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1 Group Actions Preserving Structures
Weāve said a bit about group actions, but I havenāt explained the most important part: how any
structure on a manifold gives a group action!
Given a manifold Xwith some extra structure (for example: a Riemannian metric - a very nice
distance function), you can ask: which diffeomorphisms f:XāXpreserves this extra structure?
Such diffeomorphisms form a group! We get lots of interesting groups, including a lot of Lie groups.
Example: Take X=Rnand let the extra structure be the structure of the vector space. Then
f:RnāRnpreserves this extra structure iff:
ā¢f(0) = 0
ā¢f(x+y) = f(x) + f(y)
ā¢f(Ī±x) = Ī±f (x)
We say fis linear. We get a group of diffeomorphisms f:XāXpreserving the vector space
structure:
G= GL(n)
={f:RnāRn:fis linear and invertible}
Example: Let X=Rnbut with its vector space structure and its usual metric. The group
preserving this structure is
G= O(n)
={fāGL(n) : ||f(x)|| =||x||}
If we also demand that our transformations preserve an āorientationā on Rn, we get a smaller group
that excludes reflections: SO(n).
Example: Let X=Rnwith its usual metric. The group preserving this structure is the Euclidean
group
G=E(n)
= O(n)n Rn
In other words, any fāE(n) can be written as:
f(x) = Rx +v
for RāO(n), vāRn, but
ff0(x) = R(R0x+v0) + v=RR0x+Rv0+v
with f, f 0āE(n), instead of merely RR0x+v+v0which we would have in O(n)ĆRn.
We can turn this game around: given a group (e.g. a Lie group) Gacting on a manifold X
we can ask: what structure on Xdoes this group preserve? We have seen that the Galilei group
G(n+ 1) acts on RnĆR: it is the group generated by:
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