Basic differential geometry as a sequence of interesting problems

TL;DR

This book introduces differential geometry through a sequence of interesting problems, highlighting how curvature notions emerge from solving geometric problems related to applications, accessible to students and mathematicians alike.

Contribution

It presents elementary definitions of curvature notions within a problem-based expository approach, bridging geometric concepts with practical problem solving.

Findings

Different notions of curvature naturally arise from geometric problem solutions.

Elementary definitions of curvature are accessible to students with multivariable analysis background.

The problem-based presentation makes complex geometric ideas approachable for a broad audience.

Abstract

This book is expository and is in Russian (sample English translation of two pages is given). It is shown how in the course of solution of interesting geometric problems (close to applications) naturally appear different notions of curvature, which distinguish given geometry from the 'ordinary' one. Direct elementary definitions of these notions are presented. The book is accessible for students familiar with analysis of several variables, and could be an interesting easy reading for professional mathematicians. The material is presented as a sequence of problems, which is peculiar not only to Zen monasteries but also to serious mathematical education.