**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