By Benz W.
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Extra resources for A Beckman Quarles Type Theorem for Plane Lorentz Transformations
The set P 2k of all pairs (x, y), where x ∈ M k , y ∈ M k , x = y, naturally forms a smooth manifold of dimension 2k. e. the ray of the vector f (y) − f (x) (see § 1, «Н»). Let e be an arbitrary non-zero vector from the space C 2k+1 and let πe be the projection along the one-dimensional space e∗∗ containing e. It turns out that the regular mapping πe f is typical for any selfintersection pair (see «A») if and only if the mapping σ from the manifold P 2k to the manifold S 2k is proper in the point e∗ ∈ S 2k .
Let δ be the distance √ k k between the sets E0k \ Ua1 and U a2 . Suppose that the diagonal length ε k of each cube from Ω is less than δ. Then each cube Ki from Ω lies in the k domain Ua1 and, by virtue of (1), the√ set ϕ(Ki ) is contained in some cube Li of the space E l with edge length c k · ε; the volume of the latter cube k equals cl k l/2 · εl−k . Thus the whole set ϕ(U a2 ) is contained in the union of cubes Li , whose number does not exceed c1 /εk ; thus the total volume 3rd April 2007 9:38 WSPC/Book Trim Size for 9in x 6in 14 L.
K, may serve for introducing in a neighbourhood of a the new coordinates ξ 1 , . . , ξ k of the point x. Furthermore, assuming ηj = yj , j = 1, . . , l, we see that in coordinates ξ 1 , . . , ξ k , η 1 , . . , η l the mapping f can be written as η j = ξ j , j = 1, . . , l. (2) Thus, if the manifold M k is closed and b ∈ N l is a proper point of the mapping f , then f −1 (b) is a smooth (k − l)-dimensional submanifold of the manifold M k with local coordinates ξ l+1 , . . , ξ k in the neighbourhood of a.