In this chapter, we will highlight the interesting connection between finite and infinite dimensional differential geometry. To this end, we shall consider Lie groupoids, which can be understood as elements from higher geometry. The moniker higher geometry stems from the fact that in the language of category theory, these objects form higher categories. Previously we discussed how finite-dimensional manifolds and geometric structures give rise to infinite-dimensional structures such as Lie groups (e.g. the diffeomorphims and groups of gauge transformations) and Riemannian metrics (such as the L^2-metric from shape analysis). While a manifold determines an (in general infinite-dimensional) group of diffeomorphisms, we turn this observation now on its head and investigate whether the underlying finite-dimensional geometric structure is recognisable from its associated infinite dimensional object.

In this chapter, we consider spaces of differentiable mappings as infinite-dimensional spaces. These spaces will then serve as the model spaces for manifolds of mappings, i.e. manifolds of differentiable mappings between manifolds. The resulting manifolds will allow us to construct essential examples in later chapters, such as the diffeomorphism groups. Moreover, they arise naturally in the context of many applications such as shape analysis and the geometric treatment of partial differential equations which will be discussed later. Finally, we introduce an indispensable tool for the treatment of differentiable mappings on manifolds of mappings in this chapter: the exponential law. roughly speaking, the exponential law allows to interpret a smooth map taking values in a manifold of mappings as a smooth map of two arguments. Thus differentiability questions can often be relegated to a finite dimenisonal setting.

It is well known that multidimensional calculus, aka Fréchet calculus, carries over to the realm of Banach spaces and Banach manifolds. Banach spaces are often not sufficient for our purposes. To generalise derivatives, we will, as a minimum, need vector spaces with an amenable topology (which need not be induced by a norm). This chapter presents first a notion of calculus in locally convex spaces, which requires the existence and continuity of directional derivatives. The resulting calculus is called Bastiani calculus and we compare it to some common (but inequivalent) notions of calculus such as the convenient calculus. Building on the chain rule, we then construct the basic building blocks of (infinite-dimensional) differential geometry: manifolds and their tangent spaces as well as submersions and immersions.

In this chapter, we will discuss Riemannian metrics on infinite-dimensional spaces. Particular emphasis will be placed on the new challenges which arise on infinite-dimensional spaces. One new feature is that Riemannian metrics comes in several flavours on infinite-dimensional spaces. These are not present in the finite dimensional setting. The strongest flavour (as we shall see) is the notion of a strong Riemannian metric which is treated in classical monographs on infinite-dimensional geometry. It is also the most restrictive setting as it forces one to work on Hilbert manifolds. Of greater interest are for this reason the weak Riemannian metrics which are however possibly ill behaved. As an example we will discuss at length geodesics for Riemannian metrics on infinite-dimensional spaces. The aim is to exhibit examples of Riemannian manifolds for which the finite dimensional theory breaks down and the geodesic distance vanishes.

In this chapter, we study in detail the (weak) L^2-metric on spaces of smooth mappings. Its importance stems from the fact that this metric and its siblings, the Sobolev H^s -metrics are prevalent in shape analysis. It will be essential for us that geodesics with respect to the L^2-metric can explicitely be computed. Let us clarify what we mean here by shape and shape analysis. Shape analysis seeks to classify, compare and analyse shapes. In recent years there has been an explosion of applications of shape analysis to diverse areas such as computer vision, medical imaging, registration of radar images and many more. Another typical feature in (geometric) shape analysis is that one wants to remove superfluous information from the data. For example, in the comparison of shapes, rotations, translations, scalings and reflections are typically disregarded as being inessential differences. Conveniently, these inessential differences can mostly be described by actions of suitable Lie groups (such as the rotation and the diffeomorphism groups).

In this appendix, we give a short introduction to differential forms on infinite-dimensional manifolds. The main difference between the finite dimensional (or Banach) and our setting, is that it is in general impossible to interprete differential forms as (smooth) sections into certain bundles of linear forms. The reason for this is again that the topology on spaces of linear forms breaks down beyond the Banach setting. Even worse, the many equivalent ways to define differential forms in finite dimensions become inequivalent in the infinite-dimensional setting. Most notably, there is no useful way to describe differential forms as a sum of differential forms coming from a local coordinate system. We begin with the definition of a differential form. This definition is geared towards avoiding any reference to topologies on spaces of linear mappings. Then, we shall discuss differential forms on a Lie group and in particular the Maurer–Cartan form.

In this chapter, we shall give an introduction to Euler–Arnold theory for partial differential equations (PDEs). The main idea of this theory is to reinterpret certain PDEs as smooth ordinary differential equations (ODEs) on infinite-dimensional manifolds. One advantage of this idea is that the usual solution theory for ODEs can be used to establish properties for the PDE under consideration. This principle has been successfully applied to a variety of PDE arising for example in hydrodynamics. Among these are the Euler equations for an ideal fluid, the Camassa–Holm equation, the Hunter–Saxton and the inviscid Burgers equation. Indeed there is a much longer list of physically relevant PDE which fit into this setting. We shall mainly orient ourselves along the classical exposition by Arnold and Ebin and Marsden and study the Euler equation of an incompressible ideal fluid.

This section contains some auxiliary results on topological vector spaces and locally convex spaces in particular. Note that for some of the results in this appendix, it is essential that we only consider Hausdorff topological vector spaces (which is the standing assumption in the present book). In some more specialised section, we will discuss the following topics: (1) smooth bump function (or the lack thereof) in locally convex spaces, (2) the (failure of) the inverse function theorem in locally convex spaces, (3) the breakdown of the solution theory to ordinary differential equation, (4) a comparison of Bastiani calculus and the convenient calculus

This appendix sketches the construction of a canonical manifold of mappings structure for smooth mappings between (finite-dimensional) manifolds. Before we begin, let us consider for a moment the locally convex space of smooth functions from a manifold with values in a locally convex space. The topology and vector space structure allow us to compare two smooth maps by measuring their pointwise difference on compact sets. As manifolds lack an addition we can not mimick this for manifold valued functions (albeit the topology still makes sense!). On first sight, it might be tempting to think that one could use the charts of the target manifold to construct charts for the smooth functions. However, if the target manifold does not admit an atlas with only one chart, there will be smooth mappings whose image is not contained in one chart. Thus the charts of the target manifold turn out to be not very useful. Instead one needs to find a replacement of the vector space addition to construct a way in which charts vary smoothly over the target manifold. This leads to the concept of a local addition which enables the construction of a manifold structure.

In this appendix, we recall the construction of the Lie algebra of vector fields on a smooth manifold. For a finite dimensional manifold, this Lie algebra becomes a locally convex Lie algebra, while it does not inherit a suitable topology if the underlying manifold is infinite-dimensional.

In this chapter, we will discuss the (infinite-dimensional) geometric framework for rough paths and their signature. Rough path theory originated in the 1990s with the work of T. Lyons. It seeks to establish a theory of integrals and differential equations driven by rough signals. For example, one is interested in controlled ordinary differential equations driven by a rough signal. Here, a rough signal is a Hölder continuous path of potentially low Hölder regularity. Numerical methods for equations with more regularity suggest that iterated integrals of the rough signal against itself are needed to construct solutions. However due to Youngs theorem, these iterated integrals do not exist. To compensate this problem, the notion of a rough path was developed. After a qucik introduction to the theory of rough paths, we shall see that rough paths of various flavours can be understood as certain continuous paths taking values in infinite-dimensional Lie groups. The main focus of the chapter is to present an introduction to this geometric side of the theory.

This short appendix covers several basic concepts of topology, which we review for the readers convenience. For example, the compact open topology on spaces of continuous functions is discussed together with an appropriate version of the exponential law.

In this chapter, one aim is to study spaces of mappings taking their values in a Lie group. It will turn out that these spaces carry again a natural Lie group structure. However, before we prove this, the definition and basic properties of (infinite dimensional) Lie groups and their associated Lie algebras are recalled. Infinite-dimensional Lie theory (beyond Banach spaces) is by comparison relatively young and in its modern form goes back to Milnor’s seminal works. One key feature of infinite-dimensional Lie theory is that the conncection between Lie algebra and Lie group is looser then in finite dimensions. For advanced tools in Lie theory one has to require the Lie group to be regular (in the sense of Milnor). These concepts are introduced and considered for several main classes of examples, such as the diffeomorphism groups, loop groups and gauge groups.