Lecture 7 47

Examples of symmetric spaces are given by Rn, Sn, and RP n,

as well as by their direct products, about which we will say more

momentarily. First, notice that the flat torus is symmetric, being the

direct product of two symmetric spaces

S1.

However, the embedding

of the torus into

R3

produces a space which is not symmetric, since

the isometry group does not act transitively on the points of the sur-

face. In fact, the isometry group of the embedded torus of revolution

(the bagel) in

R3

is a finite extension of a one-dimensional group of

rotations, while the isometry group of the flat torus is, as we saw last

time, a finite extension of a two-dimensional group of translations.

Hence the two surfaces are homeomorphic but not isometric.

The flat torus

R2/Z2

has no isometric embedding into

R3,

but it

is isometric to the embedded torus in

R4

given as the zero set of the

two equations

x1

2

+ x2

2

= 1,

x3

2

+ x4

2

= 1.

c. Remarks concerning direct products. Given any two sets X

and Y , we can define their direct product, sometimes called the Carte-

sian product, as the set of all ordered pairs (x, y):

X × Y = { (x, y) | x ∈ X, y ∈ Y }.

It is very often the case that if X and Y carry an extra structure,

such as that of a group, a topological space, or a metric space, then

this structure can be carried over to the direct product in a natural

way. For example, the direct product of two groups is a group under

pointwise multiplication, and the direct product of two topological

spaces is a topological space in the product topology.

If X and Y carry metrics dX and dY , then we can put a metric

on X × Y in the same manner as we put a metric on

R2,

by defining

d((x, y), (x , y )) = dX (x, x )2 + dY (y, y )2.

If there are geodesics on X and Y , we can define geodesics on X × Y ,

and hence can define the geodesic flip, which can be shown to satisfy

the formula

I(x,y)(x , y ) = (Ix(x ), Iy(y )).