Introduction to differential and Riemannian geometry

Abstract

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 allow in the sequel 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.

A Theory of Everything: When Information Geometry Meets the Generalized Brownian Motion and the Einsteinian Relativity

“…This work creates new avenues for research by establishing connections between queueing theory and a wide range of other mathematical fields, including matrix theory, differential geometry, and information theory. In particular, the addition of geometric ties between queueing theory and information allows one to revealing a ground-breaking unified relativistic and of Riemannian geometric analysis of queue manifolds [3,4,7]. This paper's contributions suggest several potential study directions for the new applications of information geometry, such as using information geometrics on different statistical manifolds to further advance many existing physical phenomena.…”

“…Applying non-Euclidean geometry approaches and techniques to probability theories and stochastic processes is, thus, the central premise of IG. A topological finite dimensional Cartesian space, ℝ , where one possesses an infinite-dimensional manifold, is called a manifold [2][3][4]. The challenge of making decisions can be applied to the parameter inference 𝜃 of a model based on data in Figure 1.…”

Subspace Detours Meet Gromov–Wasserstein

“…Let x ∈ M, then for any v ∈ T x M, there exists a unique geodesic γ (x,v) starting from x with velocity v, i.e. such that γ (x,v) (0) = x and γ ′ (x,v) (0) = v (Sommer et al, 2020). Now, we can define the exponential map as exp : T M → M which for any x ∈ M, maps tangent vectors v ∈ T x M back to the manifold at the point reached by the geodesic γ (x,v) at time t = 1:…”

Derivative Formulas and Gradient of Functions with Non-Independent Variables

“…This is also the case for models defined via functions with independent input variables and equations or inequations connecting such inputs; functions subject to constraints involving input variables and/or the model output. Knowing the key role of partial derivatives at a given point in (i) the mathematical analysis of functions and convergence, (ii) Poincaré inequalities ( [1,2]) and equalities ( [3,4]), (iii) optimization and active subspaces ( [5,6]), (iv) implicit functions ( [7,8]) and (v) differential geometry (see, e.g., [9][10][11]), it is interesting and relevant to have formulas that enable the calculations of partial derivatives of functions in the presence of non-independent variables, including the gradients. Of course, such formulas must account for the dependency structures among the model inputs, including the constraints imposed on such inputs.…”

“…In differential geometry (see, e.g., [9][10][11]), using the differential of the function f , that is, d f = 2y 2 z 3 dx + 4xyz 3 dy + 6xy 2 z 2 dz , the gradient of f is defined as the dual of d f with respect to a given tensor metric. Obviously, different tensor metrics will yield different gradients of the same function.…”