Dunajski M. Geometry. A Very Short Introduction 2022

Textbook in PDF format

The study of geometry is at least 2500 years old, and it is within this field that the concept of mathematical proof - deductive reasoning from a set of axioms - first arose. To this day geometry remains a very active area of research in mathematics.
This book provides a fresh modern introduction to geometry, an ancient branch of mathematics with important applications. It takes readers from Euclidean and non-Euclidean geometries, to curved spaces, and the geometry of space-time inside a black hole, and outlines the role geometry plays in the broader context of science and art.
What is geometry?
Euclidean geometry
Non-Euclidean geometry
Geometry of curved spaces
Projective geometry
Other geometries
Geometry of the physical world