SYMMETRIC SPACES OF THE NON COMPACT TYPE

Julien Maubon

Description

SYMMETRIC SPACES OF THE NON-COMPACT TYPE : DIFFERENTIAL GEOMETRY by Julien MAUBON Abstra t. This is an introdu tion to Riemannian symmetri spa es of the non- ompa t type from the (dierential) geometer's point of view. We start from the denition in terms of geodesi symmetries and, while our methods are as geometri as possible, we dedu e geometri but also algebrai results, su h as the semi-simpli ity of the isometry group of su h spa es. This is done by rst establishing lassi al omparison theorems on Hadamard manifolds (and more generally on CAT(0) spa es). Résumé (Espa es symétriques de type non- ompa t : géométrie diérentielle) Ce texte est une introdu tion aux espa es symétriques riemanniens de type non- ompa t du point de vue de la géométrie diérentielle. Nous partons de la dénition en terme de symétries géodésiques pour aboutir, le plus géométriquement possible, à des résultats tant géométriques qu'algébriques. Par exemple nous démontrons la semi-simpli ité du groupe des isométries d'un tel espa e en utilisant les théorèmes de omparaison lassiques sur les variétés de Hadamard (et plus généralement les espa es CAT(0)). Contents 1. Introdu tion. . . . . . . . . .

• positive urvature


• tion


• ve tor eld


• riemannian manifold


• along ?


• eld along


• spa


• espa


• unique ja

Sujets

• Riemannian manifold

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 106
• Langue: English

Extrait

.
SYMMETRIC
.
SP
.
A
of
CES
groups,
OF
.
THE
of
NON-COMP
.
A
.
CT
.
TYPE
Riemannian
:
.
DIFFERENTIAL
.
GEOMETR
.
Y
.
by
.
Julien
.
MA
Key
UBON
.
A
.
bstr
.
act
.
.
.

Riemannian
This
.
is
.
an
.
in
.
tro
.

.
to
.
Riemannian
.

.
spaces
.
of

the


.
t
.
yp
.
e
.
from
.
the
.
(dieren
spaces
tial)
.
geometer's
.
p
.
oin
.
t
.
of
.
view.
.
W
.
e
10
start

from
.
the
.
denition
18
in
e
terms
.
of
.
geo
.

29
symmetries
.
and,
.
while
.
our
.
metho
.
ds
.
are
.
as
40


as
phr
p
e
ossible,
T(0)
w
.
e
.

.

.
but
.
also
.

.
results,
.

.
h
.
as
2
the

semi-simplicit
.
y
.
of
.
the
.
isometry
.
group
.
of
.

.
h
8
spaces.

This
.
is
.
done
.
b
.
y
.
rst
.
establishing
.

.

Riemannian
theorems
ositiv
on
.
Hadamard
.
manifolds
.
(and
.
more
.
generally
.
on

CA
t
T(0)
.
spaces).
.
R
.
?sum?
.
(Espaces
.
sym?triques
.
de
.
t
.
yp
.
e
.

.
:
.
g?om?trie
.
di?ren
.
tielle)
.
Ce
.
texte
.
est
.
une
.
in
.
tro
.

.
aux
.

.
sym?triques
Mathematics
riemanniens
ation
de
53C21,
t
ds
yp
.
e
non-p

ature,
du
manifolds,
p

oin
.
t
.
de
.
vue
.
de
.
la
.
g?om?trie
.
di?ren
.
tielle.
.
Nous
.
partons
.
de
.
la
.
d?nition
.
en
.
terme
.
de
.
sym?tries
.
g?o
.
d?siques
.
p
.
our
3.
ab
lo
outir,

le
.
plus
.
g?om?triquemen
.
t
.
p
.
ossible,
.
?
.
des
.
r?sultats
.
tan
.
t
.
g?om?triques
.
qu'alg?briques.
.
P
.
ar
.
exemple
.
nous
.
d?mon
4.
trons
globally
la
spaces
semi-simplicit?
.
du
.
group
.
e
.
des
.
isom?tries
.
d'un
.
tel
.

.
en
.
utilisan
.
t
.
les
.
th?or?mes
.
de
.

.

5.
sur
manifolds
les
non-p
v
e
ari?t?s
ature
de
.
Hadamard
.
(et
.
plus
.
g?n?ralemen
.
t
.
les
.

.
CA
.
T(0)).
.
Con
.
ten
6.
ts
spaces
1.

In
yp
tro
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
References
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
2000
.
Subje
.
Classic
.
.
.
53C35,
.
22E15.
.
wor
.
and
.
ases
.

.
spaces,
.
ositiv
2

2.
isometry
Riemannian
Hadamard
preliminaries
CA
.
spaces,
.
theorems.
.(M,g)
n
TM M
∇ : Γ(TM)×Γ(TM)−→ Γ(TM)
∞f ∈C (M) X,Y ∈ Γ(TM)
∇ Y =f∇ YfX X
∇ fY = df(X)Y +f∇ YX X
∇ Y m M X mX
(M,g)
g
∇ Y −∇ X = [X,Y] X,Y ∈ Γ(TM)X Y
∇g = 0 X.g(Y,Z) =g(∇ Y,Z)+g(Y,∇ Z) X,Y,Z ∈ Γ(TM)X X
rule).
ol
tro
is
oth
non-p
P
ositiv
Riemannian
ely
either

v
ed
view.
geometries,
of
w
t
e
spaces.
ha
in
v
that,
e
elds
insisted
tro
on
the
the
of
asp
alue
ects
there
of

non-p
that
ositiv
all
e
,


ho
ature
a
whic
function
h
and


b
en
e

generalized
p
to
t
m
v
uc
p
h
only
more
tro
general
quic
settings

than
L
Riemannian
whic
manifolds,
namely

mean
h
ab
as
spaces
CA
ho


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will
text
in
is
all
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and
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ositiv
oth
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of

Man
Riemannian
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manifold
2
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for
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ery
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to
Riemannian
e
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In
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this
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review
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v
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ery
in
quic

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,
the
A

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oin
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geometer's
geometry
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that
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Since
Note
.
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alue
rest
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of
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a
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ends
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text
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,
o
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example
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onne
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In
(and
UBON
from
JULIEN
start
ys
needed
in2g(∇ Y,Z) =X.g(Y,Z)+Y.g(X,Z)−Z.g(X,Y)−g(X,[Y,Z])+g(Y,[Z,X])+g(Z,[X,Y]).X
X,Y,Z ∈ Γ(TM) R(X,Y)Z = ∇ Z −[∇ ,∇ ]Z[X,Y] X Y
m X Y Z
m R g
R(X,Y,Z,T) =g(R(X,Y)Z,T)
R(X,Y,Z,T) =−R(Y,X,Z,T) =R(Z,T,X,Y)
R(X,Y)Z +R(Y,Z)X +R(Z,X)Y = 0
K(P) P T Mm
g (u,v) P K(P) = R(u,v,u,v)
P
T M S M P mm
P S m
(M,g) κ

R(X,Y)Z =κ g(X,Z)Y −g(Y,Z)X .
c
1/c
n n 2 2κ E =R dx +...+dx1 n
n n+1κ S ⊂ R n ≥ 2
n+1R
nκ H
n+1R n≥ 2
2 2 2 n n+1(n,1) q(x,x) =x +...+x −x H ={x∈R |q(x,x) =1 n n+1
−1, x > 0} qn+1
n nH x q H
−1
(see
ature
.
on
p
a
metric
surface.
en
Namely
k
,
set
if
h
CES
as
is
dene
a
First
tangen