Kriegar 220, 516-6637 Office Hours: Mondays?

sormani@math.jhu.edu

**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

Isometries,B5.9

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