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This chapter argues that calculus, the theory of differentiation and integration that permeates modern science, can be formulated as a purely geometric theory, which does not entail the existence of mathematical objects such as numbers, sets and functions. In particular it is argued that modern differential geometry can be ‘nominalised’, i.e. be formulated in such a way that it does not entail the existence of mathematical entities, by making judicious use of the geometric structure of the fibre bundles spaces which were encountered in chapter 6.

It is shown how Weyl intertwined phenomenological analysis and mathematical construction in building the foundation of his “pure infinitesimal geometry” underlying gravitation and electromagnetism. In response to the Einstein-Pauli objection, Weyl put forth a second version of his theory, arguing on various non-empirical grounds for the conceptual superiority of his gauge-theoretic approach to general relativity. Speculation is made regarding Weyl’s philosophical reasons for opposing Élie Cartan’s “moving frame” generalization of Weyl’s notion of an affine connection, and it is suggested that the basic approach of Weyl’s transcendental phenomenological foundation of differential geometry can be accommodated within the modern fiber bundle formulation of gauge field theories.

The purpose of this chapter is to provide a historical overview of some of the contemporary infinitesimalist alternatives to the Cantor-Dedekind theory of continua. Among the theories we will consider are those that emerge from nonstandard analysis, nilpotent infinitesimalist approaches to portions of differential geometry and the theory of surreal numbers. Since these theories have roots in the algebraic, geometric and analytic infinitesimalist theories of the late nineteenth and early twentieth centuries, we will also provide overviews of the latter theories and some of their relations to the contemporary ones. We will find that the contemporary theories, while offering novel and possible alternative visions of continua, need not be (and in many cases are not) regarded as replacements for the Cantor-Dedekind theory and its corresponding theories of analysis and differential geometry.

This chapter deals with topological invariants and differential geometry. It first considers a topological space X for which singular homology and cohomology are defined, along with the Euler number e(X). The Euler number, also known as the Euler-Poincaré characteristic, is an important invariant of a topological space X. It generalizes the notion of the cardinality of a finite set. The chapter presents the simple formulas for computing the Euler-Poincaré characteristic (Euler number) of many of the spaces to be encountered throughout the book. It also discusses fundamental groups and covering spaces and some basics of the theory of complex manifolds and Hermitian metrics, including the concept of real manifold. Finally, it provides some general facts about divisors, line bundles, and the first Chern class on a complex manifold X.

This chapter presents a survey of the basic definitions of Riemannian and Lorentzian differential geometry used in this book. The first nine sections use the simplest formulations, in local coordinates, as they are needed for the first five chapters and physical applications. The later sections contain material used in the following, more advanced, chapters. Topics covered include manifolds, differential mappings, vectors and tensors, pseudo-Riemannian metrics, Riemannian connection, geodesics, curvature, geodesic deviation, maximum length and conjugate points, linearized Ricci and Einstein tensors, and second derivative of the Ricci tensor.

This chapter discusses complex algebraic surfaces, with particular emphasis on the Miyaoka-Yau inequality and the rough classification of surfaces. Every complex algebraic surface is birationally equivalent to a smooth surface containing no exceptional curves. The latter is known as a minimal surface. Two related birational invariants, the plurigenus and the Kodaira dimension, play an important role in distinguishing between complex surfaces. The chapter first provides an overview of the rough classification of (smooth complex connected compact algebraic) surfaces before presenting two approaches that, in dimension 2, give the Miyaoka-Yau inequality. The first, due to Miyaoka, uses algebraic geometry, whereas the second, due to Aubin and Yau, uses analysis and differential geometry. The chapter also explains why equality in the Miyaoka-Yau inequality characterizes surfaces of general type that are free quotients of the complex 2-ball.

This is the core chapter, introducing concepts that articulate continua, continuous substance, and continuous process. These concepts find precise and deep forms in point set topology, topological and differentiable dynamical systems (qualitative, topological, and geometrical approaches to systems of ordinary differential equations), and the much more sophisticated perspectives of differential geometry and fiber bundles. Basic poetic concepts introduced include: open (closed) set, neighborhood, map, space, continuity, connectedness, limit, convergence, compactness, and so forth. Introducing the work of Brouwer, Thom, and Petitot prepares the reader for a critical encounter with Petitot's program on ontogenesis. Articulating matter with such anexact concepts seeds the ground for an alternative, nonreductionist approach to ontogenesis. Certain terms used in earlier chapters for their intuitive senses, such as continuous, limit, dense, etc., will now be presented more rigorously, so that they can be used with more precise connotations and conceptual purchase after this chapter.

This chapter begins by examining p-forms and the exterior product, as well as the dual of a p-form. Meanwhile, the exterior derivative is an operator, denoted d, which acts on a p-form to give a (p + 1)-form. It possesses the following defining properties: if f is a 0-form, df(t) = t f (where t is a vector of Eₙ), which coincides with the definition of differential 1-forms. Moreover, d(α‎ + β‎) = dα‎ + dβ‎, where α‎ and β‎ are forms of the same degree. Moreover, the exterior calculus can be used to obtain a compact and elegant formulation of Maxwell’s equations.

The mathematical basis of general relativity is differential geometry. This chapter establishes the starting point of differential geometry: manifolds, tangent vectors, cotangent vectors, tensors, and differential forms. The metric tensor is introduced, and its symmetries (isometries) are described. The importance of diffeomorphism invariance (or “general covariance”) is stressed.

This chapter presents some elements of differential geometry, the ‘vector’ version of Euclidean geometry in curvilinear coordinates. In doing so, it provides an intrinsic definition of the covariant derivative and establishes a relation between the moving frames attached to a trajectory introduced in Chapter 2 and the moving frames of Cartan associated with curvilinear coordinates. It illustrates a differential framework based on formulas drawn from Chapter 2, before discussing cotangent spaces and differential forms. The chapter then turns to the metric tensor, triads, and frame fields as well as vector fields, form fields, and tensor fields. Finally, it performs some vector calculus.

The metric tensor uniquely defines the geometric properties of a metric space, while differential geometry is concerned with the derivatives of vectors and tensors within the metric space. Derivatives of vector quantities include the derivatives of basis vectors, which lead to Christoffel symbols that are directly connected to the metric tensor. This connection between tensor derivatives and the metric tensor provides the tools necessary to define geodesic curves. The geodesic equation is derived from variational calculus and from parallel transport. Geodesic motion is the trajectory of a particle through a metric space defined by an action metric that includes a potential function. In this way, dynamics converts to geometry as a trajectory in a potential converts to a geodesic curve through an appropriately defined metric space.

This chapter provides a brief summary of the basic aspects of Einstein’s theory of general relativity. Although not aimed to be comprehensive, a step-by-step presentation of the fundamental principles of relativity is provided. Following this spirit, a pragmatic presentation of all the relativistic concepts and quantities that are used in the rest of the book is resented. Our introduction to general relativity is based on a balance between the differential-geometry approach, which is nowadays regarded as the most elegant and physically appropriate one, and the coordinate-components approach, which is nevertheless fundamental for converting formal and compact equations into relations involving quantities that may be measured or computed.

This chapter studies vector-valued forms. Ordinary differential forms have values in the field of real numbers. This chapter allows differential forms to take values in a vector space. When the vector space has a multiplication, for example, if it is a Lie algebra or a matrix group, the vector-valued forms will have a corresponding product. Vector-valued forms have become indispensable in differential geometry, since connections and curvature on a principal bundle are vector-valued forms. All the vector spaces will be real vector spaces. A k-covector on a vector space T is an alternating k-linear function. If V is another vector space, a V-valued k-covector on T is an alternating k-linear function.

This introductory chapter first sets out the book's purpose, which is to provide a rigorous proof of the Deser–Schwimmer conjecture. This work is a continuation of the previous two papers of the author, which established the conjecture in a special case and introduced tools that laid the groundwork for the resolution of the full conjecture. The chapter then provides a formulation of the conjecture, presents some applications, and discusses its close relation with certain questions in index theory and in Cauchy–Riemann and Kähler geometry. Then, it broadly outlines the strategy of the proof and very briefly present the tasks that are undertaken in each of the subsequent chapters.

This chapter introduces the angle-action method and provides several examples of its use, including its essential role in the ‘old’ quantum theory, a precursor to the Schroedinger wave equation that sought to introduce quantum ideas into a classical model. Angle-action methods are also used in what is called canonical perturbation theory, the study of systems that deviate only slightly from cyclic ones. That vast subject is best studied using the language of differential geometry. Study of the present chapter will give the reader a firm grasp of the angle-action foundation upon which canonical perturbation theory is constructed. Certain specialised systems of importance in astronomy and quantum theory can be treated by a canonical transformation to angle-action variables.

This chapter discusses connections on a principal bundle. Throughout the chapter, G will be a Lie group with Lie algebra g. One possible definition of a connection on a principal G-bundle P is a C∞ right-invariant horizontal distribution on P. Equivalently, a connection on P can be given by a right-equivariant g-valued 1-form on P that is the identity on vertical vectors. The chapter shows the equivalence of these two definitions of a connection. A connection is one of the most basic notions of differential geometry. It is essentially a way of differentiating sections. From a connection, the notions of curvature and geodesics follow.