Higher Order Contact of Submanifolds of Homogeneous Spaces
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610 Gary R. Jensen
Higher Order Contact of Submanifolds of Homogeneous Spaces
Springer-Verlag Berlin Heidelberg New York 1977
Author Gary R. Jensen Department of Mathematics Washington University St. Louis, Missouri 63130 USA
AMS Subject Classifications (1970): 53A05, 53A 15, 53A20, 53A40, 53A55 ISBN 3-540-08433-9 ISBN 0-387-08433-9
Springer-Verlag Berlin Heidelberg New York Springer-Verlag New York Heidelberg Berlin
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© by Springer-Verlag Berlin Heidelberg 1977 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
TABLE OF CONTENTS Page Introduction • . • . . I.
The general theory 1.
Contact
2.
The Grassmann bundle
5
3.
Adapted frames of a homogeneous space
6
III.
IV.
.
4.
Zeroth order frames of a submanifold
7
5.
Lie transformation groups
8
6.
First order frames
9
7.
Second order frames
12
8.
Third and higher order frames
17
9.
II.
v
Frenet frames
22
The role of the Maurer-Cartan form of
11.
Congruence and Existence Theorems
30
12.
Homogeneity Theorem
41
Surfaces in Real curves in
. . . . .
,2
under
Holomorphic curves in
Surfaces in lli 3
44
under IE(3)
V. Holomorphic curves in
VI.
G
23
10.
SO(4) under
CG 4,2
under
58
SU(3) SU(4)
under the special affine group
66
81 115
References
151
Index
152
INTRODUCTION
These notes contain an exposition of Elie Cartan's theory of higher order invariants of submanifolds of homogeneous spaces treated by the method of moving frames. amples.
The theory is then applied to five ex-
We were introduced to Cartan's magnificent book [C, 1937} , and
our view of the theory is strongly guided, by the paper of P. Griffiths [G, 1974].
It is Cart an 's stated aim in his book (which consists of the lecture notes taken by Jean Leray of a course given by Cartan at the Sorbonne in 1931321) to develop the fundamental theorems of Lie groups together with the basic principles of the method of moving frames in differential geometry.
They are developed together because the latter
depends in such an essential way on the former.
For example, the clas-
sical equations of Darboux in the theory of surfaces in Euclidean space are just the structure equations of the group of Euclidean motions.
In
more general Kleingeometries, where the Euclidean motions are replaced by an arbitrary Lie group
G,
Darboux' s structure equations are re-
placed by the structure equations of by Maurer in 1888.
G,
which were first established
VI
The basic theory of Lie groups - the Maurer-Cartan forms and eQuations, the theorem
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