Differential Geometry
: 18.950 (Spring 2006)
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Lectures
Lectures are by
Jeff Viaclovsky on Tuesdays and Thursdays
at 9:30-11:00 AM in Room 2-143. I will have office hours on Tuesdays
and Thursdays in 2-175 from 2 - 3 PM.
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Textbooks
Differential Geometry of Curves and Surfaces by Do Carmo is the required text.
Riemannian Manifolds: An Introduction to Curvature by John Lee is recommended.
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Graders
The grader is Steven Sivek (ssivek at mit.edu).
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Examinations and Homework
There will be several homework assignments,
2 in-class tests, and no final. The homework will
count for 40% of the final grade, and the
2 exams will count for 60%.
The first exam will be held on Thursday, March 23.
The second exam wil be held on Thursday, May 11.
- HW #1: Due Thursday, March 2.
Ch 2.2: 5, 12, 13.
Ch 2.3: 1, 3, 4.
Ch 2.4: 1, 3, 12, 13, 17, 18, 21, 24.
Ch. 2.5: 1, 3, 14.
- HW #2: Due Tuesday March 21.
Ch 3.2: 8. Ch 3.3: 1, 2, 5, 6, 7, 16, 20, 21, 22.
- HW #3: Due Thursday April 13.
Ch 3.5: 12, 13, 14.
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HW #4: Due Tuesday May 9.
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Brief Lecture notes
- Lecture 1: Tuesday, February 7
- Introduction and Outline.
- Curves in R^n.
- Curvature and torsion of curves in R^3.
- Fundamental theorem of the local theory of curves.
- Lecture 2: Thursday, February 9
- Isoperimetric Inequality
- 2-dimensional surfaces in R^3.
- Lecture 3: Tuesday, February 14
- Stereographic projection.
- More on regular surfaces.
- Overlap maps are diffeomorphisms.
- Critical and regular values.
- Lecture 4: Thursday, February 16
- Surfaces of revolution.
- Tangent plane
- Differential of maps between surfaces.
- First Fundamental Form.
- Surfaces of revolution
- Holiday, no class on Tuesday, February 21 (Don't forget to
go to your Monday classes instead!)
- Lecture 5: Thursday, February 23
- Torus.
- Orientation.
- Orientable and non-orientable surfaces.
- Lecture 6: February 28
- Area of surfaces.
- Transformation rule for vector, 1-forms, and tensors.
- The Gauss Map, differential of Gauss Map.
- Lecture 7: March 2
- More on Gauss Map and its differential.
- Self-Adjoint Linear Maps.
- Normal curvature.
- Lecture 8: March 7
- Diagonalize quadratic forms.
- Diagonalize symmetric matrix by orthogonal matrices.
- Computations of normal curvature using normal sections.
- Lecture 9: March 9
- Gaussian curvature and mean curvature.
- Elliptic, hyperbolic, parabolic and planar points.
- Computations for a hyperbolic paraboloid.
- Umbilic points.
- Lecture 10: March 14
- General formula for second fundamental form, Gaussian
curvature, mean curvature in coordinates.
- Gaussian curvature of a torus.
- Differential equation for asymptotic curves.
- Lecture 11: March 16
- Differential equation for lines of curvature.
- Gaussian and mean curvature for surfaces of revolution
and graphs.
- Lecture 12: Tuesday, March 21
- Connected and umbilic -> contained in sphere or plane.
- Relation between Gaussian curvature and area of Gauss map image.
- Review for Exam.
- Thursday, March 23
- Exam I.
- Lecture 13: April 4
- Variation of area functional.
- Minimal surfaces.
- The catenoid.
- Lecture 14: April 6
- The helicoid.
- Isometries and local isometries.
- Catenoid and Helicoid are locally isometric.
- Lecture 15: April 11
- Locally conformal maps.
- The map J : T_p S -> T_p S.
- Cauchy-Riemann equations.
- Lecture 16: April 13
- More on Cauchy-Riemann equations.
- The gradient.
- The Laplacian
- Existence of isothermal coordinates.
- Patriots Day, no class on April 18
- Lecture 17: April 20
- Tensor product of vector spaces.
- Exterior product of vector spaces.
- Wedge product.
- Lecture 18: April 25
- Dual spaces.
- Differential p-forms on surfaces.
- Exterior differential d.
- Lecture 19: April 27
- Integral of 2-forms on oriented surfaces.
- Surfaces with boundary.
- Stokes' Theorem for triangles.
- Hodge star-operator.
- Lecture 20: May 2
- Stokes' Theorem for compact oriented surfaces.
- Review of inner product on \Lambda^p(T^*S).
- More on Hodge star-operator.
- Divergence operator
- Laplacian on forms.
- Integration by parts formula on surfaces.
- Lecture 21: May 4
- Cone and plane are locally isometric.
- Gaussian curvature of a cone.
- Christoffel symbols.
- Gauss' Theorema Egregium.
- Lecture 22: May 9
- Parallel Transport
- Geodesics
- May 11
- Exam II.
- Lecture 23: May 16
- The Gauss-Bonnet Theorem (local version)
- Lecture 24: May 18
- Global version of Gauss-Bonnet Theorem.
- Applications of Gauss-Bonnet.