmates but be sure to
             do most of the work yourself.
      Midterm Exam, 20%  Given in mid october.   This will
             begin in class on Thursday October 22, you may ask questions
             by email that night and I will reply to the entire class on Friday.
             After that no questions may be asked.   Spend a maximum of 5
             hours over the weekend and it will be due on Monday October 26
             before 6pm.  Do not consult with anyone regarding the problems
             but do consult the textbooks and notes and refer to lemmas and
             theorems explicitly.  Do not consult other texts.
      Final Project, 10% : This project will be due before the last
               day of class.  It consists of a poster covering a chosen topic
               (eg Gluing and connected sums, four vertex theorem,
               isoperimetric theorem, spaces of constant curvature,
               rotation indexes, isothermal coordinates, conformal
               mapping, ruled surfaces, developable surfaces,  helicoids
               and catenoids, ellipsoids, schwartzchild metric...)   A
               topic must be handed in by September 18.  I will assign
               appropriate reading and timing for the project.   The poster
               will contain mathematics as well as pictures.
     Final 40%:    This exam will be given during the finals week.  (3 hours)
                A cheat sheet will be provided with some formulas.
                * There may be one problem in which you must match
                   drawn curves with their properties (Frenet Frame at a
                   point, torsion positive or negative, curvature =0...)
                * There may be a drawn surface on which you must locate
                   points of positive and negative Gauss curvature and
                   explain with a drawing of normals.  You may also be asked
                   to suggest a coord chart for the surface near a given point
                   using projections.
                * You may need to apply the implicit functions theorem
                   and regular values to prove that a given surface is a surface.
                * Given a surface with given normals, you may 
                   be asked to take covariant derivatives and/or verify
                   that certain curves are geodesics.
                * Given a chart and a surface find an appropriate domain
                   for that chart so that it is 1:1, and be able to verify
                   that it is a chart.   Be able to compute gij.  Be able to write
                   a vector in terms of the chart vectors.
                *  Given the Christoffel symbols (gamma ijk), be able to
                   verify if a curve a geodesic and to take covariant
                   derivatives using the definitions of these symbols.
                *  Be able to use formulas like X<Y,Z>=... and [X,Y]=...
                    (as in proof of curvature formula for geod polar coords)
                *  Given a map, f, between two surfaces with charts, you
                    should be able to verify whether f is smooth, a diffeomorphism,
                    find the push forward of a vector, verify if it  is a local isomorphism.
                *  Given a surface with normals and geodesics described explicitly,
                    you should be able to find the principal directions and curvatures
                    (that is the eigenvalues and eigenvectors of  the Gauss Map's
                    push forward) and compute the Gauss curvature.  You should
                    also be able to find exp_p(v) for any given vector.
                *  You should be able to verify that a certain function is a tensor
                    or to use the fact that a certain function (like curvature) is a tensor.

Syllabus:   This syllabus will be updated as the course progresses.
         We may go faster or slower depending on your background .
        Week 1:   Curves in Euclidean space  B4.3,
                                 (handout on diffeom, homeom in 1D)
                            Tangent, Normal Binormal B4.4 ,
      Week 2:    Fundamental Theorem of ODE handout
                             Curvature and Torsion B4.5,
        Week 3:   Fund Thm of Curves B4.6,
                             Plane Curves B4.7, review S1

        Week 4:   Derivatives in higher dimensions S2 p15-33, B4.2
                             Surfaces and Coord patches B5.2 -p208, Handout
                     HW: handout, 5.2.2, 5.2.3, 2-4, 2-5, 2-29

        Week 5:   Examples 5.3 (1)-(3), Spivak 2-4 and 2-5
                             Open sets, Inverse Function Theorem, Spivak 34-40
                    HW:  5.3.1, 5.3.3, 5.3.4, 2-16,
                             Handout  5, 6, 8, 9
        Week 6:    Implicit Funct Thm S2 40-45,  (Marsden Handout 7.1, 7.2)
                             Bloch 4.2, 5.2 Example B5.3 4,
                    HW:  Marsden Handout: 7.1/1,2,  7.2/1,2
                             Bloch 4.2.4 (use implicit function thm), 5.2.6
       Week 7:   Tangents and normals B5.4
                             First Fundamental form B5.5
                     HW due Thursday Oct 22: Excerpt from Marsden : 7.2/5
                             Tangent Vectors Handout (a)-(e)
                             Bloch 5.4.1 (i), 5.4.3, 5.5.1, 5.5.2
                             Read Abstract vector spaces do problem: b
                    Final Project:  Think about possible projects.  Physics majors and other
                              students interested in manifolds (higher dimensions) should
                              read the  Manifolds Handout and hand in problems a,b.   The rest
                              of the handout can be considered as part of a final project.

        Week 8:    Midterm exam (Thursday Oct 22) (week 1-6)
                               Vector Fields
                     HW:  Midterm exam due Monday October 26 at 6pm (hand in to me in person!)
                              Bloch 5.4.4, 5.4.5, 5.5.3, more...

         Week 9:   Covariant Differentiation B5.6-5.7 (handout)
                                            (parallel vector fields)
                             Tensors, S4 p75-77 emphasis on examples
                     HW:  Cov Diff Handout problems 2-8.
                               Tensor Handout all problems.
                               Bloch 5.6.3 only prove iv (rem <Z,W> is a smooth function)
                               5.6.5 only prove iii.
         Week 10:  Area B5.8, Week 10 Handout
                       HW: Bloch 5.7.3, Cov Diff Handout 9
                               Week 10 Handout 1, 2, Bloch 5.8.3, 5.8.7
                               Week 10 Handout 5, 8,9,10, 11
                               Bloch 5.9.1, 5.9.3

         Week 11:  Curvature and Second Fundamental Form B6.2
                               Gauss Curvature  B6.3-6.4
                        HW: Week 10 Handout 6, 7, 12-16
                                then Curvature Handout 1,2, 5,6,7,8
                                and Curvature Continued 1, 4,5
                                 Extra Credit Bloch  5.9.4

       Week 12:  Theorem Egregium B6.5
                               Geodesics B7.1-B7.2
                        HW: Curvature Continued 6,7,8,9,10,12
                                Gauss Thm Proved 2 (use 1 but don't prove it)
                                Review Problem (do not hand in:
                                 look at Helicoid from p218 of Bloch
                                    a) verify g_{i,j} is as described in exer 5.5.8
                                    b) verify gamma i j k is as in Example 5.7.3 (2)
                                    c) solve for the geodesic through (0,0,0)
                                        with starting direction (0,0,1) by solving the
                                        system of diff eqn from class (Prop 7.2.4)
                                        Hint: try to just hold one component constant.
                                    d) verify your answer using the defn of a geodesic.
                                There will be no office hours this monday but
                                  instead I will hold them on Tuesday.

         Week 13:  Exponential Map B8.1-8.3
                               Exponential Polar Coords and Curvature
                               No homework , finish projects, (due Dec 10)

         Poster Session,  Examples and Overview

        Final:   Friday, December 18, Gilman 48, 2-5pm