riemannian geometry – lecture 19 · riemannian geometry – lecture 19 isotropy dr. emma carberry...
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![Page 1: Riemannian Geometry – Lecture 19 · Riemannian Geometry – Lecture 19 Isotropy Dr. Emma Carberry October 12, 2015. Recap from lecture 17: ... “Foundations of Differentiable Manifolds](https://reader033.vdocuments.mx/reader033/viewer/2022060208/5f0439817e708231d40ced15/html5/thumbnails/1.jpg)
Riemannian Geometry – Lecture 19
Isotropy
Dr. Emma Carberry
October 12, 2015
![Page 2: Riemannian Geometry – Lecture 19 · Riemannian Geometry – Lecture 19 Isotropy Dr. Emma Carberry October 12, 2015. Recap from lecture 17: ... “Foundations of Differentiable Manifolds](https://reader033.vdocuments.mx/reader033/viewer/2022060208/5f0439817e708231d40ced15/html5/thumbnails/2.jpg)
Recap from lecture 17:
![Page 3: Riemannian Geometry – Lecture 19 · Riemannian Geometry – Lecture 19 Isotropy Dr. Emma Carberry October 12, 2015. Recap from lecture 17: ... “Foundations of Differentiable Manifolds](https://reader033.vdocuments.mx/reader033/viewer/2022060208/5f0439817e708231d40ced15/html5/thumbnails/3.jpg)
Definition 17.21a Lie group G acts on a manifold M if there is a map
G ×M → M(g,p) 7→ g · p
such thate · 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 bijection)
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Definition 17.21 (continued)if for every g ∈ G,
g : M → Mp 7→ g · p
is smooth then we say that G acts smoothlythe isotropy subgroup Gp of G at p is the subgroup fixing pG acts transitively on M if for every p,q ∈ M there existsg ∈ G such that q = g · p
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Definition 17.22
A homogeneous space is a manifold M together with a smoothtransitive action by a Lie group G.
Informally, a homogeneous space “looks the same” at everypoint.
Since the action is transitive, the isotropy groups are allconjugate
Gg·p = gGpg−1
and for any p ∈ M we can identify the points of M with thequotient space G/Gp.
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Definition 17.23
A Lie subgroup H of a Lie group G is a subset of G such thatthe natural inclusion is an immersion and a grouphomomorphism (i.e. H is simultaneously a subgroup andsubmanifold).
Since the the group action on a homogeneous space is inparticular continuous, the isotropy subgroups are closed in thetopology 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 ofG.
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Theorem 17.26
If G is a Lie group and H a Lie subgroup then the quotientspace 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 Manifolds and Lie Groups”, pp120–124.
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Corollary 17.27
A homogeneous space M acted upon by the Lie group G withisotropy subgroup Gp is diffeomorphic to the quotient manifoldG/Gp where the latter is given the unique smooth structure ofthe previous theorem.
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Lecture 18 continued:
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Example 18.8 ( Real projective space, RPn)
Recall that RPn is the space of lines through the origin in Rn+1.Equivalently,
RPn =Rn+1 \ {0}∼
,
where
(X 0,X 1, . . . ,X n) ∼ λ(X 0,X 1, . . . ,X n), λ ∈ R \ {0}.
We write the equivalence class of (X 0,X 1, . . . ,X n) as[X 0 : X 1 : . . . : X n].
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Example 18.8 (Continued)
The natural projection
Rn+1 \ {0} → RPn
(X 0,X 1, . . . ,X n) 7→ [X 0 : X 1 : . . . : X n]
restricts to Sn to exhibit Sn as a double cover of RPn.That is, we have a diffeomorphism
Sn/ ∼ ∼= RPn.
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Example 18.8 (Continued)
We have
Sn ∼= SO(n + 1)/SO(n) ∼= O(n + 1)/O(n)
Sn is a double cover of RPn
O(n) is a double cover of SO(n).
This suggests trying to show that
RPn is diffeomorphic to SO(n + 1)/O(n).
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Example 18.8 (Continued)
Identify O(n) with its image under the embedding
O(n)→ SO(n + 1)
A 7→(
det AA
).
Using the identification
Sn/ ∼ ∼= RPn
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) .
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Example 18.10 (Complex projective space, CPn)
CPn is the space of complex lines through the origin in Cn+1.Equivalently,
CPn =Cn+1 \ {0}∼
,
where
(X 0,X 1, . . . ,X n) ∼ λ(X 0,X 1, . . . ,X n), λ ∈ C \ {0}.
We write the equivalence class of (X 0,X 1, . . . ,X n) as[X 0 : X 1 : . . . : X n].The proof that CPn is a smooth manifold of real dimension 2n iscompletely analogous to the argument for real projective space.
Exercise 18.11
Show that CPn is a homogeneous manifold diffeomorphic toSU(n + 1)/S(U(1)× U(n)).
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Lecture 19:
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Isometry group
Definition 19.1
A diffeomorphism ϕ : (M,g)→ (N,h) is called an isometry ifϕ∗(h) = g, equivalently if
h(dϕp(v),dϕp(w)) = g(v ,w)
for all p ∈ M, v ,w ∈ TpM.
Definition 19.2
The set of isometries ϕ : (M,g)→ (M,g) forms a group, calledthe isometry group I of the Riemannian manifold (M,g).
The isometry group is always a finite-dimensional Lie groupacting smoothly on M (see eg Kobayashi, TransformationGroups in Differential Geometry, Thm II.1.2).
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Homogeneous and isotropic Riemannian manifolds
Definition 19.3
A Riemannian manifold (M,g) is a homogeneous Riemannianmanifold if its isometry group I(M) acts transitively on M.
A homogeneous space M is just a differentiable manifold, noRiemannian metric. M is diffeomorphic to the quotient manifoldG/Gp.
A homogeneous Riemannian manifold also has a Riemannianmetric compatible with the group action.
A homogeneous Riemannian manifold “looks the same” atevery point in terms both of its smooth structure and of theRiemannian metric.
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Definition 19.4
A Riemannian manifold (M,g) is isotropic at p ∈ M if theisotropy subgroup Ip acts transitively on the set of unit vectorsof TpM, via
Ip × {X ∈ TpM | 〈X ,X 〉 = 1} → {X ∈ TpM | 〈X ,X 〉 = 1}(g,X ) 7→ dgp(X ).
The geometric interpretation is that the Riemannian manifold Mnear p looks the same in all directions.
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Definition 19.5
A homogeneous Riemannian manifold which is isotropic at onepoint must be isotropic at every point. Such a manifold is calleda homogeneous and isotropic Riemannian manifold.
A homogeneous and isotropic Riemannian manifold then looksthe same at every point and in every direction: can you think ofany examples?
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Example 19.6
PropositionThe isometry group of Sn is O(n + 1). It acts transitively on Sn.The isotropy group acts transitively on the unit vectors of thetangent space. Hence, Sn is a homogeneous and isotropicRiemannian manifold.
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Example 19.6 (Continued)
ProofSince O(n + 1) is by definition the isometry group of Rn+1 andSn inherits its Riemannian metric from Rn+1, the action ofSO(n + 1) on Sn is by isometries. Conversely, any isometry ofthe sphere can be extended linearly to give an isometry of Rn+1
(we will check this momentarily). Hence I(Sn) = O(n + 1).
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Example 19.6 (Continued)
The proofs of the next two statements are similar.Choose any point p ∈ Sn. It is unit length, so extend it to anorthonormal basis v1 = p, v2, · · · , vn+1 of Rn+1.The matrix g with columns v1, · · · , vn+1 is in O(n + 1) and takese1 to p. Thus, by going via e1, the action is transitive.To see it is isotropic, we compute at e1 ∈ Sn. We have seenthat the isotropy subgroup is just O(n) acting on the standardbasis e2, · · · ,en+1. Choose any unit vector v ∈ TpSn. We needto find g such that dge1(e2) = v .But the differential of the linear map is itself, so we may replacedge1 with g.Again, extend v to an orthonormal basis and then we may takethe columns of g to be v , v3, · · · , vn+1, similarly to what we haveseen before.
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Example 19.6 (Continued)
To see that every isometry ϕ of a sphere extends to a linearmap, we can write any point p ∈ Rn+1 as λs for λ ∈ R≥0 ands ∈ Sn.Define ϕ̃(p) = λϕ(s). This is well defined (check at p = 0).To see it is linear, take any basis (e0, . . . ,en) of Rn+1. Bychecking inner products, so too is (ϕ(e0), . . . , ϕ(en)). Wecompute
ϕ̃(p) =∑〈ϕ̃(p), ϕ(ei)〉ϕ(ei)
=∑〈λϕ(s), ϕ(ei)〉ϕ(ei)
=∑
λ〈ϕ(s), ϕ(ei)〉ϕ(ei)
=∑
λ〈s,ei〉ϕ(ei) =∑〈p,ei〉ϕ(ei).
Note the right-hand side is a linear function of p. It’s an easycheck that it is an isometry.
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Corollary 19.7
The sphere Sn has constant sectional curvature.
We could just as easily have argued above with the sphere ofradius r > 0, the special orthogonal group similarly actstransitively on it by isometries and furthermore acts transitivelyon the orthonormal frames of Sn(r).
Hence the sphere Sn(r) of radius r > 0 has constant sectionalcurvature too.