### Test 2 Study Guide

It is time for our first test in order to be sure that everyone reviews some of the fundamental concepts before we move on to geometry of space-time and applications to general relativity

This test will be closed to notes/books, but a calculator will be allowed. There will be three parts to the test.
Part 1: Fill in the blank/short answer
Part 2: Calculations and Interpretations
Part 3: Short Proofs

I suggest that you review your class notes, the class highlights page, including clicker questions, and go over ASULearn solutions to the projects, and the surfaces glossary.

Part 1: Fill in the blank/short answer There will be some short answer questions, such as providing:

• definitions and big picture ideas related to any of the topics in the glossary on surfaces (Wiki-like entries are on ASULearn for the terms), including the parametrization, shape and geometric properties of the 9 surfaces listed there (review class notes, homework and clickers for those).
• questions similar to previous clicker questions (see the class highlights page), where you fill in a blank rather than answer as a multiple choice.
• questions on material from class and homework solutions

Note: there is often more than one answer possible for fill in the blank questions: choose one response. Full credit responses demonstrate deep understanding of differential geometry. Informal responses are fine as long as they are correct.

Part 2: Calculations and Interpretations There will be some by-hand computations and interpretations
• Be able to compute Xu and Xv for a surface
• Be able to calculate the curvature vector dT/ds = T'(t)/speed, a normal to a surface Xu x Xv, the projection of the curvature onto the normal (the normal curvature), and the geodesic curvature vector (what is left over) for a curve on the plane or the cylinder (where the computations are quicker than they would be on other surfaces).
• Be able to interpret whether curves are geodesics via a given geodesic curvature as well as geometric/physical arguments relating to whether the curvature vector is completely in the normal direction or not. [For example, for a circle on a surface, we know the curvature vector of any circle points in to the center of the circle. Combine this with intuition about the normal to a surface to say whether the curvature vector is parallel to the normal and hence gives a geodesic, or not]
• Be able to compute E, F and G and interpret whether the Pythagorean theorem holds or not, or whether Xu and Xv are perpendicular.

Part 3: Short Proofs There will be some short proofs - the same as we've seen before. Review the following:

• Prove how E, F, and G and the metric equation arise from our usual definition of arc length along a curve, as on class slides
• Prove that a geodesic must be a constant speed curve as on class slides
• Prove that the determinant of the metric form gives the area of a flat Xu, Xv parallelogram as on class slides
• Assume that gamma and gamma'' have already shown to be parallel for a geodesic parametrized by arc length on the sphere. Prove that gamma had to be a great circle, as on the last bullet point of the last class slide
• In hyperbolic geometry models with radius r as the interior radius of the annuli of width delta, with delta approaching 0, prove that if two geodesics are d units apart along the base curve and we travel c units away from the base curve along each geodesic, then they are at a distance of d exp(-c/r). We filled in the details, after starting from here