Riemannian Submersion.

Riemannian Submersion.

• Let M and M be smooth manifolds, and π : M → M is a surjective submersion.
— For any p ∈ M , the fiber over y, denoted by My, is the inverse map π−1(y) ⊂ M ;

it is a closed, embedded submanifold by the implicit function theorem.

• If M has a Riemannian metric g, at each point x ∈ M the tangent space TxM
decomposes into an orthogonal direct sum

TxM = HxM ⊕ VxM ,

where VxM = Ker π∗ = TxMπ(x) is the vertical space and HxM = (VxM )⊥ is
the horizontal space.
• Any vector field W on M can be written uniquely as W = W H + W V , where
W H is horizontal, W V is vertical, and both W H and W V are smooth.

• The differential dπ = π∗ of π restricted to HpM is a vector space isomorphism
between HpM and Tπ(p)M by construction.

– But both HpM and Tπ(p)M are equipped with Euclidean inner products.
– We say that π : (M , g) → (M, g) is a Riemannian submersion when ∀q ∈ M ,

this vector space isomorphism is a Euclidean isomorphism.

Lemma. If X ia a vector field on M , there is a unique smooth horizontal vector
field X on M, called the horizontal lift of X, that is π-related to X; i.e. π∗Xq =
Xπ(q), ∀q ∈ M .

Definition. If g is a Riemannian metric on M , π is said to be a Riemannian
submersion if g(X, Y ) = g(π∗X, π∗Y ) whenever X and Y are horizontal; in other
words, ∀x ∈ M , π∗ is an isometry between Hx and Tπ(x)M .

Example. When the fiber is discrete (say of dimension 0), then we have a Rie-
mannian covering.
Example. The projections from a Riemannian product onto the factor spaces are
Riemannian submersions.
Example. A surface of revolution provides a Riemannian submersion:

the submersion goes from the surface to any meridian.
— One can define objects of revolution in any dimension n; they have the spherical

symmetry given by an action of the orthogonal group O(n − 1), namely

M = I × Sn−1

where I is ant interval of the real line.
– The base is still one dimensional.
– The metric can be written

g = dt2 + φ2(t)dsn2 −1

where ds2n−1 designates the standard metric of the sphere Sn−1.

Typeset by AMS-TEX

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Example. In Sakai 1996 a warped product is defined on any Riemannian product

manifold (M × N, g × h) by a function f : N → R forcing the modification of the

metric g × h into

(v, w) 2 = f2 v 2 + w 2 .
M N

The projection from (M × N, f 2g + h) onto (N, h) is a Riemannian submersion.

Proposition 2.38*. Let G be a Lie group acting smoothly M by isometries
θα(p) = α · p. Suppose that

(1) π(α · p) = π(p) for α ∈ G and p ∈ M, and
(2) G acts transitively on each fiber My.
Then there is a unique Riemannian metric g on M such that π is a Riemannian
submersion.

Proposition 2.38. Let (M , g) be a Riemannian manifold.
Let G be a group of isometries of (M , g) acting properly and smoothly on M ,
(hence M ∼= M /G is a smooth manifold and π : M → M/G is a fibration).
Then there exists on M ∼= M /G a unique Riemannian metric g such that π is a
Riemannian submersion.

Proof. Let p ∈ M and U, V ∈ TpM . For p ∈ π−1(p), there exist unique vectors U ,
V ∈ Hp such that

π∗U = U and π∗V = V.

To make π∗ be an isometry between Hp and TpM , we must set

gp(U, V ) = gp(U , V ).

– This metric gp does not depend on the choice of p in the fiber;
indeed, if π(p) = π(p ) = p, there exists α ∈ G such that θα(p) = α · p = p .
Since (θα)∗ is an isometry between Hp and Hp , we have

gp(U , V ) = gp ((θα)∗U , (θα)∗V ).

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The Complex Projective Space
Definition. Complex projective n-space, denoted by CPn, is defined to be the
set of 1-dimensional complex-linear subspaces of Cn+1, with the quotient topology
inherited from the natural projection

π : Cn+1 \ {0} → CP n.

Definition*. A complex linear subspace of Cn+1 of complex dimension one is
called line. Define the complex projective space CPn as the space of all lines in
Cn+1.

• Thus, CPn is the quotient of Cn+1 \ {0} by the equivalence relation

z ∼ w. ⇔ ∃λ ∈ C \ {0} w = λz.

Namely, two points of Cn+1 \ {0} are equivalent iff they are complex linearly
dependent, i.e. lie on the same line.
Denote the equivalence class of z by [z].

We also write

z = (z0, · · · , zn) ∈ Cn+1

and define

Ui = {[z] : zi = 0} ⊂ CPn,

i.e. the space of all lines not contained in the complex hyperplane {zi = 0}.
— We then obtain a bijection ϕi : Ui → Cn via

ϕi([z0, · · · , zn]) := z0 zi−1 zi+1 zn
zi , · · · , zi , zi , · · · , zi .

Thus CPn becomes a smooth manifold, because, assuming w.l.o.g. i < j, the
transition maps

ϕj ◦ ϕ−i 1 : ϕ(Ui ∩ Uj ) = {z = (z1, · · · , zn) ∈ Cn : zj = 0} → ϕ(Ui ∩ Uj )

ϕj ◦ ϕi−1(z1, · · · , zn) =ϕ([z1, · · · , zi, 1, zi+1, · · · , zn])
z1 zi 1 zi+1 zj−1 zj+1 zn

= zj , · · · , zj , zj , zj , · · · , zj , zj , · · · , zj

are diffeomorphisms.

• The vector space structure of Cn+1 induce an analogous structure on CPn by
homogenization:

– Each linear inclusion Cm+1 ⊂ Cn+1 induces an inclusion CPm ⊂ CPn.
The image of such an inclusion is called linear subspace.

– The image of a hyperplane in Cn+1 is again called hyperplane,
and the image of a two-dimensional space C2 is called line.

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• Instead of considering CPn as a quotient of Cn+1 \ {0}, we may also view it as

a compactification of Cn.

— One says that the hyperplane H at infinity is added to Cn; this means the

following: the inclusion

Cn → CPn

is given by

(z1, · · · , zn) → [1, z1, · · · , zn].

Then

CPn \ Cn = {[z] = [0, z1, · · · , zn]} =: H,

where H is a hyperplane CPn−1. It follows that

(1) CPn = Cn ∪ CPn−1 = Cn ∪ Cn−1 ∪ · · · ∪ C0.

Proposition. CP 1 is diffeomorphic to S2.

Proof. It follows from (1) that the two spaces are homeomorphic.
In order to see that they are diffeomorphic, we recall that S2 can be described via

stereographic projection from the north pole (0, 0, 1) and the south pole (0, 0, −1)
by two charts with image C, namely

ϕ1(x1, x2, x3) = x1 x2 ,
ϕ1(x1, x2, x3) = 1 − x3 , 1 − x3

x1 x2
1 + x3 , 1 + x3

and the transition map z → 1 . This, however, is nothing but the transition map
z
1 CP1.
[1, z] → [ z , 1] of

Proposition. The quotient map π : Cn+1 \{0} → CP n is smooth. The restriction
of π to S2n+1 is a surjective submersion.

Define an action of S1 on Sn+1 by
z · (w1, · · · , wn+1) = (zw1, · · · , zwn+1).

This action is smooth, free and proper. Thus, we have the following.

Proposition. CPn ∼= S2n+1/S1.

Each line in Cn+1 intersects S2n+1 in a circle S1, and we obtain the point of CPn
defined by this line by identifying all points on S1.

Proposition. CPn can be uniquely given the structure of smooth, compact, real
2n-dimensional manifold on which the Lie group U (n + 1) acts smoothly and tran-
sitively. In other words, CPn is a homogeneous U (n + 1)-space.

Proof. The unitary group U (n+1) acts on Cn+1 and transforms complex subspaces
into complex subspaces, in particular lines to lines. Therefore, U (n + 1) acts on
CPn.

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Proposition. The round metric on S2n+1 decends to a homogeneous and isotropic
Riemannian metric on CPn+1, called the Fubini-Study metric.

• The projection

π : S2n+1 → CPn

is called Hopf map. In particular, since CP1 = S2, we obtain a map

π : S3 → S2

with fiber S1.

Hopf Fibration

We have the smooth map

H : C2 \ {0} → S2

2uv |u|2 − |v|2
H : (u, v) → |u|2 + |v|2 , |u|2 + |v|2 .

• On S3(1), write the metric as
dt2 + sin2(t)dθ12 + cos2(t)dθ22, t ∈ [0, π/2],

and use the complex natation,
(t, eiθ1 , eiθ2 ) → (sin(t)eiθ1 , cos(t)eiθ2 )

to describe the isometric embedding

(0, π × S1 × S1 → S3(1) ⊂ C2.
)
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• Since the Hopf fibers come from complex scalar multiplication, we see that they

are of the form

θ → (t, ei(θ1+θ), ei(θ1+θ)).

• On S 2( 1 ) use the metric
2

dr2 + sin2(2r) dθ2, π
r ∈ [0, ],
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with coordinates

(r, eiθ) → 1 cos(2r), 1 sin(2r)eiθ .
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• The Hopf fibration in these coordinates, therefore, looks like

(t, eiθ1 , eiθ2 ) → (t, ei(θ1−θ2)).

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• Now on S3(1) we have an orthognal frame

∂θ1 + ∂θ2 , ∂t, cos2(t)∂θ1 − sin2(t)∂θ2 ,
cos(t) sin(t)

where the first vector is tangent to the Hopf fiber and the two other vectors have

unit length.

• On S 2( 1 )
2

2
∂r, sin(2r) ∂θ

is an orthonormal frame.
• The Hopf map clearly maps

∂t → ∂r,

cos2(t)∂θ1 − sin2(t)∂θ2 → cos2(r)∂θ + sin2(r)∂θ = 2 · ∂θ,
cos(t) sin(t) cos(r) sin(r) sin(2r)

thus showing that it is an isometry on vectors perpendicular to the fiber.
• Note that the map

(t, eiθ1 , eiθ2 ) → (t, ei(θ1−θ2)) → cos(t)eiθ1 − sin(t)eiθ2
sin(t)e−iθ2 cos(t)e−iθ1

gives us the isometry from S3(1) to SU(2).

• The map (t, eiθ1 , eiθ2 ) → (t, ei(θ1−θ2)) from I × S1 × S1 to I × S1 is actually always

a Riemannian submersion when the domain is endowed with the doubly warped

product metric

dt2 + ϕ2(t)dθ12 + ψ2(t)dθ22

and the target has the rotationally symmetric metric

dr2 + (ϕ(t) · ψ(t))2 dθ2 .
ϕ2(t) + ψ2(t)

.

• This submersion can be generalized to higher dimensions as follows.
On I × S2n+2 × S1 consider the doubly warped product metric

dt2 + ϕ2(t)ds22n+1 + ψ2(t)dθ2.

The unit circle acts by complex scalar multiplication on both S2n+1 and S1,
and consequently induces a free isometric action on the space: if λ ∈ S1 and
(z, w) ∈ S2n+1 × S1, then λ · (z, w) = (λz, λw).
The quotient map

I × S2n+1 × S1 → I × ((S2n+1 × S1)/S1)

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can be made into a Riemannian submersion by choosing suitable metric on the
quotient space.
— To find the metric, we split the canonical metric

ds22n+1 = h + g,

where h corresponds to the metric along the Hopf fiber and g the orthogonal

complement.
– In other words, if π : TpS2n+1 → (TpS2n+1)V is the orthogonal projection (with

respect to ds22n+1) whose image is the distribution generated by the Hopf action,
then

h(v, w) = ds22n+1(π∗v, π∗w)

and
g(v, w) = ds22n+1(v − π∗v, w − π∗w).

— We can then define

dt2 + ϕ2(t)ds22n+1 + ψ2(t)dθ2 = dt2 + ϕ2(t)g + ϕ2(t)h + ψ2(t)dθ2.

Now notice that

(S2n+1 × S1)/S1 = S2n+1

and that S1 only collapses the Hopf fiber while leaving the orthogonal component

to the Hopf fiber unchanged.
In analogy with the above example, we therefore obtain that the metric on
I × S2n+1 can be written

ds2 + ϕ2(t)g + (ϕ(t) · ψ(t))2
ϕ2(t) + ψ2(t) h.

(i) In the case when n = 0, we recapture the previous case, as g does not appear.

(ii) When n = 1, the decomposition ds23 = h + g can also be written
ds32 = (σ1)2 + (σ2)2 + (σ3)2,

where (σ1)2 = h, (σ2)2 + (σ3)2 = g, and {σ1, σ2, σ3} is the coframing coming
from the identification S3 ∼= SU (2).
– The Riemannian submersion in this case can therefore be written

(I × S3 × S1, dt2 + ϕ2(t)[(σ1)2 + (σ2)2 + (σ3)2] + ψ2(t)dθ2)


(I × S3 × S1, dt2 + ϕ2(t)[(σ2)2 + (σ3)2] + (ϕ(t)·ψ(t))2 (σ 1 )2 ).
ϕ2 (t)+ψ 2 (t)

(iii) If we let ϕ = sin(t), ψ = cos(t) and t ∈ I = [0, π/2], then we obtain the

generalized Hopf fibration

S2n+3 → CPn+1

defined by (0, π ) × (S2n+1 × S1) → (0, π ) × (S2n+1 × S1/S1)
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as a Riemannian submersion, and the Fubini-Study metric on CPn+1 can be

represented as

dt2 + sin2(t)(g + cos2(t)h).