Differential Geometry

DO

64428 CD

OUP

2273

19-1

1-7910,000 Copies.

OSMANIA UNIVERSITY LIBRARY Ctll

No.

Author

5Y6 7 P"?^ -

Accession No.

'^7 ^

3?

Title i

1 his book should be returned on or before the date last marked below

A. V.

POGORELOV

DIFFERENTIAL GEOMETRY Translated from the first Russian Edition by

LEO

F.

BORON

The Pennsylvania

P.

NOORDHOFF

N.V.

-

State University

GEONINGEN

-

THE NETHERLANDS

No part of this book may be reproduced in any form, by print, photoprint, microfilm or any other means without written permission from the publisher.

CONTENTS Page

vm

INTRODUCTION

PART ONE Theory

CHAPTER 1

.

2. 3. 4. t

5. 6.

7.

8.

of

Curves

The concept of curve Elementary curve I.

1

1

Simple curve General curve

2

-.

4

Regular curve. Analytic definition of a curve J the implicit representation of a curve

8 9

Asymptotes to curves EXERCISES FOR CHAPTER

CHAPTER

,

.

2. 3.

4. 5.

II.

18

I

Concepts for curves which are related

3.

to the

concept

22

Vector functions of a scalar argument Tangent to a curve >< The osculating plane to a curve w

26

30

Contact of curves

32

of a family of curves,

Envelope meter EXERCISES FOR CHAPTER

III.

22

-

depending on a para35 37

II

Fundamental concepts

lated to the concepts of curvature

2.

20

I

of contact

CHAPTER .

12 15

PROBLEMS AND THEOREMS FOR CHAPTER

1

6

Singular points on regular plane curves Singular points on analytic curves, defined by equations in the implicit form

PROBLEMS AND THEOREMS FOR CHAPTER

1

...

On

for curves

39

II

which are

re-

and torsion

Concept of arc length of a curve vf Arc length of a smooth curve. Natural parametrization of a curve v< Curvature of a curvet .

42

42 45

49

CONTENTS

VI

Page 4. 5. 6.

Torsion of a curve The Frenet formulas. Natural equations of a curve Plane curves

EXERCISES FOR CHAPTER III PROBLEMS AND THEOREMS FOR CHAPTER P..RT

Theory

CHAPTER

.

III

Two

of surfaces

67 67 69 73 75 78

IV. Concept of surfaces surface.

surface. General surface

2.

Simple Elementary Regular surface. Analytic definition

3.

Special parametrizations of a surface

4.

Singular points on regular surfaces

1.

52 54 57 62 65

,

of a surface ^/

.

.

.

EXERCISES AND PROBLEMS FOR CHAPTER IV

CHAPTER

V. Fundamental concepts for surfaces which are re-

80 80

lated to the concept of contact

1

.

2.

Tangent plane to a surface ^ Lemma on the distance from a point to a '.

tact of a curve with a surface

surface. Con-

^

84

3.

Osculating paraboloid. Classification of points on a

4.

Envelope of a family of surfaces, depending on one or two parameters Envelope of a family of surfaces, depending on one parameter EXERCISES FOR CHAPTER V

87

surface

5.

91

93 96 98

PROBLEMS AND THEOREMS FOR CHAPTER V CHAPTER VI.

First quadratic form of a surface

related to

and concepts

it

100

2.

on a surface Length between curves on a surface Angle

3.

Surface area

104

4.

Conformal mapping Isometric surfaces. Bending of surfaces EXERCISES FOR CHAPTER VI

109 112

PROBLEMS AND THEOREMS FOR CHAPTER VI

113

1

.

5.

of a curve

1

00

102

1

07

CONTENTS

VII

Page

CHAPTER VII. Second

quuu,rui,i,c

form

about surface theory related 1

.

2.

3. 4.

of a surface

and questions 116

to it

;' ^ Curvature of a curve lying on a surface ^ < ^v. directions. curves. Asymptotic Conjugate Asymptotic directions. Conjugate nets on a surface Principal directions on a surface. Lines of curvature v^ Relation between the principal curvatures of a surface and the normal curvature in an arbitrary direction. Mean and Gaussian curvatures of a surface ,.

.

V

under the mapping

a neighborhood of the Conversely, any neighborhood of the / is

point f(X) on the curve y. point f(X) can be obtained in this manner.

The proof of this assertion is straightforward. The image of oj under the mapping / is an elementary curve, inasmuch as CD is an open interval or an open arc of a circumference, and / is one-to-one and bicontinuous. In virtue of the bicontinuity of the mapping

/,

a sphere a(Y),

which does not contain any other points of the curve y except the points /(co), can be described about each point /(Y) belonging to

The

G consisting of

all such open spheres a(Y) is open. This only those points of the curve y which belong to the elementary curve /(co). According to the definition, /(o>) is a neighborhood of the point f(X) on the curve. This proves the first part

/(co).

open

set

set contains

of the assertion.

We

shall

borhood

now prove

the second part. Suppose f(co) is a neighon the curve y. Since f(a>) is an ele-

of the point f(X)

mentary curve, it is the image of an open interval a < r < ft under a one-to-one and bicontinuous mapping 99. Suppose for definiteness that g is the open interval a < t < b. Each point r is assigned a definite point on the curve y, and to the latter point there corresponds a definite point t on the interval. Thus, t may be considered as a function of

r,

t

=

t(r)

.

CHAPTER

4

The function of the

mapping a




*

y

= /2(0,

*

=

/s(*),

where /i, /2, /s are functions defined on the open interval a < t < b or on the half-open interval a < t < b. This system of equations are called the equations of the curve in the parametric form.

4. Regular curve. Analytic definition of a curve. It follows from the definition of a general curve that there exists a neighborhood for each of its points which is an elementary curve.

We shall say that the curve y is regular (&-times differentiable) if each of the points of this curve has a neighborhood which permits a i.e.

regular parametrization,

where

/i,

functions.

A

/a

/2,

For k

the possibility of giving

its

equations

form

in the parametric

are regular (&-times continuously differentiable) 1 the curve is said to be smooth.

=

,

said to be analytic if it permits of an analytic parametrization (the functions /i, /2, /s are analytic) in a sufficiently

curve

is

small neighborhood of each of its points. In the sequel we shall consider regular curves exclusively. As was shown in the preceding section, a curve may be given

means

= x(t),

x

where val a

x(t], y(t), z(t)

< < t

= y(t),

z

=

z(t],

are certain functions defined in

b or half-open interval a

The question naturally x

y

= x(t),

y




0,

the curve can indeed be

defined

=

y in a



I,

8

*y(t) a,

provide the

first

condition is

satisfied.

If this limit is denoted by p, then the equation of the asymptote will be fix

+

ay

p

=

0.

an asymptote to the curve and that a and ft are its direction cosines. The equation of the straight line can be written in the form PROOF. Suppose g

is

ftx

The point

+

ay

p

=

0.

on the curve tends to

infinity, coming arbitrarily close to the straight line g as t -> a. It follows from this that the ratios x(t)/p(t), y(t)/p(f) as -> a converge either to a and ft, or to Q(t)

a and

ft,

depending on which

of the

two

directions

on the

straight line g the projection of the point Q(t) tends to infinity. Suppose, for definiteness, that x(t)/p(t)

-

a,

y(t)lp(t)

->

/?.

+

The quantity ftx(t) a. This completes the proof of the necessity portion of the theorem.

We

shall

now prove

the sufficiency. Suppose

>p, as

t

->

a.

We

shall

show that the ftx

is

+

-

ftx(t)

-> p

straight line g with equation

p

ay

an asymptote to the curve. In

*y(t)

=

fact, the expression

- fix + *y(t) - p is,

to within sign, the distance from the point

straight line

as

t

->

g.

But

- px(t) +

ay (t)

+

p ->

a.

This completes the proof of the theorem.

t

on the curve to the

CHAPTER

I,

8

17

EXAMPLE. Suppose the curve y y or,

what amounts

=

When



-> oo as

t

same

->

y

t,

defined




+

-f-

f(t)

z(t)e*.

f

(t)

-At)

- y(t) + Aty

n\7-

(/(*)

+

e(t,

At)),

0.

= x(t)ei + y(t)e In fact, = x(t) + Atx + x(t + At) y(t +

At n

-

-

-

+

But At* n\

(xW(t)

At n

f

(t)

+.-.+n\

(y(n)( t )

+ ei), +

2 ),

an

CHAPTER

z(t

II,

25

1

+ At) =

z(t)

+ Atz'(t) +

+

Multiplying these equations by e, then noting that #*i yW(t)e 2

(*(*)

e%, e%

*s).

respectively, adding,

+ zM(t)ez =

+

+

/(/),

and

we obtain the

Taylor formula for the vector function f(t). The concept of integral in the Riemruin sense for vector functions introduced literally as in the case of scalar functions. The integral of a vector function possesses the usual properties. Namely, if f(t)

is

is

#

a vector function which

< < b, t

and a



and r(t) is a ^-times differentiable function Since da/dt Vr' 2 (t) of t, t is a &-times differentiable function of a. But for a close to ai, r(a)

=

r(t(cr)).

from

It follows

this that r(a) is a regular (&-times

differentiable) function and dr(a)

dr(t)

dt

df(t)

da

dt

da

dt

1

df(t)

dt

=

1.

Consequently, \r'(a)\ This concludes the proof of the theorem.

COROLLARY.

A

regular

(k-times

differentiable,

analytic)

curve

permits a regular (k-times differentiable respectively analytic) parametrization "in the large' i.e. for the entire curve. f

CHAPTER

49

3

III,

Such a parametrization is the natural parametrization r(a] and also any parametrization obtained from it by means of a regular transfor-

=

where y(t) is a regular (respectively

as

and passing

As ->

to the limit,

we obtain

This completes the proof of the theorem. Suppose the curvature does not vanish at a given point on a curve. Consider the vector n (ljki)r' (s). The vector n is a unit

=

vector and

lies in

f

the osculating plane of the curve ( 3, Chapter II). is perpendicular to the tangent vector r,

Moreover, this vector so that r 2

=

1

and, consequently r-r

r-rjki

=

0.

Thus, this

directed along the principal normal to the curve. Obvithe direction of the vector n does not change if the initial ously, point of the arc s or the direction of traversing 5 is changed. In the

vector

is

when we mention the unit vector on the principal normal to we shall have in mind the vector n. b is directed along the binormal Obviously, the vector r X n

sequel,

the curve,

of the curve. This vector will be called the unit binormal vector of

the curve.

We

shall find an expression for the curvature of a curve in the case of an arbitrary parametric representation. Suppose the curve is given by the vector equation r shall express the second r(t).

=

We

derivative of the vector function r with respect to the arc s in terms of the derivatives with respect to t. have

We

r'

=

r s s'.

It follows that r'2

c'2

CHAPTER

III,

3

51

and consequently

Differentiating this equality once

=

r ss s'

r"/Vr^

more with

-

(r'

respect to

we obtain

t,

V>'/(v9*)3.

Squaring both sides of this equality and noting that

s' 2

=/

2 ,

we

have

or,

what amounts

same

to the

,

From

this

=

* defined

*(/),

y

= y(t),

of a curve given

=

z

by the

z(t)

the curve

is

"

y

a plane curve lying in the IY"M' \Xf

the plane curve

is

given by the equation y

REMARK. The curvature it is

y-plane,

I

y"

For plane curves,

x,

\)"r'\Z -^ y

y

If

r")

by X

If

X

(''

,

we obtain that the curvature

equations

is

thing,

of a curve

y(x),

2

is,

by

definition, nonnegative.

convenient in

many cases to choose the sign cases it is positive and in others

some The tangent vector r'(t)

of curvature so that in

of the curve rotates as it moves negative. along the curve in the direction of increasing t. Depending on the

direction of rotation of the vector positive or negative. If

r'(t)

we determine

the curvature

is

considered

the sign of the curvature of a

CHAPTER

52

III,

4

plane curve by this condition, then we obtain the following formulas for it: f

%"y*

In particular,

if

=

k

the curve

y"l(\

In conclusion, we find its points.

We

is

+ y'2)

3

y"x

%"y'

y"x'

given in the form y

/i

or k

= y(x),

= _ ^"/(i + y'

2

3/2 )

.

all

the curves having curvature zero at

*i

=

all

have |r"(s)|=0.

It follows that r"(s) and, consequently, a and 6 are constant vectors.

r(s)

=

as

+

b,

where

Thus, a curve having curvature everywhere equal to zero is either a straight line or an open interval on a straight line. The converse

is

also true.

P is an arbitrary point on the We denote the angle between the osculating planes to the curve at the points P and Q by A& and we denote the length of the segment PQ on the curve by As. The of the curve y at the point P is understood to be absolute torsion Torsion of a curve. Suppose curve y and Q is a point on y near P. 4.

\ft2\

the limit of the ratio Aft I As as

THEOREM.

A

Q

-> P.

regular (three-times continuously differentiate) curve at every point where the curvature is \k%\

has a definite absolute torsion different

from

zero. If r

curve, then

r(s) is the

N=

natural parametrization of the

|(r

PROOF. If the curvature of the curve y at the point P is different from zero, then by continuity it is different from zero at all points sufficiently close to P. At every point where the curvature differs from zero, the vectors r'(s) and r"(s] are different from zero and are not parallel. Therefore, a definite osculating plane exists at each point Q near P.

and b(s + As) are unit binormal vectors at the points P and Q on the curve y. The angle A& is equal to the angle between the vectors b(s) and b(s + As). Since the vectors b(s) and b(s + As) are unit vectors and form Suppose

b(s)

CHAPTER

the angle

4

III,

53

^s

Aft, \b(s -f

\b(s

+

Aft

2 sin

Therefore

.

Aft

o

~

As)

=

6(s)|

)

.

mn

ci^rj

Aft

2

2

6(5)1

As

As

Aft

As

Aft

~2~

From

this

we

obtain, passing to the limit as

=

\k*\

The vector

V

As ->

=

perpendicular to b since b' -b not difficult to see that b' is also perpendicular to In fact,

=

But

T'||W.

(r 6'

Therefore,

perpendicular to

r.

X

n)'

=r

== r'

X

that

\b'\.

is

V

0,

X n

+t

X

whence

n',

Thus, the vector

it

(|&

2 )'

=

0. It is

r.

w'.

follows that

b'

parallel to the vector

b' is

is

n

and, consequently,

N= If we set n we obtain

(\/k)r"

and

6

N=

=

r'

\b'-n\.

X

^

r"/^i

|(rvv")|/*i

n^

this last equation,

2 -

This completes the proof of the theorem. We shall now define the torsion of a curve. It follows

from the

fact that the vectors

b'

and n are

parallel that

the osculating plane to the curve rotates about the tangent to the curve as it moves along the curve in the direction of increasing s. In this connection,

we

define the torsion of a curve

equation kz

= |

2

by means

of the

|

and we shall take the sign (+) if the rotation of the tangent plane occurs in the direction from b to n, and ( ) if the rotation occurs in the direction from n to b. If we define the torsion of a curve in this

way, we

shall

have k%

We shall now find

=

b'

-n or

the expression for the torsion of a curve in the

CHAPTER

54

when

case r

=

it

We

r(t).

defined

is

by an

5

III,

arbitrary regular parametrization

have

ys

=

=

r ss

r't',

+

r"t'*

r't" ,

r

r"} is a linear combination of the vectors r' and r" If we substitute the expressions for r s r ss and r sss just found into the 2 we obtain formula for fa and note that t' 2 l/r'

where

{r

'.

,

,

,

=

fa

=_

(

,

r 'r"r'")l(r'

X

r")

2 .

In concluding this section we shall find all the curves for which V -n 0, but the torsion vanishes at every point. We have fa as we saw, b' -r and b' -b 0. Consequently, b' 0, 6 60 constant vector.

= =

=

=

=

The vectors follows that

r

and

ro)

(/'(s)

=

Therefore r'-bo which means that the curve

6 are perpendicular.

60

=

0,

0. the plane given by the vector equation (r ro) 60 as at curve vanishes whose torsion Thus, every point curve. The converse assertion is also true.

5.

is

=

0.

It

lies in

a plane

The Frenet formulas. Natural equations

Three

half-lines,

of a curve. and having curve on the a from emanating point

the directions of the vectors

r, n, b

are edges of a trihedron. This

trihedron is called the natural trihedron.

In order to investigate the properties of the curve in a neighborof an arbitrary point P it turns out in many cases to be con-

hood

venient to choose a cartesian system of coordinates taking the point on the curve as the origin of coordinates and the edges of the

P

natural trihedron as the coordinate axes. Below

we

shall obtain the

equation of a curve with such a choice of coordinate system. We shall now express the derivatives of the vectors r, n, b with respect to arc length of the curve again in terms of r'

To obtain b'-n

=

2-

b',

let

=

r"

=

r, n, b.

We have

kin.

us recall that the vector

b' is

parallel to

n and that

It follows that b'

-

k*n.

Finally, \

bx n

=

(kir

+ fab).

CHAPTER

5

III,

The system

55

of equations

r ri

are called the Frenet formulas.

We shall find the expansion of the radius vector r(s + As) in a neighborhood of an arbitrary P, corresponding to the arc s along the axes of the natural trihedron at this point. We have r(s

+

=

As)

r(s)

But

at the point P, so on. Thus,

r(s

+ Asr'(s) + r= 0,

Z

r"(s)

r'=r, r"=kin,

+ As) = I

(~^2~

+ -r'"(s] + D

r"'=kin

k^r

kik^b,

and

(As 6

+ ~^6~- + X

We see that in order to expand the function r(s + series in

As

it is

sufficient to

curve as functions of the arc

know

^s) as a power the curvature and torsion of the

This gives the basis for assuming that the curvature and torsion determine the curve to some extent. And s.

indeed we do have the following valid theorem.

THEOREM. Suppose k\(s] and k%(s] are arbitrary regular functions with ki(s) 0. Then there exists a unique (up to position in space] curve for which ki(s] is the curvature and k%(s) is the torsion at the

>

point corresponding

to the arc s,

PROOF. Let us consider the following system of

differential

equations

where

|,

17,

Suppose

are

unknown

(s), rj(s), t(s) is

vector functions. the solution of this system satisfying the

CHAPTER

56 initial

=

conditions

y

lo,

=

C

??o,

=

=

SQ, where f o, ??o, Co whose triple product

f or s

Co

are three mutually perpendicular unit vectors

equals

We

1

:

*?o,

,

(

shall

Co)

=

1

-

show that the vectors

(s),

mutually perpendicular for arbitrary

we

end,

shall

compute

2

(I

2

fa

)',

2

If

for

)'

=

2feirf,

(f ,

??,

are unique

and

To

this

(C )', (f -1?)', (iff)',

we obtain

(f -I)'

= *nK +

=

C)

1

.

(')' Making

the following ex-

fef -17.

we consider these equations as a system of differential equations 2 2 2 I- 1/, ??, C'l, we note that it is satisfied by the set of C rj ,

values

,

,

=

2

=

q2

1,

1,

f

other hand, this system 2 r]

f(s)

r](s),

and

5

2 )',

use of the equations of the system, pressions for these derivatives:

(C

5

III,

(s),

,

=

f !

-

2

is

f(s) 'f(s).

1,

??

=

-

iff

0,

- 0. =!

0, f -f

the values | 2

satisfied

by Both these solutions coincide

On

the

= = SQ> 2

2

(s),

rj

for s

and consequently, they coincide identically according to the theorem on the uniqueness of the solution. Hence, for all 5 we have I

2 (s)

=

We shall show that

1,

1,2(5)

s

s 0>

(|,

77,

C)

I,

,

=

(1(5), rj(s), f (5))

perpendicular unit vectors,

product

=

we have

f(s)-f(s) 1

Since

.

(,

??,

depends continuously on

and therefore

it is

equal to

1

for

f)

|,

=

it

s,

-0. >y,

C are 1

.

equals

mutually

The

triple

+

when

1

all s.

We shall now consider the curve 7, defined by the vector equation

We note first of all that the parametrization of the curve y is the natural parametrization. In fact, the arc length of the segment SQS of the curve y equals 80

The curvature

SO

-

of the curve y equals |r"(s)|

torsion of the curve y equals

,

ki'f,

+ __ ki(-

= |!'(s)| =

ftif

&i(s).

The

= k2(s) ,

.

.

.

CHAPTER

III,

57

6

Thus, the curve 7 has curvature point corresponding to the arc

ki(s)

and torsion

#2(5) at

the

s.

This completes the first part of the theorem. to the proof of the second part.

We now

proceed

Suppose 71 and 72 are two curves which have the same curvature ki(s) and torsion &2(s) at the points corresponding to the arc s. We

and 72 by means of points correand with the natural trihedra at these points.

shall correspond the curves 71

sponding to the arc

SQ,

Suppose TI, tti, 61 and r%, n%, b% are unit tangent, principal normal, and binormal vectors to the curves 71 and 72 respectively. The triples of vector functions TI(S), n\(s} bi(s) and TZ(S), nz(s), y

#2(s) are solutions of the system of equations for |, 77, f. The initial values of these solutions coincide. It follows from this that

the solutions coincide identically. In particular, TI(S) == n'(s) s= r2 (s). Integrating this equality between the limits

T2(s),

or

SQ, s,

we

f

obtain ri(s)

=r

2 (s).

Thus, the curve 72 differs from 71 only by This completes the proof of the theorem.

The system

its

position in space.

of equations

are called the natural equations of the curve. According to the theoto within position

rem proved above, a curve is defined uniquely in space by its natural equations.

Plane curves. In this section we shall consider the oscuand involutes of a plane curve. is a curve and that P is a point on 7. A circumferSuppose 7 plane 6.

lating circle, evolutes,

ence K passing through the point P, is called the osculating circle to the curve 7 at the point P if the curve has, at this point, contact of the second order with the circle. The center of the osculating circle is

called the center of curvature of the curve.

We

shall find the osculating circle of a regular curve

point P, where the curvature

is

different

from

r *= r(s) is the natural parametrization of the curve.

any circumference has the form (

r

_

0)2

_ #2 ^ o,

zero.

7 at a

Suppose

The equation

of

CHAPTER

58

where a

R

is its

is

III,

6

the position vector of the center of the circumference and

radius.

According to the theorem in 4, Chapter II, a necessary and sufficient condition that the curve y have contact of the second order with the circumference at the point P is that the following conditions be satisfied at this point (r(s)

_ d2 {(r(s)

-

a}

2

a)

2

_

k(s)

=

(r

1