Appendix B

Topological transformation
groups

This section summarises the theory of topological transformation groups that

is relevant for this thesis. The discussion uses concepts from from point set
topology (explained in Appendix A). For more elaborate discussions of topo-
logical transformation groups can be found in the books by Montgomery and
Zippin [97] and Bredon [32].
A group is a set G so that for each x and y in G there is a unique product
xy in G satisfying the following axioms:

1. There is a unique element eG in G such that xeG = x = eG x for all x in

G.

2. For each x in G there is an x−1 in G such that xx−1 = eG = x−1 x.

3. For all x, y and z in G, x(yz) = (xy)z.

A group G that is also a topological space is called a topological group if

x 7→ x−1 is a continuous function from G to G and (x, y) 7→ xy is a continuous
function from G × G to G.
If G is a group, then a subset H of G is a subgroup of G if H is itself a
group (using the same product). The subset H is a subgroup of G if and only
if xy −1 belongs to H for all x and y in H. A subset H of a If G is a topological
group, and H is a subgroup of G, then H is a topological group, where H has
the subspace topology.
Each element g of a group G defines two functions from G onto G. The left
translation by g is the function h 7→ gh from G to itself. The right translation

153
154 APPENDIX B. TOPOLOGICAL TRANSFORMATION GROUPS

by g is the function h 7→ hg −1 from G to itself. In the case of a topological

group, both the left and right translations are continuous functions.
Let G be a group, let X be a set, and let f be a function from G × X to X
whose values are denoted by g(x) = f (g, x). Then, G is a transformation group
on X if it satisfies the following axioms:
1. For each x in X, eG (x) = x.
2. For each g and each h in G, (gh)(x) = g(h(x)).
In this case, for each x in X, the function x 7→ g(x) is a bijection from X onto
itself.
Let G be a topological group that is a transformation group on a topological
space X. Then, G is is a topological transformation group on X if (g, x) 7→
g(x) is a continuous function from G × X to X. In this case, for each x in
X, the function x 7→ g(x) is a homeomorphism from X onto itself, Below,
several topological transformation groups that are important in this thesis are
discussed.
In this thesis, the focus lies mostly on transformation groups that consist of
differentiable functions from Rk onto itself. Concepts are used that are related
to the total derivative of a function in a point [51]. A function f from Rk onto
itself is called differentiable in a point x ∈ Rk if there exists a linear function l
from Rk onto itself such that
lim kf (x + v) − f (x) − l(v)k/kvk = 0 (B.1)
v→o,v6=o

where o is the origin of Rk , and v ranges over Rk . If it exists, the linear function
l is uniquely defined, and is called the total derivative of f in x, denoted by
Df (x). A function f is called differentiable if it is differentiable in each point
of its domain. Expressed in the standard basis of Rk , the total derivative can
be expressed as a k × k matrix of partial derivatives of coordinate functions of
f , denoted by ∂j fi . The determinant of a linear function l from Rk onto itself
is denoted by det (l). The Jacobi determinant of a function f : Rk → Rk in a
point x ∈ Rk is the determinant of the total derivative of f in x.
The group of diffeomorphisms in Rk , denoted by Dif k , consists of all home-
omorphisms g on Rk onto Rk such that both g and g −1 have total derivatives
that are continuous as functions of Rk . The subgroup CDif k of Dif k consists
of all diffeomorphisms for which the Jacobi determinant is constant. These
transformations preserve the ratio of volumes of each two sets. The subgroup
UDif k of CDif k consists of all diffeomorphisms for which the absolute value of
the Jacobi determinant equals 1. These transformations are volume preserving.
The affine transformations Af k from a subgroup of Dif k , consisting of all
diffeomorphisms from Rk onto itself that can be written as
x 7→ (Lx) + t, (B.2)
 155

where L is a k × k matrix over R having a nonzero determinant and t is an

element of Rk . The matrix L represents a linear transformation, the vector
t represents a translation. The affine transformations are a proper subgroup
of CDif k . The affine transformations map simplices to simplices. Several sub-
groups of the affine transformations are discussed below.
The volume preserving affine transformations, denoted with UAf k are those
affine transformations that can be expressed as in Equation B.2, where L is a
matrix whose determinant has absolute value 1. The volume-preserving affine
transformations are a proper subgroup of UDif k .
The following affine subgroup has no official name, therefore it is named
here: the stretch transformations, denoted by Stretk , are affine transforma-
tions for which the matrix L in Equation B.2 is restricted to be a diagonal
matrix, that is, a matrix in which elements off the diagonal are zero. The
transformations map k-dimensional intervals onto k-dimensional intervals.
The group of homotheties (or homothetic transformations), denoted Thetk ,
is a subgroup of the stretch transformations. It consists of all affine transfor-
mations as in Equation B.2, where L is a diagonal matrix in which all nonzero
elements have the same value. Such matrices represent uniform scaling. Under
each homothety, the image of each line is parallel to the original line.
The similarity transformations, denoted Simk , consist of all affine trans-
formations as in Equation B.2, where L is a scalar multiple of an orthogonal
matrix. These transformations preserve angles.
The (Euclidean) isometries in Rk , denoted by Isok , consist of affine transfor-
mations as in Equation B.2, where L is an orthogonal matrix, that is, LLT = I.
Here, LT denotes the transpose of the matrix L. The isometries preserve Eu-
clidean distances in Rk .
The translations Latk are all affine transformations that can be put in Equa-
tion B.2, where L is the identity. These transformations preserve vectors be-
tween points.
Finally, the identity group Idk (sometimes called trivial group) is the group
consisting only of the identity transformation x 7→ x from Rk to itself.
156 APPENDIX B. TOPOLOGICAL TRANSFORMATION GROUPS