### Dr. Sarah's Differential Geometry Class Highlights Spring 2009 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 Apr 21 Finish the solutions of the geodesic equations on the sphere by hand. Geodesics on the cone and the torus in Maple via demos from John Oprea and Robert Jantzen. The geometry of Minkowski space (Lorentzian/Special Relativity). Show that free particles follow straight line geodesics. Discuss the geometry of general relativity and Einstein's field equations.

• Thur Apr 23 Define the potential function and prove the Laplace equation and discuss Einstein's derivation of the Field equations from his 1916 paper. Wormhole metric. Discuss the final project.
• Thur Apr 16 Metric form presentations. Solving the geodesic equations on the sphere by hand.
• Tues Apr 7 Test 2 on Surfaces

• Thur Apr 9 Review test 2. Continue the derivation of the geodesic equations on a surface with the Christoffel symbols.
• Tues Mar 31 Continue geodesic equations and Christoffel symbols. Derive the Christoffel symbols on the sphere.

• Thur Apr 2 Take questions on surfaces for the test. Continue geodesic equations and Christoffel symbols. Show that geodesics have constant speed,and derive the geodesic equations with the Christoffel symbols in the case of surfaces.
• Tues Mar 24 Gauss Curvature and Mean Curvature of a flat torus and Klein bottle, and in hyperbolic geometry. Geodesic equations and Christoffel symbols.

• Thur Mar 26 Presentations. Continue geodesic equations and Christoffel symbols.
• Tues Mar 17 Applications of the first fundamental form to surface area calculations using the determinant of the metric form for the strake, the hyperbolic annulus, and the cone. Begin curvature. Normal Curvature 1, Normal Curvature 2, Quotations, Gauss Curvature and Mean Curvature, helicoid.

• Thur March 19 Review principal curvatures, Gauss curvature and mean curvatures. Normal Curvature 1 on a saddle, Discuss curvatures on a cylinder. Gauss map of a catenoid. Discuss Gauss map of a cylinder. Gauss Curvature and Mean Curvature. Assign project 5. Alone or in groups of 2, prepare a short presentation for a surface -- 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). Gauss map of a cylinder. Theorema Egregium of Gauss, that the (Gauss) curvature K of a surface is invariant under isometry. Gauss-Bonnet theorem.
• Tues Mar 3 Show that distance is exponential in the hyperbolic annulus model. Begin the first fundamental form and the metric. The plane and Pythagorean theorem. The helicoid.

• Thur Mar 5 Review the first fundamental form. Applications of the first fundamental form via local isometries (catenoid and helicoid via E,F,G) deformation. Examine a saddle and Enneper's surface and use E, F, G to distinguish them even though they look the same when plotted from u=-1/2..1/2, v=-1/2..1/2. Additional applications to surface area calculations using the determinant of the metric form for the sphere, and the strake.
• Tues Feb 24 Review regular curves. Discuss regular surface by a hyperboloid. Geodesics on a hyperboloid via symmetry arguments. Maple file on geodesic and normal curvatures adapted from David Henderson. Class commands for the file for cylinder, sphere, and cone.

• Thur Feb 26 Maple file on geodesic and normal curvatures Class commands for the file for the hyperboloid, cone and helicoid.. Introduction to hyperbolic geometry.
• Tues Feb 17 Go over the first 2 pages of the test. Define manifolds, orbifolds, surfaces, and geodesics. Continue with the cylinder. Use covering arguments to answer questions about the geodesics. Review extrinsic cylindrical coordinates and define geodesic rectangular coordinates.

• Thur Feb 19 Use local coordinates to express the equations of the geodesics on the cylinder. Clarify the relationship between curvature and geodesics. Normal to a surface and a regular patch/parametrization. Normal and geodesic curvature by hand.
• Tues Feb 10 Finish the isoparametric inequality. Discuss other results from the global differential geometry of curves. Review for test 1. If time remains, then begin surfaces.

• Thur Feb 12 Test 1 on curves.
• Tues Feb 3 Review the Frenet equations. Implications of the equations. Continue the geometry of helices and torsion/curvature constant condition. Prove that curvature 0 iff a line. Prove that torsion 0 iff planar.

• Thur Feb 5 Discuss that curvature constant for a plane curve means part of a circle. Discuss the fundamental theorem of curves. Discuss results from global differential geometry, including the isoparametric inequality.

• Tues Jan 27 Begin applications of the Frenet equations, including curvature and the 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.

• Thur Jan 29 Discuss the history of Frenet Frames. Review Frenet Frames. Continue deriving the Frenet equations. Begin the geometry of helices. Maple commands:
with(VectorCalculus):
helix:=<r*cos(t), r*sin(t), h*t>
TNBFrame(helix,t)
simplify(Curvature(helix,t))
simplify(Torsion(helix,t),trig)
• Tues Jan 20 Mention solutions to Calc 3 review on ASULearn. Prove that the shortest distance between two points is a line. Take questions on 1.1. Begin 1.2 and 1.3 on arc length and Frenet frames.

• Thur Jan 22 Begin deriving the Frenet equations.
• Tues Jan 13 Fill out information sheet. Introductions. Introduction to the course. Why is a line the shortest distance path between 2 points? Our intuition might be that a curve is inefficient since it starts off pointing away from the endpoint. However this intuition is false on a sphere. Prove that a line in R3 is the shorter than such a curve. If time remains, begin working on Calc 3 review problems.

• Thur Jan 15 Students present problems from calc 3 review.