Lecture Notes for 120 - UCLA Department of Mathematics

7.5. ISOMETRIES 163
Basic examples of isometries are rotations around the z axis for surfaces of
revolution around the z axis, or mirror symmetries in meridians on a surface of
revolution. The sphere has an even larger number of isometries as it is a surface of
revolution around any line through the origin. The plane also has rotational and
mirror symmetries, but in addition translations.
Some other examples we have seen that are less obvious come from geodesic
coordinates. There we saw that if we select geodesic coordinates around a geodesic
in a space of constant Gauss curvature K, then we always get the same answer.
This means that locally any space of constant Gauss curvature must look the same
everywhere, and even that any two spaces of the same constant Gauss curvature
are locally the same.
It is possible to construct isometries that do not preserve the second fundamental
form. The simplest example is to image a flat tarp or blanket, here all points
have vanishing second fundamental form and also there are isometries between all
points. Now lift one side of the tarp. Part of it will still be flat on the ground,
while the part that’s lifted off the ground is curved. The first fundamental form
has not changed but the curved part will now have nonzero entries in the second
fundamental form.
It is not always possible to directly determine all isometries. But as with
geodesics there are some uniqueness results that will help.
Theorem 7.5.2. If F and G are isometries that satisfy F (p) = G (p) and
DF (p) =DG (p) then F = G in a neighborhood of p.
Proof. We just saw that isometries preserve geodesics. So if q (t) is a geodesic
with q (0) = p, then F (q (t)) and G (q (t)) are both geodesics. Moreover they have
the same initial values
F (q (0)) = F (p) ,
G (q (0)) = G (p) ,
d
dt F (q (t)) | t=0 = DF ( ˙q (0)) ,
d
dt G (q (t)) | t=0 = DG ( ˙q (0)) .
This means that F (q (t)) = G (q (t)) . By varying the initial velocity of ˙q (0) we
can reach all points in a neighborhood of p.
⇤
Often the best method for finding isometries is to make educated guesses based
on what the metric looks like. One general guide line for creating isometries is
the observation that if the first fundamental form doesn’t depend on a specific
variable such as v, then translations in that variable will generate isometries. This
is exemplified by surfaces of revolution where the metric doesn’t depend on µ.
Translations in µ are the same as rotations by a fixed angle and we know that such
transformations are isometries. Note that reflections in such a parameter where v
is mapped to v 0 v will also be isometries in such a case.
Here is a slightly more surprising relationship between geodesics and isometries.
Theorem 7.5.3. Let F be a nontrivial isometry and q (t) aunitspeedcurve
such that F (q (t)) = q (t) for all t, then q (t) is a geodesic.