Introduction to differential and Riemannian geometry

Stefan Sommer (1) , Tom Fletcher (2) , Xavier Pennec (3, 4)

1 UCPH - University of Copenhagen = Københavns Universitet
2 University of Virginia
3 COMUE UCA - COMUE Université Côte d'Azur (2015-2019)
4 EPIONE - E-Patient : Images, données & mOdèles pour la médeciNe numériquE

This chapter introduces the basic concepts of differential geometry: Manifolds, charts, curves, their derivatives, and tangent spaces. The addition of a Riemannian metric enables length and angle measurements on tangent spaces giving rise to the notions of curve length, geodesics, and thereby the basic constructs for statistical analysis of manifold-valued data. Lie groups appear when the manifold in addition has smooth group structure, and homogeneous spaces arise as quotients of Lie groups. We discuss invariant metrics on Lie groups and their geodesics. The goal is to establish the mathematical bases that will further allow to build a simple but consistent statistical computing framework on manifolds. In the later part of the chapter, we describe computational tools, the Exp and Log maps, derived from the Riemannian metric. The implementation of these atomic tools will then constitute the basis to build more complex generic algorithms in the following chapters.

Domaines: Imagerie médicale; Géométrie différentielle [math.DG]; Autres [stat.ML]

Stefan Sommer, Tom Fletcher, Xavier Pennec. Introduction to differential and Riemannian geometry. Riemannian Geometric Statistics in Medical Image Analysis, Chap. 1, Elsevier, pp.3-37, 2020, ⟨10.1016/b978-0-12-814725-2.00008-x⟩. ⟨hal-02341901⟩