Math 6230 Homework #7 Solutions
1. Suppose M is a two-dimensional manifold with coordinates x = (x, y) and u = (u, v),
and that : (, ) M is a curve expressed in coordinates as x (t) = x(t), y(t)
and u (t) = u(t), v(t) .
(a) Show directly using the Chain R
Math 6230 Homework #4 Solutions
1. In class we dened the torus to be the image T2 = F [R2 ] of F : R2 R3 given by
F (u, v) = (2 + cos u) cos v, (2 + cos u) sin v, sin u ,
or as the inverse image H 1 cfw_1 where H : R3 R is given by
x2 + y 2 2
H(x, y, z) =
Math 6230 Homework #2 Solutions
1. (a) Let V be a two-dimensional vector space, and consider the (2, 0) tensor g : V V
R given in a basis cfw_e1 , e2 by gij = g(ei , ej ) = ij . If a new basis cfw_f1 , f2 is given
by f1 = 3e1 4e2 , f2 = 2e1 + 3e2 , nd
Math 6230 Final Solutions
1. Let M be a smooth manifold and a 1-form on M .
(a) Show that if M is two-dimensional with d|p = 0 and p = 0, then there is a
coordinate chart (, U ) around p such that = y dx in terms of the coordinates
(x, y). (Hint: straight
Math 6230 Homework #1 Solutions
Read Introduction to Dierential Geometry up to Chapter 4.
1. Suppose you lived on a small sphere of radius R, so that spherical geometry was more
natural than planar geometry. What formulas would children be taught in schoo
Math 6230 Homework #6 Solutions
1. A simple choice of coordinates on S 2 is stereographic coordinates, often dened with
reference to the north and south poles: north pole coordinates are
(x, y, z) =
u2 + v 2 1
1 + u2 + v 2 1 + u2 + v 2 u + v 2
Math 6230 Midterm Solutions
1. Let n S 2 denote the north pole of the 2-sphere, and identify S 2 \n with the complex
plane C using stereographic coordinates.
(a) If f : C C is given by f (z) = z 3 , show that f extends to a smooth function
f : S 2 S 2 wit
Math 6230 Homework #8 Solutions
1. Suppose M is a smooth submanifold of a smooth manifold N in the sense of Denition
9.1.9, and let : M N be the inclusion. Let Y be a smooth vector eld on N such
that whenever p M we have Y (p) [Tp M ]. Prove that there is
Math 6230 Homework #12 Solutions
1. Suppose M is a compact orientable n-dimensional manifold and is an (n 1)-form
on M . Show that d must be zero somewhere on M .
Solution: Let be a volume form on M . Since every other n-form is a multiple of the
Math 6230 Homework #13 Solutions
1. Recall that the Klein bottle K is dieomorphic to the quotient of R2 by the action
of the discrete group generated by the transformations 1 (x, y) = (x, y + 1) and
2 (x, y) = (x + 1, y). Lets prove that H 2 (K) = cfw_0.
Math 6230 Homework #10 Solutions
1. For the 1-form on R4 given by
(w,x,y,z) = xy dw wz dx + y 2 dy zx dz,
Solution: Use the product rule and d2 = 0: we have
d = d(xy) dw d(wz) dx + d(y 2 ) dy d(zx) dz
= (x dy + y dx) dw (w dz + z dw) dx + 2y dy
Math 6230 Homework #3 Solutions
1. For the dierential equation
x2 + 1
x(0) = a,
nd the solution by separating the variables. For a given a, what is the largest interval
(, ) on which the solution is dened? What is the inmum of all such as
Math 6230 Homework #9 Solutions
1. Suppose X and Y are vector elds on M , with ows t and t respectively. Prove
[X, Y ]p =
t s t s (p).
t t=0 s s=0
Solution: By the Chain Rule, we have
t s t s (p) = (t )
s t s (p) .
Fix p and write (s) = s (p
Math 6230 Homework #11 Solutions
1. Let : R2 R3 be the dieomorphism (u, v) = (2v u2 , 3u, 4u + v 2 ), and let =
y dx dy z dz dx + x dx dy. Compute # .
Solution: We have x = 2v u2 , y = 3u, and z = 4u + v 2 , so that dx = 2u du + 2 dv,
dy = 3 du, and dz =
Math 6230 Homework #5 Solutions
1. Regular hexagons (with all interior angles equal to 120 ) tile the Euclidean plane E 2 .
Suppose we try to set up a group of isometries of E 2 for which the fundamental polygon
is a regular hexagon. What goes wrong?