## Spring 2014

Instructor. Michael Anderson, Math Tower 4-110.
E-mail: anderson AT math.sunysb.edu, Phone: 632-8269.
Lectures: Tu/Th: 1:00-2:20pm, in Library N4000

Office Hours. MWF 2-3pm, and by appointment.

Grader. Maryam Pouryahya, Math Tower S-240A.
E-mail: mpouryah AT math.sunysb.edu

Course Description. The foundation of differential geometry is the concept of curvature. The course will focus on understanding this and related concepts very clearly, both geometrically and computationally, for the case of surfaces in Euclidean space. For this, you'll need a solid background in multivariable calculus and linear algebra. We hope to give some idea of how curvature is understood in higher dimensions; this is the basis of Riemannian geometry and General Relativity.

Prerequisites. MAT 205 (Calc III) and MAT 210 (Linear Algebra).

Text. There is no formal text for the class. As a backbone or guideline, we will use the online text:

Differential Geometry: A First Course in Curves and Surfaces
by Theodore Shifrin.

This is a preliminary version of a text to appear and is currently available free online: here
Permission for this use has been obtained. (Shifrin is an old grad student friend).

A more advanced online text, at a level of the lectures in class, is:

Curves and Surfaces, Lecture Notes for Geometry I

by Henrik Schlichtkrull

available as PDF here

There are many other texts that you are encouraged to browse and use as you see fit. Some of these are:
• Manfredo do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, (1976).
• Andrew Pressley, Elementary Differential Geometry, Springer Verlag, 2001.
• B. O'Neill, Elementary Differential Geometry.
• Millman and Parker, Elements of Differential Geometry.
These and many others are in the Library (I believe).

Assignments and Grading. There will be one Midterm Exam,

MIDTERM: Date: Tuesday, March 25

This is a TAKE HOME EXAM, due in class on Thursday, March 27 (1pm). You may use or consult texts or course notes, but not others. The material of the exam is all of the course up to and including March 13 (the week before Spring Break).

FINAL EXAM
The FINAL EXAM will be a TAKE HOME EXAM.
It will be emailed as a PDF file to all enrolled students on Thursday, May 15, 10am, and due on Tuesday, May 20, 12 Noon.
Bring the exam to my office, Math Tower 4-110, (or put it under my office door before that time).
The material for the exam will be the full semester of course work, but with an emphasis on course work covered since the Midterm Exam.

The Final Exam as PDF is here

There will be regular homework assignments, due roughly once per week. Your grade will be determined via the following percentages:
• Homework: 40%
• Midterm: 25%
• Final: 35%

• Homework assignments will appear on this page during the course of the semester.
Problems are from the Shifrin text, unless indicated otherwise.

Week of

Topics

Problems Due

Due Date

Feb 3

Local Surfaces, Surfaces

The metric/first fundamental form

2.1: 1, 3, 10, 14

Feb 11

Feb 10

The metric/first fundamental form

Intrinsic geometry, isometry, etc

2.1: 2, 4ab, 6, 7, 8, 15

Feb 20

Feb 17

Intrinsic geometry, isometry, etc

Extrinsic geometry, 2nd fundamental form

-- --

--

Feb 24

2nd fundamental form, Gauss Map

Principal, Gauss and mean curvatures

2.2: 1, 3abc, 6, 7, 22a

Mar 4

Mar 3

More on extrinsic geometry and computation

Intrinsic geometry revisited

2.2: 8b, 10, 14, 21

Mar 11

Mar 10

Teorema Egregium and Consequences

Intrinsic Invariants

2.3: 1, 5, 6, 12

Mar 25

Mar 17

Spring Break

----

---

Mar 24

Intro to Connections/Covariant derivative

Geodesics

Midterm Exam: take home

Mar 27

Mar 31

More on Covariant derivative

Geodesics

2.4: 8, 10, 12, 13, 14, 21.

Apr 8

Apr 7

More on Geodesics

Geodesic coordinate systems

2.4: 6, 9, 24, 28

Apr 15

Apr 14

Intro to Gauss Bonnet theorem

7.6: 1, 2 (Schlichtkrull text)

Apr 22

Apr 21

The Gauss Bonnet theorem

3.1: 8, 11, 12, 13.

Apr 29

Apr 28

The Gauss Bonnet theorem II

3.1: 2, 3, 5, 6.

May 6

May 5

Final discussion on Gauss-Bonnet

Abstract Riemannian metrics

---

---

Americans with Disability Act: If you have a physical, psychological, medical or learning disability that may impact your course work, please contact Disabled Student Services in ECC (Educational Communications Center) Building, Room 128, (631) 632-6748, or at the website here. They will determine with you what accomodations are necessary and appropriate. All information and documentation is confidential.

Academic Integrity: Each student must pursue his or her academic goals honestly and be personally accountable for all submitted work. Representing another person's work as your own is always wrong. Faculty is required to report any suspected instances of academic dishonesty to the Academic Judiciary. Faculty in the Health Sciences Center (School of Health Technology & Management, Nursing, Social Welfare, Dental Medicine) and School of Medicine are required to follow their school-specific procedures. For more comprehensive information on academic integrity, including categories of academic dishonesty please refer to the academic judiciary website at here

Critical Incident Management: Stony Brook University expects students to respect the rights, privileges, and property of other people. Faculty are required to report to the Office of University Community Standards any disruptive behavior that interrupts their ability to teach, compromises the safety of the learning environment, or inhibits students' ability to learn. Faculty in the HSC Schools and the School of Medicine are required to follow their school-specific procedures. Further information about most academic matters can be found in the Undergraduate Bulletin, the Undergraduate Class Schedule, and the Faculty-Employee Handbook.