A. D. Alexandrov: Selected Works, Part II. Intrinsic Geometry of Convex Surfaces

After Carl Friedrich Gauss, Aleksandr Danilovich Alexandrov is considered the second most important contributor to surface theory. He laid the foundation of non-smooth intrinsic geometry. The second volume of his selected works presents a thorough introduction to the intrinsic theory of convex surfaces and is comprehensible to non-specialists. The key notion here is that of an intrinsic metric; given a metric space (in the topological sense), its metric is called intrinsic if the distance of any two points equals the infimum of lengths of curves joining them. One of the main results of the book is the following “realizability” theorem; a metric space homeomorphic to a two-dimensional sphere is isometric to a convex surface (in R3) if and only if the metric is intrinsic and has positive curvature (i.e. locally, the sum of interior angles of any geodesic triangle is at least π). Further, the Gauss-Bonnet formula is derived for non-smooth surfaces of positive curvature; the curvature (measure) is defined via the spherical map. All notions used are very natural and have been used and developed in non-smooth geometry over the last few decades.

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