## Description

The link between the physical world and its visualisation is geometry. This easy-to-read, generously illustrated textbook presents an elementary introduction to differential geometry with emphasis on geometric results. Avoiding formalism as much as possible, the author harnesses basic mathematical skills in analysis and linear algebra to solve interesting geometric problems, which prepare students for more advanced study in mathematics and other scientific fields such as physics and computer science.

The wide range of topics includes curve theory, a detailed study of surfaces, curvature, variation of area and minimal surfaces, geodesics, spherical and hyperbolic geometry, the divergence theorem, triangulations, and the Gauss–Bonnet theorem. The section on cartography demonstrates the concrete importance of elementary differential geometry in applications. Clearly developed arguments and proofs, colour illustrations, and over 100 exercises and solutions make this book ideal for courses and self-study. The only prerequisites are one year of undergraduate calculus and linear algebra.

Christian Bär is Professor of Geometry in the Institute for Mathematics at the University of Potsdam, Germany.

**Contents:**

**1 Euclidean geometry**

1.1 The axiomatic approach

1.2 The Cartesian model**2 Curve theory**

2.1 Curves in Rn

2.2 Plane curves

2.3 Space curves**3 Classical surface theory**

3.1 Regular surfaces

3.2 The tangent plane

3.3 The first fundamental form

3.4 Normal fields and orientability

3.5 The second fundamental form

3.6 Curvature

3.7 Surface area and integration on surfaces

3.8 Some classes of surfaces**4 The inner geometry of surfaces**

4.1 Isometries

4.2 Vector fields and the covariant derivative

4.3 Riemann curvature tensor and Theorema Egregium

4.4 Riemannian metrics

4.5 Geodesics

4.6 The exponential map

4.7 Parallel transport

4.8 Jacobi fields

4.9 Spherical and hyperbolic geometry

4.10 Cartography

4.11 Further models of hyperbolic geometry**5 Geometry and analysis**

5.1 The divergence theorem

5.2 Variation of the metric**6 Geometry and topology**

6.1 Polyhedra

6.2 Triangulations

6.3 The Gauss–Bonnet theorem

6.4 Outlook

Appendix A Hints for solutions to (most) exercises

Appendix B Formulary

Appendix C List of symbols