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.