TENSOR ANALYSIS ifJTH APPLICATIONS TO

RIEMAMN SPACES

APPROVED n

Hi 3 or

V-,Jgjo-v. Minor Professor"**

Director of the Department of Mathematics

Dean of the Graduate School

TENSOR ANALYSIS WITH APPLICATIONS TO RIEMAM SPACES

THESIS

Presented to the Graduate Council of the

Uorth Texas State University in Partial

Fulfillment of the Requirements

For the Degree of

MASTER OP SCIENCE

By-

William H. Osborne, B. S<

Denton, Texas

January, 1966

TABLE OF CONTENTS

Chapter Page

I. PROPERTIES OP TENSORS 1

II. RIEMANN SPACES 33

BIBLIOGRAPHY 48

iii

CHAPTER I

PROPERTIES OP TENSORS

In this paper three assumptions will be made. The first

assumption will be a knowledge of the real numbers. The sec-

ond will be that S is a set of points.

Definition 1.1: n-Tuple of Real Numbers A n-tuple is

1 2 n

an ordered set of n real numbers of the form (a ,a ,...,a ),

where a"*", i=1,2,...,n, is a real number.

The third assumption that will be made is that there is

a one-to-one mapping of a set of n-tuples of real numbers to

the set of points S. A one-to-one mapping will mean that

there is only one n-tuple associated with a given point and

only one point associated with a given n-tuple under the map-

ping used.

Definition 1.2: Coordinate System' The set of n-tuples

in the above paragraph will be called a coordinate system and

a n-tuple in that set will be called the coordinate of a

point.

Definition 1.3: Connected Set of Real Numbers A set of

real numbers is connected if and only if for x in the set R

of real numbers then R will be defined by one and only one of

the following sets. There is a real number a and a real num-

ber b such that,

1) a<x<b, 4) a<x, 7) x^b, or

2) a*<x<b, 5) a^x, 8) x = any real number

3) a<x^b, 6) x<b,

Definition 1.4: Connected n-TuPle A set H of n-tuples

of real numbers is a set of connected n-tuples if and only If

thirfe- exist: sets R.j ,R?, ... ,Rn such, that for each p, where

1^p^n, Ep is a connected set of real numbers and the n-tuple

*1 2 XL 1 (a ,a ,...,a ) is an element of H if and only if a is in , 2 21 ct • is in Rg * * • • > in *

Definition 1.5J Connected Coordinate System If the

coordinate system is a set of connected n-tuples then the

coordinate system is connected. All coordinate systems in

this worlc will be connected.

Definition 1.6: Open Set of Points in a Coordinate

System If S is a set of points then an open set of S with

respect to some coordinate system X is a .subset ¥ of S so H O y\

that if P is in ¥ and (x ,x ,...,x ) is the coordinate of P

then there is a positive real number 8 so that if the sum

(x1- a1 )2+ (x2- a2)2+...+(xn- an)2<62 then (a1 ,a2,...,an) is

in ¥. Definition 1.7: Vector Space The set of n-tuples which

form a coordinate system will be called a n-dimensional vec-

tor space V •

Definition 1.8: Transformations If each of X and Y is

a coordinate system, and ¥ is an open set of points with

respect to X, then a transformation T from X to Y In W is a

set of n functions y ,y , . . . , y so that if P is in ¥, the

coordinate of P in X is (x1,x2,...,xn) and the coordinate of

P in I Is ( y 1 , y 2 , . . . , y n ) then y i= y ^ x 1 ,x2, ...,xn), where

i-— 1)2)... ,n.

Definition 1.9s Permissible Transformations If I is a •"""d. -i , transformation such that for each y in T, 1=1,2,...,n, y is

a real, single-valued, reversible functional transformation

of the form y^=y^(x), where y^ (x)=y** (x1 ,x2, •.. ,xn), and if

y^(x) is continuous together with its third partial derivative

in some region R of Vn then T will be a permissible transfor-

mation. It will be assumed that these n equations will be

solvable for x3" so that xi=xi(y1 ,y2, ...,yn)=xi(y). This set

of n equations will be called the inverse of T and denoted

by T"1. All transformations used in this paper will be per-

missible transformations.

Definition 1.10: Admissible Coordinate Systems A col-

lection of coordinate systems such that if X and 1 are

elements of 0 then there exists a permissible transformation

T so that T(X)=Y will be called a set of admissible coordinate

systems. All coordinate systems in this paper will be this

type.

Definition 1.11: Range Convention A suffix which

occurs Just once in a term implies that this suffix will

range in value from 1 to n.

Definition 1.12: Summation Convention A suffix which

occurs just twice in a term implies summation with respect

to that suffix over the range 1,2,...,n.

Definition 1.13; Oovariant Tensor Let ¥ be a set of

points and let S "be a set of coordinate systems so that,

1) if X is in S then ¥ is an open set with respect to X

2) if X is in S and I is in S then there is a permissible

transformation T of X to I in ¥

3) there is a positive integer r so that if X is in S

then there exists nr real valued functions A. * ,(x) X,

associated with X whose domain contains the X coordi-

nate of the points in ¥

4)-a if X is in S and Y is in S, I is the transformation

of X to I in ¥, |Ai( i s tiie se,fc functions

associated with X, {B., ^(y)} is "the of func-«Ji Ja. * • • or J

tions associated with I, and P is- a point in ¥ so

that x is the X coordinate of P, y is the coordinate

in Y of P, then ax1' ox1*

Bdi 3 a..Ory ) = ay3' sy*5A"*By^Ai|iA...i/J

x)

Let C be the class of sets of functions so that a set

{AJ. * . (x) h belongs to 0 if and only if there is a s 2.j ZL • • • JLy* -> X in S so that ^ ^ (x)} is associated with the

.f . • mXf,

coordinate system X. Then 0 will be called a covari-

ant tensor of rank r with respect to the set S. The

set {a^ ^ associated with the X coordinate

system will "be called the components of the tensor C

in X,

4-b if the first three of the above conditions are true

and if X is in S and I is in S, T is the transforma-

tion of X to Y in ¥, A(x) is a function associated

with X, B(y) is a function associated with Y, and P

is a point in ¥ so that x is the X coordinate of P,

y is the Y coordinate of P, then B(y) = A(x). Let 6

be the set of functions so that A(x) belongs to 0 if

and only if there is a X in S so that A(x) is asso-

ciated with X. Then 0 will be called a covariant

tensor of rank zero with respect to the set S or a

tensor of rank; zero. The function A(x) will be

called the component of 0 in X.

Definition 1.14: Oontravariant Tensor Let I be a set

of points and let S be a set of coordinate systems so that,

1) if X is in S then ¥ is an open set with respect to X

2) if X is in S and Y is in S then there is a permissi-

ble transformation T of X to Y in ¥

3) there is a positive integer r so that if X is in S

then there exist' n2"* real valued functions A"' * #^(X)

associated with X whose domain contains the X coordi-

nate of the points in ¥

4)-a if X is in S and Y is in S, T is the transformation

of X to Y in ¥, jA^ "*"A* * ,if,(x )J- Is the set of functions

associated with. X, •k***^r(y)} is the set of func-

tions associated with Y, and P is a point in ¥ so

that x is the X coordinate of P, and y is the Y co-

ordinate of P, then

ax1' sx a ixlr

Let 0 be the class of sets of functions so that a set

A"3"' * * '^(x )| belongs to 0 if and only if there is a

X in S so that -[a1' * * *i|r(x)} Is associated with the

coordinate system X. Then 0 will be called a contra-

variant tensor of ranlc r with respect to the set S.

The set -[a1' ^ " * '\x)j associated with the X coordin-

ate system will be called the components of the tensor

0 in X.

4)-b a contravariant tensor of ranlc zero will be as

stated in 4)-b of definition 1.13.

Definition 1.15: Mixed Tensor Let ¥ be a set of points

and let S be a set of coordinate systems so that,

1) if X is in S then ¥ is an open set with respect to X

2) if X is in S and Y is in S then there is a permis-

' sible transformation T of X to Y in ¥

3) there is a positive integer r+s, r and s are positive

integers, so that if X is in S then there are n

real valued functions ^ ' ^ " ' ^ ( x ) associated with X (Xj U.£» • •U»f*

whose domain contains the X coordinates of the points

in ¥

4) if X is in S and Y is in S, T is the transformation

of X to I in If, *]] sr(x)} i s se-fc associ-

ated with X, {b|' i *** (y)j" is the set of functions s ' X| • • • X|* J

associated with Y, and P is a point in W so that x is

the X coordinate of P, y is the Y coordinate of P,

then . , Sy ' Sy^a 5y^ dxa' dxaa bxa.r R R

B?' •••ky) = — B — b — i — i ' " — " ' a < x > l< **'xr 3xp< Bx * Bx * Sy 1 dy 4 Sy 1^ * * *a

Let 0 he the class of sets of functions so that a set

f J3, "belongs to 0 if and only if there is a X W CCj • • •OCjf J

in S so that ]]*^(x)} i s associated with the X

coordinate system. Then C will he called a mixed

tensor contravariant of rank s and covariant of rank

r with respect to the set S. The set -[a^

will he called the components of the tensor 0 in X.

A mixed tensor is of rank zero if it satisfies 4)-h

of definition 1.13.

Definition 1.16; Krone eke r Delta and Kronecker Delta

Function 6^(x) S'jj (x) = 6^ = 1, if i = j

S j(x) = 6* = 0, if i / 3

Definition 1.17: Invar lance If A?' ***^(x), 1 , i = 1 LI-': '•jn 1 1JL 1 " XI • • • (X U#

1,2,...,lc, is a set of functions in the X coordinate system,

b3, •••i(y), i 1 = 1,2,...,k, is a set of functions in the Y J • • * 1 ^ (X cx

8

coordinate system, and T(x) = y, where T is a permissible

transformation, then if jJ1 *"Mx) - B?' the sets of

JLj • • •Xf

functions will be said to be invariant.

1 2 21

Lemma 1.1 If x ,x , ...,x are n independent variables then = 6"L

a x ' dx1 Sx1

Proof: If i = J then — . = — . = 1. If i / j then ± 1 5xJ dx

since x and xJ are n independent variables it follows that

ax1 ax1 , -—j = 0. Therefore * = 6,, S x 3 x i i 1 2 n

Lemma 1.2 If T(x) = y and y = y (x ,x ,...,x ), i=1,...,n,

then = 6*

^ , Proof: By lemma 1.1 — . = 6Jt By the chain rule of

i i h7a J i a differentiation ^ so that ^ = 6^.

hyt 3xa Sy3 Sxa dy3

Theorem 1.1 If the components of a tensor are zero for

all x in their domain in one coordinate system then they are

zero in all other coordinate systems.

Proof: Let A(i,x), i=1,2,...,k, denote the components

of some'tensor in the X coordinate system such that A(i,x) is

zero for every i. Then if B(i,y) are the components in the I

coordinate system, B(i,y) = K A(a,x), where K are the appro-vX UL

priate partial derivatives. But A(a,x) is zero for each a so

that B(i,y) = Ka(0) = 0. Therefore each component in the Y

coordinate system is zero. A tensor whose components are all

zero will be called a zero tensor.

Definition 1.18: Sum of Two Tensors Let a|' and ~'~"r " r m J«i • • • Jup

"""Hx) be the components of two tensors in the X coordi-If • • • Jup

nate system such, that both tensors have the same covariant

ranlc and contravariant rank and a common open set in their

domains. Then the sum of the two tensors in the X coordinate

system, denoted by a|' **"3s(x)+A$' #***'(x), will be the set Jmf • • • Xfn JLj # • • JL[*

{a«' * * '^(x)+a|' ***^(x)}, where the suffixes are the same. V Jmf • • •Jmf* X J • • • Xjj* <*

Definition 1.19: Difference of Two Tensors The dif-

ference of two tensors will be defined in the same way as the

sum of two tensors, except wherever there is a plus a minus

will be substituted. The difference will be denoted by

XI • • • JLj* JL I • • • JLjn

Theorem 1.2 The sum (or difference) of two tensors with

the same contravariant rank and covariant rank is a tensor of

the same type and ranlc as the given tensors.

P;roo'f: Let H and H be two tensors of the same type and

rank defined over the same open set of points. The law of

transformation is given by

, , Sxa' 3xa* Sy1' dy1' B B BV * • * (y ) = — •••—, — 3 ; * *—b A®' • • 7(x)

J$ Sy ' dyk ax3' 5xPr ai#,*as

3xa' Sxa^ 3v "' 3v^r

" ayJ, ay3,dxB, Sx^r ' * * *a5

where A > '"^(x) and B^' * **V(y) are components of H and (Xj • • • &£ J i • • • J $

'"ir(x) and B^,#*^r(y) are components of H. Then by di • • •ctj ji • • •us

adding (or subtracting) and factoring yields

10

1 1 - 1 1 a*a' ^"say1' ayir _ „ Bt' •••hy)+B^ •*7(y) = —<•••—, —,•••—= JI * • • SyJ, ay^j Sx ' 3x r

S' vXj • • •cx

Therefore the sum (or difference) of two tensors Is a tensor »

of the same rank as the given tensors.

Theorem 1.3 If A%' * * *?s(x) and are the com-CXj • « •U,» X j • • • Xj*

ponents of a tensor in the X and Y coordinate systems respec-

tively and if A^' "'^(x) and Bj' ,#*^s(y) are the components CXj • • •CCi X| • • •A.p

of a tensor in the X and Y coordinate systems so that they

have a common domain and A ' **"^(x) = A^' ***fJ(x) in the X CX j • • •CC CX | • • «CX

coordinate system then the tensor components in the Y coordi-

nate system are also equal.

Proof: Since **fs(x) = if1 #"^(x) then CX | • • •CX f» CX | • • •CXp

A?' "#Es(x) - 2 ' ""^(x) = 0. By theorem 1.2 A*?' '"^Cx) " CX j • • •CXj. CX| • • •CXf CX | • • •Qyt •H3 B ir1 is a tensor component in the X coordinate system CX j mm •CXy» and Bj' "*ja(y) - Bi®' ***i®4(y) is a tensor component in the Y

Xj • • • J-p X| • • #Xf*

coordinate system. Since the components of this tensor is

zero in the X coordinate system by theorem 1.1 they are zero

in all other coordinate systems. Hence b|' ***A(y) - b|' *#*|s(y) Xj • • • Xf* X | • • *x

is zero or b|' ***|J(y) = B^' Xj • • •JLp X j • • *X |f

Definition 1.19: Outer Product Let A*1 "'k(x) and " Xi • • »x

I 0 0 AJ **V(x) be the components of two tensors in the X coordi-a i • • »cxy-»

nate system. Then the components of the outer product of the

two tensors in the X coordinate system will be the set of

functions obtained by multiplying each component of the first

tensor by each component of the second. The outer product

11

will be denoted by A,' * * * 5(x)*A.f' * **j^(x). Prom here on when X| • • •xr ex| • # »(Xy

operations on two or more tensors are talked about it will be

understood that the tensors have a common domain.

Theorem. 1 . 4 L e t A ^ ' * * * | s ( x ) a n d l ? ' * * i M x ) be t h e c o m -a. j • • «>Xjr II | • • •Xiy

ponents of the tensors T and T in the X coordinate system.

Then there is a tensor S so that the elements of the outer

product of T and T are the components of S and so that S is

covariant of rank s+v and contravariant of rank r+u.

Proof: If b1' "'^(y) and B?1 ' * * "(y) are the components J j • • • 11/ • • •XLy

of T and T respectively in the Y coordinate system then, let-

ting P_ denote the product of the and P r denote the pro-ay1 ' ay3

duct of the , 3xp

Bi| = p pr/, Jj • • • » U| • •

and letting Py and Pu be defined as above

wa, ...%( ) = p p W , •. . V x j H j • * •Iky V 0 j • • #Oy

Let ri' •••Jtyy) = jjii ---^(y) and Jj • • * 0$ **| • • •*V Jl •••Js • • •U-V

0P, •••PrY, = fit " A j J , ^en by substituting CX( . * #a3 oi . • *0 al ». •ai Of # • »0y w ^

jji, .. .irxa, .. .m^ ) _ p p • • • P j ^ x ) p puTf/ ***}y(x)

• • • $$ i •• «JQy 3 CC| • * •OCjj V 6 ( • • #§y

= P P P^P^O^' *** (pYi * * *Yk(x )

V v * r a ...aj6, . • .6i/

Therefore the outer product of two tensors is a tensor that

is contravariant of rank r+u and covariant of rank s+v.

TJhenever the notation P_ and P r can be used for the product

of the partial derivatives in the covariant or contravariant

transformation without ambiguity they will be used.

12

Theorem 1.5 If, in a mixed tensor contravariant of rank

s and covariant of rank r, a covariant and contravariant

index is equated, then the result is a mixed tensor contra-

variant of rank s-1 and covariant of rank r-1.

Proof: Let ;; Afr) and — f a ) b, the components

of a mixed tensor in the X and I coordinate systems respec-

tively. Then

1 * Bxa' bx^by1' by1* B^' — ••• — JL ••• -I a3I ...0,/-) ' * * • r By^r Sx ' ax8* a' '' ,ar

Let i = 3 where 1^y^s and Kri^r, D:' ^ '"ls(y) = df • • • Jy./ • • •3r

Bl, ^ „S, ...9,., |3„, ...g o, 3, • • .lr.. .jr ,a, .. ,ay.( aw| .. .ct/

x' a. .. .0r .. ,aJx '•

Then

Di,.. .i,., ir„ • • -i (y) = na'.. .af:. . ^ n 1 : . M 1 : . V s .

1 1-1 V'i" dy3l ay1* 8y^ Sx5j

6 Bxa< bxar-i 3xa»' Bx^By1' By3"*-'

— '•••—. -Jl •••.JL. • By i Byk' By *w By " Bx0( bx®*-l

by1*! by1* R „ • • • .P| • « 'Pi/yN

5x |r+l Bx 5 a/ • • ,a5

Bxa' Bxajr-' bxa™ bx^by1' by1*-' »mmm * # - • • • .. • • • •

By i Byk-' By^w By * 3^1 ax®*"-'

V. A ' * * ,^5r * * * s(x)

Bx^' Bx^a'

r3xa,# # # Bxa*-' Bxa^ #Bx

ar By1' By1*"'

by^i by^r-i 3y«5f« By«k Bx0' ax0*"-'

13

~"~o * * a/ " * *ar~i art) • • * ai

3xP*' dXP*

Thus the result is a mixed tensor contravariant of rank s-1

and covariant of ranlc r-1. •

Definition 1.20: Contraction The process of theorem

1.5 is called contraction.

Definition 1.21 : Inner Product If it is possible to

apply the operation of contraction to the outer product then

the result is called an inner product and each will "be de-

noted by T«l", where T and T are tensors.

Theorem 1.6 The' inner product of two tensors is a ten-

sor. Proof: Let i?» * * '^(x) and S 1 Y' \,,Yy(x) be the compon-

Pi • • , 01 • • •Oy

ents of the tensors T and T in the X coordinate system. By

theorem 1.4 the outer product of two components is a tensor

component and by theorem 1.5 the result of contraction of

tensor components is a tensor component. Therefore T«T is a

tensor. Theorem 1.7 If in two coordinate systems X and I

Bi" a, •••3i(y)= — _ • - " — , * - AS'. :: •!&*> Sy^s 3xa' dxar

ii . * »i»« -n,,P| Cti • • *0,f •I • * Sx 1 SX'J 3yX| Sy r

T> 3 I • • . .i

3xP' 3xPi dy1' SyirBd,

• •• , • • » -WJ| A \J 4

Aa, =:dy^« By^ Sxa' Sxar

Proof:

14

Prom the hypothesis,

3x3' 3:xPs dyi( 5yir 1 ^ — • • • • — • 3(1 ' " h y ) =

5y i By-s dx 1 5x r 1 * * * r

bx^ bx®s ay3-' dyirdy3' ay'3* 3xa' Sxtt|r g B

By i 5y^Sxa' Sxaf SxP| dx3* By1' Syi|r a' " M r

By rearranging and using Lemma 1.2 it follows that

d2c"'» • • Sx s Sy . .dy r j ^ o g g o «• .Bt/ \ — — — — b|/ • * * Js(y)= 6p' 6^. . .6pS A ' „5(x) ay3i ayibxa' 3xar 1i-,,ir p-> a| " M r

= £• Li i 09 »U* f

Theorem 1.8 Let ,AY(x) and B^Cy) be the components in

the X and Y coordinate systems respectively of a tensor I.

Also let A3' ***05(x) "be a set of functions in the X coordinate 01 j • • •Ct f

system. Let O1^' #"£j|4(y) "be a set of functions in the Y co-

ordinate system so that AY(X)A3' *'j^(x) and C^1 *#^s(y) are v CX| • • •U,y -L/ • • • JL |f«

the components in the X and Y coordinate systems respectively

of a tensor I1 • If Bk(y) 0, let B ' ' ^ (y) = C^'^l'^Cy)/ Bk(y), no summation on k. If B^(y) = 0, let B?i *" *2s(y) = Aa " a^x^* T h e n ' " " a ^

i, .. .irvjr

Sx3, ax3ff 3yi, ayir a, .. a, .. .ar

and Bj' * *#Jj(y) are components in the X and Y coordinate X | • • #Xf*

systems respectively of a tensor T. Proof: If B^Cy) i- 0 then .

15

ay*5' ay^s axa' axap&y* - R •

— b ' " — f i — i ' " — a . — v ±Y(*>?' , #*a ( x )

B?i •*,^(y) = 5 x 5 y Sylr5xY a' ' * r

i, .. .ir ' s lc

~ A Y < X > axY

_< < &y3' sy^s*®' sxa' „ Thus, B,3' • • • k y ) = —=*•'—» —1*"—i V • " ? ( * ) • If for

ii .. «ir ^xp, ^xpj By ' by r '

some lc, B^(y) = 0 then as given B^' ***^(y) Is the same as •L| t • *JLp

the equation immediately above. In either of the above cases

by theorem 11.7 ax9' hyp* ay1' ayir

o H ...H 1L •••-! f3i " I ' M = * a . a,Bi, ...i,K> Att, !I!ar

(x) = ay3' ay3*axa« a x ^ 1 ' •* , i^

Therefore "a?1 ***?s(x) and Bj?' * * *h(j) are components of a Ct j • • •QLf* i| • • •ip

tensor.

Corollary 1.8-1 By a similiar proof it can be shown

that theorem 1.8 is also valid when the components of a co-

variant tensor of rank one is used in place of the components

of a contravariant tensor of rank one.

Definition 1.23: Scalar A scalar is a tensor of zero

rank.

Theorem 1.9 If A^(x) are the components of a mixed ten-A

sor then A^(x) is a scalar invariant. 4

Proof: Since Alj(x) are the components of a mixed tensor

in the X coordinate system, the components in the Y coordi-

nate system will be given by

Am- -^ ( x ) J ayJ axp

16

, ax ay1

Let a = 3 so that Bt(y) = — , — A (x). By theorem 1.7, D dyJ axa a

5yJ 3xa , rr 1 1 ^ ( x ) = — a — ^ B^(y) and by lemma 1.2 = 6^B^(y).

Bx By*

Summing on I and 3 yields that 6*B^(y) = b| (yJ+BgCyJ+.-.+B^Cy).

Therefore A^'(x) = B^(y) and A^(x) is a scalar invariant.

Theorem 1.10 If X and I are admissible coordinate sys-

tems such that &|(x) is a set of functions in X and &^(y) is

a set of functions in X then 6 £(x) and 5^(y) are the compon-

ents of a tensor that is covariant and contravariant of rank

one.

Proof: Let x* = x3" (y1 ,y2,... fjr31) be a permissible

transformation. Then by definition 1.16,

3xa + dxa

+ 3X1 + ax2 t 3xn

t

— i 6 c c ^ = — i 6a ^ — i 61 + ~"~i 62 + * " + — i 6n ay1 ay ay ay ay1

axa t ax1 ax2 ax11

Khcn % zz 1,2, • • •, n "fctiGix ***"*» 6 s t • • • > rospoc*" S xa ay1 ag ay1 ay1 ay1

tively. Therefore — 1 6^ = . Now in the Y coordinate ay1 a ay1

system

, , ay3 ax3 ay3 axa p ay^ axa

p . 6Y (y ) = 6? = — 0 — 3 ^ = — o — i = — B — ± 61 (x) 1 • i ax0 ay1 axp ay1 a ax0 ay1 a

Thus 6i®(x) and 6^ (y) are the components of a tensor. 1 cc

Theorem 1.11 If ^ y ( x ) and B^t(y) are components of a

tensor T, and if Aa3(x) and Bi;^(y) are sets of functions so

that Aa0(x)i^y(x) = 6* and B1 3 (y)B^(y) = 6*, then Aa®(x)

and B1^(y) are the components of a tensor covariant of rank

17

two.

Proof: Let TY(x) and T^y) be the components of an

arbitrary non-zero tensor V. By theorem 1.6, ^y(x)Ty(x) =

Sg(x) and Sg(x) are tensor components. Then since

Aai? (x)A^y (x) = 6® it follows that IY(x)6^ = Aag (x)!^ (x)TY (x)

or

0?(x) = Aa3(x)Apy(x)TY(x) = Aa0(x)Sg(x)

Similarly,

T^yJ&J = B^tyjB^ly^Cy)

f^y) = b1^ Cy)s^ (y)

Thus since the products Aa^(x)Sg(x) and B^(y)Sj(y) are com-

ponents of a tensor, by corollary 1.8-1 it follows that

Aa^(x) and B^(y) are the components of a tensor that is

contravariant of rank two.

Definition 1.24: Reciprocal Tensors If T and T are

tensors so that If A1J(x) and A^tx) are components of I and

T respectively and A ^ (x)A^ (x) = 6^ tlien T and T are recip-

rocal tensors.

Definition 1.25: Symmetric Tensor When an interchange

of two contravariant (or covariant) indices in the components

of a tensor does not alter the value of the components then

the tensor is said to be symmetric with respect to those two

indices.

Definition 1.26: Skew-Symmetric Tensor When an inter-

change of two contravariant (or covariant) Indices in the

18

components of a tensor merely changes the sign of the com- .

ponents then the tensor is said to be skew-symmetric with

respect to those two indices.

Theorem 1.12 If the components of a tensor are symmetric

with respect to two indices in one coordinate system then they

are symmetric in all other coordinate systems.

Proof: Let Ai'1 ***i®s(x) be the components of a tensor A

in the X coordinate system so that these components are sym-

metrical with respect to two indices. Without loss in

generality assume these two indices are ^ and Then

aJi = AjM' V'^Cx) or AjMa • • ) - a-Mi •••^(x)=0. lj • • • X p JL | • • • JL X • X fr «*- / • • • X

By theorem 1.2 this difference is the component of a zero

tensor in the X coordinate system. If ^*!*^(y) and Xj • • • X

b3j 3, are the components of A in the I coordinate sys-X I • • • X yt tem then by theorem 1.1 it follows that

b1 ^*;*3s(y) - b W *;*^(y) = 0 Xj«««Xy» X | • • • X|»

BjJ/ •;,i(y) = B?a^ \"^(y) X | • • • Xp% X | • • • Xy» I

Corollai y 1.12-1 If the components of a tensor are skew-

symmetric with respect to two indices in one coordinate system

then they are skew-symmetric in all other coordinate systems.

Proof: The proof is similiar with the proof of theorem

1.12 when skew-symmetry is used instead of symmetry.

Theorem 1.13 If A^Cx) and 3 ^ (y) are sets of symmetric

functions so that if R^x) and Sa(y) are the components of an

arbitrary tensor then A ^ (x = Ba3 ^ ^ ^ ^ ^

19

then A^Cx) and Ba^(y) are the components of a covariant

tensor of rank two.

Proof: Since (x)Ri(x)R^ (x) = Ba^ (y )Sa (y )S (y) and

i * Zx ,

R (x) = — S (y) It follows by substituting for R (x) that

ax1 bx}

Ba(3 (y)Sa(y)SS (y) = ~ a 3 Sa (y)SP (yjA^ (x)

dx " Sx^ [Ba3(y> " ~ a ^ Ai3(x)]S

a(y)S9 (y) = 0

Let Ca^(y) equal the term enclosed in the above set of

brackets. Then (yjs" (y )Se (y) = 0. Summing on a and f) and

collecting terms yields

0,, (y)Cs' (y)]2+022(y)[S2(y)]2+...+0ml(y)[S

11(y)]2+

C 0aB ( y ) + ° B a( y ) ] s a ( y ) S 0 ( y ) = 0

where a / 3 and a,3 = l,2,...,n. Since this sum must be

zero for any tensor T it must be that the coefficients are

zero. Thus * dxA SxJ

Oa0(y) + C8a(y) = CBae(y) - - ^ A^tx)] +

1 i dxJ 5x

[ V y ) " ^ - a V x ) ] = °

3x " 3x^ ' CBa3(y) + Bga(y)3 = ~ a ^ [ A ^ x ) + A^Cx)]

But A^Cx) and Bap(y) are symmetric functions so that

bx1 Bx3

Baf3(y) = — — p Ai;j(x). Therefore A^Cx) and Bag(y) are By 3y

components of a covariant tensor of rank two.

20

Theorem 1.14 Let an& (y) b e '®ie components of

a tensor so that the determinant of the g^teJ's a-*1*3-

determinant of the hag(y)'s, denoted by g(x) and h(y) respec-

tively, are not zero. Also let gi^(x) "be the cofactor of

g^(x) divided by g(x). Then g3" (x)g^(x) = 6^ and g^(x)

are the components of a tensor whose components in the I co-

ordinate system are ha^(y), where ha^(y) is the cofactor of

hga(y) divided by h(y).

Proof: Summing on 3 gives that

g1^ (x)g^lc(x) = gi1 (x)g1lc(x) + ...+ g^CxJg^Cx)

cofactor g,,(x) cofactor g.(x) — J +...+ g^Cx)

g(x) g(x)

If k 4 i then g1^ (x)g^(x) = l/g(x)[0] = 0. If i = lc then

g1^ (x)g^(x) = l/g(x)[g(x)] = 1. Therefore g1^ (x)gjk(x) = 6£.

By a similar argument it follows that ha^(y)hgY(y) = 6^ if

ha^(y) is as defined in the hypothesis. Then by theorem 1.11

it follows that g^(x) and ha|3 (y) are tensor components.

Definition 1.27: Fundamental Tensors The two tensors

of theorem 1.14 with components in the X coordinate system of

g^fx) and gj^(x) will ke called fundamental tensors and

g^(x) will be called the conjugate of g^(x).

Theorem 1.15 Let g±j(x) and \ 0 ( y ) be the components

of a symmetric tensor so that the determinants of the Sjy(x)'s

and hag(y)'s, denoted by g(x) and h(y) respectively, are not

zero. Also let g1^ (x) be the cofactor of g-^fe) divided by

21

g(x). Then g^(x) are the components of a symmetric tensor.

Proof: By theorem 1.14 g^(x) are the components of a

tensor. Let (x) = g ^ and g^(x) = g"^. Then by defi-

nition g1^ = cofactor g^/g(x). Similarly g^= cofactor g^^/g(x).

Since the g ^ are symmetric g"^ = g ^ and g^(x) are the com-

ponents of a symmetric tensor.

Note: When no trouble in notation will arise A*' *** 5 X | • • • J.

will be used for A*' ***;js(x) x/ # • «Xf

Theorem 1.16 If A^Cx) and Ai;j(x) are the components of

symmetric fundamental tensors, if u^(x) are the components of i i n

a covariant tensor, and if u (x) = A (x)ua(x) then

A ^ (x)u^(x )u (x) = A ^ (x )u^ (x )u.. (x).

Proof: Since A"^ and A^j are fundamental tensor compon-

ents by theorem 1.11 AaiAi;5 = 6^ and. u^CA®1^) = u ^ = u^.

Also since A ^ is symmetric A ^ A ^ = A ^ A ^ = Thus

6^Ai^ = A^Ai;JAi;I = A0^ and

AP 3 (uaAaiAi;j) = (6^A

i3 )u3

• y A P V a l V s VtiAllV u^u^A^ = u^u.jA^, where u1 = uaA

ai.

Definition 1.28: Ohrlstoffel Symbols" Let S-jj(x) be 1^Le

components of a symmetric covariant tensor such that the de-

terminant of the g_y(x)'s is not zero, and let g^(x) be the

conjugate of gjj(x). Then the sets denoted by [i;J»k] and

{i^jl defined as follows,

22

/ v r Sg,, (X) Bg. ,(x) (a) [id,k]=>|[— LL1—]

BxJ ax1 5xIC

0>) { A } = glk(x)[i^,lc] Z"

The symbols defined In (a) and (b) denote the Ohrlstoffel

symbols of the first and second kind respectively.

Theorem 1.17 If g^(x) are the components of a covariant

symmetric tensor then = +

axD

Proof: By definition 1.28

&S11r . ag,, ag., 2[ij,k] = — ^

3xJ Sxx ax*

SgiT . aS1lr Sg, 1 — ~ = 2[i3,k] f2-+ — e 1

SxJ ax 3x

3slk Bgik 5Si 4 ag1lr 5g, , = [ij,lc] + i-[—p- + —

3xJ ax ax ax ax

Bgj, agj j agji = Ci3,kJ + ft—-P- + - P - ft

axJ ax* ax

= Cij.ic] + CJcj.i]

Theorem 1.18 If g^(x) is the conjugate of the symmetric

components g^te) of a tensor then —r— = -g"*"a{a ki -g^aia lei*. . J axc

Proof: Prom theorem 1.14 g^agai = so that by differ-

entiating this expression with respect to x yields

la^ai 8 S ? " + ^ a i = 0

Since g s l = gia

23

_ 1a Ssia

W 5s^a - - -1P-3® 5 g l a ® % a " ' k"' *~ a k " S ^ ]c l a bx dx Sx

Then by theorem 1.17

5P-0;5 JL_ = _ _ _ = -g

lB Sialic,a] + [ale,i] )

5 X S X = -glp (g3a[Ik,a]) - g^a(g1B[alc,i])

= - S ^ J k } -S3a{aSk)

By replacing the summation index in the first term with a and

replacing 3 with i in the second part of the above relation

nothing will be changed since the same operations are still

indicated. Thus

p r = - s ^ i c }

OX Theorem 1.19 If g.Jx) are the components in the X co-

' 1 1

ordinate system of a symmetric tensor T'and if y = y (x) is

a nonlinear permissible transformation of X into the I co-

ordinate system then the Ohristoffel symbol, of the first kind

is not a tensor. *

Proof: Let k^CyJ.be 'b^Le components of T in the Y co-

ordinate system. Then

Sxa 3xP

^ By3" 3y^ gaP

and taking the partial derivative of h ^ with respect to

24

yields

ah^ axa 3x0 S^g 3xy a2xa bx® Sxa a2x0

Sylc ay1 ay"3 axY Sy^ ay^y* ay"5 S(Xp ay1 dy^dy*

Similarly,

ah,, Sxa SxY ag Sx0 a2xa SxY 3xa a2xY tmmmmSm •— t n m w * mmmmm mmmmmmX mmmmm «£• mmmmmmrnmmmm mm*mt £>» *|» mmmmm mmmmmrnmmmmm

By*3 ay1 a/1 axa ay3 ay^y3 a/^*7 ay1 ay^ayk

ah;Uc axp axY ag8v axa a2x0 axY ax3 a2xY

By 1 ay-3 a/1 axa ay1 ay^ay1 ay^ § 0 Y ay3 ay^ay1 S g Y

Let [i3,3c]y. indicate the Ghristoffel symbol in the I coordi-

nate system and [i ,1c] indicate the Ohristoffel symbol in >A>

the X coordinate system. By definition [i ,lc] will then be «y

given, using the above relations, by

axa axY ax3 agay a2xa axY axa a2xY

Cid»lc]y a ^ y i Syls: Syd ax3 ay1ay^ ayk S(XY ay1 byhy^ SccY

ax® axY axa ag9Y a2x^ axY ax0 a2xY

ay15 ay^ ay1 axa ay^ay1 ayk S0Y ay^ ay^ay1 S g Y

axa ax0 axY ag^ a2xa ax3 axa a2x3

ay1 ay3 ay12 axY ay -ay* ay3 ga0 ay1 ay3ayk Sa0^

axa axY ax0 ag axa axY ax0 ag a = 4(P- 7? 7? J + ^ P S* ' • axa axY ax3 aga3 a2xa axY axa a2xY

ay1 ay^ ay3 axY ^ + S^ayisyi3 ayk SaY * ay1 ay^ayk °aY

a2xp axY ax@ a2xY a2xa axp

ay^ay1 ay^ S 0 Y ay3 a^ay1 S g Y ayiayk ay3 Sap

25

5za bS7?

ay1 ay^ay^ ^ ^

!2?2i Y i f r H ! % Y + ^ Y . i S t f n +

ay1 Sy^ Sy*3 * ax0 axa axY

a2xa axY

a y V a / b a Y

axa axY ax0 a2xa axY

- — _ _ Cap,Y3x + — 7 — 1 — v g„v

ay ay ay"3 x ayHy 0 ay* a Y

Therefore [ij,k] is not a tensor.

Theorem 1.20 If g ^ C x ) and h ^ C y ) are the components in

the X and Y coordinate systems respectively of a symmetric

tensor T and if y " = y^(x) is a nonlinear permissible trans-

formation of X into Y then the Christoffel symbol of the sec-

ond kind is not a tensor.

Proof: Let denote the Ohristoffel symbol of the

second kind in the Y coordinate system and

Christoffel symbol in the X coordinate system. Since

a n d ki^Cy) are the components of a symmetric tensor, then by

theorem 1.19

axa a 2x a s xy •

C i 3 » k] — 4 V '"•< I- » Y ] y- < 4 V Sff v 7 ay ay^ ay*3 x ayiayi3 ay* a Y

-lkr lk ^ 5 x Y

h^CiJ.k] = h 1 ] £(- - - Ca3,Y]x + * ), ay1 ayc ayJ x ay ay3 ay a Y

T 1y

where h is the conjugate of h ^ . By theorem 1.11 h l k are

tensor components so that

26

1 ic ^ 2 1 g5v

Sx ax

where g6 Y i s the conjugate of g 6 y . Therefore h l i c [ i 3 , k ] y =

{ i 13 } y a a i >

, ay 1 a / 6 axa axY ^ s V a*Y

{ j j L = ~~~c ~~v g — k •"""* Ccc0,v] + —r—•. , ®av *4 S x

6 3 x v ay a / 1 ay3 ay ay3 a j r

ay 1 ay k axY

ax6 axY ay^

ay 1 axa axe

ax6 By1 ay3

_ S y l axa ax0

ax6 ay 1 ay3

s , Y Cae,Y] , +

3 2 x a by 1 ay^ 3xY 6 y

i ^ V 5 ? ^ Y S ? ® ®aY

a x ay 6

( a M + ~ 6

I

x .-8y l3y3 3x6 a

, 1 ^ + i 1 i ay ayJ ax

Thus { j ^ } i s 3ao"t a "tensor.

Theorem 1.21 I f A, , i s def ined as X9 J

SA j (x ) f a L / \

• A i - 3 = £ 3 ^ M C x ) >

where A^Cx) are the components of a tensor A i n the X coordi-

nate system, then A^ ^ are the components of a tenso r .

P roo f : Let B^(y) be the components of A i n the Y co-

ord ina te system. Then ax

B - —. 1 ay 1 a

27

Taking the partial derivative of B± vrith respect to y^ yields

5B, dxa Sx0 a2xa

rnmmmJmm MM* mmmmm mmmmWWWMW «|* mmmmmmmmmm*

By^ ay1 axP 3y^ ay^y^ a

Prom theorem 1.20 it follows that

3 x 6 r 1 | _ Sx6 By1 Bxa 3x0 r 5 "i S2x6 ay1 ax6

ta J J y " Z i T 3 3 i 7T.1 la ei": ay1 U J ; y = ay1 ax6 ay1 ay* t a ^ + a y V ax6 ay1

a2x6 ax6 r ! i 5xa ax0 , 6

ay^y^ ay1 * y ay1 ay* 0

^2 a a x

Thus by substituting for 11 • .1 % ay ayJ

aB ^ _ axa ax0 aAa axa , 1 , axY ax0 , a .

ay* ay1 ay* axg aay 1 * y ^ay3* ay^ Y 0

ax aBi J . _ -S

Ihen noting that J a — x = B x and B ± > } = — 3 - ^ {j1.,} ay ,J ay

S Bi J 1 I 9x<X d x P d Aa S x Y 3 x 0 r a I

ay3 1 ^ y ay1 ay* ax0 "^ay1 ay* " Y p x

- S-fct f V L ^ ^

^ ay3" ay^ax 0 \ la pj" Sjri a yj Aa,p

since the changes in suffixes still indicate the same

operations. Therefore A^ ^ are tensor components.

Definition 1.29: Oovariant Derivative If a|' " * *^5(x) L L' JLj • • «lLp

are the conponents of a tensor A in the X coordinate system

and if g ^ ( x ) are the components of a fundamental tensor in X

then the covariant x 1 derivative of A*' denoted by X j • •

28

jJi-'i', (x), is JLt«*«JLy*yJU jt .i

3 Ai " i t x ) = a x r - (i"J. -

{ 4 } - ••• - {i°J *2; :::£*> +

• W • - •

W

Theorem 1.22 If Aa^ and B"^ are the components of a ten-

sor A in the X and I coordinate system respectively and if

Ha£i

3x AaB = — + ( M A69 + ( 3 1 A< , Y ...Y U YJXa 16 yJx

Bi3 - — + f 1 \ £3 + S i I *,1 a yl

+ Ik l/y Ik liy

then Aa^ and B3*-, are the components of a tensor. > Y f -k

Proof: Since B3" are the components of A in I then

Bij _ El1 ^ &

' zJ ^ 11 1

Taking the derivative of B J with respect to y yields

SBi3 S yi 5y3 3Aa3 axY+ a2yi 5xY ^

Sy1 dxa dxP 9xy. Sy1 axa5xY ay1 Sx^

By1 J * V SxY a 0

dxa dxgdxY ay1

As shown in theorem 1.21

a V _ ^ 1 s 1 _ r 1 1 axaaxp ax6 1 3 x dxa axe *,i 3 7

29

Then on substitution,

3^ ivY N.ttP > 1 ! £ = i r iT" r ! T + , ! £ ( . ) a B

Sy bxa SxB ay1 axV 5xa U yJx 5xa Sxy Ik l/y' iyX ~ 8 A

+ ? 4 { 8 J. . 22" J 1 } j!!1 !iY Aae . axB \f> yJx axB 3 x y to l i y ^ a 3yi

A

By using different indices to indicate summation, the tensor

relation between B ^ and Aap, and lemma 1.2, it follows that

SB1^ By1 dyj dxY 3Aa3 , , •>

s 7 = sla ^ ( + {aaY}x

A<1 S ' V y I x ^ > - l^y^3

h,im •y

aB1^ ^3 „,,Y --r - f 1 I + / n b1* - — s r S x ,-M 3y l k ^ " 3xa & a.Y

B 4 = 22* !iV Aae ' dxa Bx^ by1 ,Y

Therefore A ^ are components in the X coordinate system. A ^

represents the covariant x1 derivative of the contravariant

tensor A"^.

Corollary 1.22-1 If Aag and B ^ are the components of a

tensor A in the X and Y coordinate system respectively and if

SA

^P,Y = " (a y}xA63 " {0%}* Aa6

3B. . . -

*13,1 = ^ - { i V y - { 3\}y Bi*

then Aag^Y and B ^ ^ are the components of a tensor.

30

Proof; The proof is similar to that of theorem 1.22

with the result that A ^ ^ is a covariant tensor component of

rank three.

Corollary 1.22-2 If A^g and B ^ are the components of a

tensor A in the X and. Y coordinate system respectively and if

a.y6

$ . C = ^ 8 - W . < - W x < +

W x $ + V J x $ a Bkm

iJ. f n I Jem / n 1 ^km . Bij,l - S yl li liy \ ) - {j liy in +

f lc 1 rsHni , f m 1 -p.3oi In l/y Bij + in l/y \ j

then ^^ and 3^? ^ are the components of a tensor-

Proof: The proof is very similar to the proof of theorem

1.22, the only difference being the length of the terms in-

volved.

Theorem 1.23 If A^^ and A ^ are tensor components then

(Aij + A ^ ) ^ = A i ^ 1 + A i ^ 1 .

Proof: Let A ^ and A^^ be the components of the tensors

in the X coordinate system. By corollary 1.22-1

IN* ^ »4 Jt ^ JLj» jl - _

^ 3 , 1 + A U , 1 = + ^ " ^ 3 + M i l)*--

'Aik + f u c K / l l x

bince A ^ and A^^ are tensor components by theorem 1.2

(Aij + A ^ ) are tensor components covariant of rank two. Thus

by corollary 1.22-1

31

a

dx"*". (A^ + ~

+ ^ij^ " ^ 3 * "

W k + ^ikK^llx

a A j j j S A j j ^ r v •

" a ? + a ? " + " '

* fucKj^lIx

Therefore, U ±^ + = Aij,i + Ai;),l

Theorem 1.24 If A ^ and A*031 are tensor components in the

X coordinate system then (A^A1™)^ =: A^ifj + A ^ ^A1011.

Proof: Since A ^ and A are tensor components, by using

theorem 1.22 and corollary 1.22-1, multiplying, and adding the

results it follows that

al1™

h f t * = V - r + W • ( A l * t a > • 3r

3 A,

ax jtoiflhj _ i" a 1 A - i a } A ) A \„1 li 1J Aaj 13 1J ia'

Now A^l?011 are the components of a mixed tensor, denoted by

B^, that is covariant and contravariant of rank two. Thus by

corollary 1.22-2

(A A1™) - 1 ^ 5 — L I a -icmx - { a W A A1™) + tAiJA ,1 dxl ti 1J (A^A ) 13 1J U±a A *

la^l} + {ami} 1^°" ^

s~km

= V i a + 1 A } ^ * { +

32

Therefore, U 1^a t L) t l =

A ± ^ l + h ^ l ^ '

Theorem 1.25 If an<i SX^(x) are the components in

the X coordinate system of fundamental tensors then g ^ ^ = 0

and = 0. I J-Proof: By corollary 1.22-1

si3,l = H P " (iai}sa3 " {^ai}sia

.a

By theorem 1.17 — ^ = [il»3] + Ul»i]> and by definition

{i^j} = so that

%c3"{i 3} = slcps " 6gti^»a] - [ij»3]

Therefore

siJ,l = + ^ 1» i^ ~ {^il^ad " it J"Sia

= {iailsa^ + {^lKxi " {iai}ga3 " {/l}si<

Since gai = gia it then follows that = °*

By theorem 1.14 by theorem 1.24

= gi;5s3ic»i + = 6^,i

But g 1 3 ( 1 = 0 so that g^g 3 k = How .

6k,l = ^ + = ^ ^ = °

Hence S ^ g ^ = 0» From the definition of g3" it is required

that the determinant of the g^'s not be zero so that it must

be that g1^ = 0.

CHAPTER II

RIEMAM SPACES

Definition 2.1 : Simple Arc A simple arc is a set A of

points with respect to a vector space Vn such that

1) every point (x1,x2,...,xn) in A is given by x3" = f^t),

where f (t) are n functions that are continuous, have

a continuous first derivative, and a^t^b with a<b

2) if P is the point given by x^ = f^Ca) and P1 is the

i. i point given by x = f (b) then P is the first element

of A and P' is the last element of A and if a<t^<...<b

1 i then P precedes the point P represented by x = f (t^),

i i P1 precedes the point P2 represented by x = f (tg),

i i

..., and the point Pn represented by x = f (t^) pre-

cedes the point P1, and

3) If a«t,<t2sb then there is a positive integer iSn so

that f^t. ) 4 fi(t2)

It will be assumed that associated with each ordered pair

of points is a non-negative real number having the properties

1 ) if P is a point and Q is a point such that P = Q then

the number associated with P and Q is zero,

2) if P is not Q then there is a positive number associated

with CP,Q) and this same positive number is also asso-

ciated with (Q,P), and

33

34

3) if P,Q, and R are points then the number associated

with P and Q plus the number associated with Q and R

is greater than or equal to the number associated

with P and R.

Definition 2.2: Distance Between Two Points The number

associated with a pair of points will be called the distance

between the two points.

Definition 2.3: Arc Length Let A be the arc given by

i 1 x = f (t) with a^t^b and a<b, let P be an element of A given

i i 1 JL by x = f (c), let Q be an element of A given by x = f (d),

where c<d, and let D be any subdivision of the interval [c,d]

of the form c = tQ<ti<t2<.. .<tn = d. For each t^, i = 0,1,...,n,

a point P^ in A is obtained. Let. d^ be the distance between

the points P^ and P ^ and let S =Xd^. For each subdivision

D a S is obtained. Let B be the set of all such S. If B has

an upper bound then the arc length between P and Q along A will

be the least upper bound of B. If B has no least upper bound

then the arc length will be infinite. All arcs considered in

this paper will have a finite length.

Definition 2.4: Ourve A curve in a vector space Vn is

an arc.

Definition 2 .5 : Riemann Space A set of points R^ is a

Riemann space if and only if there is a set S of coordinate

systems for Rn, a covariant tensor T of rank: two, and a class

H of contravariant tensors of rank one such that

35

1) if R is in H then R and T have components in each

element of S

2) if X is in S and g.y(x) are the components of T in X

then g-j^Cx) = the determinant of the Sj^(x)

i

is not zero, and for any non-zero components R (x) of

a contravariant tensor in X g^(x)R^(x)R^£x)>0, and

3) if P and Q are two points of R^ with coordinates a

and b, P Q, and ds is the distance between P and Q

then there are in X tensor components H*(x) and H "(x)

of two tensors from H such that if ds is the distance

from P to Q then

(ds)2 = g ^ (a/H1 (a)H (a) = (bJH1 (bJH3 (b)

The tensor T will be called a metric tensor for R^.

Lemma 2.1 : If P and Q are two points in a Riemann space

Rn, X and Y are two coordinate systems for Rn such that g-j^fc)

and h ^ (y) are the components of a metric tensor in X and I 1 1

respectively, and R (x) and S (y) are tensor components in X

and Y respectively such that the distance ds between P and Q

in X is (ds)2 = g ^ (xjR^CxjR^ (x) and the distance ds' between

P and Q in Y is (ds1)2 = h ^ (y JS1 (y )S (y) then ds = ds'.

dxa 3x^ Proof: Since ^ ( y ) = — ± Sap(x) and

• dy Sy S (y) = — Ra(x) by substituting into the expression for (ds1)'

3xa it follows that

36

p 3xa ax0 Sy1 By3 fi (ds') = [—, — 1 &,8(x)][— R (x)][—X R3£x)]

dyx dyJ a p Sxa axp

Sxa By1 Sx® By3 B

- (— —* )(—1 ~ B ) SaB(x)R (x)R (x) by dxa SyJ dxp a p

= 6 a 6 3 Sap (x)Ra(x)RP (x)

= Sa8 (x)Ra(x)RP (x)

Therefore (ds )2 = (ds')2 and ds = ds1

Definition 2.6: Euclidean Space Let 6 ^ = 0 if i / j

or 6^^ = 1 if i = J and if (a1 ,a2, ..., a11) and (b1,b2,...,bn)

i i i

are the coordinates of two points let dx = b -a . If there

is a coordinate system X for a Riemann space Rn so that if

(a1 ,a2,... ,an) and (b1,b2,..., bn) are the coordinates in X of

two points then the distance ds between the two points is

given in X by (ds)2 = S^dx^dx^ "the*1 "the Riemann space is an

Euclidean space E .

Definition 2.71 Cartesian Coordinate System If Y is a

coordinate system for the Riemann space R^ in which the com-

ponents gj_j(y) of a metric tensor are constants then Y is a

Cartesian coordinate system.

(Theorem 2.1 The components S;y(x) a metric tensor

reduce to constants k^(y) i n some coordinate system T if and

only if the Chrlstoffel symbols of the second kind in the Y

coordinate system are zero.

• Proof: If the ^^(y) are constant in the Y coordinate

system, then

37

{ikj}y = = 4*kl A 1 * ^ 3 ) = ihkl(0) = 0.

' dyJ by by

Thus if the jCy) are constants in I then in this coordinate

system the Ohristoffel symbols of the second kind are zero.

Talcing the covariant derivative of h. . (y) with respect 1

to y gives that

= Ir3- {AIAJ " {j'IW But if the Ohristoffel symbols of the second kind are zero in

1; .1

the Y coordinate system then h. . , = ° i.1 _ ,, „ __ iO»l — r . By theorem 1.25

Ah h. . , = 0. Therefore i;5 _ A i +-u ^ j. A ij»l ~~r - 0 and the iu are constants. ay

Definition 2.8: Tensor Length Let A (x) he the com-

ponents in the X coordinate system of a tensor A in a Riemann

space Rn and let g-y(x) be the components in X of a metric

tensor for B^. Then the length L of A at x in X is given by 2 i 1

I» = Sj_jA AJ. Similarly for the components A^(x) in X of the

tensor I, the length L is given by L2 = gi^A1Aj, where g1^ is

the conjugate of g^.

Definition 2.9i Associated Tensor Let X be a coordinate

system for a Riemann space R^, g - (x) be the components in X

of a metric tensor for R^, and A^(x) be the components in X of

a tensor A. Then the components A^(x) of the tensor X, given i —»

by A^ = g^A , will be called the components of the tensor A associated with A. Similarly if A^(x) are the components in

X of a tensor A and if g^(x) are the conjugates of g^Cx)

then the components AiCx), where A^x) = g1^ (x)Aj (x), will be

38

called the components of the tensor associated with A.

Theorem 2.2 If L is the length of the contravariant

tensor A of ranlc one in the X coordinate system for a Riemann

space then L is the length of the tensor A associated with A

in X.

Proof: Let A be the components of A in the X coordinate

system and g ^ the components in X of a metric tensor. By

definition 2.8 the length of A at x is L2 = g^A^A^ and by

definition 2.9 the components of the tensor associated with A

in X are A^ = g^A^. gia are the conjugates of g^a then

= g i aS l / = = Aa

Therefore the substitution of the above into the expression

for the length of A in X yields

*2 = = Vp and L is also the length of the tensor associated with A.

Definition 2.10: Ooslne of the, An^le Between Two Tensors

1 1

Let A (x) and B (x) be the non-zero components of the tensors

A and B respectively in the X coordinate system for a Riemann

space R^ and let g^(x) be the components in X of a metric

tensor for The cosine of the angle 9 between A and B in X *

will be given by

g,, ix)A1ix)Bi (x) COS© = "•

[g^ (x)A1 (x)A^ Cx)]^

i i

Theorem 2.3 If A and B are the non-zero components of

the tensors A and B respectively in the X coordinate system

39

for a Riemann space R^ and if the cosine of the angle © be-

tween A and B in X is defined as in definition 2.10 then

|cos©| 1 .

Proof: Let g ^ be the components of a metric tensor for

i i in X. If for any real number a, aA + B / 0 then since i 1 i

gijC 0d>0 for any non-zero tensor components G

gi^aAX+ Bi)(aA3+ B3 )>0

gjy (a^AiA^+ 2aB^Ai+ BiB^ )>0

gi^AiA;5a2+ 2gi^A

iB'5a + gi^BiB^>0

The last expression above is a quadratic in a so that the

discriminant of this quadratic must be negative. If it is

not negative then there will be some value for which the

quadratic will be zero and thus not positive. Therefore

Since g,-A BJ>0,

teg^AV )2- 4 (gi;.AiAi3) Cgi BiB'3 )<0

gi^AiB^< CgijA^ )^(g1;JB

1B;3)«

and |cos0j<1.

i i If there is a real number a so that aA + B = 0 then

i 1 B = - aA . Upon substituting this in the formula for cos©,

-a COS© = - 'W = +1

[(-a)2]* " Therefore J cos ©I =£1.

Definition 2.11 : Hypersurface in a Riemann Space A

hypersurface in a Riemann space Rn is a set of points S such

that S is contained in R^ and S has the following properties

1) there is one coordinate system X for R^, an interval

40

1 2 a<u <b, an interval e<u <d such, that every point *1 P V J

(x ,x ,...,x ) in can be represented by the func-i l l p 1

tions x = x (u ,u ). The functions x will be real

valued, continuous along with its first partial de-

rivative, and reversible over the above intervals of

1 2 definition of the parameters u and u .

2) if H is a set of two dimensional coordinate systems

1 P

for S containing (u ,u ) then S is a Riemann space Eg*

It will be assumed that 1 = 1,2,3 for the hypersurfaces

throughout the rest of this paper. 1 2

Definition 2.12: u and u Curves of a Hypersurface Let S be a hypersurface as defined in definition 2.11. Let

1 1 ?

a<h<b and c<lc<d. The curve x = x (h,u ) will be called the

u^ curve with respect to h. The curve x"*" = x Cu'',]!) will be

called the u1 curve with respect to k.

Definition 2.13J Coordinate Curves of a Hypersurface

1 2

The u and u curves as defined in definition 2.12 will be

called coordinate curves for the hypersurface S with respect

to h and 3c.

Theorem 2 A If S is a hypersurface in the Euclidean

space E^ and X is the coordinate system in E^ so that the

distance ds between two points represented in X by a and b is

(ds)2 = S^dx^dx^ then there is a metric tensor with compo-

nents Bxa 3xa hi3 ( u ) = ~ 1 J 3u dUJ

41

so that the distance ds in X is equal to the distance ds1 for

the hypersurface coordinate system containing (u1,u2).

Proof: Let each point be represented in X by (x1,x2,x"^),

where x3" = x^u^u 2). Krom calculus, letting a = 1,2,

, ax3" dx = —• dua

3u

Since in X it was given that (ds )2 = 6^dx^dx^ for any two

points, by substituting for dx1 it follows that

o dx1 „ Sx^ dx1 dx3, „ Q

(ds) = 6 ± ( — du )(—6 du0 ) = — — duadu3

± z dua dup du dup

dxa dxa

Let hj. i (u) = — . —~. Prom lemma 2.1 it was shown that ds is XJ. dux duJ

invariant for any contravariant tensor from H. Therefore by

theorem 1.13 it follows that h^(u) are the components of a

covariant tensor of ranic two and thus are the components of a

metric tensor for S.

Theorem 2.5 If S is a hypersurface in the Euclidean

space Ej, if X is a coordinate system so that the distance

between two points represented by a and b in X is given by

2 i 1 (ds) = S^^dx dxJ, and h^(u) are the components of a metric

tensor for.S as defined in theorem 2.4 then the components of

1 2 i the unit tangents to the u and u curves are 1 dx and

i 1 respectively. 1 )asu

(h22 )* du

Proof: Let x1'= xl(u1,u2) be the representation of x1

in X. In Ej every point will be determined by an ordered

4-2

4 p 4 4 4 p triple of real numbers (x ,x ,x ). Since x = x (u ,u )

these ordered triples can "be written as I I P P I P ^ I P 1

Cx (u ,u ),x (u ,u ),x (u ,u )] and every point on the u 1 1 P 1 ^ 1

curve will be given by [x (u , c),x (u ,c ),x (u ,c )]. Denote 1 1 2 5 1

the points of the u curve by (r ,r ,r ). Then r , i =' 1,2,3,

are the components of a radius vector r. Using the result

/dr dl*2 . from Vector Analysis that ), where s is arc

1 1 P

length, is a unit tangent to the u curve at (r ,r ,r ) gives

dr * that , i = 1,2,3, are the components of a unit tangent. Prom calculus

i i 1 i 2 dr 3x du1 dx - du

ds du1 ds du2 ds 2 1 2 du Por the u curve u is constant so that = 0 and thus ds

1 i 1 dr dxx du1

—. = — —. . prom theorem 2.5 ds is given for the hyper-ds 3u ds

2 1 1 1 surface by (ds) = h^du duJ. However along the u curve this

reduces to (ds )2 = h ^ (du1 )2. Thus ds = (h^ ) "du1 and

J — Therefore 4 s " Ch,,)** i i 1 i

dr1 bxx du1 1 3x

ds du1 ds (h 1 Su1

1 Sx1 < and t — 1 are the components of a unit tangent to the u

(h11 P 5u1 1 dx1

curve. By a similar proof it follows that — — - x —•_ are the (h22)s 5u^

2 components of a unit tangent to the u curve.

Definition 2.14: Angle Between a u1 and u2 Curve

43

S be a hypersurface in a Hiemann space and let X be a

coordinate system for so that every point P of S is repre-1 1 1 P

sented in X by x = x (u ,u )• Let x be tlie point of

1 2 2 intersection of the u and u curves formed by letting u = c 1 1 p and u = 3c. The angle between the u and u curves will be

i the angle that the tangent to the u curve at x and in the

1 2 u -u plane must be rotated counter-clockwise so that it lies

2 1 2 on the tangent to the u curve at x in the u -u plane. If

the two curves do not intersect then the angle between them

will be zero.

Theorem 2.6 If S is a hypersurface in the Euclidean

space E > X is a coordinate system for E^ so that the distance P "1 i

ds between two points is (ds) = &.ydx dxJ, are the

components of a metric tensor for S as defined in theorem 2.4,

1 2

the components of the tangents to the u and u curve at the

point x are the components in U, where u is a coordinate sys-

tem for S, of a contravariant tensor of rank one, and S is

Euclidean then the cosine of the angle 9 between the u1 and 2 ^12 u curve is cosG = 1 i 1"' t»

Proof: By theorem 2.5 the components of the tangent to

1 2 1 dx1 1 ax1 the u and u curve are s — , and —•——± — « respectively.

Chn)* au1 (h22)i du

By definition for an Euclidean space

u

1 ax1 1 6 [ LM - r — ] [ l" Ji 0] iJ (hn)

e au1 (h22)s au

COS0

, 1 ax1 . . . 1 ax^ 1 ax1 1 ax3

(n11 )* au1"^ (hn )» du (h22)^ au

2""" (h, ^ ^"2 22 au

1 ax1 ax^ ~ T7~~rx i o

- 1 1 22 d U 5 u

1 W a & i

(h. F(n22)8 au1 au1 au2 au2

axa axa By theorem 2.4 h, . = — , — , so that substituting above yields au1 auJ

h12 cos© = vi, 1

(h^ )8(ii22^S

Theorem 2.7 If X and I are two coordinate systems for an

Euclidean space En, x5* = x^Cy1 y*1) is a permissible trans-

formation, X is a coordinate system so that the distance ds o 1 1 1 1

between two points is (ds) = 6. .dx dxJ, and x = x (s) where i J i

rbr d.v

s indicates arc length then and are the components of a

contravariant tensor of rant one and length one.

Proof; Since the transformation is permissible it is

possible to express y1 as j1 = y1 (x1,... ,xG). Then from cal-

dy1 ay1 dxa dx1 dy* cuius —- = — — . Then — and — are the components of a

ds axa ds ds ds contravariant tensor of rank one.

By definition the length L of a tensor with components

45

dr^" 2

in X is L = 6 ^ ^ 2s" • Also in X the distance bet-ween

two points is given by (ds)2 = S^dx^dx^. .Dividing this last O

expression by (ds) yields 1 - A dx3" dx^ ~ ij ds ds

2 Therefore L =1 and L=1 and the length in X of a tensor with

dx1

components is one. By lemma 2.1 the length will be the same in all other coordinate systems for E . Thus the length

i dx

of a tensor with components ^ is one.

Definition 2.15: Curvilinear Coordinate System Let A X ! p *z

x = x (y ,y ,y ) be a permissible transformation from the

coordinate system I to a coordinate system X for a Riemann

i i i space Rj. If x equals some constant c and c is allowed to

1 1

vary then x =c determines a set of one parameter surfaces in

Y. Thus for a fixed c , i = 1,2,3, three surfaces are deter-

mined and these surfaces will be called the coordinate surfaces 1

for c . Since the transformation is permissible the surfaces

will intersect at only one point and the surfaces x1 .(y1 ,y2,y"^) = c1 and x2(y1,y2,y^) = c2 will intersect in a

•3 line. This line will be called the x^ coordinate curve.

1 2

Similarly the x and x coordinate curve is formed. These

curves, intersecting at the same point the surfaces do, will

be a curvilinear coordinate system for R^.

Definition 2.16: Angle Between Two Coordinate Curves

The angle 9 between the x3" and x^ coordinate curve is the

angle that the tangent to the x 1 coordinate curve at the point

46

of intersection and in the plane determined by the x3" and x^

curves must be rotated counter-clockwise so that it lies on

the tangent to the coordinate curve at the point of inter-

i 1

section and in the plane determined by the x and x° curves.

Theorem 2.8 Let R^ be a Riemann space, be the com-

ponents in the X coordinate system for R^ of a metric tensor,

and the distance ds between two points in X be given by

(ds)2 = g^dx^dx^. Let the permissible transformation

x3" = xi(y1,y2,y^) and (c1,c2,c^) determine a curvilinear co-

ordinate system for R^. If x3" = f"*"(s), if are the

components of a tangent to the x3" coordinate curve as in def-dx1

inition 2.16, and if are the components in X of a

contravariant tensor of rank one and length one then ~ '—, and the cosine of the angle 9 between the x3" and d S

1 ®i.1 xJ coordinate curve is cos© = —u—-r x •

(Sii) a%^) 8

Proof: Let (x1,x2,x^) and (xVdx1 ,x2+dx2,x^+dx^) be two

points on the x3" coordinate curve. Then the distance ds be-

2 i i v - * i

tween these two points is (ds) - gj^dx i an(^ = (g^^cLx •

dx1 I Therefore - i

cSiir ds 8

Since are the components of a contravariant tensor

of rank: one and length one, then by definition the cosine of

the angle © between ^ and ^ is

i

47

dx^ 0 0 3 0 - SiJ ds dF

dx1

Then substituting for yields

_ncjfl si.1

C O S 6 * — • IJ.ri.rrf "I- ..

(Sil)»cSj3)«

Theorem 2.9 Let x3" = c3" be the coordinate curves of a

curvilinear coordinate system in a Riemann space R^ as de-

fined in theorem 2.8. Then the curvilinear coordinate system

will be orthogonal if and only if g ^ - 0 for every point

where i / 3.

Proof: Prom theorem 2.8 the cosine of the angle © be-

i 1 tween the x and x° coordinate curve is

sii COS© = t'1"" -1

Csi3.)s Cs33>

e

If the curves are orthogonal then cos© = 0 and therefore

the above expression for cos© is zero. Thus it must be that

g ^ = 0 for i 4 3*

If g ^ = 0 for i 5 3 then from the above expression for

i 1

cos©, cos© = 0. Hence the x and xJ coordinate curves are

orthogonal. Since and x^ are arbitrary the curvilinear

coordinate system is orthogonal.

BIBLIOGRAPHY

Books

Eisenhart, Luther Pfatiler, Riemannian Geometry. Princeton, Princeton University Press, 1926".

Lass, Harry, Vector and Tensor Analysis, Hew York, McGraw-Hill Inc., 1950"

Sokolnikoff, I. S., Tensor Analysis, New York, John Wiley and Sons Inc., 1951.

48