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TitleÉlie Cartan (1869-1951)
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Table of Contents
Translations of Mathematical Monographs 123
S Title
Elie Cartan (1869-1951)
Copyright (C) 1993 by the American Mathematical Society.
	ISBN 0-8218-4587-X
	QA29.C355A6613 1993  16.3' 76' 092-dc20
	LCCN 93-6932 CIP
CHAPTER1  The Life and Work of E. Cartan
	§1.1. Parents' home
	§ 1.2. Student at a school and a lycee
	§1.3. University student
	§ 1.4. Doctor of Science
	§1.5. Professor
	§ 1.6. Academician
	§ 1.7. The Cartan family
	§1.8. Cartan and the mathematicians of the world
CHAPTER 2  Lie Groups and Algebras
	§2.1. Groups
	§2.2. Lie groups and Lie algebras
	§2.3. Killing's paper
	§2.4. Cartan's thesis
	§2.5. Roots of the classical simple Lie groups
	§2.6. Isomorphisms of complex simple Lie groups
	§2.7. Roots of exceptional complex simple Lie groups
	§2.8. The Cartan matrices
	§2.9. The Weyl groups
	§2.10. The Weyl affine groups
	§2.11. Associative and alternative algebras
	§2.12. Cartan's works on algebras
	§2.13. Linear representations of simple Lie groups
	§2.14. Real simple Lie groups
	§2.15. Isomorphisms of real simple Lie groups
	§2.16. Reductive and quasireductive Lie groups
	§2.17. Simple Chevalley groups
	§2.18. Quasigroups and loops
CHAPTER 3  Projective Spaces and Projective Metrics
	§3.1. Real spaces
	§3.2. Complex spaces
	§3.3. Quaternion spaces
	§3.4. Octave planes
	§3.5. Degenerate geometries
	§3.6. Equivalent geometries
	§3.7. Multidimensional generalizations of the Hesse transfer principle
	§3.8. Fundamental elements
	§3.9. The duality and triality principles
	§3.10. Spaces over algebras with zero divisors
	§3.11. Spaces over tensor products of algebras
	§3.12. Degenerate geometries over algebras
	§3.13. Finite geometries
CHAPTER 4  Lie Pseudogroups and Pfaffian Equations
	§4.1. Lie pseudogroups
	§4.2. The Kac-Moody algebras
	§4.3. Pfaflian equations
	§4.4. Completely integrable Pfaffian systems
	§4.5. Pfaffian systems in involution
	§4.6. The algebra of exterior forms
	§4.7. Application of the theory of systems in involution
	§4.8. Multiple integrals, integral invariants, and integral geometry
	§4.9. Differential forms and the Betti numbers
	§4.10. New methods in the theory of partial differential equations
CHAPTER 5  The Method of Moving Frames and Differential Geometry
	§5.1. Moving trihedra of Frenet and Darboux
	§5.2. Moving tetrahedra and pentaspheres of Demoulin
	§5.3. Cartan's moving frames
	§5.4. The derivational formulas
	§ 5.5. The structure equations
	§5.6. Applications of the method of moving frames
	§5.7. Some geometric examples
	§5.8. Multidimensional manifolds in Euclidean space
	§5.9. Minimal manifolds
	§5.10, "Isotropic surfaces"
	§ 5.11. Deformation and projective theory of multidimensional manifolds
	§5.12. Invariant normalization of manifolds
	§5.13. "Pseudo-conformal geometry of hypersurfaces"
CHAPTER 6  Riemannian Manifolds. Symmetric Spaces
	§6.1. Riemannian manifolds
	§6.2. Pseudo-Riemannian manifolds
	§6.3. Parallel displacement of vectors
	§6.4. Riemannian geometry in an orthogonal frame
	§6.5. The problem of embedding a Riemannian manifoldin to a Euclidean space
	§6.6. Riemannian manifolds satisfying "the axiom of plane"
	§6.7. Symmetric Riemannian spaces
	§6.8. Hermitian spaces as symmetric spaces
	§6.9. Elements of symmetry
	§6.10. The isotropy groups and orbits
	§6.11. Absolutes of symmetric spaces
	§6.12. Geometry of the Cartan subgroups
	§6.13. The Cartan submanifolds of symmetric spaces
	§6.14. Antipodal manifolds of symmetric spaces
	§6.15. Orthogonal systems of functions on symmetric spaces
	§6.16. Unitary representations of noncompact Lie groups
	§6.17. The topology of symmetric spaces
	§6.18. Homological algebra
CHAPTER 7  Generalized Spaces
	§7.1. "Affine connections" and Weyl's "metric manifolds"
	§7.2. Spaces with af'ine connection
	§7.3. Spaces with a Euclidean, isotropic, and metric connection
	§7.4. Afllne connections in Lie groups and symmetric spaces with an af'ine connection
	§7.5. Spaces with a projective connection
	§7.6. Spaces with a conformal connection
	§7.7. Spaces with a symplectic connection
	§7.8. The relativity theory and the unified field theory
	§7.9. Finsler spaces
	§7.10. Metric spaces based on the notion of area
	§7.11. Generalized spaces over algebras
	§7.12. The equivalence problem and G-structures
	§7.13. Multidimensional webs
Dates of Cartan's Life and Activities
List of Publications of 1lie Cartan
APPENDIX A  Rapport sur les Travaux de M. Cartan
	Groupes continus et finis
	Groupes discontinus et finis
	Groupes continus et infinis
	Equations aux derivees partielles
APPENDIX B  Sur une degenerscence de la geometrie euclidienne
APPENDIX C  Allocution de M. Elie Cartan
APPENDIX D  The Influence of France in the Development of Mathematics
Back Cover
Document Text Contents
Page 1

Translations of


Volume 1.23

Elle Cartan.

M. A. Akivis
B, A. Rosenfeld

American Mathematical society'

Page 2

Translationsof MathematicalMonographs 123

Page 167


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:


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


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



dp - 2prvi = -2gcv1.

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

(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

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