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Home , Homogeneous space, Isometry, Isotropy

Riemannian – Lecture 19

Dr. Emma Carberry

October 12, 2015

Recap from lecture 17:

Definition 17.21. • a Lie G acts on a M if there is a

G × M → M (g, p) 7→ g · p

such that e · p = p for all p ∈ M

and (gh) · p = g · (h · p) for all g, h ∈ G, p ∈ M.

(these force the action of each element to be a )

Definition 17.21 (continued). • if for every g ∈ G,

g : M → M p 7→ g · p

is smooth then we say that G acts smoothly

• the isotropy subgroup Gp of G at p is the subgroup fixing p

• G acts transitively on M if for every p, q ∈ M there exists g ∈ G such that q = g · p

Definition 17.22. A homogeneous is a manifold M together with a smooth transitive action by a G.

1 Informally, a “looks the same” at every point.

Since the action is transitive, the isotropy groups are all conjugate

−1 Gg·p = gGpg and for any p ∈ M we can identify the points of M with the quotient space G/Gp.

Definition 17.23. A Lie subgroup H of a Lie group G is a subset of G such that the natural inclusion is an immersion and a group (i.e. H is simultaneously a subgroup and submanifold).

Since the the on a homogeneous space is in particular continuous, the isotropy subgroups are closed in the topology of G.

Theorem 17.24 (Lie-Cartan). A closed subgroup of a Lie group G is a Lie subgroup of G.

Corollary 17.25. An isotropy subgroup Gp of a Lie group G is a Lie subgroup of G.

Theorem 17.26. If G is a Lie group and H a Lie subgroup then the quotient space G/H has a unique smooth structure such that the map

G × G/H → G/H (g, kH) 7→ gkH is smooth.

We omit the proof; it is given for example in Warner, “Foundations of Differentiable and Lie Groups”, pp 120–124.

Corollary 17.27. A homogeneous space M acted upon by the Lie group G with isotropy sub- group Gp is diffeomorphic to the quotient manifold G/Gp where the latter is given the unique smooth structure of the previous theorem.

Lecture 18 continued:

2 Example 18.8 ( Real , RPn). Recall that RPn is the space of lines through the origin in Rn+1. Equivalently, n+1 \{0} n = R , RP ∼ where 0 1 n 0 1 n (X ,X ,...,X ) ∼ λ(X ,X ,...,X ), λ ∈ R \{0}. We write the of (X0,X1,...,Xn) as [X0 : X1 : ... : Xn]. Example 18.8 (Continued). The natural projection

n+1 n R \{0} → RP (X0,X1,...,Xn) 7→ [X0 : X1 : ... : Xn] restricts to Sn to exhibit Sn as a double cover of RPn. That is, we have a

n ∼ n S / ∼ = RP .

Example 18.8 (Continued). We have

Sn ∼= SO(n + 1)/SO(n) ∼= O(n + 1)/O(n)

n n S is a double cover of RP O(n) is a double cover of SO(n).

This suggests trying to show that

n RP is diffeomorphic to SO(n + 1)/O(n).

Example 18.8 (Continued). Identify O(n) with its image under the

O(n) → SO(n + 1) ! det A A 7→ . A

Using the identification n ∼ n S / ∼ = RP we have a smooth transitive action of SO(n + 1) on RPn.

Exercise 18.9. Show that the isotropy group of [1 : 0 ... : 0] is O(n) .

3 Example 18.10 (, CPn). CPn is the space of complex lines through the origin in Cn+1. Equivalently, n+1 \{0} n = C , CP ∼ where 0 1 n 0 1 n (X ,X ,...,X ) ∼ λ(X ,X ,...,X ), λ ∈ C \{0}. We write the equivalence class of (X0,X1,...,Xn) as [X0 : X1 : ... : Xn].

The proof that CPn is a smooth manifold of real dimension 2n is completely analogous to the argument for real projective space.

Exercise 18.11. Show that CPn is a homogeneous manifold diffeomorphic to SU(n+1)/S(U(1)× U(n)).

Lecture 19:

Isometry group

Definition 19.1. A diffeomorphism ϕ :(M, g) → (N, h) is called an isometry if ϕ∗(h) = g, equivalently if

h(dϕp(v), dϕp(w)) = g(v, w) for all p ∈ M, v, w ∈ TpM.

Definition 19.2. The of isometries ϕ :(M, g) → (M, g) forms a group, called the I of the (M, g).

The isometry group is always a finite-dimensional Lie group acting smoothly on M (see eg Kobayashi, Transformation Groups in , Thm II.1.2).

4 Homogeneous and isotropic Riemannian manifolds

Definition 19.3. A Riemannian manifold (M, g) is a homogeneous Riemannian manifold if its isometry group I(M) acts transitively on M.

A homogeneous space M is just a , no Riemannian . M is diffeo- morphic to the quotient manifold G/Gp.

A homogeneous Riemannian manifold also has a Riemannian metric compatible with the group action.

A homogeneous Riemannian manifold “looks the same” at every point in terms both of its smooth structure and of the Riemannian metric.

Definition 19.4. A Riemannian manifold (M, g) is isotropic at p ∈ M if the isotropy subgroup

Ip acts transitively on the set of unit vectors of TpM, via

Ip × {X ∈ TpM | hX,Xi = 1} → {X ∈ TpM | hX,Xi = 1}

(g, X) 7→ dgp(X).

The geometric interpretation is that the Riemannian manifold M near p looks the same in all directions.

Definition 19.5. A homogeneous Riemannian manifold which is isotropic at one point must be isotropic at every point. Such a manifold is called a homogeneous and isotropic Riemannian manifold.

A homogeneous and isotropic Riemannian manifold then looks the same at every point and in every direction: can you think of any examples? Example 19.6. Proposition

The isometry group of Sn is O(n + 1). It acts transitively on Sn. The isotropy group acts transitively on the unit vectors of the tangent space. Hence, Sn is a homogeneous and isotropic Riemannian manifold.

Example 19.6 (Continued). Proof

Since O(n+1) is by definition the isometry group of Rn+1 and Sn inherits its Riemannian metric from Rn+1, the action of SO(n + 1) on Sn is by isometries. Conversely, any isometry of the sphere can be extended linearly to give an isometry of Rn+1 (we will check this momentarily). Hence I(Sn) = O(n + 1).

5 Example 19.6 (Continued). The proofs of the next two statements are similar.

n Choose any point p ∈ S . It is unit length, so extend it to an orthonormal basis v1 = n+1 p, v2, ··· , vn+1 of R .

The matrix g with columns v1, ··· , vn+1 is in O(n + 1) and takes e1 to p. Thus, by going via e1, the action is transitive.

n To see it is isotropic, we compute at e1 ∈ S . We have seen that the isotropy subgroup is just n O(n) acting on the standard basis e2, ··· , en+1. Choose any unit vector v ∈ TpS . We need to

find g such that dge1 (e2) = v.

But the differential of the is itself, so we may replace dge1 with g.

Again, extend v to an orthonormal basis and then we may take the columns of g to be v, v3, ··· , vn+1, similarly to what we have seen before. Example 19.6 (Continued). To see that every isometry ϕ of a sphere extends to a linear map, we can write any point p ∈ Rn+1 as λs for λ ∈ R≥0 and s ∈ Sn. Define ϕ˜(p) = λϕ(s). This is well defined (check at p = 0).

n+1 To see it is linear, take any basis (e0, . . . , en) of R . By checking inner products, so too is

(ϕ(e0), . . . , ϕ(en)). We compute X ϕ˜(p) = hϕ˜(p), ϕ(ei)iϕ(ei) X = hλϕ(s), ϕ(ei)iϕ(ei) X = λhϕ(s), ϕ(ei)iϕ(ei) X X = λhs, eiiϕ(ei) = hp, eiiϕ(ei).

Note the right-hand side is a linear of p. It’s an easy check that it is an isometry.

Corollary 19.7. The sphere Sn has constant sectional .

We could just as easily have argued above with the sphere of radius r > 0, the special similarly acts transitively on it by isometries and furthermore acts transitively on the orthonormal frames of Sn(r).

Hence the sphere Sn(r) of radius r > 0 has constant too.

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