Differential Geometry, Lie Groups, and Symmetric Spaces

635 Pages·1978·10.105 MB·iii-xv, 1-629\635

by  Sigurdur Helgason

Differential Geometry,
Lie Groups,
Symmetric Spaces
Sigurdur Helgason
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, Massachusetts
ACADEMIC PRESS New York San Francisco London 1978
A Subsidiary of Harcourt Brace Jovanovich, Publishers
COPYRIGHT 8 1978, BY ACADEMPIRCE SS, hC.
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ACADEMIC PRESS, INC.
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United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
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Library of Congress Cataloging in Publication Data
Helgason, Sigurdur, Date
Differential geometry, Lie groups, and symmetric
spaces.
(Pure and applied mathematics, a series of mono-
graphs and textbooks ; )
Includes bibliographies and index.
1. Geometry, Differential. 2. Lie groups.
3. Symmetric spaces. I. Title. 11. Series.
QA3.P8 [QA641] 510l.8~ [516’.36] 78-17916
ISBN 0-12-338460-5
PRINTED IN THE UNITED STATES OF AMERICA
8182 9 8 7 6 5 4 3 2
PREFACE
The present book is intended as a textbook and a reference work on
the three topics in the title. Together with a volume in progress on
“Groups and Geometric Analysis” it supersedes my “Differential
Geometry and Symmetric Spaces,” published in 1962. Since that time
several branches of the subject, particularly the function theory on
symmetric spaces, have developed substantially. I felt that an expanded
treatment might now be useful.
This first volume is an extensive revision of a part of “Differential
Geometry and Symmetric Spaces.” Apart from numerous minor
changes the following material has been added:
Chapter I, $15; Chapter 11, 57-98; Chapter 111, $8; Chapter VII,
$$7, 10, 11 and most of $2 and of $8; Chapter VIII, part of 57;
all of Chapter IX and most of Chapter X. Many new exercises
have been added, and solutions to the old and new exercises are
now included and placed toward the end of the book.
The book begins with a general self-contained exposition of dt@rential
and Riemannian geometry, discussing affine connections, exponential
mapping, geodesics, and curvature. Chapter I1 develops the basic
theory of Lie groups and Lie algebras, homogeneous spaces, the adjoint
group, etc. The Lie groups that are locally isomorphic to products of
simple groups are called semisimple. These Lie groups have an extremely
rich structure theory which at an early stage led to their complete
classification, and which presumably accounts for their pervasive
influence on present-day mathematics. Chapter I11 deals with their
preliminary structure theory with emphasis on compact real forms.
Chapter IV is an introductory geometric study of symmetric spaces.
According to its original definition, a symmetric space is a Riemannian
manifold whose curvature tensor is invariant under all parallel transla-
a.
tions. The theory of symmetric spaces was initiated by Cartan in
1926 and was vigorously developed by him in the late 1920s. By their
definition, symmetric spaces form a special topic in Riemannian geom-
ix
X PREFACE
etry; their theory, however, has merged with the theory of semisimple
Lie groups. This circumstance is the source of very detailed and
extensive information about these spaces. They can therefore often
serve as examples on the basis of which general conjectures in differential
geometry can be made and tested.
The definition above does not immediately suggest the special nature
of symmetric spaces (especially if one recalls that all Riemannian
manifolds and all Kahler manifolds possess tensor fields invariant under
parallelism). However, the theory leads to the remarkable fact that
symmetric spaces are locally just the Riemannian manifolds of the form
Rn x GIK where Rn is a Euclidean n-space, G is a semisimple Lie
group that has an involutive automorphism whose fixed point set is the
(essentially) compact group K, and GIK is provided with a G-invariant
Riemannian structure. I?. Cartan’s classification of all real simple Lie
algebras now led him quickly to an explicit classification of symmetric
spaces in terms of the classical and exceptional simple Lie groups. On
the other hand, the semisimple Lie group G (or rather the local iso-
morphism class of G) above is completely arbitrary; in this way valuable
geometric tools become available to the theory of semisimple Lie groups.
In addition, the theory of symmetric spaces helps to unify and explain
in a general way various phenomena in classical geometries. Thus the
isomorphisms that occur among the classical groups of low dimensions
are geometrically interpreted by means of isometries ; the analogy
between the spherical geometries and the hyperbolic geometries is a
special case of a general duality for symmetric spaces.
In Chapter V we give the local decomposition of a symmetric space
into Rn and the two main types of symmetric spaces, the compact type
and the noncompact type. These dual types are already distinguished
by the sign of their sectional curvature. In Chapter VI we study the
symmetric spaces of noncompact type. Since these spaces are completely
determined by their isometry group, this chapter is primarily a global
study of noncompact semisimple Lie groups. In Chapter IX this study
is carried quite a bit further in the form of Cartan, Iwasawa, Bruhat, and
Jordan decompositions.
In Chapter VII we derive topological and differential geometric
properties of the compact symmetric space UIK by studying the isotropy
action of K on UIK and on its tangent space at the origin. Chapter VIII
deals with Hermitian symmetric spaces ; we are primarily concerned
with the noncompact ones and the Cartan-Harish-Chandra representa-
tion of these as bounded domains.
The book concludes with a classification of symmetric spaces by
means of the Killing-Cartan classification of simple Lie algebras over C
PREFACE xi
and Cartan’s classification of simple Lie algebras over R. The latter is
carried out by means of Ka?s classification of finite-order automorphisms
of simple Lie algebras over C.
Each chapter begins with a short summary and ends with references
to source material. Given the enormity of the subject, I am aware that
the result is at best an approximation as regards completeness and
accuracy. Nevertheless, I hope that the notes will help the serious
student gain a historical perspective, particularly as regards Cartan’s
magnificent papers on Lie groups and symmetric spaces, which are
found in the two first volumes of his collected works. For example, he
can witness Cartan’s rather informal arguments in his climactic paper
[lo], written at the age of 58, leading him to a global classification of
symmetric spaces; an example of a different kind is Cartan’s paper in
Leipziger Berichte (1893) where he indicates models of the exceptional
groups as contact transformations or as invariance groups of Pfaffian
equations and which, to my knowledge, have never been verified in
print. Being the first such models, they have distinct historical interest
although simpler models are now known.
In addition to papers and books utilized in the text, the bibliography
lists many items on topics that are at best only briefly discussed in the
text, but are nevertheless closely related, for example, pseudo-
Riemannian symmetric spaces, trisymmetric spaces, reflexion spaces,
homogeneous domains, discrete isometry groups, cohomology and
Betti numbers of locally symmetric spaces. This part of the bibliography
is selective, and no completeness is intended; in particular, papers on
analysis and representation theory related to the topics of “Groups and
Geometric Analysis” are not listed unless used in the present volume.
This book grew out of lectures given at the University of Chicago in
1958, at Columbia University in 1959-1960, and at various times at
MIT since then. At Columbia I had the privilege of many long and
informative discussions with Harish-Chandra, to whom I am deeply
grateful. I am also indebted to A. Korhyi, K. de Leeuw, E. Luft,
H. Federer, I. Namioka, and M. Flensted-Jensen, who read parts of
the manuscript and suggested several improvements. I want also to
thank H.-C. Wang for putting the material in Exercise A.9, Chapter VI
at my disposal. Finally, I am most grateful to my friend and colleague
Victor Kai: who provided me with an account of his method for classifying
automorphisms of finite order (95, Chapter X) of which only a short
sketch was available in print.
With the sequel to this book in mind I will be grateful to readers who
take the trouble of bringing errors in the text to my attention.
SUGGESTIONS TO T H E READER
Since this book is intended for readers with varied backgrounds, we
give some suggestions for its use.
Introductory Differential Geometry. Chapter I, Chapter IV, $1, and
Chapter VIII, $1-93 can be read independently of the rest of the book.
These 120-odd pages, including the offered exercises, have on occasion
served as the text for a one-term course on differential geometry, with
only advanced calculus and some point set topology as prerequisites.
Introduction to Lie Groups. Chapter I, §1-$6, Chapter 11, and
Chapter I11 could similarly be used for a one-term course on Lie groups,
assuming some familiarity with topological groups.
The chapters are rather independent after the fourth one and could
for the most part be read in any order.
Exercises. Each chapter ends with a few exercises. Some of these
furnish examples illuminating the theory developed in the text, while
others produce extensions and ramifications of the theory. With a few
possible exceptions (indicated with a star) the exercises can be worked
out with methods from the text. Since the exercises present additional
material and since some exercise groups furnish suitable topics for
student seminars, I felt that leaving out the solutions would be counter-
productive and might turn the exercises into unnecessary obstacles.
Accordingly, solutions are provided at the end of the book which each
reader can use to the extent he wishes.
S. Helgason
...
Xlll
TENTATIVE CONTENTS OF T H E SEQUEL
Groups and Geometric Analysis
I Integral Geometry and Radon Transforms
Integration on Manifolds. Radon Transform on R". A Duality in Integral Geometry
II Invariant Differential Operators
Geometric Operations of Differential Operators. Differential Operators on Homo-
geneous Spaces. Invariants and Harmonic Polynomials
Ill Spherical Functions and Spherical Transforms
Elementary Properties. The Euclidean Case. The Noncompact Type. Harish-
Chandra's Expansion. The Compact Type
IV Analysis on Compact Symmetric Spaces
Representations of Compact Lie Groups. Expansions on Compact Groups and
Compact Homogeneous Spaces
V Duality for Symmetric Spaces
The Spaces of Horocycles. Radon Transform and Its Dual. Spherical and Conical
Representations. Conical Distributions and Intertwining Operators
VI The Fourier Transform on a Symmetric Space
Plancherel Formula. The Paley-Wiener Theorem. Generalized Spherical Func-
tions. The Case of K-Finite Functions
VI I Differential Equations on Symmetric Spaces
Solvability. Eigenfunctions. Integral Representations. Mean-Value Thecrems.
Wave Equations. Huygens Principle
Vlll Eigenspace Representations
Generalities. Irreducibility Criteria for the Compact Type U/K, Euclidean Type,
Noncompact Type G/K, the Horocycle Space GIMN, and the Complex Space GIN
xv
CHAPTER I
ELEMENTARY DIFFERENTIAL GEOMETRY
This introductory chapter divides in a natural way into three parts: $1 -93 which
deal with tensor fields on manifolds, 54-38 which treat general properties of
affine connections, and $9-914 which give an introduction to Riemannian geometry
with some emphasis on topics needed for the later treatment of symmetric spaces.
91-93. When a Euclidean space is stripped of its vector space structure and
only its differentiable structure retained, there are many ways of piecing together
domains of it in a smooth manner, thereby obtaining a so-called differentiable
manifold. Local concepts like a differentiable function and a tangent vector can
still be given a meaning whereby the manifold can be viewed “tangentially,” that
is, through its family of tangent spaces as a curve in the plane is, roughly
speaking, determined by its family of tangents. This viewpoint leads to the
study of tensor fields, which are important tools in local and global differential
geometry. They form an algebra D(M), the mixed tensor algebra over the
manifold M. The alternate covariant tensor fields (the differential forms) form
a submodule %(M)o f D(M) which inherits a ‘multiplication from D(M), the
exterior multiplication. The resulting algebra is called the Grassmann algebra
of M. Through the work of 8. Cartan the Grassmann algebra with the exterior
differentiation d has become an indispensable tool for dealing with submanifolds,
these being analytically described by the zeros of differential forms. Moreover,
the pair (%(M)d, ) determines the cohomology of III via de Rham’s theorem,
which however will not be dealt with here.
84-98. The concept of an affine connection was first defined by Levi-Civita
for Riemannian manifolds, generalizing significantly the notion of parallelism for
Euclidean spaces. On a manifold with a countable basis an affine connection always
exists (see the exercises following this chapter). Given an affine connection on
a manifold M there is to each curve y(t) in M associated an isomorphism between
any two tangent spaces A&fvct,, and Thus, an affine connection makes it
possible to relate tangent spaces at distant points of the manifold. If the tangent
vectors of the curve y(t) all correspond under these isomorphisms we have the
analog of a straight line, the so-called geodesic. The theory of affine connections
mainly amounts to a study of the mappings Exp, : M, + M under which straight
lines (or segments of them) through the origin in the tangent space M, correspond
to geodesics through p in M. Each mapping Exp, is a diffeomorphism of a neigh-
borhood of 0 in M, into M, giving the so-called normal coordinates at p. Some
other local properties of Exp, are given in 96, the existence of convex neigh-
borhoods and a formula for the differential of Exp,.
An affine connection gives rise to two important tensor fields, the curvature
tensor field and the torsion tensor field which in turn describe the affine connec-
tion through 8.C artan’s structural equations [(6) and (7), 98)].
1
2 ELEMENTARY DIFFERENTIAL GEOMETRY [Ch. I
59-814. A particularly interesting tensor field on a manifold is the so-called
Riemannian structure. This gives rise to a metric on the manifold in a canonical
fashion. It also determines an affine connection on the manifold, the Riemannian
connection; this affine connection has the property that the geodesic forms the
shortest curve between any two (not too distant) points. The relation between the
metric and geodesics is further developed in $9-510. The treatment is mainly
based on the structural equations of E. Cartan and is independent of the Calculus
of Variations.
The higher-dimensional analog of the Gaussian curvature of a surface was
discovered by Riemann. Riemann introduced a tensor field which for any pair
of tangent vectors at a point measures the corresponding sectional curvature,
that is, the Gaussian curvature of the surface generated by the geodesics tangent
to the plane spanned by the two vectors. Of particular interest are Riemannian
manifolds for which the sectional curvature always has the same sign. The irreduci-
ble symmetric spaces are of this type. Riemannian manifolds of negative curvature
are considered in $13 owing to their importance in the theory of symmetric spaces.
Much progress has been made recently in the study of Riemannian manifolds
whose sectional curvature is bounded from below by a constant > 0. However,
no discussion of these is given since it is not needed in later chapters. The next
section deals with totally geodesic submanifolds which are characterized by the
condition that a geodesic tangent to the submanifold at a point lies entirely in it.
In contrast to the situation for general Riemannian manifolds, totally geodesic
submanifolds are a common occurrence for symmetric spaces.
$1. Manifolds
Let Rm and Rn denote two Euclidean spaces of m and n dimensions,
respectively. Let 0 and 0’ be open subsets, 0 C Rm, 0’ C Rn and
suppose q.~i s a mapping of 0 into 0’. The mapping T is called dareren-
tiable if the coordinates yj(~(po)f) p )(p) are differentiable (that is, inde-
finitely differentiable) functions of the coordinates x&), p E 0. The
mapping is called analytic if for each point p E 0 there exists a neigh-
< <
borhood U of p and n power series Pi (1 j n) in m variables such
that Yj(dd)= pj(xl(q) - Xl(P), “‘2 xn,(q) - Xm(P)) (1 < j d n) for
q E U. A differentiable mapping T : 0 + 0’ is called a daJeomorphism of
0 onto 0’ if ~ ( 0=) 0 ’, T is one-to-one, and the inverse mapping q~-l
is differentiable. In the case when n = 1 it is customary to replace the
term “mapping” by the term “function.”
An analytic function on Rm which vanishes on an open set is identically
0. For differentiable functions the situation is completely different. In
fact, if A and B are disjoint subsets of Rm,A compact and B closed,
then there exists a differentiable function q~ which is identically 1 on A
and identically 0 on B. The standard procedure for constructing such
a function q is as follows:
§ 11 Manifolds 3
Let 0 < a < b and consider the function f on R defined by
( 0 otherwise.
Then f is differentiable and the same holds for the function
which has value 1 for x < a and 0 for x 3 b. The function t+h on Rm
given by
4(x1, ...) Xm) = F(x; + ... + X:,)
is differentiable and has values 1 for x: 4- ... + x:~ < a and 0 for
.Y: + ... + x$ 3 b. Let S and S' be two concentric spheres in Rm,
S' lying inside S. Starting from $ we can by means of a linear trans-
formation of RJTc1o nstruct a differentiable function on Rm with value 1
in the interior of S' and value 0 outside S. Turning now to the sets A
and B we can, owing to the compactness of A, find finitely many spheres
< < < <
Si( 1 i fz), such that the corresponding open balls Bi( 1 i n),
form a covering of A (that is, A C U:!l RJ and such that the closed
balls Bi (1 < i < n) do not intersect B. Each sphere Sic an be shrunk
to a concentric sphere Si such that the corresponding open balls Bi
still form a covering of A. Let t+hi be a differentiable function on Rm
which is identically 1 on B; and identically 0 in the complement of Bi.
Then the function
(P = 1 - (1 - #l(1) - #d .*a (1 - +??I
is a differentiable function on Rm which is identically 1 on A and iden-
tically 0 on B.
Let M be a topological space. We assume that M satisfies the Hausdorff
separation axiom which states that any two different points in M can be
separated by disjoint open sets. An open chart on M is a pair (U,p ')
where U is an open subset of M and (P is a homeomorphism of U onto
an open subset of Rm.
Definition. Let M be a HausdorfT space. A difJerentiable structure
on M of dimension m is a collection of open charts (Urnq, JcrsAon M
where (P(, U,) is an open subset of Rm such that the following conditions
are satisfied:
u urn.
(Ml) M =
,€A