Metric Affine Geometry by Ernst Snapper

By Ernst Snapper

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Since dark­ ness dilates the pupils of our eyes and does not dilatate them, we see no reason for the extra " t a . " Until we have discussed dilations of lines, it is understood that n ^ 2 when­ ever we mention dilations. The identity mapping of X is clearly a dilation. However, there are also interesting dilations. 1. Every translation is a dilation. Proof. 1). If S(x,U)is an affine subspace of X and T is a translation, then T(S(x,U)) = S(Tx,U) whence S || T(S). So, in particular, if S is a line, 51| T(S).

Let x0, Al9 . . , An be a coordinate system for X. Suppose that the hyperplanes S(x, U) and S(y, W)of Xhave, respectively, the equations Z1OL1 + -" + znocn = a and z1ß1 + · · · + znßn = b. Prove that S(x, U) = S(y> W) if and only if there is a nonzero scalar y e k such that ßt = o^y for / = 1, . . , n and b = ay. In short, the equation of a hyperplane is unique except for a nonzero right factor in k. In case k is a field, the coefficients a l5 . . , a„ in the equation zlal + · · · + znocn = a 24 1.

Corresponding to an affine space, we have the group of the dilations and the group of the translations. Moreover, to each dilation we can associate an element of the multiplicative group k*, namely, the dilation ratio of the given dilation. We are now ready to see how these groups are interrelated. 1. The mapping/: Di->k* which associates with each dilation its dilation ratio is a group homomorphism from Di onto k*. 52 1. AFFINE GEOMETRY Proof First we show t h a t / i s a group homomorphism. Let Du D2e Di have dilation ratios rx and r 2 , respectively.

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