### Dr. Sarah's Differential Geometry Class Highlights Fall 2006 Page The following is NOT HOMEWORK unless you miss part or all of the class. See the main class web page for ALL homework and due dates.

• Tues Dec 5 Go over test 3. Einstein's field equations continued. Evaluations.

• Tues Nov 28 Discuss geodesics on the cone. Mention p. 262 of Oprea on numerical issues for the torus. Mention p. 269 about the cylinder uncovering, and the application to plastic wrap production on p. 282. Revisit spacetime via the Wormhole metric. Revisit spacetime via the geometry of Minkowski space (Lorentzian/Special Relativity) and showing that free particles follow straight line geodesics, and discuss the geometry of general relativity and begin Einstein's field equations. Hand out review sheet and take questions on the test.

• Thur Nov 30 Test 3
• Tues Nov 21 Presentations from Thursday wrap up. Discuss p. 134-136 that the Gauss curvature only depends on the metric, and why the geodesic equations yield that the geodesic curvature is 0 (p. 230).
• Tues Nov 14 Finish the Geodesic worksheet.

• Thur Nov 16 Meet in the computer lab. Highlight the methods used on p. 231-232 to compute the geodesics on the sphere. Discuss the relationship between the Christoffel symbols and E, F, G. Discuss the 2nd fundamental form and Gauss curvature. Look over Maple worksheet on Gauss curvature and the sphere, helicoid, and catenoid. Discuss the cylinder. In groups of 2, students prepare a short presentation:
1) Parametrization used for the Maple worksheet
2) Sketch a picture of their surface on the white board
3) Give the Gauss curvature
4) Discuss instrinsic Gauss curvature arguments (positive, negative, or zero)
5) Specify if the mean curvature is 0 or not
Problems: p. 123-126 3.2.11 (hyperboloid of 2 sheets), 3.2.12 (hyperboloid of 1 sheet), 3.2.13 (elliptic paraboloid), 3.2.14 (hyperbolic paraboloid), 3.2.16 (saddle), 3.2.17 (Kuen's Surface ), 3.2.19 (Cone), p. 130-132 3.3.3 (torus), 3.3.11 (pseudosphere), p. 86 2.2.4 (Mobius strip).
• Tues Nov 7 Presentations

• Thur Nov 9 Presentations
• Tues Oct 31 Continue with Applications of the first fundamental form via local isometries (catenoid and helicoid via E,F,G and an .avi deformation). Go over test 2.

• Thur Nov 1 Continue with applications of the first fundamental form to surface area calculations using the determinant of the metric form for the sphere, the cone, the strake, and the hyperbolic annulus. If time remains, begin the geodesic worksheet: Christoffel symbols the geodesic equation, Euler-Lagrange equations, and curvature.
• Tues Oct 24 Take questions on test 2. Continue with the first fundamental form.

• Thur Oct 26 Test 2.
• Tues Oct 17 Revisit surfaces and regularity. First fundamental form.
• Tues Oct 10 Revisit the cylinder and discuss the curvature vector and curves on the cylinder. Geodesic and normal curvature and the relationship to geodesics from an intrinsic point of view. Maple file on geodesic and normal curvatures adapted from David Henderson. Discuss the sphere and the helicoid.

• Thur Oct 12 Lab: Maple file on geodesic and normal curvatures adapted from David Henderson. Class work and links. Discuss the problem set.
• Tues Oct 3 Finish cylinders via coordinate systems. Revisit the strake and the helicoid. Begin the sphere and geodesics from an intrinsic point of view.

• Thur Oct 5 Finish the sphere. Begin hyperbolic geometry from an intrinsic point of view and show that distance is exponential.
• Tues Sep 26 Go over test 1. Begin surfaces by revisiting the strake and the cylinder from an intrinsic viewpoint.

• Thur Sep 28 Continue surfaces and geodesics from an intrinsic viewpoint via the cylinder.
• Tues Sep 19 Finish applications of the Frenet equations. Review.

• Thur Sep 21 Test 1
• Tues Sep 12 Continue deriving the Frenet equations.

• Thur Sep 14 Revisit torsion. Finish deriving the Frenet equations. Discuss the fundamental theorem of curves. Applications of the Frenet equations.
• Tues Sep 5 Revisit curvature and the strake problem, and begin deriving the Frenet equations.

• Thur Sep 7 Finish strake problem. Discuss the calculation of curvature when parametrizing by arc length is impractical. Discuss and prove the formula for curvature for twice-differentiable function of one variable.
• Tues Aug 29 Students share something from Chapter 1. Take questions. Finish section 1.1. Grad student presentations on arc length and Frenet frames.

Thur Aug 31 If necessary, finish presentation on Frenet frames. Time to work on Project 2 in the computer lab 205 with Dr. Rhoads.
• Tue Aug 22 Intro to course. History and Applications of Differential Geometry ( powerpoint). Fill out information sheet. Introductions. If time remains, begin working on Calc 3 review problems.

• Thur Aug 24 Take questions on the syllabus. Students present calc 3 review. Begin section 1.1.