# Prof. Mikhail Katz

Analytic and Differential geometry 88-201 (formerly 88-526)

Course notes/Choveret shel hakurs: Analytic and differential geometry course notes in .pdf format

Important information for students: Class participation is obligatory.

Targil grade is 10% of the final grade, and the final exam is 90%.

Old homework for the year '10: homework 0 , homework 1 , homework 2 , homework 3 , homework 4 , homework 5 , homework 6

More old homework: friedman1 , friedman2 , friedman3 , friedman4 , friedman5 , friedman6 , friedman7 , friedman8 ,

Suggestions for the metargel:

1. The mathwiki site for the course is here.

2. The bochan should be scheduled during shaot machlaka only. In '17 there was a problem with many students being unable to come to the bochan because of an exam in a different course. This creates an insoluble problem for assigning a grade for tirgul. The problem can only be solved in advance by only scheduling the bochan during shaot machlaka.

3. In july '17 many students reported that problems involving the coefficients of the Weingarten map and the second fundamental form were solved in targil using multiplication of matrices. This is a defective approach because passing from index notation to matrices involves loss of information, as I emphasized several times in the course. Students report that they tried doing it with matrices several times and often the answers come out different. Formulas in index notation should be used to solve such problems.

4. The metargel should devote the entire first targil to manipulations of indices following Einstein summation convention, including distinction between free index and summation index, and transformation and simplification of formulas involving indices. The Einstein notation is a technical tool that is the basis of the entire course, essential for proof of results such as the theorema egregium of Gauss. Experience shows that during the final exam many students are still having trouble doing such manipulations correctly. The exam in july '16 showed again that many students still can't do these manipulations properly inspite of our efforts.

5. Emphasize distinction between free indices and summation indices.

6. To construct surfaces of revolution, we use functions (r(φ), z(φ)) to parametrize the generating curve. After rotating around the z-axis, we list the parameters of the surface as (θ, φ) where θ is the polar angle. Therefore the matrix of the first fundamental form has the function g11=r2 in the upper left corner. This is the convention adopted in the choveret of the course. Switching the order as compared to the choveret could be confusing to the students. If the initial curve is arclength then g22=1.

7. Is not helpful, as has been sometimes done in targil, to denote the matrix (gij) by capital letter G , as is any additional notation related to the second fundamental form and the Weingarten map. Please refrain from introducing superfluous notation not found in the choveret. Avoiding index notation puts the students at a disadvantage during the exam where they are expected to be able to handle transformations of indices (see item 1 above).

8. the notation (θ, t) is an inappropriate notation for the pair of variables for surfaces of revolution, and should not be used. This is because the variable t is routinely used for the parametrisation of a curve. More precisely, the choveret denotes the parameter of an arbitrary regular curve by t, and the arclength parameter usually by s. Therefore using t also for surfaces of revolution can cause confusion, since we also treat curves ON surfaces of revolution. Therefore I would suggest sticking to (θ, φ) as parameters of a surface of revolution, as in the choveret.

9. A specific coefficient of the first fundamental form is denoted gij. The corresponding matrix is denoted (gij) namely with parentheses. Therefore there is no need for special notation for the 2 by 2 matrix of the metric.

10. New theoretical material should not be presented in targil. An example is the theorem that a totally umbilic surface is a portion of the plane or a sphere. This is an elegant result but the proof is time-consuming. The time can be better spent to treat examples such as finding the point of maximal curvature on a conic. The exam in july '16 showed that a majority of students are unable to solve simple problems like this one.