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 )).
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