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Transcript
```Symmetric Spaces
Xinghua Gao
May 5, 2014
Notations
M
N0
Np
sp
fΦ
XΦ
K(S)
Drs
I(M )
1
Riemannian manifold.
normal neighborhood of the origin in Tp M .
normal neighbourhood of p, Np = exp N0 .
geodesic symmetry with respect to p.
dΦ f = f ◦ Φ.
dΦ X.
sectional curvature of M at p along the section S.
set of tensor ﬁelds of type (r, s).
the set of all isometries on M
Aﬃne Locally Symmetric Space, Isometry Group
and Others
Deﬁnition 1 (normal neighborhood) A neighborhood Np of p in M is called
a normal neighbourhood if Np = exp N0 , where N0 is a normal neighborhood of
the origin in Tp M , i.e. satisfying: (1)exp is a diﬀeomorphism of N0 onto an
open neighborhood Np ; (2)if X ∈ N0 , 0 ≤ t ≤ 1, then tX ∈ N0 (star shaped).
Deﬁnition 2 (geodesic symmetry) ∀q ∈ Np , consider the geodesic t → γ(t) ⊂
Np passing through p and q s.t. γ(0) = p, γ(1) = q. Then the mapping
q → q ′ = γ(−1) of Np onto itself is called geodesic symmetry w.r.t p, denoted
by sp .
Remark: sp is a diﬀeomorphism of Np onto itself and (dsp )p = −I.
Deﬁnition 3 (Aﬃne locally symmetric) M is called aﬃne locally symmetric if each point m ∈ M has an open neighborhood Nm on which the geodesic
−1
symmetry sm is an aﬃne transformation. i.e. ∇X (Y ) = (∇X sm (Y sm ))sm ,
∀X, Y ∈ X(M ).
Deﬁnition 4 (Pseudo-Riemannian structure) Let M be a C ∞ -manifold.
A pseudo-Riemannian structure on M is a symmetric nondegenerate (as bilinear
form at each p ∈ M ) tensor ﬁeld g of type (0, 2).
Remark: A pseudo-Riemannian manifold is a connected C ∞ -manifold with
1
a pseudo-Riemannian structure. g is called a Riemannian structure iﬀ gp is
positive deﬁnite ∀p ∈ M .
Deﬁnition 5 (Isometry) Let M and N be two C ∞ manifolds with pseudoRiemannian structures g and h, respectively. Let ϕ be a mapping of M into
N.
Then ϕ is called an isometry if ϕ is a diﬀeomorphism of M onto N and ϕ∗ h =
g. ϕ is called a local isometry if for each p ∈ M there exist open neighborhoods
U of p and V of ϕ(p) s.t. ϕ is an isometry of U onto V .
Deﬁnition 6 (Riemannian locally symmetric space) M is called a Riemannian locally symmetric space if for each p ∈ M ∃ a normal neighborhood Np
of p on which the geodesic symmetry sp is an isometry, i.e. sp (g) = g, where g
is the pseudo-Riemannian structure on Np ⊂ M . (sp is a local isometry from
M to itself ).
2
Deﬁnition of Symmetric Space
Deﬁnition 7 (Riemannian Globally Symmetric Space) Let M be an analytic Riemannian manifold, M is called Riemannnian globally symmetric if each
p ∈ M is an isolated ﬁxed point of an involutive (its square but not the mapping
itself is the identity) isometry sp of M . Or equivalently ∀p ∈ M there is some
sp ∈ I(M ) with the properties: sp (p) = p, (dsp )p = −I.
Example 1: Euclidean Space
Let M = Rn with the Euclidean metric. The geodesic symmetry at any point
p ∈ Rn is the point reﬂection sp (p + v) = p − v. The isometry group is the
Euclidean group E(n) generated by translations and orthogonal linear maps;
the isotropy group of the origin is the orthogonal group O(n). Note that the
symmetries do not generate the full isometry group E(n) but only a subgroup
which is an order-two extension of the translation group.
Example 2: The Sphere
Let M = Sn be the unit sphere with the standard scalar product. The symmetry
at any x ∈ Sn is the reﬂection at the line Rx in Rn+1 , i.e. sx (y) = −y + 2⟨y, x⟩x
(the component of y in x-direction, ⟨y, x⟩x, is preserved while the orthogonal
complement y⟨y, x⟩x changes sign). In this case, the symmetries generate the
full isometry group which is the orthogonal group O(n + 1). The isotropy group
of the last standard unit vector en+1 = (0, ..., 0, 1)T is O(n) ⊂ O(n + 1).
Example 3: Compact Lie groups
Let M = G be a compact Lie group with biinvariant Riemannian metric, i.e.
left and right translations Lg , Rg : G → G acts as isometries for any g ∈ G.
Then G is a symmetric space where the symmetry at the unit element e ∈ G is
the inversion se (g) = g −1 . Then se (e) = e and dse v = −v for any v ∈ g = Te G,
so the involutive condition is satisﬁed. We have to check that se is an isometry,
2
i.e. (dse )g preserves the metric for any g ∈ G. This is certainly true for g = e,
and for arbitrary g ∈ G we have the relation se ◦ Lg = Rg−1 ◦ se which shows
(dse )g ◦ (dLg )e = (dRg−1 )e ◦ (dse )e . Thus (dse )g preserves the metric since so
do the other three maps in the above relation.
Example 4: Projection model of the Grassmannians
Let S = Gk (Rn ) be the set of all k-dimensional linear subspaces of Rn . The
group O(n) acts transitively on this set. The symmetry sE at any E ∈ Gk (Rn )
will be the reﬂection sE with ﬁxed space E, i.e. with eigenvalue 1 on E and −1
on E ⊥ .
But what is the manifold structure and the Riemannian metric on Gk (Rn )?
One way to see this is to embed Gk (Rn ) into the space S(n) of symmetric
real n × n matrices: We assign to each k-dimensional subspace E ∈ Rn the
orthogonal projection matrix pE with eigenvalues 1 on E and 0 on E ⊥ . Let
P (n) = {p ∈ S(n)| p2 = p} denote the set of all orthogonal projections. This
set has several mutually disconnected subsets, corresponding to the trace of the
elements which here is the same as the rank:
P (n)k = P (n) ⊂ S(n)k ,
S(n)k = {x ⊂ S(n)| trace x = k}
Now we may identify Gk (Rn ) with P (n)k ⊂ S(n), using the embedding E 7→ pE
which is equivariant in the sense gpE g T = pgE for any g ∈ O(n). In fact, each pE
lies in this set, and vice versa, a symmetric matrix p satisfying p2 = p has only
eigenvalues 1 and 0 with eigenspaces E = im p and E ⊥ = ker p, hence p = pE ,
and the trace condition says that E has dimension k. P (n)k is a submanifold
of the aﬃne space S(n)k since it is the conjugacy class of the matrix
(
)
Ik 0
p0 =
0 0
i.e. the orbit of p0 under the action of the group O(n) on S(n) by conjugation.
The isotropy group of p0 is O(k)×O(n−k) ⊂ O(n). A complement of TI (O(k)×
O(n − k)) in TI O(n) is the space of matrices of the type
(
)
0 −LT
L
0
with arbitrary L ∈ R(n−k)×k , thus P (n)k = Gk (Rn ) has dimension k(n − k).
Deﬁne F : S(n) → S(n), F (p) = p2 − p, then P (n)k = Gk (Rn ) ⊂ F −1 (0), the
kernel ker dFp is contained in Tp Gk (Rn ). But the subspace ker dFp = {v ∈
S(n)| vp + pv = v} is isomorphic to Hom(E, E ⊥ ) since it contains precisely
the symmetric matrices mapping E = im p into E ⊥ and vice versa. Thus
ker dFp = Tp Gk (Rn ).
Now we equip P (n)k ⊂ S(n) with the metric induced from the trace scalar
product < x, y >= trace(xT y) = trace(xy) on S(n). The group O(n) acts
isometrically on S(n) by conjugation and preserves P (n)k , hence it acts isometrically on P (n)k . In particular, let sE ∈ O(n) be the reﬂection at the subspace
3
E and let ŝE be the corresponding conjugation, ŝE (x) = sE xsE . This is an
isometry ﬁxing pE , and since sE ﬁxes E and reﬂects E ⊥ , the conjugation ŝE
maps any x ∈ Tp Gk (Rn ) into −x (x is a linear map from E to E ⊥ and vice
versa). Thus ŝE is the symmetry at pE .
3
Homogeneous description
For a symmetric space, we have theorem:
Theorem 1 Let M be a Riemannian symmetric space and p0 any point in M .
Let G = I(M )0 be the identity component of the isometry group and K be the
isotropy group of G at p0 . Then K is a compact subgroup of the connected group
G and G/K is analytically diﬀeomorphic to M under the mapping gK → g(p0 ),
g ∈ G.
So we can get think of symmetric spaces as the homogeneous space of the
isometry group G. A natural question to ask is what group G and subgroup K
will lead to a symmetric space. To answer this question, we need the deﬁnition
of Riemannian symmetric pairs.
3.1
From Symmetric space to Symmetric Pairs
Deﬁnition 8 (symmetric pair) Let G be a connected Lie group and K a
closed subgroup. The pair (G, K) is called a symmetric pair if there exists an
involutive analytic automorphism σ of G s.t. (Gσ )0 ⊂ K ⊂ Gσ , where Gσ is
the set of ﬁxed points of σ in G and (Gσ )0 is the identity component of Gσ .
If in addition, AdG (K) (the adjoint group of K in G) is compact, (G, K) is said
to be a Riemannian symmetric pair.
We can get symmetric pairs from symmetric spaces:
Theorem 2 Let M be a symmetric space with a ﬁxed point p0 , G = I(M )0 be
the identity component of the isometry group and let K be the isotropy group of
G at p0 . Then the map G/K → M with K 7→ g(p0 ) is a bijection. The group
G has an involutive automorphism σ given by σ : G → G, g 7→ sp0 ◦ g ◦ sp0 with
stabilizer (Gσ )0 ⊂ K ⊂ Gσ .
Ad(K) is compact since K is closed and bounded and Ad is a homeomorphism.
So (G, K) is a Riemannian symmetric pair.
3.2
From Symmetric Pairs to Symmetric Space
In fact symmetric pairs lead to symmetric spaces.
Theorem 3 Let M be a Riemannian manifold and I(M) the set of all isometries of M. Then
(1) The compact open topology of I(M ) turns I(M ) into a locally compact topological transformation group.
4
(2) Let p ∈ M and let K̃ denote the subgroup of I(M ) which leaves p ﬁxed. Then
K̃ is compact.
Theorem 4 Let (G, K) be a Riemannian symmetric pair. Let π denote the
natural mapping of G onto G/K and put o = π(e). Let σ be any analytic,
involutive automorphism of G on M = G/K s.t. (Gσ )0 ⊂ K ⊂ Gσ . Then
there is a G-invariant Riemannian structure Q on M that makes (M, Q) a
Riemannian globally symmetric space. The geodesic symmetry so satisﬁes
so ◦ π = π ◦ σ,
τ (σ(g)) = so τ (g)so ,
g∈G
where τ is the parallel translation. In particular, so is independent of the choice
of Q.
So we get a Riemannian symmetric space M = G/K from a symmetric pair
(G, K).
Example 4: The Compact Grassmannian
First consider the Grassmannian of oriented k-planes in Rk+l , denoted by M =
G̃k (Rk+l ). Thus, each element in M is a k-dimensional subspace of Rk+l together with an orientation. We shall assume that we have the orthogonal splitting
Rk+l = Rk ⊕ Rl , where the distinguished element p = Rk takes up the ﬁrst k
coordinates in Rk+l and is endowed with its natural positive orientation.
Let us ﬁrst identify M as a homogeneous space. Observe that O(k+1) acts on
Rk+l . As such, it maps k-dimensional subspaces to k-dimensional subspaces, and
does something uncertain to the orientations of these subspaces. We therefore
get that O(k + 1) acts transitively on M. This is, however, not the isometry
group as the matrix −I ∈ SO(k + l) acts trivially if k and l are even.
The isotropy group consists of those elements that keep Rk ﬁxed as well as
preserving the orientation. Clearly, the correct isotropy group is then: SO(k) ×
O(l) ⊂ O(k + 1).
The tangent space at p = Rk is naturally identiﬁed with the space of k × l
matrices M atk×l , or equivalently, with Rk ⊗ Rl . The isotropy action of SO(k) ×
O(l) on M atk×l now acts as follows:
SO(k) × O(l) × M atk×l → M atk×l ,
(A, B, X) 7→ AXB −1 = AXB T
The representation, when seen as acting on Rk ⊗Rl , is denoted by SO(k)⊗O(l).
To see that M is a symmetric space, we have to show that the isotropy
group contains the required involution. On the tangent space Tp M = M atk×l
it is supposed to act by multiplication by −1. Thus, we have to ﬁnd (A, B) ∈
SO(k) × O(l) such that for all X, AXB T = −X.
Clearly, we can just set: A = Ik , B = −Il . Depending on k and l, other
choices are possible, but they will act in the same way.
We have now exhibited M as a symmetric space, although we didn’t use the
isometry group of the space. Instead, we used a ﬁnite covering of the isometry
group and then had some extra elements that acted trivially.
5
Remark: If we deﬁne X to be the matrix that is 1 in the (1, 1) entry and otherwise zero, then AXB T = A1 (B 1 )T , where A1 is the ﬁrst column of A and B1 is
the ﬁrst column of B. Thus, the orbit of X, under the isotropy action, generates
a basis for M atk×l but does not cover all of the space. This is an example of
an irreducible action on Euclidean space that is not transitive on the unit sphere.
Note: Example 2(sphere) and 4(Grassmannian) arises as so are called extrinsic
symmetric spaces: A submanifold S ⊂ RN is called extrinsic symmetric if it
is preserved by the reﬂections ar all of its normal spaces. More precisely, let
sp be the isometry of RN ﬁxing p whose linear part dsp acts as identity I on
the normal space νp S and as −I on the tangent space Tp S, then S is extrinsic
symmetric if sp (S) = S for all p ∈ S. Extrinsic symmetric spaces are classiﬁed.
Figure 1: some classiﬁcation
References
 Sigurdur Helgason, Diﬀerential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, 1978
 J.-H. Eschenburg, Lecture Notes on Symmetric Spaces
 Peter Holmelin, Symetric spaces, 2005
 Peter Peterson, Riemannian Geometry, Springer, 2006
6
```