Introduction

Abstract

The need to generalize concepts and tools used in “flat spaces” such as the real line, the plane, or more generally ${\mathbb {R}}^n$, to more general spaces (such as a sphere) arises naturally. Such concepts and tools include

• 1. Defining functions.

• 2. Computing derivatives of functions.

• 3. Finding minima or maxima of functions.

• 4. More generally, solving optimization problems.

• 5. Computing the length of curves.

• 6. Finding shortest paths between two points.

• 7. Solving differential equations.

• 8. Defining a notion of average or mean.

• 9. Computing areas and volumes.

• 10. Integrating functions.