holds, with (h) small relative to h. The linear map T is the derivative of u at x.

An important special case of this is when m = 1. If f : ⁿ → is differentiable at x, then the derivative of f at x is a linear map from ⁿ to . The matrix of T is a row vector of length n, which is often denoted ∇ f(x) and referred to as the gradient of f at x. This vector points in the direction in which f increases most rapidly and its magnitude is the rate of change in that direction.

5.4 Partial Differential Equations

Partial differential equations are of immense importance in physics, and have inspired a vast amount of mathematical research. Three basic examples will be discussed here, as an introduction to more advanced articles later in the volume (see, in particular, PARTIAL DIFFERENTIAL EQUATIONS [IV.12]).

The first is the heat equation, which, as its name suggests, describes the way the distribution of heat in a physical medium changes with time:

Here, T (x,y,z,t) is a function that specifies the temperature at the point (x,y,z) at time t.

It is one thing to read an equation like this and understand the symbols that make it up, but quite another to see what it really means. However, it is important to do so, since of the many expressions one could write down that involve partial derivatives, only a minority are of much significance, and these tend to be the ones that have interesting interpretations. So let us try to interpret the expressions involved in the heat equation.

The left-hand side, ∂T/∂t, is quite simple. It is the rate of change of the temperature T(x,y,z,t) when the spatial coordinates x, y, and z are kept fixed and t varies. In other words, it tells us how fast the point (x,y,z) is heating up or cooling down at time t. What would we expect this to depend on? Well, heat takes time to travel through a medium, so although the temperature at some distant point (x’,y’,z’) will eventually affect the temperature at (x,y,z), the way the temperature is changing right now (that is, at time t) will be affected only by the temperatures of points very close to (x,y,z): if points in the immediate neighborhood of (x,y,z) are hotter, on average, than (x,y,z) itself, then we expect the temperature at (x,y,z) to be increasing, and if they are colder then we expect it to be decreasing.

The expression in brackets on the right-hand side appears so often that it has its own shorthand. The symbol Δ, defined by

is known as the Laplacian. What information does Δf give us about a function f? The answer is that it captures the idea in the last paragraph: it tells us how the value of f at (x,y,z) compares with the average value of f in a small neighborhood of (x,y,z), or, more precisely, with the limit of the average value in a neighborhood of (x,y,z) as the size of that neighborhood shrinks to zero.

This is not immediately obvious from the formula, but the following (not wholly rigorous) argument in one dimension gives a clue about why second derivatives should be involved. Let f be a function that takes real numbers to real numbers. Then to obtain a good approximation to the second derivative of f at a point x, one can look at the expression (f’(x) - f’(x - h))/h for some small h. (If one substitutes -h for h in the above expression, one obtains the more usual formula, but this one is more convenient here.) The derivatives f’(x) and f’(x - h) can themselves be approximated by (f(x+h) - f(x))/h and (f(x) - f(x-h))/h, respectively, and if we substitute these approximations into the earlier expression, then we obtain

which equals (f(x+ h) - 2 f(x) + f(x-h))/h². Dividing the top of this last fraction by 2, we obtain (f(x + h) + f(x - h)) - f(x): that is, the difference between the value of f at x and the average value of f at the two surrounding points x + h and x - h.

In other words, the second derivative conveys just the idea we want—a comparison between the value at x and the average value near x. It is worth noting that if f is linear, then the average of f(x - h) and f(x + h) will be equal to f(x), which fits with the familiar fact that the second derivative of a linear function f is zero.

Just as, when defining the first derivative, we have to divide the difference f(x + h) - f(x) by h so that it is not automatically tiny, so with the second derivative it is appropriate to divide by h². (This is appropriate, since, whereas the first derivative concerns linear approximations, the second derivative concerns quadratic ones: the best quadratic approximation for a function f near a value x is f(x + h) ≈ f(x) + hf′(x) + h² f″ (x), an approximation that one can check is exact if f was a quadratic function to start with.)

It is possible to pursue thoughts of this kind and show that if f is a function of three variables then the value of Δf at (x,y,z) does indeed tell us how the value of f at (x,y,z) compares with the average values of f at points nearby. (There is nothing special about the number 3 here—the ideas can easily be generalized to functions of any number of variables.) All that is left to discuss in the heat equation is the parameter κ. This measures the conductivity of the medium. If κ is small, then the medium does not conduct heat very well and ΔT has less of an effect on the rate of change of the temperature; if it is large then heat is conducted better and the effect is greater.

A second equation of great importance is the Laplace equation, Δf = 0. Intuitively speaking, this says of a function f that its value at a point (x,y,z) is always equal to the average value at the immediately surrounding points. If f is a function of just one variable x, this says that the second derivative of f is zero, which implies that f is of the form ax + b. However, for two or more variables, a function has more flexibility—it can lie above the tangent lines in some directions and below it in others. As a result, one can impose a variety of boundary conditions on f (that is, specifications of the values f takes on the boundaries of certain regions), and there is a much wider and more interesting class of solutions.

A third fundamental equation is the wave equation. In its one-dimensional formulation it describes the motion of a vibrating string that connects two points A and B. Suppose that the height of the string at distance x from A and at time t is written h(x, t). Then the wave equation says that

Ignoring the constant 1 / υ² for a moment, the left-hand side of this equation represents the acceleration (in a vertical direction) of the piece of string at distance x from A. This should be proportional to the force acting on it. What will govern this force? Well, suppose for a moment that the portion of string containing x were absolutely straight. Then the pull of the string on the left of x would exactly cancel out the pull on the right and the net force would be zero. So, once again, what matters is how the height at x compares with the average height on either side: if the string lies above the tangent line at x, then there will be an upwards force, and if it lies below, then there will be a downwards one. This is why the second derivative appears on the right-hand side once again. How much force results from this second derivative depends on factors such as the density and tautness of the string, which is where the constant comes in. Since h and x are both distances, υ² has dimensions of (distance/time) ², which means that υ represents a speed, which is, in fact, the speed of propagation of the wave.

Similar considerations yield the three-dimensional wave equation, which is, as one might now expect,

or, more concisely,

One can be more concise still and write this equation as ² h = 0, where ² h is shorthand for

The operation ² is called the d’Alembertian, after D’ALEMBERT [VI.20], who was the first to formulate the wave equation.

5.5 Integration

Suppose that a car drives down a long straight road for one minute, and that you are told where it starts and what its speed is during that minute. How can you work out how far it has gone? If it travels at the same speed for the whole minute then the problem is very simple indeed—for example, if that speed is thirty miles per hour then we can divide by sixty and see that it has gone half a mile—but the problem becomes more interesting if the speed varies. Then, instead of trying to give an exact answer, one can use the following technique to approximate it. First, write down the speed of the car at the beginning of each of the sixty seconds that it is traveling. Next, for each of those seconds, do a simple calculation to see how far the car would have gone during that second if the speed had remained exactly as it was at the beginning of the second. Finally, add up all these distances. Since one second is a short time, the speed will not change very much during any one second, so this procedure gives quite an accurate answer. Moreover, if you are not satisfied with this accuracy, then you can improve it by using intervals that are shorter than a second.

If you have done a first course in calculus, then you may well have solved such problems in a completely different way. In a typical question, one is given an explicit formula for the speed at time t—something like at + u, for example—and in order to work out how far the car has gone one “integrates” this function to obtain the formula at² + ut for the distance traveled at time t. Here, integration simply means the opposite of differentiation: to find the integral of a function f is to find a function such that ′(t) = f(t). This makes sense, because if (t) is the distance traveled and f(t) is the speed, then f(t) is indeed the rate of change of (t).

However, antidifferentiation is not the definition of integration. To see why not, consider the following question: what is the distance traveled if the speed at time t is ? It is known that there is no nice function (which means, roughly speaking, a function built up out of standard ones such as polynomials, exponentials, logarithms, and trigonometric functions) with as its derivative, yet the question still makes good sense and has a definite answer. (It is possible that you have heard of a function Φ(t) that differentiates to , from which it follows that Φ(t ) / differentiates to . However, this does not remove the difficulty, since Φ(t) is defined as the integral of .)

In order to define integration in situations like this where antidifferentiation runs into difficulties, we must fall back on messy approximations of the kind discussed earlier. A formal definition along such lines was given by RIEMANN [VI.49] in the mid nineteenth century. To see what Riemann’s basic idea is, and to see also that integration, like differentiation, is a procedure that can usefully be applied to functions of more than one variable, let us look at another physical problem.

Suppose that you have a lump of impure rock and wish to calculate its mass from its density. Suppose also that this density is not constant but varies rather irregularly through the rock. Perhaps there are even holes inside, so that the density is zero in places. What should you do?

Riemann’s approach would be this. First, you enclose the rock in a cuboid. For each point (x,y,z) in this cuboid there is then an associated density d(x,y,z) (which will be zero if (x,y,z) lies outside the rock or inside a hole). Second, you divide the cuboid into a large number of smaller cuboids. Third, in each of the small cuboids you look for the point of lowest density (if any point in the cuboid is not in the rock, then this density will be zero) and the point of highest density. Let C be one of the small cuboids and suppose that the lowest and highest densities in C are a and b, respectively, and that the volume of C is V. Then the mass of the part of the rock that lies in C must lie between aV and bV. Fourth, add up all the numbers aV that are obtained in this way, and then add up all the numbers bV. If the totals are M₁ and M₂, respectively, then the total mass of rock has to lie between M₁ and M₂. Finally, repeat this calculation for subdivisions into smaller and smaller cuboids. As you do this, the resulting numbers M₁ and M₂ will become closer and closer to each other, and you will have better and better approximations to the mass of the rock.

Similarly, his approach to the problem about the car would be to divide the minute up into small intervals and look at the minimum and maximum speeds during those intervals. For each interval, this would give him a pair of numbers a and b for which he could say that the car had traveled a distance of at least a and at most b. Adding up these sets of numbers, he could then say that over the full minute the car must have traveled a distance of at least D₁ (the sum of the as) and at most D₂ (the sum of the bs).

With both these problems we had a function (density/speed) defined on a set (the cuboid/a minute of time) and in a certain sense we wanted to work out the “total amount” of the function. We did so by dividing the set into small parts and doing simple calculations in those parts to obtain approximations to this amount from below and above. This process is what is known as (Riemann) integration. The following notation is common: if S is the set and f is the function, then the total amount of f in S, known as the integral, is written _s f(x) dx. Here, x denotes a typical element of S. If, as in the density example, the elements of S are points (x,y,z), then vector notation such as _s f(x) dx can be used, though often it is not and the reader is left to deduce from the context that an ordinary “x” denotes a vector rather than a real number.

We have been at pains to distinguish integration from antidifferentiation, but a famous theorem, known as the fundamental theorem of calculus, asserts that the two procedures do, in fact, give the same answer, at least when the function in question has certain continuity properties that all “sensible” functions have. So it is usually legitimate to regard integration as the opposite of differentiation. More precisely, if f is continuous and F(x) is defined to be f(t) dt for some a, then F can be differentiated and F′(x) = f(x). That is, if you integrate a continuous function and differentiate it again, you get back to where you started. Going the other way around, if F has a continuous derivative f and a < x, then f(t) dt = F(x) - F(a). This almost says that if you differentiate F and then integrate it again, you get back to F. Actually, you have to choose an arbitrary number a and what you get is the function F with the constant F(a) subtracted.

To get an idea of the sort of exceptions that arise if one does not assume continuity, consider the so-called Heaviside step function H(x), which is 0 when x < 0 and 1 when x ≥ 0. This function has a jump at 0 and is therefore not continuous. The integral J(x) of this function is 0 when x < 0 and x when x ≥ 0, and for almost all values of x we have J′(x) = H(x). However, the gradient of J suddenly changes at 0, so J is not differentiable there and one cannot say that J′(0) H(0) = 1.

5.6 Holomorphic Functions

One of the jewels in the crown of mathematics is complex analysis, which is the study of differentiable functions that take complex numbers to complex numbers. Functions of this kind are called holomorphic.

At first, there seems to be nothing special about such functions, since the definition of a derivative in this context is no different from the definition for functions of a real variable: if f is a function then the derivative f′(z)at a complex number z is defined to be the limit as h tends to zero of (f(z + h) - f(z)) / h. However, if we look at this definition in a slightly different way (one that we saw in section 5.3), we find that it is not altogether easy for a complex function to be differentiable. Recall from that section that differentiation means linear approximation. In the case of a complex function, this means that we would like to approximate it by functions of the form (w) = λw + μ, where λ and μ are complex numbers. (The approximation near z will be (w) = f(z) + f′(z)(w - z), which gives λ = f′ (z) and μ = f(z) - zf′(z).)

Let us regard this situation geometrically. If λ ≠ 0 then the effect of multiplying by λ is to expand z by some factor r and to rotate it by some angle θ. This means that many transformations of the plane that we would ordinarily consider to be linear, such as reflections, shears, or stretches, are ruled out. We need two real numbers to specify λ (whether we write it in the form a + bi or re^iθ), but to specify a general linear transformation of the plane takes four (see the discussion of matrices in section 4.2). This reduction in the number of degrees of freedom is expressed by a pair of differential equations called the Cauchy-Riemcinn equations. Instead of writing f(z) let us write u(x + iy) + iv(x + iy), where x and y are the real and imaginary parts of z and u(x + iy) and υ(x + iy) are the real and imaginary parts of f(x + iy). Then the linear approximation to f near z has the matrix

The matrix of an expansion and rotation always has the form , from which we deduce that

These are the Cauchy-Riemann equations. One consequence of these equations is that

(It is not obvious that the necessary conditions hold for the symmetry of the mixed partial derivatives, but when f is holomorphic they do.) Therefore, u satisfies the Laplace equation (which was discussed in section 5.4). A similar argument shows that υ does as well.

These facts begin to suggest that complex differentiability is a much stronger condition than real differentiability and that we should expect holomorphic functions to have interesting properties. For the remainder of this subsection, let us look at a few of the remarkable properties that they do indeed have.

The first is related to the fundamental theorem of calculus (discussed in the previous subsection). Suppose that F is a holomorphic function and that we are given its derivative f and the value of F(u) for some complex number u. How can we reconstruct F? An approximate method is as follows. Let w be another complex number and let us try to work out F(w). We take a sequence of points z₀, z_l, . . . , z_n with z₀ = u and z_n = w, and with the differences | z₁ - z₀ |, | z₂ - z₁ | , . . . , | z_n - z_{n - 1} | all small. We can then approximate F(z_{i + 1}) - F(zi) by (z_{i + 1} - z_i) f(z_i). It follows that F(w) - F(u), which equals F(z) - F(z₀), is approximated by the sum of all the (z_{i + l} - z_i) f(z_i). (Since we have added together many small errors, it is not obvious that this approximation is a good one, but it turns out that it is.) We can imagine a number z that starts at u and follows a path P to w by jumping from one z_i to another in small steps of δz = z_{i + 1} - z_i. In the limit as n goes to infinity and the steps δz go to zero we obtain a so-called path integral, which is denoted ∫p f(z) dz.

The above argument has the consequence that if the path P begins and ends at the same point u, then the path integral ∫P f(z) dz is zero. Equivalently, if two paths P₁ and P₂ have the same starting point u and the same endpoint w, then the path integrals ∫P₁ f(z) dz and ∫P₂ f(z) dz are the same, since they both give the value F(w) - F(u).

Of course, in order to establish this, we made the big assumption that f was the derivative of a function F. Cauchy’s theorem says that the same conclusion is true if f is holomorphic. That is, rather than requiring f to be the derivative of another function, it asks for f itself to have a derivative. If that is the case, then any path integral of f depends only on where the path begins and ends. What is more, these path integrals can be used to define a function F that differentiates to f, so a function with a derivative automatically has an antiderivative.

It is not necessary for the function f to be defined on the whole of for Cauchy’s theorem to be valid: everything remains true if we restrict attention to a simply connected domain, which means an OPEN SET [III.90] with no holes in it. If there are holes, then two path integrals may differ if the paths go around the holes in different ways. Thus, path integrals have a close connection with the topology of subsets of the plane, an observation that has many ramifications throughout modern geometry. For more on topology, see section 6.4 of this article and ALGEBRAIC TOPOLOGY [IV.6].

A very surprising fact, which can be deduced from Cauchy’s theorem, is that if f is holomorphic then it can be differentiated twice. (This is completely untrue of real-valued functions: consider, for example, the function f where f(x) = 0 when x < 0 and f(x) = x² when x ≥ 0.) It follows that f′ is holomorphic, so it too can be differentiated twice. Continuing, one finds that f can be differentiated any number of times. Thus, for complex functions differentiability implies infinite differentiability. (This property is what is used to establish the symmetry, and even the existence, of the mixed partial derivatives mentioned earlier.)

A closely related fact is that wherever a holomorphic function is defined it can be expanded in a power series. That is, if f is defined and differentiable everywhere on an open disk of radius R about w, then it will be given by a formula of the form

f(z) = a_n (z - w)ⁿ,

valid everywhere in that disk. This is called the Taylor expansion of f.

Another fundamental property of holomorphic functions, one that shows just how “rigid” they are, is that their entire behavior is determined just by what they do in a small region. That is, if f and g are holomorphic and they take the same values in some tiny disk, then they must take the same values everywhere. This remarkable fact allows a process of analytic continuation. If it is difficult to define a holomorphic function f everywhere you want it defined, then you can simply define it in some small region and say that elsewhere it takes the only possible values that are consistent with the ones that you have just specified. This is how the famous RIEMANN ZETA FUNCTION [IV.2 §3] is conventionally defined.

Finally, we mention a theorem of LIOUVILLE [VI.39], which states that if f is a holomorphic function defined on the whole complex plane, and if f is bounded (that is, if there is some constant C such that |f(z)| ≤ C for every complex number z), then f must be constant. Once again, this is obviously false for real functions. For example, the function sin(x) has no difficulty combining boundedness with very good behavior: it can be expanded in a power series that converges everywhere. (However, if you use the power series to define an extension of the function sin(x) to the complex plane, then the function you obtain is unbounded, as Liouville’s theorem predicts.)

6 What Is Geometry?

It is not easy to do justice to geometry in this article because the fundamental concepts of the subject are either too simple to need explaining—for example, there is no need to say here what a circle, line, or plane is—or sufficiently advanced that they are better discussed in parts III and IV of the book. However, if you have not met the advanced concepts and have no idea what modern geometry is like, then you will get much more out of this book if you understand two basic ideas: the relationship between geometry and symmetry, and the notion of a manifold. These ideas will occupy us for the rest of the article.

6.1 Geometry and Symmetry Groups

Broadly speaking, geometry is the part of mathematics that involves the sort of language that one would conventionally regard as geometrical, with words such as “point,” “line,” “plane,” “space,” “curve,” “sphere,” “cube,” “distance,” and “angle” playing a prominent role. However, there is a more sophisticated view, first advocated by KLEIN [VI.57], that regards transformations as the true subject matter of geometry. So, to the above list one should add words like “reflection,” “rotation,” “translation,” “stretch,” “shear,” and “projection,” together with slightly more nebulous concepts such as “angle-preserving map” or “continuous deformation.”

As was discussed in section 2.1, transformations go hand in hand with groups, and for this reason there is an intimate connection between geometry and group theory. Indeed, given any group of transformations, there is a corresponding notion of geometry, in which one studies the phenomena that are unaffected by transformations in that group. In particular, two shapes are regarded as equivalent if one can be turned into the other by means of one of the transformations in the group. Different groups will of course lead to different notions of equivalence, and for this reason mathematicians frequently talk about geometries, rather than about a single monolithic subject called geometry. This subsection contains brief descriptions of some of the most important geometries and their associated groups of transformations.

6.2 Euclidean Geometry

Euclidean geometry is what most people would think of as “ordinary” geometry, and, not surprisingly given its name, it includes the basic theorems of Greek geometry that were the staple of geometers for over two millennia. For example, the theorem that the three angles of a triangle add up to 180° belongs to Euclidean geometry.

To understand Euclidean geometry from a transformational viewpoint, we need to say how many dimensions we are working in, and we must of course specify a group of transformations. The appropriate group is the group of rigid transformations. These can be thought of in two different ways. One is that they are the transformations of the plane, or of space, or more generally of ⁿ for some n, that preserve distance. That is, T is a rigid transformation if, given any two points x and y, the distance between Tx and Ty is always the same as the distance between x and y. (In dimensions greater than 3, distance is defined in a way that naturally generalizes the Pythagorean formula. See METRIC SPACES [III.56] for more details.)

It turns out that every such transformation can be realized as a combination of rotations, reflections, and translations, and this gives us a more concrete way to think about the group. Euclidean geometry, in other words, is the study of concepts that do not change when you rotate, reflect, or translate, and these include points, lines, planes, circles, spheres, distance, angle, length, area, and volume. The rotations of ⁿ form an important group, the special orthogonal group, known as SO(n). The larger orthogonal group O(n) includes reflections as well. (It is not quite obvious how to define a “rotation” of n-dimensional space, but it is not too hard to do. An orthogonal map of ⁿ is a linear map T that preserves distances, in the sense that d(Tx, Ty) is always the same as d(x,y). It is a rotation if its DETERMINANT [III.15] is 1. The only other possibility for the determinant of a distance-preserving map is -1. Maps with determinant -1 are like reflections in that they turn space “inside out.”)

6.3 Affine Geometry

There are many linear maps besides rotations and reflections. What happens if we enlarge our group from SO(n) or O(n) to include as many of them as possible? For a transformation to be part of a group it must be invertible and not all linear maps are, so the natural group to look at is the group GL_n() of all invertible linear transformations of ⁿ, a group that we first met in section 4.2. These maps all leave the origin fixed, but if we want we can incorporate translations and consider a larger group that consists of all transformations of the form x Tx + b, where b is a fixed vector and T is an invertible linear map. The resulting geometry is called affine geometry.

Since linear maps include stretches and shears, they preserve neither distance nor angle, so these are not concepts of affine geometry. However, points, lines, and planes remain as points, lines, and planes after an invertible linear map and a translation, so these concepts do belong to affine geometry. Another affine concept is that of two lines being parallel. (That is, although angles in general are not preserved by linear maps, angles of zero are.) This means that although there is no such thing as a square or a rectangle in affine geometry, one can still talk about a parallelogram. Similarly, one cannot talk of circles but one can talk of ellipses, since a linear map transformation of an ellipse is another ellipse (provided that one regards a circle as a special kind of ellipse).

6.4 Topology

The idea that the geometry associated with a group of transformations “studies the concepts that are preserved by all the transformations” can be made more precise using the notion of EQUIVALENCE RELATIONS [I.2 §2.3]. Indeed, let G be a group of transformations of ⁿ. We might think of an n-dimensional “shape” as being a subset S of ⁿ, but if we are doing G-geometry, then we do not want to distinguish between a set S and any other set we can obtain from it using a transformation in G So in that case we say that the two shapes are equivalent. For example, two shapes are equivalent in Euclidean geometry if and only if they are congruent in the usual sense, whereas in two-dimensional affine geometry all parallelograms are equivalent, as are all ellipses. One can think of the basic objects of G-geometry as equivalence classes of shapes rather than the shapes themselves.

Topology can be thought of as the geometry that arises when we use a particularly generous notion of equivalence, saying that two shapes are equivalent, or homeomorphic, to use the technical term, if each can be “continuously deformed” into the other. For example, a sphere and a cube are equivalent in this sense, as figure 1 illustrates.

Because there are very many continuous deformations, it is quite hard to prove that two shapes are not equivalent in this sense. For example, it may seem obvious that a sphere (this means the surface of a ball rather than the solid ball) cannot be continuously deformed into a torus (the shape of the surface of a doughnut of the kind that has a hole in it), since they are fundamentally different shapes—one has a “hole” and the other does not. However, it is not easy to turn this intuition into a rigorous argument. For more on this kind of problem, see INVARIANTS [I.4 §2.2], ALGEBRAIC TOPOLOGY [IV.6], and DIFFERENTIAL TOPOLOGY [IV.7].

Figure 1 A sphere morphing into a cube.

6.5 Spherical Geometry

We have been steadily relaxing our requirements for two shapes to be equivalent, by allowing more and more transformations. Now let us tighten up again and look at spherical geometry. Here the universe is no longer ⁿ but the n-dimensional sphere Sⁿ, which is defined to be the surface of the (n + 1)-dimensional ball of radius 1, or, to put it more algebraically, the set of all points (x₁, x₂, . . . , x_n+1) in ⁿ⁺¹ such that + + · · · + = 1. Just as the surface of a three-dimensional ball is two dimensional, so this set is n dimensional. We shall discuss the case n = 2 here, but it is easy to generalize the discussion to larger n.

The appropriate group of transformations is SO(3): the group that consists of all rotations about axes that go through the origin. (One could allow reflections as well and take O(3).) These are symmetries of the sphere S², and that is how we regard them in spherical geometry, rather than as transformations of the whole of ³.

Among the concepts that make sense in spherical geometry are line, distance, and angle. It may seem odd to talk about a line if one is confined to the surface of a ball, but a “spherical line” is not a line in the usual sense. Rather, it is a subset of S² obtained by intersecting S² with a plane through the origin. This produces a great circle, that is, a circle of radius 1, which is as large as it can be given that it lives inside a sphere of radius 1.

The reason that a great circle deserves to be thought of as some sort of line is that the shortest path between any two points x and y in S² will always be along a great circle, provided that the path is confined to S². This is a very natural restriction to make, since we are regarding S² as our “universe.” It is also a restriction of some practical relevance, since the shortest sensible route between two distant points on Earth’s surface will not be the straight-line route that burrows hundreds of miles underground.

The distance between two points x and y is defined to be the length of the shortest path from x to y that lies entirely in S². (If x and y are opposite each other, then there are infinitely many shortest paths, all of length π, so the distance between x and y is π.) How about the angle between two spherical lines? Well, the lines are intersections of S² with two planes, so one can define it to be the angle between these two planes in the Euclidean sense. A more aesthetically pleasing way to view this, because it does not involve ideas external to the sphere, is to notice that if you look at a very small region about one of the two points where two spherical lines cross, then that portion of the sphere will be almost flat, and the lines almost straight. So you can define the angle to be the usual angle between the “limiting” straight lines inside the limiting plane.

Spherical geometry differs from Euclidean geometry in several interesting ways. For example, the angles of a spherical triangle always add up to more than 180°. Indeed, if you take as the vertices the North Pole, a point on the equator, and a second point a quarter of the way around the equator from the first, then you obtain a triangle with three right angles. The smaller a triangle, the flatter it becomes, and so the closer the sum of its angles comes to 180°. There is a beautiful theorem that gives a precise expression to this: if we switch to radians, and if we have a spherical triangle with angles α, β, and γ, then its area is α + β + γ - π. (For example, this formula tells us that the triangle with three angles of π has area π, which indeed it does as the surface area of a ball of radius 1 is 4π and this triangle occupies one-eighth of the surface.)

6.6 Hyperbolic Geometry

So far, the idea of defining geometries with reference to sets of transformations may look like nothing more than a useful way to view the subject, a unified approach to what would otherwise be rather different-looking aspects. However, when it comes to hyperbolic geometry, the transformational approach becomes indispensable, for reasons that will be explained in a moment.

The group of transformations that produces hyperbolic geometry is called PSL₂ (), the projective special linear group in two dimensions. One way to present this group is as follows. The special linear group SL₂ () is the set of all matrices with DETERMINANT [III.15] ad - bc equal to 1. (These form a group because the product of two matrices with determinant 1 again has determinant 1.) To make this “projective,” one then regards each matrix A as equivalent to -A:for example, the matrices and are equivalent.

To get from this group to the geometry one must first interpret it as a group of transformations of some two- dimensional set of points. Once we have done this, we have what is called a model of two-dimensional hyperbolic geometry. The subtlety is that there is no single model of hyperbolic geometry that is clearly the most natural in the way that the sphere is the most natural model of spherical geometry. (One might think that the sphere was the only sensible model of spherical geometry, but this is not in fact the case. For example, there is a natural way of associating with each rotation of ³ a transformation of ² with a “point at infinity” added, so the extended plane can be used as a model of spherical geometry.) The three most commonly used models of hyperbolic geometry are called the half-plane model, the disk model, and the hyperboloid model.

The half-plane model is the one most directly associated with the group PSL₂ (). The set in question is the upper half-plane of the complex numbers , that is, the set of all complex numbers z = x + iy such that y > 0. Given a matrix (), the corresponding transformation is the one that takes the point z to the point (az + b)/(cz + d). (Notice that if we replace a, b, c, and d by their negatives, then we get the same transformation.) The condition ad - bc = 1 can be used to show that the transformed point will still lie in the upper half-plane, and also that the transformation can be inverted.

What this does not yet do is tell us anything about distances, and it is here that we need the group to “generate” the geometry. If we are to have a notion of distance d that is sensible from the perspective of our group of transformations, then it is important that the transformations should preserve it. That is, if T is one of the transformations and z and w are two points in the upper half-plane, then d(T(z), T(w)) should always be the same as d(z, w). It turns out that there is essentially only one definition of distance that has this property, and that is the sense in which the group defines the geometry. (One could of course multiply all distances by some constant factor such as 3, but this would be like measuring distances in feet instead of yards, rather than a genuine difference in the geometry.)

This distance has some properties that at first seem odd. For example, a typical hyperbolic line takes the form of a semicircular arc with endpoints on the real axis. However, it is semicircular only from the point of view of the Euclidean geometry of : from a hyperbolic perspective it would be just as odd to regard a Euclidean straight line as straight. The reason for the discrepancy is that hyperbolic distances become larger and larger, relative to Euclidean ones, the closer you get to the real axis. To get from a point z to another point w, it is therefore shorter to take a “detour” away from the real axis, and the best detour turns out to be along an arc of the circle that goes through z and w and cuts the real axis at right angles. (If z and w are on the same vertical line, then one obtains a “degenerate circle,” namely that vertical line.) These facts are no more paradoxical than the fact that a flat map of the world involves distortions of spherical geometry, making Greenland very large, for example. The half-plane model is like a “map” of a geometric structure, the hyperbolic plane, that in reality has a very different shape.

One of the most famous properties of two-dimensional hyperbolic geometry is that it provides a geometry in which Euclid’s parallel postulate fails to hold. That is, it is possible to have a hyperbolic line L, a point x not on the line, and two different hyperbolic lines through x, neither of which meets L. All the other axioms of Euclidean geometry are, when suitably interpreted, true of hyperbolic geometry as well. It follows that the parallel postulate cannot be deduced from those axioms. This discovery, associated with GAUSS [VI.26], BOLYAI [VI.34], and LOBACHEVSKII [VI.31], solved a problem that had bothered mathematicians for over two thousand years.

Another property complements the result about the angle sums of spherical and Euclidean triangles. There is a natural notion of hyperbolic area, and the area of a hyperbolic triangle with angles α, β, and γ is π - α - β - γ. Thus, in the hyperbolic plane α + β + γ is always less than π, and it almost equals π when the triangle is very small. These properties of angle sums reflect the fact that the sphere has positive CURVATURE [III.13], the Euclidean plane is “flat,” and the hyperbolic plane has negative curvature.

The disk model, conceived in a famous moment of inspiration by POINCARÉ [VI.61] as he was getting into a bus, takes as its set of points the open unit disk in , that is, the set D of all complex numbers with modulus less than 1. This time, a typical transformation takes the following form. One takes a real number θ, and a complex number a from inside D, and sends each z in D to the point e^iθ(z - a)/(1 - z). It is not completely obvious that these transformations form a group, and still less that the group is isomorphic to PSL₂ (). However, it turns out that the function that takes z to -(iz + 1)/(z+ i) maps the unit disk to the upper half-plane and vice versa. This shows that the two models give the same geometry and can be used to transfer results from one to the other.

Figure 2 A tessellation of the hyperbolic disk.

As with the half-plane model, distances become larger, relative to Euclidean distances, as you approach the boundary of the disk: from a hyperbolic perspective, the diameter of the disk is infinite and it does not really have a boundary. Figure 2 shows a tessellation of the disk by shapes that are congruent in the sense that any one can be turned into any other by means of a transformation from the group. Thus, even though they do not look identical, within hyperbolic geometry they all have the same size and shape. Straight lines in the disk model are either arcs of (Euclidean) circles that meet the unit circle at right angles, or segments of (Euclidean) straight lines that pass through the center of the disk.

The hyperboloid model is the model that explains why the geometry is called hyperbolic. This time the set is the hyperboloid consisting of all points (x,y,z) ∈ ³ such that z > 0 and x² + y² + 1 = z². This is the hyperboloid of revolution about the z-axis of the hyperbola x²+1 = z² in the plane y = 0. A general transformation in the group is a sort of “rotation” of the hyperboloid, and can be built up from genuine rotations about the z-axis, and “hyperbolic rotations” of the xz-plane, which have matrices of the form

Just as an ordinary rotation preserves the unit circle, one of these hyperbolic rotations preserves the hyperbola x² + 1 = z², moving points around inside it. Again, it is not quite obvious that this gives the same group of transformations, but it does, and the hyperboloid model is equivalent to the other two.

6.7. Projective Geometry

Projective geometry is regarded by many as an old-fashioned subject, and it is no longer taught in schools, but it still has an important role to play in modern mathematics. We shall concentrate here on the real projective plane, but projective geometry is possible in any number of dimensions and with scalars in any field. This makes it particularly useful to algebraic geometers.

Here are two ways of regarding the projective plane. The first is that the set of points is the ordinary plane, together with a “line at infinity.” The group of transformations consists of functions known as projections. To understand what a projection is, imagine two planes P and P′ in space, and a point x that is not in either of them. We can “project P onto P′ as follows. If a is a point in P, then its image (a) is the point where the line joining x to a meets P′. (If this line is parallel to P′, then (a) is a point on the line at infinity of P′.) Thus, if you are at x and a picture is drawn on the plane P, then its image under the projection will be the picture drawn on P′ that to you looks exactly the same. In fact, however, it will have been distorted, so the transformation has made a difference to the shape. To turn into a transformation of P itself, one can follow it by a rigid transformation that moves P′ back to where P is.

Such projections clearly do not preserve distances, but they do preserve other interesting concepts, such as points, lines, quantities known as cross-ratios, and, most famously, conic sections. A conic section is the intersection of a plane with a cone, and it can be a circle, an ellipse, a parabola, or a hyperbola. From the point of view of projective geometry, these are all the same kind of object (just as, in affine geometry, one can talk about ellipses but there is no special ellipse called a circle).

A second view of the projective plane is that it is the set of all lines in ³ that go through the origin. Since a line is determined by the two points where it intersects the unit sphere, one can regard this set as a sphere, but with the significant difference that opposite points are regarded as the same—because they correspond to the same line.

Under this view, a typical transformation of the projective plane is obtained as follows. Take any invertible linear map, and apply it to ³. This takes lines through the origin to lines through the origin, and can therefore be thought of as a function from the projective plane to itself. If one invertible linear map is a multiple of another, then they will have the same effect on all lines, so the resulting group of transformations is like GL₃ (), except that all nonzero multiples of any given matrix are regarded as equivalent. This group is called the projective special linear group PSL₃ (), and it is the three-dimensional equivalent of PSL₂ (), which we have already met. Since PSL₃ () is bigger than PSL₂ (), the projective plane comes with a richer set of transformations than the hyperbolic plane, which is why fewer geometrical properties are preserved. (For example, we have seen that there is a useful notion of hyperbolic distance, but there is no obvious notion of projective distance.)

6.8. Lorentz Geometry

This is a geometry used in the theory of special relativity to model four-dimensional spacetime, otherwise known as Minkowski space. The main difference between it and four-dimensional Euclidean geometry is that, instead of the usual notion of distance between two points (t, x, y, z) and (t′, x′, y′, z′), one considers the quantity

-(t - t′)² + (x - x′)² + (y - y′)² + (z - z′)²,

which would be the square of the Euclidean distance were it not for the all-important minus sign before (t - t′)². This reflects the fact that space and time are significantly different (though intertwined).

A Lorentz transformation is a linear map from ⁴ to ⁴ that preserves these “generalized distances.” Letting g be the linear map that sends (t, x, y, z) to (-t, x, y, z) and letting G be the corresponding matrix (which has -1, 1, 1, 1 down the diagonal and 0 everywhere else), we can define a Lorentz transformation abstractly as one whose matrix Λ satisfies Λ^TΛ = G, where I is the 4 × 4 identity matrix and Λ^T is the transpose of Λ. (The transpose of a matrix A is the matrix B defined by B_ij = A_ji.)

A point (t, x, y, z) is said to be spacelike if -t² + x² + y² + z² > 0, and timelike if -t² + x² + y² + z² < 0. If -t² + x² + y² + z² = 0, then the point lies in the light cone. All these are genuine concepts of Lorentzian geometry because they are preserved by Lorentz transformations.

Lorentzian geometry is also of fundamental importance to general relativity, which can be thought of as the study of Lorentzian manifolds. These are closely related to Riemannien manifolds, which are discussed in section 6.10. For a discussion of general relativity, see GENERAL RELATIVITY AND THE EINSTEIN EQUATIONS [IV.13].

6.9 Manifolds and Differential Geometry

To somebody who has not been taught otherwise, it is natural to think that Earth is flat, or rather that it consists of a flat surface on top of which there are buildings, mountains, and so on. However, we now know that it is in fact more like a sphere, appearing to be flat only because it is so large. There are various kinds of evidence for this. One is that if you stand on a cliff by the sea then you can see a definite horizon, not too far away, over which ships disappear. This would be hard to explain if Earth were genuinely flat. Another is that if you travel far enough in what feels like a straight line then you eventually get back to where you started. A third is that if you travel along a triangular route and the triangle is a large one, then you will be able to detect that its three angles add up to more than 180°.

It is also very natural to believe that the geometry that best models that of the universe is three-dimensional Euclidean geometry, or what one might think of as “normal” geometry. However, this could be just as much of a mistake as believing that two-dimensional Euclidean geometry is the best model for Earth’s surface.

Indeed, one can immediately improve on it by considering Lorentzian geometry as a model of spacetime, but even if there were no theory of special relativity, our astronomical observations would give us no particular reason to suppose that Euclidean geometry was the best model for the universe. Why should we be so sure that we would not obtain a better model by taking the three-dimensional surface of a very large four- dimensional ball? This might feel like “normal” space in just the way that the surface of Earth feels like a “normal” plane unless you travel large distances. Perhaps if you traveled far enough in a rocket without changing your course then you would end up where you started.

It is easy to describe “normal” space mathematically: one just associates with each point in space a triple of coordinates (x,y,z) in the usual way. How might we describe a huge “spherical” space? It is slightly harder, but not much: one can give each point four coordinates (x, y, z, w) but add the condition that these must satisfy the equation x² + y² + z² + w² = R² for some fixed R that we think of as the “radius” of the universe. This describes the three-dimensional surface of a four-dimensional ball of radius R in just the same way that the equation x² + y² + z² = R² describes the two-dimensional surface of a three-dimensional ball of radius R.

A possible objection to this approach is that it seems to rely on the rather implausible idea that the universe lives in some larger unobserved four-dimensional space. However, this objection can be answered. The object we have just defined, the 3-sphere S³, can also be described in what is known as an intrinsic way: that is, without reference to some surrounding space. The easiest way to see this is to discuss the 2-sphere first, in order to draw an analogy.

Let us therefore imagine a planet covered with calm water. If you drop a large rock into the water at the North Pole, a wave will propagate out in a circle of ever- increasing radius. (At any one moment, it will be a circle of constant latitude.) In due course, however, this circle will reach the equator, after which it will start to shrink, until eventually the whole wave reaches the South Pole at once, in a sudden burst of energy.

Now imagine setting off a three-dimensional wave in space—it could, for example, be a light wave caused by the switching on of a bright light. The front of this wave would now be not a circle but an ever- expanding spherical surface. It is logically possible that this surface could expand until it became very large and then contract again, not by shrinking back to where it started, but by turning itself inside out, so to speak, and shrinking to another point on the opposite side of the universe. (Notice that in the two-dimensional example, what you want to call the inside of the circle changes when the circle passes the equator.) With a bit of effort, one can visualize this possibility, and there is no need to appeal to the existence of a fourth dimension in order to do so. More to the point, this account can be turned into a mathematically coherent and genuinely three-dimensional description of the 3-sphere.

A different and more general approach is to use what is called an atlas. An atlas of the world (in the normal, everyday sense) consists of a number of flat pages, together with an indication of their overlaps: that is, of how parts of some pages correspond to parts of others. Now, although such an atlas is mapping out an external object that lives in a three-dimensional universe, the spherical geometry of Earth’s surface can be read off from the atlas alone. It may be much less convenient to do this but it is possible: rotations, for example, might be described by saying that such-and-such a part of page 17 moved to a similar but slightly distorted part of page 24, and so on.

Not only is this possible, but one can define a surface by means of two-dimensional atlases. For example, there is a mathematically neat “atlas” of the 2-sphere that consists of just two pages, both of them circular. One is a map of the Northern Hemisphere plus a little bit of the Southern Hemisphere near the equator (to provide a small overlap) and the other is a map of the Southern Hemisphere with a bit of the Northern Hemisphere. Because these maps are flat, they necessarily involve some distortion, but one can specify what this distortion is.

The idea of an atlas can easily be generalized to three dimensions. A “page” now becomes a portion of three- dimensional space. The technical term is not “page” but “chart,” and a three-dimensional atlas is a collection of charts, again with specifications of which parts of one chart correspond to which parts of another. A possible atlas of the 3-sphere, generalizing the simple atlas of the 2-sphere just discussed, consists of two solid three- dimensional balls. There is a correspondence between points toward the edge of one of these balls and points toward the edge of the other, and this can be used to describe the geometry: as you travel toward the edge of one ball you find yourself in the overlapping region, so you are also in the other ball. As you go further, you are off the map as far as the first ball is concerned, but the second ball has by that stage taken over.

The 2-sphere and the 3-sphere are basic examples of manifolds. Other examples that we have already met in this section are the torus and the projective plane. Informally, a d-dimensional manifold, or d-manifold, is any geometrical object M with the property that every point x in M is surrounded by what feels like a portion of d-dimensional Euclidean space. So, because small parts of a sphere, torus, or projective plane are very close to planar, they are all 2-manifolds, though when the dimension is two the word surface is more usual. (However, it is important to remember that a “surface” need not be the surface of anything.) Similarly, the 3-sphere is a 3-manifold.

The formal definition of a manifold uses the idea of atlases: indeed, one says that the atlas is a manifold. This is a typical mathematician’s use of the word is and it should not be confused with the normal use. In practice, it is unusual to think of a manifold as a collection of charts with rules for how parts of them correspond, but the definition in terms of charts and atlases turns out to be the most convenient when one wishes to reason about manifolds in general rather than discussing specific examples. For the purposes of this book, it may be better to think of a d-manifold in the “extrinsic” way that we first thought about the 3-sphere: as a d-dimensional “hypersurface” living in some higher-dimensional space. Indeed, there is a famous theorem of Nash that states that all manifolds arise in this way. Note, however, that it is not always easy to find a simple formula for defining such a hyper- surface. For example, while the 2-sphere is described by the simple formula x² + y² + z² = 1 and the torus by the slightly more complicated and more artificial formula (r - 2)² + z² = 1, where r is shorthand for, it is not easy to come up with a formula that describes a two-holed torus. Even the usual torus is far more easily described using quotients, as we did in section 3.3. Quotients can also be used to define a two-holed torus (see FUCHSIAN GROUPS [III.28]), and the reason one is confident that the result is a manifold is that every point has a small neighborhood that looks like a small part of the Euclidean plane. In general, a d-dimensional manifold can be thought of as any construction that gives rise to an object that is “locally like Euclidean space of d dimensions.”

An extremely important feature of manifolds is that calculus is possible for functions defined on them. Roughly speaking, if M is a manifold and f is a function from M to , then to see whether f is differentiable at a point x in M you first find a chart that contains x (or a representation of it), and regard f as a function defined on the chart instead. Since the chart is a portion of the d-dimensional Euclidean space ^d and we can differentiate functions defined on such sets, the notion of differentiability now makes sense for f. Of course, for this definition to work for the manifold, it is important that if x belongs to two overlapping charts, then the answer will be the same for both. This is guaranteed if the function that gives the correspondence between the overlapping parts (known as a transition function) is itself differentiable. Manifolds with this property are called differentiable manifolds: manifolds for which the transition functions are continuous but not necessarily differentiable are called topological manifolds. The availability of calculus makes the theory of differentiable manifolds very different from that of topological manifolds.

The above ideas generalize easily from real-valued functions to functions from M to ^d, or from M to M′, where M′ is another manifold. However, it is easier to judge whether a function defined on a manifold is differentiable than it is to say what the derivative is. The derivative at some point x of a function from ⁿ to ^m is a linear map, and so is the derivative of a function defined on a manifold. However, the domain of the linear map is not the manifold itself, which is not usually a vector space, but rather the so-called tangent space at the point x in question.

For more details on this and on manifolds in general, see DIFFERENTIAL TOPOLOGY [IV.7].

6.10. Riemannian Metrics

Suppose you are given two points P and Q on a sphere. How do you determine the distance between them? The answer depends on how the sphere is defined. If it is the set of all points (x,y,z) such that x² + y² + z² = 1 then P and Q are points in ³. One can therefore use the Pythagorean theorem to calculate the distance between them. For example, the distance between the points (1, 0, 0) and (0, 1, 0) is .

However, do we really want to measure the length of the line segment PQ? This segment does not lie in the sphere itself, so to use it as a means of defining length does not sit at all well with the idea of a manifold as an intrinsically defined object. Fortunately, as we saw earlier in the discussion of spherical geometry, there is another natural definition that avoids this problem: we can define the distance between P and Q as the length of the shortest path from P to Q that lies entirely within the sphere.

Now let us suppose that we wish to talk more generally about distances between points in manifolds. If the manifold is presented to us as a hypersurface in some bigger space, then we can use lengths of shortest paths as we did in the sphere. But suppose that the manifold is presented differently and all we have is a way of demonstrating that every point is contained in a chart—that is, has a neighborhood that can be associated with a portion of d-dimensional Euclidean space. (For the purposes of this discussion, nothing is lost if one takes d to be 2 throughout, in which case there is a correspondence between the neighborhood and a portion of the plane.) One idea is to define the distance between the two points to be the distance between the corresponding points in the chart, but this raises at least three problems.

The first is that the points P and Q that we are looking at might belong to different charts. This, however, is not too much of a problem, since all we actually need to do is calculate lengths of paths, and that can be done provided we have a way of defining distances between points that are very close together, in which case we can find a single chart that contains them both.

The second problem, which is much more serious, is that for any one manifold there are many ways of choosing the charts, so this idea does not lead to a single notion of distance for the manifold. Worse still, even if one fixes one set of charts, these charts will overlap, and it may not be possible to make the notions of distance compatible where the overlap occurs.

The third problem is related to the second. The surface of a sphere is curved, whereas the charts of any atlas (in either the everyday or the mathematical sense) are flat. Therefore, the distances in the charts cannot correspond exactly to the lengths of shortest paths in the sphere itself.

The single most important moral to draw from the above problems is that if we wish to define a notion of distance for a given manifold, we have a great deal of choice about how to do so. Very roughly, a Riemannian metric is a way of making such a choice.

A little less roughly, a metric means a sensible notion of distance (the precise definition can be found in [III.56]). A Riemannian metric is away of determining infinitesimal distances. These infinitesimal distances can be used to calculate lengths of paths, and then the distance between two points can be defined as the length of the shortest path between them. To see how this is done, let us first think about lengths of paths in the ordinary Euclidean plane. Suppose that (x,y) belongs to a path and (x + δx, y + δy) is another point on the path, very close to (x,y). Then the distance between the two points is . To calculate the length of a sufficiently smooth path, one can choose a large number of points along the path, each one very close to the next, and add up their distances. This gives a good approximation, and one can make it better and better by taking more and more points.

In practice, it is easier to work out the length using calculus. A path itself can be thought of as a moving point (x(t), y(t)) that starts when t = 0 and ends when t = 1. If δt is very small, then x(t + δt) is approximately x(t) + x′(t)δt and y(t + δt) is approximately y(t) + y′(t)δt. Therefore, the distance between (x(t), y(t)) and (x(t + δt), y(t + δt)) is approximately δt, by the Pythagorean theorem. Therefore, letting δt go to zero and integrating all the infinitesimal distances along the path, we obtain the formula

for the length of the path. Notice that if we write x′(t) and y′(t) as dx/dt and dy/dt, then we can rewrite dt as , which is the infinitesimal version of the expression that we had earlier. We have just defined a Riemannian metric, which is usually denoted by dx² + dy². This can be thought of as the square of the distance between the point (x,y) and the infinitesimally close point (x + dx,y + dy).

If we want to, we can now prove that the shortest path between two points (x₀, y₀) and (x₁, y₁) is a straight line, which will tell us that the distance between them is . (A proof can be found in VARIATIONAL METHODS [III.94].) However, since we could have just used this formula to begin with, this example does not really illustrate what is distinctive about Riemannian metrics. To do that, let us give a more precise definition of the disk model for hyperbolic geometry, which was discussed in section 6.6. There it was stated that distances become larger, relative to Euclidean distances, as one approaches the edge of the disk. A more precise definition is that the open unit disk is the set of all points (x,y) such that x² + y² < 1 and that the Riemannian metric on this disk is given by the expression (dx² + dy²)/(1-x²-y²). This is how we define the square of the distance between (x,y) and (x + dx,y + dy). Equivalently, the length of a path (x(t), y(t)) with respect to this Riemannian metric is defined as

More generally, a Riemannian metric on a portion of the plane is an expression of the form

E(x,y) dx² + 2F(x,y) dx dy + G(x,y) dy²

that is used to calculate infinitesimal distances and hence lengths of paths. (In the disk model we took E(x,y) and G(x,y) to be 1/(1 - x² - y²) and F(x,y) to be 0.) It is important for these distances to be positive, which will turn out to be the case provided that E(x,y)G(x,y) - F(x,y)² is always positive. One also needs the functions E, F, and G to satisfy certain smoothness conditions.

This definition generalizes straightforwardly to more dimensions. In n dimensions we must use an expression of the form

to specify the squared distance between the points (x₁, . . . ,x_n) and (x₁ + dx₁, . . . , x_n + dx_n). The numbers F_ij (x₁, . . . , x_n) form an n × n matrix that depends on the point (x₁, . . . , x_n. This matrix is required to be symmetric and positive definite: that is, F_ij(x₁, . . . ,x_n) should always equal Fji(x₁, . . . , x_n), and the expression that determines the squared distance should always be positive. It should also depend smoothly on the point (x₁, . . . ,x_n).

Finally, now that we know how to define many different Riemannian metrics on portions of Euclidean space, we have many potential ways to define metrics on the charts that we use to define a manifold. A Riemannian metric on a manifold is a way of choosing compatible Riemannian metrics on the charts, where “compatible” means that wherever two charts overlap the distances should be the same. As mentioned earlier, once one has done this, one can define the distance between two points to be the length of a shortest path between them.

Given a Riemannian metric on a manifold, it is possible to define many other concepts, such as angles and volumes. It is also possible to define the important concept of curvature, which is discussed in RICCI FLOW [III.78]. Another important definition is that of a geodesic, which is the analogue for Riemannian geometry of a straight line in Euclidean geometry. A curve C is a geodesic if, given any two points P and Q on C that are sufficiently close, the shortest path from P to Q is part of C. For example, the geodesics on the sphere are the great circles.

As should be clear by now from the above discussion, on any given manifold there is a multitude of possible Riemannian metrics. A major theme in Riemannian geometry is to choose one that is “best” in some way. For example, on the sphere, if we take the obvious definition of the length of a path, then the resulting metric is particularly symmetric, and this is a highly desirable property. In particular, with this Riemannian metric the curvature of the sphere is the same everywhere. More generally, one searches for extra conditions to impose on Riemannian metrics. Ideally, these conditions should be strong enough that there is just one Riemannian metric that satisfies them, or at least that the family of such metrics should be very small.