In this chapter we introduce some differential geometrical aspects of shape and size-and-shape. After a brief review of Riemannian manifolds, we define what is meant by the pre-shape, shape and size-and-shape of a configuration.
Throughout this text the spaces of interest are primarily Riemannian manifolds, and we begin with some informal discussion about the topic. There are many treatments of differential geometry at various levels of formalism, and an excellent introduction is given by Bär (2010).
A manifold is a space which can be viewed locally as a Euclidean space. We first consider tangent spaces for a manifold M in general. Consider a differentiable curve in M given by with γ(0) = p. The tangent vector at p is given by:
and the unit tangent vector is ξ = γ′(0)/||γ′(0)||. The set of all tangent vectors γ′(0) for all curves passing through p is called the tangent space of M at p, denoted by Tp(M). If we consider a manifold M then if it has what is called an affine connection (a way of connecting nearby tangent spaces) then a geodesic can be defined.
A Riemannian manifold M is a connected manifold which has a positive-definite inner product defined on each tangent space Tp(M), such that the choice varies ...