By Herbert Busemann

Starting with a quick evaluate of notations and terminology, the textual content proceeds to convex curves, the theorems of Meusnier and Euler, extrinsic Gauss curvature, and the impact of the curvature at the neighborhood form of a floor. A bankruptcy at the Brunn-Minkowski conception and its purposes is by means of examinations of intrinsic metrics, the metrics of convex hypersurfaces, geodesics, angles, triangulations, and the Gauss-Bonnet theorem. the ultimate bankruptcy explores the stress of convex polyhedra, the conclusion of polyhedral metrics, Weyl's challenge, neighborhood recognition of metrics with non-negative curvature, open and closed surfaces, and smoothness of realizations.

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Material which can be found in elementary books should be gone over, and also a look forwards could be had to the results in this book. 10 Preliminaries Ch. 1 As part of the preliminary programme, we now include a review of the basic concepts of geometry. Geometry should be thought of as arising from an initial experimental and observational stage, where the ﬁgures are looked at and there is a great emphasis on a visual approach. 2 The basic shapes 1. The plane Π is a set, the elements of which are called points.

Degree-measure | |◦ of angles has the following properties:(i) In all cases |α |◦ ≥ 0; (ii) If α is a straight-angle, then |α |◦ = 180; (iii) If ∠BAC is a wedge-angle and the point D = A lies in the interior region I R(|BAC), then |∠BAD|◦ + |∠DAC|◦ = |∠BAC|◦ , while if |BAC is a straight angle-support and D ∈ AB, then |∠BAD|◦ + |∠DAC|◦ = 180; (iv) If B = A, if H1 is a closed half-plane with edge AB and if the half-lines [A,C and [A, D in H1 are such that |∠BAC|◦ = |∠BAD|◦ , then [A, D = [A,C ; (v) If B = A, if H1 is a closed half-plane with edge AB and if 0 < k < 180, then there is a half-line [A,C in H1 such that |∠BAC|◦ = k.

I) l ∩ G1 = 0, / (ii) G1 ∩ G2 = 0. (iii) If P ∈ G1 and [P, Q] ∩ l = 0/ where Q ∈ l, then Q ∈ G2 . (iv) Each line l determines a unique pair of open half-planes. Proof . / The (i) By A3 (i), l ∩ (G1 ∪ G2 ) = 0/ and as G1 ⊂ G1 ∪ G2 it follows that l ∩ G1 = 0. other assertion is proved similarly. (ii) If G1 ∩ G2 = 0, / there is some point R in both G1 and G2 . By A3 (iii) with P = R, Q = R, we have that [R, R] ∩ l = 0. / But R is the only point in [R, R] so R ∈ l. This / contradicts the fact that l ∩ G1 = 0.