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:
Assignments and Grading. There will be one Midterm Exam,
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 |
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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 |
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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.
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