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By S.S. Kutateladze

A.D. Alexandrov's contribution to the sphere of intrinsic geometry used to be unique and intensely influential. this article is a vintage that continues to be unsurpassed in its readability and scope. It provides his middle fabric, initially released in Russian in 1948, starting wth an overview of the most techniques after which exploring different issues, reminiscent of common propositions on an intrinsic metric; angles and curvature; lifestyles of a convex polyhedron with prescribed metric; curves on convex surfaces; and the position of particular curvature. this article offers Adefinitive resource for the improvement of intrinsic geometry and is essential for graduate scholars who desire a greater realizing of this topic.

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Consider two plane triangles Oi Xi Yi (i = 1, 2) with sides Oi Xi = xi , Oi Yi = yi , and Xi Yi = zi = z(xi , yi ). The angles at their vertices O1 and O2 are γ1 = γ(x1 , y1 ) and γ2 = γ(x2 , y2 ); by the convexity condition, γ(x1 , y1 ) ≥ γ(x2 , y2 ).

For example, take a cone of revolution together with its base. Let O be the © 2006 by Taylor & Francis Group, LLC 7. Basic Concepts of Intrinsic Geometry 33 vertex of this cone, and let A be a point of the base circle. There are no shortest arcs on this cone tangent to the base circle; the fact that there are no such arcs on the base is obvious, and we confirm that they are absent on the lateral surface by unfolding the latter on the plane. Therefore, there is no shortest arc making the right angle with OA at point A.

The greatest lower bound of these sums over all possible partitions of the closed interval [0, 1] is called the length of the curve X(t). If this greatest lower bound does not exist or, which is the same, equals infinity, then this curve has no length or, say, has infinite length. This definition of length literally repeats the definition of length accepted in Euclidean geometry. ) Also, it is easy to prove that the so defined length has all properties of the usual length in Euclidean space; this will be done in Sec.

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