MTG 6256: Differential Geometry I
Homework Assignments, Fall 2004


Last update made by D. Groisser Tue Dec 7 12:33:10 EST 2004

This page will be updated frequently, and students should check it at least every other day. You will get the most out of this course if you complete the homework assignment before the next class meeting, since I may use results or concepts from the homework in the next class. I realize that your other time commitments may sometimes prevent you from completing the homework within one or two days of its assignment, but try not to be more than one class behind, and never as much as a week behind.

Although only a subset of the homework will be collected and graded (probably a small subset), you will not learn the material without completing essentially 100% of the homework. Occasionally, I may assign difficult problems that not every student will be able to complete, but every student should attempt every problem.

Always read the corresponding section of the book, completely, before working the problems, even if I don't say so explicitly below. As you will discover, this is a very readable book that does not assume you know much math beyond Calculus 3 and Linear Algebra. At the beginning it may even strike you as too elementary, but this impression may change as we move further along.

Date assigned
Section # / problem #'s
M 8/23/04
Read the Introduction and Section 1.1. Look at (but don't do) the exercises. If they do not appear completely trivial to you, DROP THIS CLASS. ("Completely trivial" does not mean that you should be able to do the problems in your head; it means that the steps involved are obvious to you and that, with pencil and paper [and no technological aids] you could complete them mechanically, accurately, and quickly .)
W 8/25/04
Read Sections 1.2-1.3.
  • Look at the exercises for Section 1.2, and if they do not appear completely trivial to you, DROP THIS CLASS.

  • Do problems 1.3/{1 or 2}, 3-5. There are typographical errors in problems 1 and 5: in problem 1, the integral sign should be an f, and in problem 5, U1 should be Ui. Based on this, I think it's safe to assume that we will find a good number of typographical errors in this book.

F 8/27/04
  • For Monday, do the new version of non-book problem 1 as corrected 8/27/04. The only changes since the last version are in the definition of "Leibnizian" and in part (d), where you no longer are asked to show linearity (since it's now part of the definition).
  • Do problems 1.5/1c, 2-5, 6a, 7, 11a.

M 8/30/04
Although we have not yet covered in class the material on pp. 30-31 (which is needed for almost all the problems in Section 1.6) and will cover it on Wednesday, you should read these pages and work the problems below before next class anyway. It won't be harmful when I re-do on Wednesday some of what you've already read.
  • 1.6/1,2,3,4ac,5,6 (typographical error: "phi" should be "theta"),9. You may find the last sentence in #9 confusing. What the author means is: assume that wedge-product has not been previously defined, take the formula in this problem as the definition of df  ^ dg, treat his "dx   dy" just as a formal symbol, and from this new definition deduce that dxj ^ dxk = - dxk ^ dxj.

W 9/1/04
  • 1.6/8.

  • Read Section 1.7 (in class we have covered everything except Corollary 7.7 and from Definition 7.9 to the end of the section) and do problems 1.7/3, 5, 6, 7.

  • Read Note on handing in homework below.

W 9/8/04
  • 1.7/10

  • 2.1/ Look at exercises 1-11. Do any exercise containing a fact with which you are unfamiliar or that strikes you as nontrivial.
F 9/10/04
  • Finish reading Section 2.2 (pp. 54-55).

  • 2.2/ 1,2,6,8,10,11. What I called "Groisser's Theorem" in class I really should have called "Groisser's Conjecture". Problem 1 and the unassigned problems 3 and 4 provide evidence for this conjecture.

  • 2.2/5 is false as stated. Fix it and do the corrected problem.

  • 2.2/9: Required for Computer Science students, optional for everyone else, but it wouldn't hurt the math students to locate and use a numerical integrator.

M 9/13/04
  • 2.1/12. There is a typographical error in this problem; the upper limit of the integral should be t.

  • 2.3/5,7,8

  • Do non-book problem 2 as corrected 9/18/04.

W 9/15/04
  • Read Corollary 3.5 and Lemma 3.6.
  • 2.3/1,6,11
  • Do non-book problem 3. This is intended primarily as a computational problem: you should compute the matrix exponential, not look for a way to avoid computing it.
First collection of homework: On Wednesday, Sept. 22 (I changed the date because of a mistake in non-book problem 2--the omission of an orthogonal projection--which has now been corrected), I will collect the following homework problems: 1.5/5; 1.6/1, 4c, 6; 2.3/6; and non-book problems 1,2 (as corrected 9/18/04),3. The homework you hand in must be neat and easy to read. Please use 8.5''x 11''paper, not torn out from a spiral-bound notebook, and staple the sheets together. Please do not hand in anything that is messy, that has been erased and written over, that has shreds of paper dangling from it, or that has the potential to come apart when I turn pages.
F 9/17/04
No new homework was assigned. The reason was not that I wanted to give you a vacation, but that I thought it would be fun for you to do some of the problems in Section 2.4 involving computer graphics--specifically, 11c (which is assigned as part of Monday's homework) and 12, 13, 14 (not assigned, but instructive)--until I tried to do the graphics myself using Maple, and was unable to do the problems myself in time to decide whether to assign them. For those of you don't know Maple and want to learn it, there's an appendix in the textbook that gives an introduction. However, I found that O'Neill's Maple programs for the Frenet apparatus did not work well for me, either because I mis-typed something (or made some similar dumb mistake) or because Maple has changed so many times since the textbook was written. I eventually gave up on O'Neill's programs and wrote my own, which worked better for me but still were not up to the task handling certain curves that can be handled quickly and easily without a computer, such as the cycloid in problem 14. But undoubtedly you are all better programmers than I am, so you may be able to accomplish in minutes what took me hours, and may want to have some fun with these problems. (Note: if you choose to use Maple, in the current version you need to use the "spacecurve" program in the "plots" package in order to plot a curve in 3D.)

Remark on 2.4/14a: A point on a simple closed plane curve at which the curvature has a relative extremum is called a vertex. If you do this problem, you'll see why this name was chosen. It is obvious that every simple closed plane curve C has at least two vertices, since the curvature must achieve a maximum and a minimum value on the compact set C. Much less obvious is the Four-Vertex Theorem (which we will not prove), which asserts that every such curve has at least four vertices.


M 9/20/04
  • See note above concerning the homework for 9/17/04.
  • 2.4/3, 11, 16 (as corrected below), 18 (as corrected below), 19 (as modified below). 11(c) and 19(a) are optional if you have no interest at all in using the computer to do graphs. Note: in #11, "Ex. 3.6" means Exercise 2.3/6.
    • #16: Assume the number a in part (a) satisfies a2 < 2/3, and use the same interval (-a,a) in parts (b) and (c).
    • #18: (i) The definition given of convexity applies only to simple (i.e. non-self-intersecting) closed curves. (ii) Replace "2*pi" by "a positive multiple of 2*pi". To obtain this result, you don't need to assume that the curve is simple, just that the curvature is positive (or, less stringently, that the plane curvature--the one with the tilde over it--does not change sign). O'Neill's assertion that the total curvature of a convex simple closed curve is 2*pi is true, but I don't think it can be proven using only the tools we've developed in this course. All the proofs I know of require algebraic topology (specifically, the Jordan Curve Theorem), differential topology, or complex analysis (the relevant parts of which are differential topology in disguise); you need to make use of "intuitively obvious", but not so easy to prove, consequences of the curve being simple. If you come up with a proof that requires none of this, let me know! (iii) I'm not sure why O'Neill chose the hint he gave (for "spherical image", see the last paragraph on p. 71 and first paragraph on p. 72) instead of the simpler hint "Use problem 2.3/8b". Note: as you probably observed when you did 2.3/8, the kappa in part (b) should have a tilde over it.
    • #19: (i) Remark: a knot is a simple closed curve in R3 (this includes curves such as a circle that aren't knotted in the conventional sense; in knot theory these "un-knots" are considered to be special cases of knots). (ii) Remark: "tau" in this problem is just O'Neill's name for this curve; it has nothing to do with torsion. (iii) After doing part (a), also plot the projected curve given in part (b); you may find that this helps you interpret the picture you got in part (a). (iv) In part (b) you'll get to use the famous "tan(theta/2) substitution" that we (sometimes) teach in Calculus 2, which reduces any integral of a rational function of sine and cosine to a rational algebraic function. (At least, that's how I did the integral; maybe you will come up with another way.) (v) In part (c), in place of what O'Neill says to do, prove that the curve in part (b) can be deformed in R3 to a knot with total curvature arbitrarily close to 4*pi. (Hint: interpolate between the curve in part (a) and the curve in part (b). Note that you will have to show that your deformed curve is simple.) Optional: write a program to compute curvature (or use O'Neill's if it works for you), numerically compute the total curvature of the trefoil knot in (a), and compare the answer to 4*pi.
  • 2.5/1b, 2ab, 4.
  • 2.7/2.

W 9/22/04
  • Read the book's proof of the Second Structural Equation (I've decided just to let you read this instead of spending class time on it on Friday).
  • Do non-book problems 4 and 5.

F 9/24/04
3.2/3,4.
W 9/29/04
3.3/4,5. Sorry for not getting these posted till Thursday afternoon.
F 10/1/04
3.4/5.

Clarification of non-book problem 4. Part (c), in the sentence "Apply part (a) ..." what I meant to say was: First, figure out dy in terms of dx, then use this result, plus part (a), to write the triple with partial-with-respect-to-y's in term of the triple with partial-with-respect-to-x's.


M 10/4/04
3.5/3,7,8.
F 10/8/04
  • 4.1/4,5,6,8,9,10,11
  • 4.2/2, 5abc (part (d) optional), 6a (part (b) optional)

W 10/13/04
4.3/1, 3b (notation as on p. 149), 4, 6, 13, 14, 15. At first it may strike you that #14 contradicts the existence of non-proper patches, of which I showed (pictorially, not with a formula) an example in class. The reason there is no contradiction is that in #14 one has the additional assumption that the image of the patch lies in a smooth surface. The example given in class does not have this property.
F 10/15/04
  • No new homework.

  • Midterm. I am tentatively planning to hand out the take-home midterm on Fri. Oct. 29 and collect it on Mon. Nov. 1. These dates could change, but the earliest I can conceive of handing out (or making available) the exam is Tues. Oct. 26; the latest is Fri. Nov. 5. (The dates would be the same for everyone in the class.) Please let me know, as soon as possible, any potential problems with dates in this range.

    I have not yet started to prepare the exam, so I don't know how long it will be. As far as content goes, my general rule is that anything covered in class, in homework, or in the accompanying reading is fair game (including problems you have never seen before but that can be done by synthesizing material from class, homework, textbook, and prerequisite classes).


M 10/18/04
4.4/1, 3
W 10/20/04
4.5/1, 3, 4, 5, 13. In #3 there's very little to show, and it can be done in a line or two (but remember that "mapping" means smooth mapping); I'm assigning it because it's used in #4.

In #13, it is more conventional to say that G descends to a (smooth) map F from M to N, or that the formula in the book determines a well-defined function F, than to say that the formula is consistent; however, there is nothing wrong with O'Neill's terminology. The issue underlying this problem is that in general G may not be one-to-one, so that unless the consistency condition is met there will not be a unique map from the set M to the set N--here by "map" I just mean function in the set-theoretic sense, not in the differentiable (or even continuous) sense--such that {F composed with G}=F-tilde. In fact the consistency condition is equivalent to the uniqueness of such a set-map (existence of such a set-map is guaranteed because G is assumed to be onto). In essence, what you are showing in part (a) of this problem is that, given smooth maps G and F-tilde as in the problem (with G surjective), uniqueness of a set-map F from M to N such that {F composed with G}=F-tilde implies smoothness of that map.


F 10/22/04
  • No new homework.
  • Correction to statement in lecture. The following is true. Let M, N be surfaces, with M compact and N connected. If F is a one-to-one map from M to N whose derivative is an isomorphism at each point, then F is a diffeomorphism. In class, I omitted the compactness hypothesis, without which the statement is false. (For the non-math students: a subset of Rn is compact if and only if it is closed and bounded. That's not the definition of compactness; it's a theorem--the Heine-Borel Theorem--that makes the concept of compactness much easier to understand.) A counterexample if "M compact" is omitted: take M to be the open unit disk in R2, take N=R2, and take F to be the inclusion map. The mistake I made in class was the ridiculous statement that, under a continuous map, the image of a closed set is always closed. However, it is true that, under a continuous map, the image of a compact set is always compact, hence closed. The rest of the argument I gave is correct.

    Another true statement related to the one above is this: Let M, N be surfaces, and let F be a one-to-one map from M to N whose derivative is an isomorphism at each point. Then F is a diffeomorphism onto its image. (This is true whether or not M is compact and whether or not N is connected. However, the price you pay by weakening the hypotheses this way is that the conclusion is weaker; you cannot conclude that F is onto.)


M 10/25/04
  • 4.5/10c
  • 4.6/5, 9

F 10/29/04
For the problems below, assume Stokes' Theorem (which we'll prove Monday).
  • 4.6/3, 7 (easy corollary of facts proven in class).

M 11/1/04
  • 4.7/3, 8
  • Reminder about midterm exam: As announced in class, this take-home exam will be handed out on Wednesday 11/3/04 and will be due at the beginning of class on Monday 11/8/04. Warning: I will be ending class a few minutes early on Friday to get to the airport to attend a weekend conference. This means that you will have no opportunity to ask me questions about the midterm, not even by email, after Thursday mid-afternoon; consider 3:00 p.m. Thursday the cutoff time. Email about the exam received after that time will not be answered.
W 11/3/04
In view of the midterm, the due-date for the problems below is Wed. 11/10/04.
  • 5.1/2,4,5,7,9

(posted 11/4/04)
One more typographical error on midterm: in problem 3, equation (1), the "g" on the right-hand side should be "Q".
M 11/8/04
5.2/1
W 11/10/04
  • 5.3/1,3,7. Clarification of #7:
    • In the first line of part (a), note that the formula given for F*(v) expresses only the "vector" part of this tangent vector; the basepoint is F(p), not p. (S, of course, is the shape operator of M at p).

    • A related comment applies to the first displayed equation (the one involving cross-products) in part (a): the left-hand side of the equation gives only the vector part of a tangent vector in TF(p)R3, while the right-hand side is the vector part of a tangent vector based at p. (In the second displayed equation, H, K, and k1, k2 are, of course, the mean, Gaussian, and principal curvatures of M.)

    • A similar comment applies again to part (b): given a point p in M, (i) the statement about unit normals should really be that the vector part of the unit normal to \bar{M} (M with a bar over it) at F(p) equals the vector part of U at p, and (ii) in the equation relating the two shape operators, the left-hand side denotes the vector part of a tangent vector based at F(p), while the right-hand side denotes the vector part of a tangent vector based at p. If you do not take this into account, you will end up doing a formal calculation that makes no sense and yields the wrong answer.

    • In part (c) "M" should be "\bar{M}".

    • Also in part (c), in the formulas that you are asked to derive, "\bar{K}(F)" should be what would be written in LaTeX as $\bar{K}\circ F$, the composition of \bar{K} with F. A similar comment applies to "\bar{H}(F)".

  • 5.4/2,3,6,7

M 11/15/04
  • See clarification of 5.3/7 on previous assignment. Today's lecture should make this problem easier to do than it was when originally assigned.
  • 5.3/8 (note that this has parts (a) and (b); (b) is on the next page)
  • 5.4/12,13,15

W 11/17/04
  • Do non-book problems 6 and 7. (The version of #7 posted Wed. night was incomplete--my apologies--but I've now completed it and also made some minor modifications to wording and notation.)
  • 5.6/2 (see pp. 233-234 for "asymptotic curve), 3, 9. The situation in #9 is related to non-book problem 7c, case (i): the monkey saddle is preserved by an isometry of R3 of order 3 that fixes the origin.

W 11/24/04
  • 5.6/7
  • 6.1/1,2
  • 6.2/1

W 12/1/04 (not posted till 12/2/04)
  • 6.3/1,2,4,5. An additional hint for #2 and #5: Theorem 2.6, p. 262.

F 12/3/04
  • 6.3/3

M 12/6/04 (posted 12/7/04)
  • 6.4/1,2
  • 6.5/1,2,4
  • Read section 6.6 and do problems 6.6/1,2

Note on handing in homework.

When I'm ready to start collecting homework, I'll give you fair warning as to when I want it handed in; you do not have to worry that one day I'll make a surprise announcement that I'm collecting homework that day. The minimum warning I would give you is a little less than two calendar days (the time between when I post your assignment and the next class). This does not affect the due-dates for doing the homework; it only affects the dates by which I want you to neatly and carefully write up those problems that I tell you I'll be collecting. You should always do all the homework before the next class meeting after the date the homework is assigned, unless I say otherwise. When you do the problems, you should always write up the solutions neatly enough so that they serve as useful notes to you, and so that if and when I ask you to hand in a particular problem that you did several assignments ago, you don't have to work out the problem from scratch all over again. However, I am sparing you the time-consuming process of writing up every single problem neatly and clearly enough to be read by me.

Some students, especially those not in the math department, may be unsure of what level of detail I'll expect in handed-in problems, especially proofs. I recommend that within the next week or so, most of you choose one or more problems about which you have such a question, write it (or them) up in a manner that you think would be acceptable, and come to one of my office hours to ask me whether your write-up would have been acceptable to me. Non-math students, who may have not taken many "theorem-and-proof" courses before, may benefit from reading my handouts What is a proof? and Mathematical grammar and correct use of terminology. These handouts were written for undergraduates, so please do not be insulted by their level.


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