let $\displaystyle T^{2}=\mathcal{R}^{2}/2\pi\mathcal{Z}^{2}$
can we show that $\displaystyle T^{2}$ is compact?
i will describe it by words, you write down the expression. call the parameters in the domain a and b. call the corordinate of R^3 as x, y, z.
in the x-z plane draw a circle of radius 2. call this circle C. parametrize its points by a, which is to say that you choose a reference line segment radius. consider a plane perpendicular to x-z plane, passing through y-axis, such that its intersection with x-z plane makes an angle of a with the refernce radius. this plane rotates along the y-axis, the angle of rotation being a.
when you have rotated this plane by angle a, draw a circle of radius 1 around the point (2cos a, 0, 2sin a) in this plane. parametrize the points of this circle by b ...
try to visualize what i am saying ... it is really easy if seen pictotrially.
i think u can fill in the details. ... convince yourself that this map is smooth embedding.
about the y-axis. ok i will give you the final answer. try to work it out. and also if you have access to 3d plots, try to plot it. maybe gnuplot cud help you
$\displaystyle (a,b) \longmapsto (2(1/2 +\cos{b})\cos{a}, \sin{b}, 2(1/2+\cos{b})\sin{a})$
edit :: i am not very sure i have done the computations correctly. plz check it.
See the LaTeX help section of the forums (in particular, the tutorials).