Mathematical notions used in the theory of statistical shape analysis

We review key mathematical concepts used in the theory of Statistical Shape Analysis (SSA). The treatment is elementary and aims at providing a brief guide to a large number of results and ideas which are dispersed over an also very large literature on Non-Euclidean Geometry, Differential Geometry, and Topology. The goal is to provide an introduction to the ideas SSA touches in these areas to researchers wishing to apply SSA in practice. 0.1 Relations, equivalence relations and equivalence classes Definition 1. A relation on a set A is a subset, R, of A = A×A. Usually, relations are defined by providing a statement that singles out a collection of elements of A× A for membership in the relation. A relation R on a set A is: • reflexive if for all x ∈ A, xRx. • symmetric if, for all x, y ∈ A, xRy implies yRx. • transitive if, for all x, y, z ∈ A, xRy and yRz imply xRz. • an equivalence relation if R is reflexive, symmetric and transitive. Example 1. Let F be the set of fractions of integers. Define a/b ≡ c/d if ad = bc. Then ≡ (equality of fractions) is a relation on F . Thus, e.g., the pair (1/2, 2/4) is in the subset of F defined by ≡. Furthermore, ≡ is an equivalence relation since: 1. it is reflexive: for each a/b ∈ F , a/b ≡ a/b; 2. it is symmetric: for each a/b, c/d ∈ F , if a/b ≡ c/d, then c/d ≡ a/b;