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

Summary

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.

Table of Contents

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

Reviews

“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