Hilbert’s fourth problem, which involves finding all geometries where “ordinary lines” are “geodesics,” is both accessible and profound. While the problem can be appreciated by beginning graduate students, its solution requires tools from various branches of mathematics, including geometry, analysis, and the calculus of variations.

A partial solution was provided by Georg Hamel in 1901. Later, A. V. Pogorelov, inspired by Herbert Busemann’s idea presented at the 1966 International Congress of Mathematicians in Moscow, offered an elegant and comprehensive solution. Pogorelov’s approach, which slightly reformulates Hilbert’s problem, is celebrated for its clarity and mathematical depth.

The book is well-written, introducing necessary mathematical concepts as needed, making it accessible to readers with a foundation in advanced calculus. The English translation, reviewed by Eugene Zaustinsky, includes helpful notes guiding readers to further literature.

Pogorelov’s work is a valuable contribution to the mathematical literature, particularly for those interested in geometry and its foundations.

You can get the book here and here.

INTRODUCTION 5

SECTIONS

1. Projective Space 9

2. Projective Transformations 13

3. Desarguesian Metrizations of Projective Space 19

4. Regular Desarguesian Metrics in the Two-Dimensional Case 24

5. Averaging Desarguesian Metrics 31

6. The Regular Approximation of Desarguesian Metrics 38

7. General Desarguesian Metrics in the Two-Dimensional Case 46

8. Funk’s Problem 54

9. Desarguesian Metrics in the Three-Dimensional Case 61

10. Axioms for the Classical Geometries 68

11. Statement of Hilbert’s Problem 75

12. Solution of Hilbert’s Problem 82

NOTES 88

BIBLIOGRAPHY 93

INDEX 95