By Ralph Stocker, Heiner Zieschang

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**Example text**

Before beginning with the study of rigid transformations in 3 , let us remember that, since in 3 we have a way of measuring the distance between a ¯ and ¯b, the isomorphism between the additive group 2 and the group of translations of the plane (under composition) allows us to speak of “nearby” translations, or more precisely of the neighborhood of radius of a translation Ta¯ : it is formed by all the translations T¯b such that ||¯ a − ¯b|| < . An analogous property holds for the group of rotations around the origin: we can identify the rotation by an angle θ with the point of the unit circle in 2 : Ê Ê 1 S = {(x, y) ∈ Ê |x 2 2 Ê 2 + y = 1} determined by the oriented line through the origin that forms an angle θ (measured counter-clock-wise) with the the positive X axis.

Is − {0} a group with respect to multiplication? 2. Prove that GL(2, Ê), the set of 2 × 2 matrices with real entries and determinant diﬀerent from zero, forms a group under multiplication called the general linear group of order 2. Is this group commutative? 3. 4) is commutative. 4. Prove that a reﬂection on the plane with respect to a straight line through the origin is a rigid transformation. 5. Prove that the inverse transformation of a reﬂection of the plane with respect to one line through the origin is that same reﬂection.

18. There are as many normal sections as diameters in a small circle centered at P0 of the tangent plane, in other words, one for each θ ∈ [0, π]. Therefore, the curvature of a normal section, which is called sectional curvature, can be seen as a function of the angle θ, k(θ). A very important result in calculus establishes that a continuous function deﬁned in a closed interval takes its maximum and its minimum. Therefore, for some θ1 , θ2 ∈ [0, π], we have k1 = k(θ1 ) as the maximal curvature and k2 = k(θ2 ) as the minimal curvature of the normal sections.