Introduction to Riemannian Geometry

This chapter is intended to help those with little previous exposure to differential geometry by providing a rather informal summary of background for our purposes in the sequel and pointers for those who wish to pursue more geometrical features of the spaces of probability density functions that are our focus in the sequel. In fact, readers who are comfortable with doing calculations of curves and their arc length on surfaces in 葷³ could omit this chapter at a first reading.

We need rather little geometry of Riemannian manifolds in order to provide background for the concepts of information geometry. Dodson and Poston [70] give an introductory treatment with many examples, Spivak [194, 195] provides a six-volume treatise on Riemannian geometry while Gray [99] gave very detailed descriptions and computer algebraic procedures using Mathematica[215] for calculating and graphically representing most named curves and surfaces in Euclidean E³ and code for numerical solution of geodesic equations. Our Riemannian spaces actually will appear as subspaces of 葷nso global properties will not be of particular significance and then the formulae and Gray's procedures easily generalize to more variables.