An introduction to the geometry of homogeneous spaces
Takashi Koda
Department of Mathematics, Faculty of Science, University of Toyama, Gofuku, Toyama 930-8555, Japan
e-mail : koda@sci.u-toyama.ac.jp
Abstract. This article is a survey of the geometry of homogeneous spaces for students. In particular, we are concerned with classical Lie groups and their root systems. Then we describe curvature tensors of the Riemannian connection and the canonical connection of a homogeneous spaceG/H in terms of the Lie algebrag.
1 Lie groups
1.1 Definition and examples
Definition 1. A set Gis called a Lie groupif (1) Gis an (abstract) group,
(2) Gis aC∞ manifold, (3) the group operations
G×G→G, (x, y)7→xy, G→G, x7→x−1 are C∞ maps.
Example 1 Rn is an (abelian) Lie group under vector addition:
(x,y)7→x+y, x7→ −x.
Example 2(Classical Lie groups) LetM(n,R) be the set of all realn×nmatrices.
We may identify it as
M(n,R)∼=Rn2.
The subset GL(n,R) = {A ∈ M(n,R) | detA 6= 0} is an open submanifold of M(n,R), and becomes a Lie group under the multiplication of matrices. GL(n,R) is called thegeneral linear group.
Definition 2. Let Gbe a Lie group. A Lie group H is aLie subgroupof Gif (1) H is a submanifold of the manifoldG(H need not have the relative topology), (2) H is a subgroup of the (abstract) groupG.
121
Example 3
SL(n,R) ={A∈GL(n,R)|detA= 1}, dim =n2−1 O(n) ={A∈GL(n,R)|tAA=AtA=In}, dim = n(n−1)
2 SO(n) ={A∈O(n)| detA= 1}, dim = n(n−1)
2
U(n) ={A∈GL(n,C)|A∗A=A∗A=In}, dim =n2 SU(n) ={A∈U(n)| detA= 1}, dim =n2−1
Sp(n) ={A∈GL(n,H)| AA∗=A∗A=In}, dim = 2n2+n
where, H = {a+bi+cj+dk | a, b, c, d ∈ R} denotes the set of all quaternions (i2=j2=k2=−1, ij=−ji=k, jk=−kj =i, ki=−ik=j), and A∗=tA=tA is the conjugate transpose of a matrixA.
Theorem 1.1. (Cartan) LetH be a closed subgroup of a Lie group G. ThenH has a structure of Lie subgroup ofG. (The topology of H coincides with the relative topology and the structure of Lie subgroup is unique).
1.2 Left translation
For any fixed a∈G, the map La:G→Gdefined by La(x) =ax forx∈G
is called the left translation. La is a C∞ map on G, and furthermore it is a diffeomorphism ofG.
La−1=La−1.
Similarly, the right translationRa bya∈Gis a diffeomorphism defined by Ra(x) =xa forx∈G.
Theinner automorphismia:G→G
ia:=La◦Ra−1, ia(x) =axa−1 (x∈G) isC∞map onG.
Definition 3. Let Gbe a Lie group. A vector fieldX on Gisleft invariantif (La)∗X=X,
where
((La)∗X)f :=X(f◦La) (f ∈C∞(G)).
More precisely,
(La)∗xXx = Xax forx∈G
((La)∗xXx)f = Xx(f◦La) forf ∈C∞(G) We denote by gthe set of all left invariant vector fields onG.
Proposition 1.2. (1) For X, Y ∈gandλ, µ∈R, we haveλX+µY ∈g.
(2) ForX, Y ∈g, we have[X, Y]∈g.
Remark. For vector fieldsX, Y on a manifold, bracket [X, Y] is defined by [X, Y]f :=X(Y f)−Y(Xf) (f :C∞function).
1.3 Lie algebras
Definition 4. An n-dimensional vector space g over a field K = R or C with a bracket [, ] :g×g→gis called aLie algebra if it satisfies
(1) [, ] is bilinear,
(2) [X, X] = 0for∀X∈gand
(3) (Jacobi identity)[X,[Y, Z]] + [Y,[Z, X]] + [Z,[X, Y]] = 0 for∀X, Y, Z∈g.
Example M(n,R), M(n,C) andM(n,H) with a commutator [, ] defined by [X, Y] =XY −Y X
are Lie algebras over R.
Definition 5. A vector subspacehof a Lie algebragis called a Lie subalgebraif [h,h]⊂h.
Example
sl(n,R) = {X ∈M(n,R)| traceX = 0} ⊂M(n,R) o(n) = {X ∈M(n,R)| X+tX = 0} ⊂M(n,R) so(n) = {X ∈o(n)|traceX = 0} ⊂o(n)
u(n) = {X ∈M(n,C)| X+X∗=O} ⊂M(n,C) su(n) = {X ∈u(n)|traceX= 0} ⊂u(n)
1.4 Lie algebra of a Lie group
LetX, Y be left invariant vector fields on a Lie groupG, then (La)∗X=X, (La)∗Y =Y fora∈G, hence we get
(La)∗[X, Y] = [X, Y].
In general, ifϕ:M →N is aC∞ map and vector fieldsX, Y onM andX0, Y0 on N satisfy
ϕ∗X =X0, ϕ∗Y =Y0, then we have
ϕ∗[X, Y] = [X0, Y0].
Proposition 1.3. The setgof all left invariant vector fields on a Lie group Gis a Lie algebra with respect to the Lie bracket[X, Y] of vector fieldsX, Y on G.
gis called the Lie algebra of a Lie groupG.
Proposition 1.4. The Lie algebra gof a Lie group G may be identified with the tangent spaceTeGat the identitye∈G.
g3X 7→Xe∈TeG (isomorphism between vector spaces) and conversely
TeG3Xe7→X ∈gdefined byXx:= (Lx)∗eXe (x∈G).
Example An elementxin GL(n,R)⊂M(n,R)∼=Rn2 may be expressed as x= (xij),
thenxijare global coordinates ofGL(n,R). A left invariant vector fieldX ∈gl(n,R) onGL(n,R) may be written as
X= Xn
i,j=1
Xji ∂
∂xij.
ThenXe∈TeGL(n,R) may be identified with then×nmatrix (Xji(e))∈M(n,R).
gl(n,R)3X = Xn
i,j=1
Xji ∂
∂xij ←→(Xji(e))∈M(n,R)
Then we may see that the Lie algebragl(n,R) ofGL(n,R) is isomorphic toM(n,R) as Lie algebras:
(gl(n,R),[, ])∼= (M(n,R),[, ])
Theorem 1.5. LetGbe a Lie group and gits Lie algebra.
(1) IfH is a Lie subgroup ofG,his a Lie subalgebra of g
(2) Ifhis a Lie subalgebra, there exists a unique Lie subgroupH ofGsuch that the Lie algebra of H is isomorphic toh.
1.5 Exponential maps
Definition 6. A homomorphismϕ:R→Gis called a1-parameter subgroupof a Lie group G. ϕ(t)is a curve in Gsuch that
ϕ(0) =e, ϕ(s+t) =ϕ(s)ϕ(t).
For any 1-parameter subgroupϕ:R→G, we consider its initial tangent vector ϕ0(0) =ϕ∗0
µd dt
¶
0
∈TeG This is a one-to-one correspondense.
For each left invariant vector field X ∈ g on G, we denote by ϕX(t) the 1- parameter subgroup with the initial tangent vectorϕX0(0) =Xe∈TeG.
Definition 7. The map exp :g→Gdefined by expX :=ϕX(1) is called theexponential map of a Lie groupG.
Example For eachX ∈M(n,R), the series of matrix eX=I+X+X2
2! +· · ·+Xn n! +· · · converges in M(n,R)∼=Rn2. The map
exp :M(n,R)→M(n,R), X 7→expX =eX is called theexponential mapof matrices.
If [X, Y] = 0,
exp(X+Y) = expX expY and we may see that
expO=I, exp(−X) = (expX)−1. Hence
exp :M(n,R)→GL(n,R).
This coincides with the exponential map of the Lie groupGL(n,R).
1.6 Adjoint representations
LetGbe a Lie group andia denotes the inner automorphism bya∈G.
ia=La◦Ra−1:G→G, x7→axa−1 We denote its differential by
Ad(a) = AdG(a) := (ia)∗.
For any left invariant vector fieldX ∈g, Ad(a)X= (ia)∗X is also left invariant.
Ad(a) :g→g (linear) satisfies
[Ad(a)X,Ad(a)Y] = Ad(a)[X, Y].
Thus we have a homomorphism
Ad :G→GL(g), a7→Ad(a) which called theadjoint represenationofG.
Proposition 1.6. For any a∈GandX ∈g,
exp(Ad(a)X) =ia(expX).
Letgbe a Lie algebra with a bracket [, ]. For anyX, the linear map ad(X) :g→g, Y 7→ad(X)Y := [X, Y]
satisfies
ad(X)[Y, Z] = [ad(X)Y, Z] + [Y,ad(X)Z]
(Jacobi identity). The homomorphism
ad :g→gl(g), X 7→ad(X) is called theadjoint representationofg.
Proposition 1.7. For any X, Y ∈g,
Ad(expX)Y =ead(X)Y = X∞
k=0
1
k!ad(X)kY.
Definition 8. Let gbe a Lie algebra. The bilinear form B(X, Y) :=tr(ad(X)◦ad(Y)) (X, Y ∈g)
is called the Killing form of g, where the right-hand-side means the trace of the linear map
ad(X)◦ad(Y) :g→g.
Example
sl(n,R) B(X, Y) = 2ntr(XY) o(n) B(X, Y) = (n−2) tr(XY) su(n) B(X, Y) = 2ntr(XY) sp(n) B(X, Y) = (2n+ 2) tr(XY) 1.7 Toral subgroups and maximal tori
Definition 9. Let Gbe a connected Lie group. A subgroup S of G which is iso- morphic to S|1× · · · ×{z S1}
k
is called a toral subgroupofG, or simply atorus.
A toral subgroupS is an arcwise-connected compact abelian Lie group.
Definition 10. A torus T is amaximal torusof Gif
T0 is any other torus withT ⊂T0 ⊂G=⇒T0 =T Theorem 1.8. Any torusS of Gis contained in a maximal torusT.
Theorem 1.9. (Cartan–Weil–Hopf) Let G be a compact connected Lie group.
Then
(1) Ghas a maximal torusT. (2) G= [
x∈G
xT x−1.
(3) Any other maximal torus T0 of Gis conjugate toT, i.e.,
∃a∈G s.t. T0=aT a−1
(4) Any maximal torus of Gis a maximal abelian subgroup ofG.
The dimension`= dimT ofT is called therankofG.
Example G=U(n).
T :=
eiθ1
0 0
. .. eiθn
¯¯
¯¯
¯¯
¯¯
θ1,· · ·, θn∈R
∼=S|1× · · · ×{z S1}
n
is a toral subgroup ofU(n).
Proposition 1.10. Any unitary matrixA∈U(n)can be diagonalized by a unitary matrixP ∈U(n),
P−1AP =
eiθ1
0
. ..
0
eiθn
=:diag[eiθ1,· · ·, eiθn]
Example G=SO(n). We denote by Rθ the rotation matrix
Rθ=
µ cosθ sinθ
−sinθ cosθ
¶
and ifn= 2mput
T =
Rθ1
0
0
. .. Rθm
¯¯
¯¯
¯¯
¯¯
θ1,· · · , θm∈R
∼=S|1× · · · ×{z S1}
m
and ifn= 2m+ 1 we put
T =
Rθ1
0
. ..
Rθm
0
1
¯¯
¯¯
¯¯
¯¯
¯¯
θ1,· · ·, θm∈R
∼=S|1× · · · ×{z S1}
m
thenT is a maximal torus ofSO(n).
Proposition 1.11. Any orthogonal matrix A∈O(n)may be put into the following
form by an orthogonal matrixQ∈O(n),
tQAQ=
1
0
. ..
1
−1 . ..
−1
cosθ1 sinθ1
−sinθ1 cosθ1
. ..
0
cosθr sinθr−sinθr cosθr
Remark.
R0=
µ 1 0 0 1
¶
=I2, Rπ=
µ −1 0 0 −1
¶
=−I2, 1.8 Root systems
LetGbe a compact connected Lie group.
Let (ρ, V) be any representation ofG.
ρ:G→GL(V) (homomorphism)
(V is a vector space overRor C). We may choose a ρ(G)-invariant inner product onV. For anyρ(G)-invariant subspaceW ⊂V, we have
V =W⊕W⊥ (ρ(G)-invariant decomposition) Therefore, (ρ, V) is completely reducible.
(ρ, V) = (ρ1, V1)⊕ · · · ⊕(ρk, Vk) where (ρi, Vi) is irreducible.
In particular, the adjoint representation (Ad,g) of G is completely reducible, and we have a decomposition
g=g0⊕g1⊕ · · · ⊕gm
where
g0={X∈g|ad(Y)X= 0 for∀Y ∈g} (center ofg) andgi is an ideal ofg(i.e., [g,gi]⊂gi) andgi is irreducible.
Proposition 1.12. LetA, B be two diagonalizable matrices. If[A, B] = 0, thenA andB are simultaneously diagonalizable, i.e.,
∃x6=0 s.t. Ax=λx, Bx=µx (λ, µ :eigenvalues)
Let G be a compact connected Lie group, and S a torus of G. We choose an Ad(S)-invariant inner product on g. Then the representation (Ad,g) of S is completely reducible. If we consider the complexification gC = g⊗RC and the representation (Ad,gC) ofS, the eigenvalues of Ad(s) are complex numbers with absolute value 1, and corresponding eigenspaces ingCare of dimC= 1. Thengmay be decomposed into irreducible subspaces.
g=V0⊕V1⊕ · · · ⊕Vm
where
Ad(s)|V0 =id on V0
and for some linear formθk :s→R Ad(expv)|Vk=
µ cosθk(v) −sinθk(v) sinθk(v) cosθk(v)
¶
onVk forv∈s
w.r.t. an orthonormal basis onVk. Theseθk are unique up to sign and order, and do not depent on the inner product and an orthonormal basis.
Definition 11. Linear forms±θk (k= 1,· · ·, m) are calledroots ofGrelative to S. If S is a maximal torusT ofG, they are called simply theroots of G. The set
∆ of all roots ofGis called theroot systemof G.
Remark. Sometimes we define roots as
± 1 2πθk
so as to takeZ-values on the unit lattice exp−1(e)⊂t.
Example G=SU(n) andT is the maximal torus defined as before.
T ={diag[eix1,· · ·, eixn]|xk ∈R, Xn
k=1
xk = 0}
The Lie algebratofT is
t={diag[ix1,· · ·, ixn]|xk ∈R, Xn
k=1
xk = 0}
x1,· · · , xn∈t∗ are deined in natural way.
linear form diag[ix1,· · · , ixn]7→xk is denoted byxk We take the inner product (, ) ongdefined by
(X, Y) :=−1
2tr (XY) and for 1≤j < k≤n,1≤`≤n, put
Ujk=
1
−1
, Ujk0 =
i i
, √
2E`=√ 2
i
which form an orthonormal basis of g. Then we may see that for any v = diag[ix1,· · ·, ixn]∈t
Ad(expv) =
µ cos (xj−xk) −sin (xj−xk) sin (xj−xk) cos (xj−xk)
¶
on spanR{Ujk, Ujk0 }.
Thus the root system ∆ ofSU(n) is
∆ ={±(xj−xk)|1≤j < k≤n}
Definition 12. Let {λ1,· · · , λ`} be a basis of t∗. The lexicographic order λ1 >
λ2>· · ·> λ`>0 is defined by v=
X`
k=1
viλi>0⇐⇒
v1=· · ·=vk−1= 0 vk >0 for somek
A root α∈ ∆ is positive (resp. negative) if α > 0 (resp. α < 0) relative to the lexicographic order w.r.t. the basis.
∆ = ∆+∪∆−
A positive rootαis called asimple rootif it can not be expressed as a sum of two positive roots. The set {α1,· · ·, α`} of all simple roots of G is called the simple root system ofG(where `=rankG).
Example G=SU(n) andTa maximal torus of all diagonal matrices. {x1,· · · , xn−1} is a basis of t∗. Consider the lexicographic order x1> x2>· · ·> xn−1>0. Then
∆+={xj−xk |1≤j < k≤n} positive root system
∆−={−(xj−xk)|1≤j < k≤n} negative root system and the simple root system is
{αj=xj−xj+1 |1≤j ≤n−1}
1.9 Root systems – again –
Here, we shall calculate the root system in terms of Lie algebra.
LetGbe a compact connected semisimple Lie group, T a maximal torus ofG, gthe Lie algebra ofG, tthe Lie algebra ofT.
gC:=g⊗RC, tC:=t⊗RC
We extend [, ] complex-linearly ongC, thengCbecomes a complex Lie algebra.
tC becomes a maximal abelian Lie subalgebra ofgC.
The Killing formBCofgC is the natural extension of the Killing formB ofg.
If we take an Ad(G)-invariant inner product (, ) ong, we can make an Ad(G)- invariant Hermitian inner product (, ) ongCin natural way.
(ad(v)X, Y) + (X,ad(v)Y) = 0
forX, Y ∈gC, v∈tC, that is, ad(v) is a skew-Hermitian transformation. Thus for anyv∈tC, ad(v) is diagonalizable.
Hence,tCis a Cartan subalgebraofgC.
Definition 13. A Lie aubalgebra h is called a Cartan subalgebra of a complex Lie algebra gif
(1)his maximal abelian subalgebra, (2) ad(H)is diagonalizable for anyH ∈h.
1.10 SU(n)
G=SU(n),T is the maximal torus defined by
T ={diag [eiθ1,· · · , eiθn]|θ1+· · ·+θn= 0}
su(n) ={X ∈gl(n,C)|X+X∗=O, trX= 0}
A basis ofgis given by
Ujk=
1
−1
, Ujk0 =
i i
, E`=
i
−i
.
ThengCis spanned (overR) by
Ujk, Ujk0 , E`, iUjk, iUjk0 , iE`. Hence gC=sl(n,C) and
t = {diag[iθ1,· · ·, iθn]| θk∈R, θ1+· · ·+θn = 0}, tC = {diag[λ1,· · ·, λn] |λk ∈C, λ1+· · ·+λn= 0}.
Forv= diag[λ1,· · ·, λn]∈tC, if we put
Ejk=
1
then we have
ad(v)Ejk= [v, Ejk] = (λj−λk)Ejk. Letxj:tC→Cbe a linear form defined by
xj(diag[λ1,· · ·, λn]) =λj, then we have
ad(v)Ejk= [v, Ejk] = (xj−xk)(v)Ejk. Thus if we putα=xj−xk ∈(tC)∗,
ad(v)X =α(v)X forX ∈gCα=CEjk. The root system of g=su(n) relative totis
∆ ={±(xj−xk)|1≤j < k≤n}, and we have the decomposition
gC=tC+X
α∈∆
,gCα, where
gCα={X ∈gC|ad(v)X =α(v)X forv∈tC} is theroot spacecorresponding to a rootα. (dimCgCα= 1).
In usual, we denote byha Cartan subalgebra, so in our case,h=tC. and then hR:={v∈h|α(v)∈R(∀α∈∆)}
is justit.
We take a basis{v1,· · ·, v`} ofhR as
vj:= diag[0,· · ·,1,· · ·,0]∈it
and put{λ1,· · ·, λ`}its dual basis ofhR∗, λj=xj.
We consider the lexicographic order in hR∗. The root system is a union of the positive root system and the negative root system.
∆ = ∆+∪∆− If we put
αj =xj−xj+1 (1≤j≤n−1 =`)
then they are the simple roots andα ∈∆+ may be written as aZ≥0-coefficients linear combination of simple roots.
The restriction tohof the Killing formBofgCis nondegenerate, and we denote it by (, ).
Forλ∈h∗, we definevλ∈hby
(vλ, v) =λ(v) forv∈h, and we define an inner product onh∗ by
(λ, µ) := (vλ, vµ).
We put
(, )R:= (, )|hR, (, )R:= (, )|h∗R,
which are positive definite. For convenience, we denote these simply by (, ).
Forα, β∈∆ (α6=±β), we consider the sequence
β−pα, · · ·, β−α, β, β+α,· · · , β+qα∈∆ whereβ−(p+ 1)α6∈∆ and β+ (q+ 1)α6∈∆.
{β+mα| −p≤m≤q}
is called theα-sequence containingβ. We may see that 2(β, α)
(α, α) =p−q∈Z.
In particular, ifp= 0, i.e.,β−α6∈∆, 2(β, α)
(α, α) ≤0.
Therefore for simple rootsαj, αk
2(αj, αk) (αj, αj) ≤0.
On the other hand
2(αj, αk) (αj, αj)
2(αj, αk)
(αk, αk) = 4 cos2θ, where θis the angle ofαj andαk.
cos2θ= 0,1 4,1
2,3
4 (or 1).
Sinceπ/2≤θ < π, we have four cases.
(1) θ= π 2, (2) θ=2π
3 , kαjk=kαkk (3) θ=3π
4 , kαjk:kαkk= 1 :√ 2 or√
2 : 1 (4) θ=5π
6 , kαjk:kαkk= 1 :√ 3 or√
3 : 1 Then we draw a diagram as follows:
(1) we do not joinαj andαk ◦ ◦ (2) join by single line ◦−−◦
(3) join by double line ◦==>◦ (arrow from longer root to shorter one) (4) join by triple line ◦≡≡>◦ (arrow from longer root to shorter one)
Thus theDynkin diagramhas vertices{α1,· · ·, α`}and edges (1) – (4). The highest root µ (that is,µis a positive root such that µ≥αfor anyα∈∆) can be written as
µ=m1α1+· · ·+m`α`
in the Dynkin diagram, the integersmk are written on eachαk. The highest rootµofsu(n) is
µ=x1−xn=α1+· · ·+α`, m1=· · ·=m`= 1 2 Geometry of homogeneous spaces
2.1 Homogeneous spaces
Definition 14. A C∞ manifoldM ishomogeneousif a Lie groupGacts transi- tively onM, i.e.,
(1) There exists aC∞ map : G×M →M, (a, x)7→ax such that (i)a(bx) = (ab)x
(ii) ex=xfor any x∈M
(2) For any x, y∈M, there exists an elementa∈Gsuch thaty=ax
For an arbitraryly fixed point x0∈M, the closed subgroup H :={a∈G|ax0=x0}
is called anisotropy subgroupofGat x0.
ThenM is diffeomorphic to the coset spaceG/H.
M ∼=G/H, x7→aH, where,a∈Gis an element ofGs.t. x=ax0.
Conversely, if H is a closed subgroup of a Lie group G, then the coset space G/H becomes aC∞ manifold, andGacts onM by
G×G/H→G/H, (g, aH)7→(ga)H, and the map
τg :G/H→G/H, aH 7→(ga)H is called aleft translationbyg.
τg◦π=π◦Lg
2.2 G-invariant Riemannian metrics
LetM =G/Hbe a homogeneous space of a Lie groupGand a closed subgroup H. A Riemannian metrich, ionM isG-invariantif
h(τg)∗X,(τg)∗Yi=hX, Yi for∀g∈G
for anyX, Y ∈T M. Thenτg is an isometry of the Riemannian manifold (M,h, i).
Definition 15. A homogeneous space M = G/H is reductive if there exists a decomposition
g=h+m (direct sum) such that
Ad(H)m⊂m, i.e., Ad(h)m⊂m (∀h∈H).
Ifghas an Ad(H)-invariant inner product (, ), then the orthogonal complement m:=h⊥ satisfies the above property.
π:G→M =G/H is the natural projection,o=eH ∈G/H.
π∗e:TeG→ToM onto linear
induces an isomorphism
ToM ∼=TeG/kerπ∗e∼=g/h∼=m For anyXgH, YgH∈TgHM
hXgH, YgHigH = h(τg−1)∗gHXgH,(τg−1)∗gHYgHieH
= ((τg−1)∗gHXgH,(τg−1)∗gHYgH).
LetG/Hbe a reductive homogeneous space,
g=h⊕m (Ad(H)-invariant decomposition).
For anyh∈H
π(ih(g)) =τh(π(g)) (g∈G).
Hence for anyX ∈m⊂g
π∗e(Ad(h)X) = (τh)∗eX (π∗e:m∼=ToM).
Then aG-invariant Riemannian metrich, ionM =G/Hinduces an inner product (, ) onm.
(X, Y) =hX, Yio
Then (, ) is Ad(H)-invariant.
(Ad(h)X,Ad(h)Y) = (X, Y), for ∀h∈H, X, Y ∈m.
Hence, we have the one-to-one correspondence:
h, i: G-inv. Riem. metric onG/H l
(, ) : Ad(H)-inv. inner product onm
In the sequel, we use the same notation h, ifor an Ad(H)-inv. inner product on m.
Definition 16. A reductive homogeneous spaceM =G/Hwith a Riemannian met- rich , iisnaturally reductiveif there exists an Ad(H)-invariant decomposition
g=h+m (direct sum), Ad(H)m⊂m such that
h[X, Y]m, Zi+hY,[X, Z]mi= 0
for any X, Y, Z∈m, where[X, Y]m denotes them-component of[X, Y].
Remark. Accoring to the direct sum decompositiong=h⊕m, [X, Y] = [X, Y]h+ [X, Y]m
Definition 17. A homogeneous space M =G/H with a Riemannian metrich , i isnormal homogeneousif there exists a bi-invariant Riemannian metric(, )on Gsuch that, if we identify
m:=h⊥ (with respect to( , )) withToM, then
h, i= (, )|m×m.
Remark. IfM =G/H is normal homogeneous,M is naturally reductive.
Example If the Kiling formBof a compact Lie groupGis negative definite (then Gissemisimple), using the inner product−Bong,G/His normal homogeneous.
2.3 Riemannian connection and the canonical connection LetG/H be a homogeneous space. ForX∈g,
γ(t) =τexptX(gH)
is a curve inM through a pointgH. The vector fieldX∗ defined by XgH∗ :=γ0(0) = d
dt
¯¯
¯¯
t=0
τexptX(gH) satisfies at the origino=eH
Xo∗= d dt
¯¯
¯¯
t=0
π(exptX) =π∗e(X).
IfG/H is reductive, the isomorphismπ∗e:m∼=ToM may be expressed as m3X↔Xo∗∈ToM.
Proposition 2.1. ForX, Y ∈g,
[X∗, Y∗] =−[X, Y]∗.
Let G/H be a reductive homogeneous space with a G-invariant Riemannian metrich, i. We denote by∇ the Riemannian connection of (M/H,h, i).
Proposition 2.2. For any X ∈g, the vector field X∗ is a Killing vector field on G/H, i.e.,
X∗hY, Zi=h[X∗, Y], Zi+hY,[X∗, Z]i for any vector fields Y, Z ∈X(G/H).
Since
X∗hY, Zi=h∇X∗Y, Zi+hY,∇X∗Zi, we have
h∇YX∗, Zi+hY,∇ZX∗i= 0.
Proposition 2.3. Let (G/H,h ,i) be a naturally reductive homogeneous space.
Then
∇vX∗=
[X, v] ifX ∈h
1
2[X, v]m ifX ∈m forv∈m∼=ToM.
Definition 18. Let(G/H,h, i)be a naturally reductive homogeneous space. Then there exists a metric connection D with torsion tensor T and the curvature tensor B such that
DT = 0, DB= 0.
D is called the canonical connection.
The canonical connectionD is given by DvX∗=
½ [X, v] ifX ∈h [X, v]m ifX ∈m and
DXY = ∇XY +1
2T(X, Y), T(X, Y) = −[X, Y]m,
B(X, Y)Z = −[[X, Y]h, Z], forX, Y, Z∈m.
2.4 Curvature tensor
Theorem 2.4. Let(G/H,h, i)be a naturally reductive homogeneous space. Then the Riemannian curvature tensor Ris given by
(R(X∗, Y∗)Z∗)o = −[[X, Y]h, Z]−1
2[[X, Y]m, Z]m
−1
4[[Y, Z]m, X]m−1
4[[Z, X]m, Y]m
forX, Y, Z∈m.
Then we have
(R(X∗, Y∗)Y∗)o=−[[X, Y]h, Y]−1
4[[X, Y]m, Y]m
and the sectional curvature is given by hR(X∗, Y∗)Y∗, X∗io= 1
4h[X, Y]m,[X, Y]mi+h[[X, Y]h, X]m, Yi.
If (G/H,h, i) is normal homogeneous hR(X∗, Y∗)Y∗, X∗io= 1
4h[X, Y]m,[X, Y]mi+h[X, Y]h,[X, Y]hi ≥0.
2.5 Symmetric spaces
Definition 19. A connected Riemannian manifold(M, g)is called aRiemannian (globally) symmetric space if
∀p∈M,∃sp∈I(M, g)s.t. sp2=id., pis an isolated fixed point ofsp
sp is the geodesic symmetry:
sp(γv(t)) =γv(−t), whereγv(t) is the geodesic s.t.
γv(0) =p, γv0(0) =v∈TpM.
The connected Lie group G=I0(M, g) (identity component) acts transitively on M, and M becomes a Riemannian homogeneous space G/H, where a closed subgroupH is an isotropy subgroup at p0∈M. The map
σ:G→G, a7→sp0◦a◦sp0
is an involutive automorphism ofG, and
Gσ0 ⊂H ⊂Gσ,
where
Gσ={a∈G|σ(a) =a}, Gσ0 is the identity component ofGσ. Such a pair (G, H) is called asymmetric pair.
Examples
M =G/H G H
Sn SO(n+ 1) SO(n) sphere
Grk(Rn) O(n) O(k)×O(n−k) real Grassmann mfd.
Grk(Cn) U(n) U(k)×U(n−k) complex Grassmann mfd.
The Lie algebragmay be decomposed into±1-engenspaces of the differential dσ:g→g.
g=h⊕m, dσ|h=id, dσ|m=−id Then we see that
[h,h]⊂h, [h,m]⊂m, [m,m]⊂h
and hence the canonical connection D coincides with the Riemannian connection
∇, thus
∇R= 0,
R(X, Y)Z =−[[X, Y], Z]
forX, Y, Z∈m.
2.6 The root system of a symmetric space
LetM =G/H be a Riemannian symmetric space. Then as before, we have the Ad(H)-invariant decomposition
g=h⊕m.
We take a maximal abelian subspacea⊂m. For anyH ∈a, we consider the linear maps ad(H) :g→g, which are simultaneously diagonalizable. If there existsX ∈g such that
ad(H)X =α(H)X
for allH ∈a for some nonzero linear formα∈a∗,αis called arootofM relative to a. The set Σ of all roots ofM relative toa is called a restricted root system of M.
Σ can be obtained as follows: take a Cartan subalgebra hC of gC such that hC ⊃ a, let ∆ be the root system of gC relative to hC. Then Σ coinsides with {α|a |α∈∆} (as well as their multiplicities).
3 Homogeneous structures
When does a Riemmanian manifold (M, g) become a Riemannian homogeneous manifold?
W. Ambrose and I. M. Singer characterized it by the existense of some tensor fieldT of type (1,2) onM, which is called ahomogeneous structure.
Theorem 3.1. ([1]) Let (M, g) be a connected, simply connected, complete Rie- mannian manifold. (M, g)is a Riemannian homogeneous space if and only if there exists a tensor field T of type(1,2)on M satisfying
(1)g(T(X)Y, Z) +g(Y, T(X)Z) = 0 (2)∇XR=T(X)·R
(3)∇XT =T(X)·T
where∇andR denotes the Riemannian connection and the Riemannian curvature tensor of(M, g), respectively.
∇˜ :=∇ −T
is called a A–S connection. (1)–(3) are equivalent to ˜∇g= 0, ˜∇R˜= 0, ˜∇T˜= 0.
Remark. Here, we consider the tensor field T :X(M)×X(M)→X(M) of type (1,2) as, forX∈X(M),
T(X) :X(M)→X(M), Y 7→T(X)Y and
(T(X)·R)(Y, Z)W := T(X)(R(Y, Z)W)−R(T(X)Y, Z)W
−R(Y, T(X)Z)W−R(Y, Z)T(X)W (T(X)·T)(Y)Z := T(X)T(Y)Z−T(T(X)Y)Z−T(Y)T(X)Z About the homogeneity of almost Hermitian manifold (M, g, J), (J is an almost complex structure;J2=−I), the following result was shown by K. Sekigawa.
Theorem 3.2. ([6]) Let (M, g, J) be a connected, simply connected, complete almost Hermitian manifold. M is a homogeneous almost Hermitian manifold if and only if there exists a tensor field T of type(1,2)on M satisfying
(1)g(T(X)Y, Z) +g(Y, T(X)Z) = 0 (2)∇XR=T(X)·R
(3)∇XT =T(X)·T (4)∇XJ =T(X)·J
Analmost contact metric manifold(M, φ, ξ, η, g) is aC∞manifoldM with an almost contact metric structure (φ, ξ, η, g) such that
(1)φis a tensor field of type (1,1),
(2)ξ∈X(M) is called a characteristic vector field, (3)η is a 1-form,
(4)g is a Riemannian metric satisfying
φ2X =−X+η(X)ξ, η(ξ) = 1,
g(X, ξ) =η(X),
g(φX, φY) =g(X, Y)−η(X)η(Y)
About the homogeneity of (M, φ, ξ, η, g), the following result was shown.
Theorem 3.3. ([5]) Let(M, φ, ξ, η, g)be a connected, simply connected, complete almost contact metric manifold. M is a homogeneous almost contact metric mani- fold if and only if there exists a skew-symmetric tensor fieldT of type(1,2) onM satisfying
(1) g(T(X)Y, Z) +g(Y, T(X)Z) = 0 (2) ∇XR=T(X)·R
(3) ∇XT =T(X)·T (4) ∇Xη=−η◦T(X) (5) ∇Xφ=T(X)·φ References
[1] W. Ambrose and I. M. Singer,On homogeneous Riemannian manifolds, Duke Math. J., 25 (1958), 647–669.
[2] A. Arvanitoyeorgos, An Introduction to Lie Groups and the Geometry of Ho- mogeneous Spaces, Amer. Math. Sci., 2003.
[3] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Aca- demic Press, New York, 1978.
[4] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vols. I and II, Interscience, New York, 1963 and 1969.
[5] T. Koda and Y. Watanabe, Homogeneous almost contact Riemannian mani- folds and infinitesimal models, Bollettino U. M. I. (7) 11-B (1997), Suppl. fasc.
2, 11–24.
[6] K. Sekigawa, Notes on homogeneous almost Hermitian manifolds, Hokkaido Math. J., 7 (1978), 206–213.
[7] W. Ziller, The Jaacobi equation on naturally reductive compact Riemannian homogeneous spaces, Comment. Math. Helvetici, 52(1977), 573–590.
