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Introduction to Riemannian Geometry

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1953)

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.

Keywords

  • Riemannian Manifold
  • Tangent Space
  • Tangent Vector
  • Tangent Bundle
  • Riemannian Geometry

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© 2008 Springer-Verlag Berlin Heidelberg

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(2008). Introduction to Riemannian Geometry. In: Information Geometry. Lecture Notes in Mathematics, vol 1953. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69393-2_2

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