1. Differentiable Manifolds 15
Examples and Exercises
1.20. Discuss the following submanifolds of E
n
; in particular make drawings
of them:
The unit sphere
S71'1
= { x G l
n
: (x, x) = 1} C
W1.
2
The ellipsoid {x G
Rn
: f(x) := Y^i=i ^ ~ 1}
a*
^ ^'
w
^ ^ principal axis
ai,.. . , an.
The hyperboloid {x e
W1
: f(x) := E l U ^ S = !} ^ =
±]L a*
^ °
w i t h
principal axis a; and index — J ] ^ .
The saddle {x G
M3
: xs = X1X2}.
The foras: the rotation surface generated by rotation of (y —
R)2
+
z2
—
r2^
0 r R, with center the z-axis, i.e.,
{(x,y,z):(y/^T^-R)2
+
z2=r2}.
1.21. A compact surface of genus g. Let f(x) := x(x —
l)2(x
—
2)2
...{x - (g -
l))2(x
- g). For small r 0 the set {(x,y,z) :
(y2
+
f(x))2
+
z2
=
r2}
describes a surface of genus g (topologically a sphere with
g handles) in
IR3.
Visualize this:
1.22. The Moebius strip. It is not the set of zeros of a regular function
on an open neighborhood of
Rn.
Why not? But it may be represented by
the following parameterization:
(cos
ip(R + r cos((p/2))
sin ip(R + r cos((p/2))
rsm((p/2)
(r,y)€ (-1,1) x [0,2*0,
where R is quite big.
1.23. Describe an atlas for the real projective plane which consists of three
charts (homogeneous coordinates) and compute the chart changings.
Then describe an atlas for the n-dimensional real projective space
Pn(M)
and compute the chart changes.
