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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 topological 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 v→o,v6=o kf (x + v) − f (x) − l(v)k/kvk = 0 (B.1) 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 homeomorphisms 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 subgroups 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 transformations 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 transformations 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 transformations 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 transformations 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 Euclidean distances in Rk . The translations Latk are all affine transformations that can be put in Equation B.2, where L is the identity. These transformations preserve vectors between 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