1st Edition

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

Edited By S.S. Kutateladze Copyright 2006
    444 Pages 100 B/W Illustrations
    by Chapman & Hall

    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

    Biography

    S.S. Kutateladze

    “This classic is quite readable and opens a deeper understanding of this field also through self-study without any special prerequisites …”
    — H. Rindler, Wien, in Monatshefte für Mathematik, Vol. 149, No. 4, 2006