By Artstein-Avidan S.

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**Extra resources for A Bernstein-Chernoff deviation inequality, and geometric properties of random families of operators**

**Sample text**

Since the metric on the cylinder and the torus that we shall construct come from the Euclidean metric in 2 , and the distances between “nearby points” in the torus and the cylinder are just the distances between the corresponding “nearest” Ê Ê Ê 38 Chapter 1. Euclidean geometry Ê points in 2 , one has that the geodesics in these surfaces are the image of the lines in 2 under the natural projection. When we were children, to construct a cylinder (rather, a portion of a cylinder) we used to take a rectangle made out of pasteboard and glue two of its parallel sides.

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. The curvature of a surface at one of its points, K(P ), was deﬁned by Leonhard Euler (1707–1783) as the product of the maximal and minimal curvatures K(P ) = k1 (P )k2 (P ). Notice that the value of K(P ) does not change even if we choose as normal vector −N (P0 ). At present, K(P ) is called Gaussian curvature of the surface at a point P , after Karl Friedrich Gauss (1777–1865), who identiﬁed this as one of the most important concepts in diﬀerential geometry (see [DoC]).

Just as the ﬁrst condition leads us to a vertical strip as fundamental domain, the second condition leads us to a fundamental region which is a square of side 1. This square is the intersection of a vertical strip as before, with a horizontal strip. 21). The reader can easily see that the resulting object is a torus S 1 × S 1 , although if he or she tries to construct it with a square made out of paper, he or she will confront the problem that when gluing the last two edges, the paper gets crumpled.