# Math 346 - Differential Geometry II

### (See Math 345 - Differential geometry I)

[Top]     [Home]             Instructor: Alex Degtyarev     Office: SA 130     Phone: x2135     Mail: no spam

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Textbook:     Manfredo do Carmo, Differential Geometry of Curves and Surfaces. Prentice-Hall, 1976.
• Intrinsic geometry of surfaces (continued):
- review of Math 345;
- the Gauss-Bonnet theorem;
- the exponential map;
- shortest geodesics.
• Global differential geometry of surfaces:
- rigidity of the sphere;
- completeness {\it vs.} geodesic completeness;
- the Bonnet theorem (compactness of surfaces of positive curvature);
- Jacobi fields and conjugate points; Jacobi's theorem;
- Hadamard theorem (on surfaces of negative curvature);
- surfaces of zero Gaussian curvature;
- abstract surfaces;
- Hilbert's theorem on the hyperbolic plane;
- ... (whatever time permits).

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Midterm I     (25%)
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important remarks

Topics covered (tentative):
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Midterm II     (25%)
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important remarks

Topics covered (tentative):
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Final     (40%)
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important remarks

Topics covered* (tentative):
All previous material is fully included!
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Homeworks     (10%)   Approximately weekly
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Remarks

During the exams please keep in mind the following:
• Calculators are  not  allowed
• Identical solutions (especially identically wrong ones) will  not  get credit. I reserve the right to decide what "identical" means. You still have the right to complain
• Do not argue about the distribution of the credits among different parts of a problem. I only accept complaints concerning my misunderstanding/misreadung your solution
• Show all your work. Correct answers without sufficient explanation might  not  get full credit
• Indicate clearly and unambiguously your final result. In proofs, state explicitly each claim
• Do not misread the questions or skip parts thereof. If you did, do not complain
• If you believe that a problem is misstated, do not try to solve it; explain your point of view instead
• Each problem has a reasonably short solution. If your calculation goes completely out of hands, something must be wrong