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DIFFERENTIAL GEOMETRY

Dover Publications, Inc.

INTRODUCTION

1. The nature of differential geometry. Differential geometry may be roughly described as the study of curves and surfaces of general type by means of the calculus. In contrast to it, there is algebraic geometry, which employs algebra as its principal tool and restricts itself to the consideration of a much narrower class of curves and surfaces. Thus, the theory of conic sections or quadric surfaces, with which the reader is familiar from analytic geometry, belongs to algebraic geometry, whereas that of the curvature of a general curve, or that of the tangent plane to a general surface, pertains to differential geometry.

A geometric configuration has two different kinds of properties, those which pertain to the configuration as a whole, and those which are definable for restricted portions of it. Thus, the order of a plane algebraic curve—the number of points in which it is cut by a straight line—, is a property of the curve in its entirety. On the other hand, the curvature of a curve at a point depends only on the shape of the curve in the neighborhood of the point.

Generally speaking, algebraic geometry is concerned with properties of the whole of a configuration, whereas differential geometry deals with properties of a restricted portion of it. Algebraic geometry is essentially a geometry of the whole or a geometry in the large, and differential geometry, a geometry in the small.

Euclidean geometry, either in the synthetic form of the preparatory school or the analytic form of the college, is algebraic geometry. So also is the ordinary projective geometry with which the reader is perhaps conversant. But these two geometries differ essentially in content. Euclidean geometry deals with properties of figures which are unchanged by rigid motions, for example, with distance, angle, and area. Projective geometry deals only with properties of figures which are unchanged by projections, such as the property that a point lie on a line, or that a number of lines be concurrent. The former is a quantitative, or metric, geometry, whereas the latter is concerned with properties of position and has nothing to do with measurement.

The distinction between metric and projective geometry is applicable, also, to differential geometry. Thus, there is a metric, or Euclidean, differential geometry and a projective differential geometry. In this book we shall be concerned only with metric differential geometry. In other words, we shall study, by means of the calculus, properties of curves and surfaces which are unchanged when the curves and surfaces are subjected to rigid motions.

We shall begin by recalling, in perhaps a somewhat novel form, a few facts from solid analytic geometry.

2. Directed line-segments. Vectors. As the basis of rectangular coordinates in space, we choose a system of coordinate axes which is right-handed, as shown in Fig. 1, and denote the coordinates of a point referred to this system by (x1, x2, x3).

Let [bar.PP]' be the directed line-segment joining the point P with the coordinates (x1, x2, x3) to the point P': (x'1, x'2, x'3). If α1, α2, α3 are the measures of the projections of [bar.PP'] on the axes of x1, x2, x3, we know that

(1) α1 = x'1 - x1, α2 = x'2 - x2, α = x'3 - x3.

In terms of α1, α2, α3, the length a of [bar.PP'] is

(2) a = [square root of α21 + α22 + α23].

Finally, if A1, A2, A3 are the angles (between 0 and π inclusive) which [bar.PP'] makes with the positive axes of x1, x2, x3, then

(3) α1 = acosA1, α2 = acosA2, α3 = acosA3

By definition, cos A1, cos A2, cos A3 are the direction cosines of the line L on which [bar.PP'] lies, when L is directed in the same sense as [bar.PP']. Hence α1, α2, α3 are, themselves, direction components of the line L. We shall call them the direction components, or simply, the components, of the directed line-segment [bar.PP'].

When a = 1, that is, when [bar.PP'] is of unit length, α1, α2, α3 become the direction cosines of the directed line L, or, as we shall say, the direction cosines of the unit directed line-segment

Vectors. It is evident geometrically that two directed line-segments which lie on the same line, or on parallel lines, and have the same sense and the same length, have the same components. Conversely, if α1, α2, α3 are three numbers, not all zero, there are infinitely many directed line-segments, in fact, one issuing from each point, which have α1, α2, α3 as their projections on the axes. Each two of these directed line-segments have the same direction, the same sense, and the same length.

It follows that, in order to fix the position in space of a directed line-segment [bar.PP'], we must know, not only its components α1, α2, α3, but also the coordinates x1, x2, x3 of its initial point P. If the components alone are known, the directed line-segment is free to move throughout space, provided merely that it keeps the same direction and sense. The directed line-segment is then called a vector. In other words, whereas a directed line-segment has precise position as well as direction, sense, and length, a vector has only direction, sense, and length. It takes all six quantities x1, x2, x3, α1, α2, α3 to determine a directed line-segment, and only the three quantities α1, α2, α3 (not all zero) to determine a vector.

We shall call α1, α2, α3 the direction components, or simply, the components, of the vector; a, its length; and cos A1, cos A2, cos A3, its direction cosines. In short, we shall apply to vectors the terminology which we should naturally use for directed line-segments.

In order that every number triple α1, α2, α3 shall be the components of a vector, we introduce the null vector with the components 0, 0, 0. To distinguish other vectors from it, we shall call them proper vectors.

The vectors, proper and null, which we have thus far discussed are known as free vectors. In addition to them, we shall have occasion to use fixed, or localized, vectors. A localized vector is a vector with fixed initial point, that is, a directed line-segment. If the initial point is P, we shall speak of the vector as localized at P, or, more simply, as a vector at the point P.

3. Parallel and perpendicular vectors. Consider now a number of vectors α, β, γ, ..., with the components α1, α2, α3, β1, β2, β3, γ1, γ2, γ3, ..., and think of them as being either all free vectors or all localized at the same point.

We think of two vectors as parallel when they have the same direction, provided they are proper vectors. When one is the null vector, the definition is inapplicable, since the null vector has no direction. To cover this case, we extend the definition by agreeing to regard the null vector as parallel to every vector.

If α is a proper vector and β is a vector parallel to α, the components of β are a multiple of those of α:

(4)

β1 = kα1, β2 = kα2, β3 = kα3.

According as β is a proper vector with the same direction as α or the null vector, k ≠ 0 or k = 0.

Conversely, equations (4) say that β is a vector parallel to α, the null vector if k = 0, and a proper vector if k ≠ 0. Hence, we have proved the proposition:

THEOREM 1. A necessary and sufficient condition that the vector β be parallel to the proper vector a is that there exist a constant k such that equations (4) are satisfied.

In particular, if k = —1, β has the same length as α but the opposite sense; and, if k = 1, β is identical with α.

THEOREM 2. The two vectors α and β are parallel if and only if

(5)

α2β3 - α3β2 = 0, α3β1 - α1β3 = 0, α1β2 - α2β1 = 0.

In proving the theorem, we distinguish two cases, according as α is, or is not, the null vector.

If a is the null vector, equations (5) are obviously satisfied, on the one hand, and, on the other hand, α is, by definition, parallel to β.

If α is not the null vector, it suffices to show that equations (5) are equivalent to equations (4). That equations (5) follow from equations (4) is clear. Vice versa, equations (4) follow from equations (5). For, since α is not the null vector, at least one of its components is not zero. If α3 ≠ 0, for example, the first two equations in (5) may be solved for β2 and β1, respectively. The resulting equations, β1 = (β3/α3)α1, β2 = (β3/α3)α2, together with β3 = (β3/α3)α3, are then precisely equations (4), when k = β3/α3.

If we had not agreed to consider the null vector as parallel to every vector, the foregoing theorems would have been subject to exceptions which would later prove bothersome.

If α and β are both proper vectors, the angle θ between them, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is given by the familiar formula:

(6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The proper vectors α and β are, then, perpendicular if and only if α1β3 + α2β2 + α3β3 = 0. Evidently, this equation is also satisfied if one of the vectors is the null vector. Accordingly, we agree to regard the null vector as perpendicular to every vector. We may, then, say:

THEOREM 3. The vectors α and β are perpendicular if and only if

(7)

α1β1 + α2β2 + α3β3 = 0.

We have found it expedient to regard the null vector as parallel, and also perpendicular, to every vector. We shall also think of it as parallel, and perpendicular, to every line and every plane. We shall not, however, think of it as making an angle with any other vector, line, or plane, inasmuch as formula (6) is meaningless if the length of either of the vectors in question is zero.

THEOREM 4. If the vectors α and β are not parallel, then

(8)

α2β3 - α3β1 - α1β3, α1β2 - α2β1

are the components of a proper vector which is perpendicular to each of them.

Since α and β are not parallel, the expressions (8) are, according to Theorem 2, not all zero, and hence are the components of a proper vector. That this vector is perpendicular to each of the vectors α and β may be proved by application of Theorem 3.

In this connection we note the identity of Lagrange:

(9)

(α2β3)2 + (α3β1 - α1β3)2 + (α1β2 - α2β1)2 = (α21 + α22 + α23)(β21 + β22 + β23) - (α1β1 + [alpha]2β2 + α3β3)2

which is readily verified by expanding and comparing the two members.

Unit vectors. The vector α is a unit vector, that is, a vector of unit length, if and only if

β21 + α22 + α23 = 1.

The components α1, α2, α3 are, then, actually the direction cosines of the vector.

If α and β are unit vectors which are mutually perpendicular, α21 + α22 + α23 = 1, β21 + β22 + β23 = 1, and α1β1 + α2β2 + α3β3 = 0. Then the right-hand member of Lagrange's identity is equal to unity, and hence the vector with the components (8) is a unit vector. This result we may state as follows:

THEOREM 5. If α and β are mutually perpendicular unit vectors, then

(10)

γ1 = α2β3 - α3β2, γ2 = α3β1 - α1β3, γ3 = α1β2 - α2β1

are the components of a unit vector, γ, which is perpendicular to each of them.

There is, of course, a second unit vector perpendicular to both α and β, namely, the vector with the components -γ1, -γ2, -γ3. We shall call this the vector - γ.

The three vectors α, β, γ of Theorem 5 are mutually perpendicular unit vectors. We shall say that they have, in the order given, the same disposition as the coordinate axes if, after they have been localized at a point, there exists a rigid motion which carries α into OX1, β into OX2, and γ into OX3, where OX1, OX2, OX3 are the unit vectors at the origin of coordinates in the positive directions of the axes of x1, x2, x3.

Consider, in addition to the triple of vectors a, β, γ, the triple α, β, -γ. Only one of these two ordered triples can have the same disposition as the axes. This will be the triple α, β, γ or the triple α, β, -γ according as, after α and β have been varied continuously so as to become respectively OX1 and OX2, it is γ or -γ which then becomes OX3. But, when α and β have respectively the components 1, 0, 0 and 0, 1, 0, it is γ which has the components 0, 0, 1, as may readily be verified by means of formulas (10). Hence, it is the triple α, β, γ which has the same disposition as the axes.

This result we may state in the following way.

THEOREM 6. If α, β, γ are three mutually perpendicular unit vectors which have, in the order given, the same disposition as the axes, then

(10)

γ1 = α2β3 - α3β2, γ2 = α3β1 - α1β3, γ3 = α1β2 - α2β1

4. Algebra of number triples. The numbers with which we have been dealing, the coordinates of points and the components of vectors, appear in the form of ordered triples, and certain combinations of these triples enter frequently. It will be worth while to give to these combinations names, and to discuss their properties. In this connection we shall call a single number, to distinguish it from a triple of numbers, a scalar.

The two ordered triples of numbers a: (a1, a2, a3) and b: (b1, b2, b3) are identical if and only if ai = bi (i = 1, 2, 3). More often than not we shall write, instead of these three equations, the single symbolic equation a = b. Similarly, we shall use the symbolic equation a = 0 to stand for the three equations ai = 0 (i = 1, 2, 3) which say that the triple a is the triple (0, 0, 0).

The scalar a1b1 + a2b2 + a3b3 is known as the inner or scalar product of the two triples a and b. It shall be denoted by (a|b), -read "a into b":

(11)

(a|b) = a1b1 + a2b2 + a3b3.

Evidently,

(b|a) = (ab),

and

(a|a) = a21 + a22 + a23.

The triple of two-rowed determinants a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1 is called the outer or vector product of the triples a and b. We represent it by the symbol -read "a b with a roof":

(12)

[??]:(a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1).

Here,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The first of these equations stands for the three equations which state that the corresponding components of [??] and [??] are negatives of one another. Similarly, the second equation is symbolic for the fact that [??] is the triple (0, 0, 0).

The determinant

(13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a combination of the three number triples a, b, c.

A fundamental relation. The development of the determinant (a b c) according to the signed minors of the c's is the scalar product of the triple [??] with the triple c:

For, the signed minors of c1, c2, c3 are precisely the components a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1, of the triple [??] and hence the development of (a b c) according to them is

(abc) = (a2b3 - a3b2)c1 + (a3b1 - a1b3)c2 + (a1b2 - a2b1)c3 = ([??]|c).

Similarly, (a|[??]) is the development of (a b c) by the signed minors of the a's: Also, (a|[??]) = (a b c). Also. ([??]|b) = (cab) = (abc). Hence,

(14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

5. Applications to vectors. The results of § 3 may now be put in a strikingly simple form.

The vector α, with the components α1, α2, α3, is a unit vector if and only if (α|α) = 1.

The angle [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], between the two proper vectors α and β is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or, if the vectors are unit vectors, by

cos θ = (α|β).

The vectors α and β are perpendicular if and only if (α|β) = 0. They are parallel if and only if [??] = 0; in particular, if α is a proper vector, β is parallel to α when and only when a scalar k exists so that β = k α, where k α is the vector whose components are k times those of α.

If α and β are not parallel, a proper vector perpendicular to them is [??], and all the vectors perpendicular to them are given by k]??] where k is an arbitrary scalar and k [??] is the vector whose components are k times the components of [??].

The foregoing statements cover Theorems 1-4 of § 3. Theorems 5 and 6 say that, if α and β are mutually perpendicular unit vectors, γ = [??] is the unit vector perpendicular to each of them and so oriented that α, β, γ have the same disposition as the axes. But, then, these vectors, arranged in the order β, γ, α, also have the same disposition as the axes, and hence α = [??]. Similarly, β = [??] Thus, the theorems in question may be extended, as follows:

THEOREM 1. If α, β, γ are mutually perpendicular unit vectors which have, in the order given, the same disposition as the axes, then

(15)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [??] = γ, we have

(αβγ) = ([??]|γ) = (γ|γ) = 1.

On the other hand, if α, β, γ had the disposition opposite to that of the axes, we should have [??] = -γ and (α β γ) would have the value—1. In other words:

THEOREM 2. A necessary and sufficient condition that the three mutually perpendicular unit vectors α, β, γ have, in the order given, the same disposition as the axes is that (α β γ) = 1.

We turn now to some propositions of a different type.

THEOREM 3. The three vectors α, β, γ are parallel to a plane if and only if (α β γ) = 0.

If α, β, γ are parallel to a plane, [??] is a vector perpendicular to the plane, and hence perpendicular to γ. Therefore ([??]|γ) = 0 and so (α β γ) = 0.

Suppose, conversely, that (α β γ) = 0 or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] If is a proper vector perpendicular to ?, as well as to α and β, and hence α, β, γ are parallel to any plane perpendicular to [??]. If [??] = 0, a and β are parallel vectors and a plane parallel to one of them and γ is parallel to the other.

THEOREM 4. The vectors α, β, γ are parallel to a plane if and only if scalars k, l, m, not all zero, exist so that

(16)

kα + lβ + mγ = 0.

For, the three equations for which (16) is symbolic are homogeneous linear equations in k, l, m and have a solution for k, l, m, other than the solution 0, 0, 0, when and only when the determinant of their coefficients vanishes. But this is the determinant (α β γ), whose vanishing is the condition that the three vectors be parallel to a plane.

COROLLARY. If the vectors α, β, γ are parallel to a plane and α and β are not parallel to one another, then scalars A and B exist so that

(17)

γ = Aα + Bβ

The vector γ is then said to be a linear combination of the vectors α and β.

It is evident that, if the scalar m is not zero, the symbolic relation (16) can be rewritten in the form (17) by dividing each of the equations for which it stands through by m and setting -k/m = A and -l/m = B. But, if m were zero, the relation (16) would become kα = -lβ and would say that the vectors α and β were parallel. Thus, the corollary is established.

THEOREM 5. If α, β, γ are three vectors which are not parallel to a plane and δ is an arbitrarily chosen vector, three scalars, A, B, C, exist so that

δ = Aα + Bβ + Cγ.

The theorem says that any vector can be written as a linear combination of three given vectors which are not parallel to a plane. We prove it by writing down the three equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

whose validity follows from the fact that the first row in each of the three determinants is identical with a subsequent row.

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Excerpted from DIFFERENTIAL GEOMETRY by William C. Graustein. Copyright © 1962 Mary Curtis Graustein. Excerpted by permission of Dover Publications, Inc..
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