##### Document Text Contents

Page 1

Translations of

MATH EMATICAL

MONO G "PHS

Volume 1.23

Elle Cartan.

[1869-1951)

M. A. Akivis

B, A. Rosenfeld

American Mathematical society'

Page 2

Translationsof MathematicalMonographs 123

Page 167

154 5. THE METHOD OF MOVING FRAMES AND DIFFERENTIAL GEOMETRY

a frame of order n - 1. The nonvanishing differential forms in the deriva-

tional formulas for the Frenet frames have the form co'+1 = - i+1 = -kids ,

and the quantities ki form a complete system of invariants defining a curve

in the space Rn up to a motion.

The Darboux frames for a hypersurface in the space Rn are also canonical

frames. These frames are formed by the vector en parallel to the normal to

the hypersurface and the vectors e1, e2, ... , en_ 1 parallel to its principal

directions. These frames are frames of order two.

§5.7. Some geometric examples

Cartan noted that simple geometric considerations do not always lead to

the construction of a canonical frame. In such cases the construction may

be conducted purely analytically by means of the structure equations of the

space. We show how this can be done for an isotropic curve of the space

CR3. These curves were considered by E. Vessiot in 1905. Cartan consid-

ered them in the book The theory of finite continuous groups and differential

geometry considered by the method of moving frames [157] and in his lectures

on The method of moving frames, the theory of finite continuous groups and

generalized spaces [144] which he delivered in Moscow in 1930.

An isotropic curve x = x(t) in the space CR3 is said to be a curve each

tangent vector x of which is isotropic, i.e., (x')2 = 0. The latter equation

implies x'x = 0 . The arc length of such a curve is equal to zero, and the

normal and tangent planes coincide. Thus, it is impossible to construct the

Frenet frame for such a curve. For studying an isotropic curve, Cartan used

the cyclic frames in the space CR3 whose vectors satisfy the relations

(5.14) e2=e2ele2=e2e =0, e2=e1e = I.3=

3 3

Only three out of the nine forms w determining the infinitesimal displace-

ments of this frame are independent. Differentiating equations (5.14) and

using equations (5.5), we easily find that they are connected by the relations

(5.15) w1=w=o2=0,

3 3 l 3

While constructing a canonical frame, we save one step by immediately

associating with the curve the frames of order one. For this, we place the

origin of a frame at the point x of the curve and take its isotropic tangent

vector (x)' as the vector el .Since now we have d x = o 1 eI, on the curve

the following equations hold:

(t)2=(t)3=0.

The form cvl is called the basis form. It contains the differential of the

parameter t defining the location of a point x on the curve. If we apply

Page 168

§5.7. SOME GEOMETRIC EXAMPLES 155

exterior differentiation to equations (5.16) with the help of the structure equa-

tions (5.11), then, by (5.15) and (5.16), we obtain only one exterior quadratic

equation c01 A rvi = 0. This equation implies that

(5.17) w1 = pcOl.

The form cvi is principal since it vanishes when the point x is fixed. More-

over, there will be only two nonvanishing independent forms on the curve,

namely, the forms to iand to . They determine the admissible transfor-

mations of frames of order one. Thus, the family of frames of order one

depends on one principal and two secondary parameters.

For further specialization of frames, we apply exterior differentiation to

equation (5.17). This gives

Ldp - 2pw1)

from which it follows that

(5.18)

Aw'=0,

dp - 2prvi = -2gcv1.

If we fix a point x on the curve, then rv 1 =0, and equation (5.18) takes the

form

(5.19) op - 2pii = 0,

where 6 denotes differentiation with respect to secondary parameters and

1 1

7r1 = (01

(0 '=0

In equation (5.19) we distinguish two cases. If p = 0 for all points of the

curve x = x(t) , then further specialization is impossible, and the family of

frames of order two coincides with the family of frames of order one. Since

in this case equation (5.16) implies that a =0, it follows from equations

(5.5) that

dx = coIel, de1 = cve1.

It follows from this that, in the case p = 0, a curve x = x(t) is an isotropic

straight line.

If p 0, equation (5.19) can be written in the form

6lnp -2n = 0.

It is easy to check that dirt = 0 if w1 = 0. Thus, the secondary form nl is

a total differential: n i = a In cp .Substituting this value and integrating the

previous equation, we obtain p = C9 2. Here (p is a secondary parameter

which determines the magnitude of the vector e1 . By an appropriate choice

of this parameter, we can reduce the quantity p to + 1or -1 . Let us take

Page 334

ISON 978 0-8218-5355-9

9'780821 11853559

MMONO/ 123.5

AMS on.the Web

www.ams.or

Translations of

MATH EMATICAL

MONO G "PHS

Volume 1.23

Elle Cartan.

[1869-1951)

M. A. Akivis

B, A. Rosenfeld

American Mathematical society'

Page 2

Translationsof MathematicalMonographs 123

Page 167

154 5. THE METHOD OF MOVING FRAMES AND DIFFERENTIAL GEOMETRY

a frame of order n - 1. The nonvanishing differential forms in the deriva-

tional formulas for the Frenet frames have the form co'+1 = - i+1 = -kids ,

and the quantities ki form a complete system of invariants defining a curve

in the space Rn up to a motion.

The Darboux frames for a hypersurface in the space Rn are also canonical

frames. These frames are formed by the vector en parallel to the normal to

the hypersurface and the vectors e1, e2, ... , en_ 1 parallel to its principal

directions. These frames are frames of order two.

§5.7. Some geometric examples

Cartan noted that simple geometric considerations do not always lead to

the construction of a canonical frame. In such cases the construction may

be conducted purely analytically by means of the structure equations of the

space. We show how this can be done for an isotropic curve of the space

CR3. These curves were considered by E. Vessiot in 1905. Cartan consid-

ered them in the book The theory of finite continuous groups and differential

geometry considered by the method of moving frames [157] and in his lectures

on The method of moving frames, the theory of finite continuous groups and

generalized spaces [144] which he delivered in Moscow in 1930.

An isotropic curve x = x(t) in the space CR3 is said to be a curve each

tangent vector x of which is isotropic, i.e., (x')2 = 0. The latter equation

implies x'x = 0 . The arc length of such a curve is equal to zero, and the

normal and tangent planes coincide. Thus, it is impossible to construct the

Frenet frame for such a curve. For studying an isotropic curve, Cartan used

the cyclic frames in the space CR3 whose vectors satisfy the relations

(5.14) e2=e2ele2=e2e =0, e2=e1e = I.3=

3 3

Only three out of the nine forms w determining the infinitesimal displace-

ments of this frame are independent. Differentiating equations (5.14) and

using equations (5.5), we easily find that they are connected by the relations

(5.15) w1=w=o2=0,

3 3 l 3

While constructing a canonical frame, we save one step by immediately

associating with the curve the frames of order one. For this, we place the

origin of a frame at the point x of the curve and take its isotropic tangent

vector (x)' as the vector el .Since now we have d x = o 1 eI, on the curve

the following equations hold:

(t)2=(t)3=0.

The form cvl is called the basis form. It contains the differential of the

parameter t defining the location of a point x on the curve. If we apply

Page 168

§5.7. SOME GEOMETRIC EXAMPLES 155

exterior differentiation to equations (5.16) with the help of the structure equa-

tions (5.11), then, by (5.15) and (5.16), we obtain only one exterior quadratic

equation c01 A rvi = 0. This equation implies that

(5.17) w1 = pcOl.

The form cvi is principal since it vanishes when the point x is fixed. More-

over, there will be only two nonvanishing independent forms on the curve,

namely, the forms to iand to . They determine the admissible transfor-

mations of frames of order one. Thus, the family of frames of order one

depends on one principal and two secondary parameters.

For further specialization of frames, we apply exterior differentiation to

equation (5.17). This gives

Ldp - 2pw1)

from which it follows that

(5.18)

Aw'=0,

dp - 2prvi = -2gcv1.

If we fix a point x on the curve, then rv 1 =0, and equation (5.18) takes the

form

(5.19) op - 2pii = 0,

where 6 denotes differentiation with respect to secondary parameters and

1 1

7r1 = (01

(0 '=0

In equation (5.19) we distinguish two cases. If p = 0 for all points of the

curve x = x(t) , then further specialization is impossible, and the family of

frames of order two coincides with the family of frames of order one. Since

in this case equation (5.16) implies that a =0, it follows from equations

(5.5) that

dx = coIel, de1 = cve1.

It follows from this that, in the case p = 0, a curve x = x(t) is an isotropic

straight line.

If p 0, equation (5.19) can be written in the form

6lnp -2n = 0.

It is easy to check that dirt = 0 if w1 = 0. Thus, the secondary form nl is

a total differential: n i = a In cp .Substituting this value and integrating the

previous equation, we obtain p = C9 2. Here (p is a secondary parameter

which determines the magnitude of the vector e1 . By an appropriate choice

of this parameter, we can reduce the quantity p to + 1or -1 . Let us take

Page 334

ISON 978 0-8218-5355-9

9'780821 11853559

MMONO/ 123.5

AMS on.the Web

www.ams.or