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The concept of a manifold is in essence a generalization of the idea, first formulated in mathematical terms by Gauss, underlying the usual procedure used in cartography (i.e. the drawing of maps of the earth’s surface, or portions of it).

We begin by discussing some of the concepts fundamental to geometry. In school geometry—the so-called “elementary Euclidean” geometry of the ancient Greeks—the main objects of study are various metrical properties of the simplest geometrical figures. The basic goal of that geometry is to find relationships between lengths and angles in triangles and other polygons. Knowledge of such relationships then provides a basis for the calculation of the surface areas and volumes of certain solids. The central concepts underlying school geometry are the following: the length of a straight line segment (or of a circular arc); and the angle between two intersecting straight lines (or circular arcs).

A surface in a 3-dimensional space is the simplest object having its own, internal or “intrinsic,” geometry. What exactly do we mean by this?

Geometrically speaking the fundamental problem of GTR is that of finding 4-dimensional manifolds M4 endowed with a metric g ab of signature (+ − − −), satisfying Einstein’s equation (see Part I, §§37.4, 39) $${R_{ab}} - \frac{1}{2}R{g_b} = \frac{{8\pi G}}{{{c^4}}}{T_{ab,}}$$ (1) where Tab is the energy-momentum tensor of the medium. From the most general point of view there is only one restriction on the tensor Tab, namely that the “energy density” be non-negative, which comes down to the condition that for every time-like vector ξ = (ξa) (i.e. such that gab ξ,a ξb > 0) at each point of M4, we have Tabξa ξb ≥ 0. However in the present chapter we shall for the most part concern ourselves with the so-called “hydrodynamical energy-momentum tensor” of an isotropic dense medium (forming a “perfect fluid”), already introduced in Part I, §39.4, important in the investigation of the gravitational fields of macroscopic bodies: $${T_{ab}} = (p + \varepsilon ){u_a}{u_b} - p{g_{ab}},$$ (2) where u = (u0) is the 4-velocity ( u, u) = 1), p is the (isotropic) pressure, and ɛ is the energy density (see loc. cit.).

The fact that many physical entities find mathematical expression as numerical functions of points in space, will by now be familiar to the reader; the distance from a fixed point to a variable point is one among many examples. If we have several such entities, then their mathematical counterparts form a collection of functions from the points of space to the numbers (or, in other words, a (single) vector-valued function on the points). Thus for instance to fully determine the position of a point in 3-dimensional space, we need the values at that point of (at least) three functions (and of course we call these values “co-ordinates” of the point): each co-ordinate x i is a function of the point, and together they form an ordered triple (x 1, x 2 , x 3) which specifies the point completely. In Chapter 1 we encountered various kinds of coordinate systems (i.e. triples of functions); for example in the plane we introduced Cartesian co-ordinates x 1, x 2, and polar co-ordinates r, φ, where x 1 = r cos φ, x 2 = r sin φ; and in space Cartesian co-ordinates, cylindrical co-ordinates r, z, φ, and spherical co-ordinates r, θ, φ.

Let OD denote a region with piecewise smooth boundary D, of the Euclidean space I1’ with Euclidean co-ordinates x1,..., xn. Consider the linear space F of smooth vector-functions f (x1,..., xn) = (f1,..., fk) defined on D, i.e. with domain D. Let L(x β;p j ;q a i )be a smooth real-valued function of the three arguments xβ, 1 ≤β ≤ n; p j , 1 ≤ j ≤ k; q a i , 1 ≤ i ≤ k, 1 ≤ a ≤n (making altogether n + k + nk real arguments); we call such a function a Lagrangian, and from a given such Lagrangian we construct a functional I[f] defined on F, as follows: $$I\left[ f \right] = \int_D {L\left( {{x^\beta };{f^j}\left( {{x^\beta }} \right);f_{{x^\alpha }}^i\left( {{x^\beta }} \right)} \right)} d{x^1} \wedge \cdots \wedge d{x^n}$$ where the integral is the multiple integral (see §26) over the region D (which we shall later assume to be bounded), and where \(f_{{x^\alpha }}^i\left( {{x^\beta }} \right) = \left( {\partial /\partial {x^\alpha }} \right){f^i}\left( {{x^\beta }} \right)\).

The present chapter is devoted to foundational questions in the theory of smooth manifolds. The proofs of the theorems will play no role whatever in the development of the basic topology and geometry of manifolds contained in succeeding chapters. Consequently in this chapter the reader may, if he wishes, acquaint himself with the definitions and statements of results only, without thereby sacrificing anything in the way of comprehension of the later material.

In §29.2 we defined the geodesics x i = x i (t) relative to a given connexion Γ jk i by means of the equation $${{\nabla }_{T}}\left( T \right) = 0$$ , where T i = dx i /dt is the velocity vector of the curve; in other words the geodesics are the solutions of the equations $$\frac{{{d^2}{x^i}}}{{d{t^2}}} + \Gamma _{jk}^i\frac{{d{x^j}}}{{dt}}\frac{{d{x^k}}}{{dt}} = 0$$ .

Let M and N be two smooth (for simplicity of class C∞) manifolds and let f: M → N, be a smooth map between them.

According to the simplest of the definitions of an orientation on a manifold given above (see Definition 1.1.3), a manifold M is oriented if the local coordinate systems x α j given on the members U j of a covering collection of local co-ordinate neighbourhoods (or charts) for M, are such that the transition functions from one local co-ordinate system to another on the regions of overlap U j ∩ U k , have positive Jacobian: $$\det \left( {\frac{{\partial x_j^\alpha }}{{\partial x_k^\beta }}} \right) > 0.$$

Most physical laws find mathematical expression as relationships between derivatives of one kind or another of various physical quantities. Many of these physical quantities are most appropriately represented as tensor (in particular vector) fields throughout space, or a region of space. We are therefore naturally interested in the question of what differential operations on tensors there exist, which are (in a sense to be made precise) independent of the co-ordinate system in terms of which we calculate their effect.