A.D. Alexandrov is considered by many to be the father of intrinsic geometry, second only to Gauss in surface theory. That appraisal stems primarily from this masterpiece--now available in its entirely for the first time since its 1948 publication in Russian. Alexandrov's treatise begins with an outline of the basic concepts, definitions, and results relevant to intrinsic geometry. It reviews the general theory, then presents the requisite general theorems on rectifiable curves and curves of minimum length. Proof of some of the general properties of the intrinsic metric of convex surfaces follows. The study then splits into two almost independent lines: further exploration of the intrinsic geometry of convex surfaces and proof of the existence of a surface with a given metric. The final chapter reviews the generalization of the whole theory to convex surfaces in the Lobachevskii space and in the spherical space, concluding with an outline of the theory of nonconvex surfaces.
Alexandrov's work was both original and extremely influential. This book gave rise to studying surfaces "in the large," rejecting the limitations of smoothness, and reviving the style of Euclid. Progress in geometry in recent decades correlates with the resurrection of the synthetic methods of geometry and brings the ideas of Alexandrov once again into focus. This text is a classic that remains unsurpassed in its clarity and scope.
BASIC CONCEPTS AND RESULTS The General Concept of Intrinsic Geometry and Its Problems Gaussian Intrinsic Geometry A Polyhedral Metric Development Passage from Polyhedra to Arbitrary Surfaces A Manifold with Intrinsic Metric Basic Concepts of Intrinsic Geometry Curvature Characteristic Properties of the Intrinsic Metric of a Convex Surface Some Special Features of Intrinsic Geometry Theorems of Intrinsic Geometry of Convex Surfaces General Propositions about the Intrinsic Metric General Theorems on Rectifiable Curves General Theorems on Shortest Arcs Nonoverlapping Condition for Shortest Arcs A Convex Neighborhood General Properties of Convex Domains Triangulation CHARACTERISTIC PROPERTIES OF THE INTRINSIC METRIC Convergence of Metrics of Convergent Convex Surfaces Convexity Condition for a Polyhedral Metric Convexity Condition for the Metric of a Convex Surface Consequences of the Convexity Condition ANGLE General Theorems on Addition of Angles Theorems on Addition of Angles on Convex Surfaces The Angle of a Sector Bounded by Shortest Arcs On Convergence of Angles The Tangent Cone The Spatial Meaning of the Angle Between Shortest Arcs CURVATURE Intrinsic Curvature Area of the Spherical Image Generalization of the Gauss Theorem Curvature of Borel Sets Set of Directions in Which It Is Impossible to Draw a Shortest Arc Curvature as a Measure of Non-Euclidicity of Space EXISTENCE OF A CONVEX POLYHEDRON WITH A GIVEN METRIC On Determination of a Metric from a Development The Idea of the Proof of the Realization Theorem Small Deformations of a Polyhedron Deformation of a Convex Polyhedral Angle Rigidity Theorem Realizability of the Metrics That Are Close to the Realized Metrics Smooth Passage from a Given Metric to a Realizable Metric Proof of the Realization Theorem EXISTENCE OF A CLOSED CONVEX SURFACE WITH A GIVEN METRIC The Result and the Method of Proof The Main Lemma on Convex Triangles Consequences of the Main Lemma on Convex Triangles The Complete Angle at a Point Curvature and Two Related Estimates Approximation of a Metric of Positive Curvature Realization of a Metric of Positive Curvature Given on the Sphere OTHER EXISTENCE THEOREMS Glueing Theorem Application of the Glueing Theorem to the Realization Theorems Realizability of a Complete Metric of Positive Curvature Manifolds on Which a Metric of Positive Curvature Can Be Given Uniqueness of a Convex Surface Various Definitions of a Metric of Positive Curvature CURVES ON CONVEX SURFACES The Direction of a Curve The Swerve of a Curve General Glueing Theorem Convex Domains Quasigeodesics A Circle AREA The Intrinsic Definition of Area The Extrinsic-Geometric Meaning of Area Extremal Properties of Pyramids and Cones THE ROLE OF SPECIFIC CURVATURE Intrinsic Geometry of a Surface Intrinsic Geometry of a Surface of Bounded Specific Curvature Shape of a Convex Surface in Dependence on Its Curvature GENERALIZATION Convex Surfaces in Spaces of Constant Curvature Realization Theorems in Spaces of Constant Curvature Surfaces of Indefinite Curvature BASICS OF CONVEX BODIES Convex Domains and Curves Convex Bodies. A Supporting Plane A Convex Cone Topological Types of Convex Bodies A Convex Polyhedron and the Convex Hull On Convergence of Convex Surfaces