Math 345 - Differential Geometry I
Letter grades
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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
*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
Grading policy
I will take off a few (2-3) points for arithmetical mistakes. However,
a lot of points will be taken off for `obvious'
mistakes, i.e., either those that you can easily avoid or those
showing a deep misunderstanding of the subject.
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