# Math 345 - Differential Geometry I

[Top]     [Home]             Instructor: Alex Degtyarev     Office: SA 130     Phone: x2135     Mail: no spam
Monday  15:40-17:30    Thursday  15:40-16:30                Office hours:     Monday  14:40-15:30    Thursday  14:40-15:30
 Midterm I Fall 1998* Fall 1999* Fall 2000 Midterm II Fall 1998* Fall 1999* Fall 2000 Final Fall 1998* Fall 1999* Fall 2000

 Homework assignments [NEW] Class syllabus Detailed weekly schedule Topics causing problems Exam rules and terms A problem with surface area [NEW]
*By Prof. Gürses. Save the file as *.tex, remove HTML tags, and TeX it.
(That is, if you know what I mean ; if you don't, you'll have to stick to the horrible HTML .)

Textbook:     Manfredo do Carmo, Differential Geometry of Curves and Surfaces. Prentice-Hall, 1976.
 Week Topic Notes 1 (1.2, 1.3) Curves, regular curves. Arc length 2 (1.4, 1.5) Curves parametrized by arc length 3 (1.6, 1.7) Global properties 4 (2.2, 2.3) Regular surfaces. Change of parameters Midterm I 5 (2.4, 2.5) Tangent plane. First fundamental form 6 (1.6, 2.7, 2.8) Orientation. Definition of area 7 Review and problems 8 (3.2) The Gauss map 9 (3.3) The Gauss map in local coordinates Midterm II 10 (3.4) Vector fields 11 (3.5) Ruled and minimal surfaces 12 (4.2, 4.3) Isometries, conformal maps, the Gauss theorem 13 (4.4) Parallel transport, geodesics 14 (4.5, 4.6) The Gauss-Bonnet theorem. exponential map 15 Review and problems Finals week Final

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Midterm I     (25%)   October 21, 2009, 8:40 am     See   important remarks

Topics covered (tentative):
• Parametrized curves, regular curves, arc length
• Tangent vector, principal normal, binormal; curvature, torsion
• Frenet formulas; intrinsic equation, the existence and uniqueness theorem
• The local canonical form
• Global properties (isoperimetric inequality, four-vertex theorem)
• Classical locus problems
• Regular surfaces/parametrizations; change of parameters; surfaces as graphs; level surfaces
• The tangent plane; tangent vectors as differentiations
• Differentiable maps and their differentials (various approaches)
• The first fundamental form; arc length, area

Chapters:     1, 2 (2.1--2.5)
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Midterm II     (30%)   December 2, 2009, 8:40 am     See   important remarks

Topics covered (tentative):
• Orientation and orientability; tubular neighborhoods
• The Gauss map and its differential; the second fundamental form (properties, relations, in coordinates)
• Principal directions/curvatures, H, K, asymptotic directions, lines of curvature, asymptotic lines, etc.
• Vector/line fields and differential equations; existence theorems, special coordinate systems
• Principal examples: graphs, surfaces of revolution, ruled surfaces, minimal surfaces

Chapters:     2, 3
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Final     (35%)   December 31, 15:30 at SAZ--18     See   important remarks

Topics covered* (tentative):
• The concept of (local) isometry; relation to the first fundamental form
• The covariant derivative: external/internal definitions and properties
• The derivatives of the basis vectors; the compatibility equations
• The parallel transport; properties and related tricks; the relation to angles
• Geodesics and geodesic curvature: simplest properties, existence, basic examples (cylinder, cone, sphere, surfaces of revolution)
• The Gauss-Bonnet theorem and applications
• The exponential map; geodesic polar coordinates; surfaces of constant curvature
• Completeness vs. geodesic completeness; the Hopf--Rinow theorem
• Jacobi vector fields; the (non-)commutativity of covariant derivatives; critical points of the exponential map
• The Hadamard theorem on surfaces of non-positive curvature; compactness of surfaces of positive curvature (bounded from zero)

Chapters:     4 (all),  5 (1, 3)
All previous material is fully included!
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Homeworks     (5--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