Riemannian Submersion. • Let M and M be smooth manifolds, and π : M → M is a surjective submersion. f f — For any p ∈ M, the fiber over y, denoted by M , is the inverse map π−1(y) ⊂ M; fy f 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 T M f f x f decomposes into an orthogonal directe sum

T M = H M ⊕ V M, x f x f x f ⊥ where V M = Ker π∗ = T M is the vertical space and H M =(V M) is x f x fπ(x) x f x f the horizontal space. • Any vector field W on M can be written uniquely as W = W H + W V , where f 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 H M is a vector space isomorphism p f between H M and T M by construction. p f π(p) – But both H M and T M are equipped with Euclidean inner products. p f π(p) – We say that π :(M,g) → (M,g)isaRiemannian submersion when ∀q ∈ M, f f this vector space isomorphisme 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∗X = e f eq X , ∀q ∈ M. π(q) f 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 e words, ∀x ∈ M, π∗ is an isometry between H and T M. f x π(x) 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

2 2 2 g = dt + φ (t)dsn−1

2 n−1 where dsn−1 designates the standard metric of the sphere S .

Typeset by AMS-TEX

1 2

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 2 2 2 2 k(v, w)k = f kvkM + kwkN . The projection from (M × N,f2g + h) onto (N,h) is a Riemannian submersion. Proposition 2.38*. Let G be a Lie group acting smoothly M by isometries f θα(p)=α · p. Suppose that e e (1) π(α · p)=π(p) for α ∈ G and p ∈ M, and f (2) G actse transitivelye on each fibere M . fy 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. f Let G be a group of isometriese of (M,g) acting properly and smoothly on M, f f (hence M =∼ M/G is a smooth manifolde and π : M → M/G is a fibration). f f f Then there exists on M =∼ M/G a unique Riemannian metric g such that π is a f Riemannian submersion. −1 Proof. Let p ∈ M and U, V ∈ TpM.Forp ∈ π (p), there exist unique vectors U, e e V ∈ He such that e p π∗U = U and π∗V = V. e e To make π∗ be an isometry between Hpe and TpM, we must set

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

– This metric gp does not depend on the choice of p in the fiber; 0 0 indeed, if π(p)=π(p )=p, there exists α ∈ G esuch that θα(p)=α · p = p . e e e e e Since (θα)∗ is an isometry between Hpe and Hpe0 , we have

gpe(U,V )=gpe0 ((θα)∗U,(θα)∗V ).  e e e e e e 3

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}3w = 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 i n Ui = {[z]:z =06 }⊂CP , i.e. the space of all lines not contained in the complex hyperplane {zi =0}. n — We then obtain a bijection ϕi : Ui → C via

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

Thus CPn becomes a smooth manifold, because, assuming w.l.o.g. i

−1 1 n n j ϕj ◦ ϕi : ϕ(Ui ∩ Uj )={z =(z , ··· ,z ) ∈ C : z =06 }→ϕ(Ui ∩ Uj )

−1 1 n 1 i i+1 n ϕj ◦ ϕi (z , ··· ,z )=ϕ([z , ··· ,z , 1,z , ··· ,z ]) 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. 4

• 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) 7→ [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 x1 x2 ϕ (x1,x2,x3)= ,  1 1 − x3 1 − x3 x1 x2 ϕ (x1,x2,x3)= , , 1 1+x3 1+x3 7→ 1 and the transition map z z . This, however, is nothing but the transition map 7→ 1 1 [1,z] [ z , 1] of CP . 

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.  5

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) 7→  , . |u|2 + |v|2 |u|2 + |v|2

• On S3(1), write the metric as

2 2 2 2 2 dt + sin (t)dθ1 + cos (t)dθ2 ,t∈ [0,π/2],

and use the complex natation,

(t, eiθ1 ,eiθ2 ) 7→ (sin(t)eiθ1 , cos(t)eiθ2 )

to describe the isometric embedding π (0, ) × S1 × S1 ,→ S3(1) ⊂ C2. 2

• Since the Hopf fibers come from complex scalar multiplication, we see that they are of the form θ 7→ (t, ei(θ1+θ),ei(θ1+θ)). • 2 1 On S ( 2 ) use the metric

sin2(2r) π dr2 + dθ2,r∈ [0, ], 4 2 with coordinates 1 1 (r, eiθ) 7→  cos(2r), sin(2r)eiθ. 2 2 • The Hopf fibration in these coordinates, therefore, looks like

(t, eiθ1 ,eiθ2 ) 7→ (t, ei(θ1−θ2)). 6

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

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

where the first vector is tangent to the Hopf fiber and the two other vectors have unit length. • 2 1 On S ( 2 ) 2 ∂ , ∂  r sin(2r) θ is an orthonormal frame. • The Hopf map clearly maps

∂t 7→ ∂r, cos2(t)∂ − sin2(t)∂ cos2(r)∂ + sin2(r)∂ 2 θ1 θ2 7→ θ θ = · ∂ , cos(t) sin(t) cos(r) sin(r) sin(2r) θ

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

iθ1 iθ2 iθ iθ i(θ −θ )  cos(t)e − sin(t)e  (t, e 1 ,e 2 ) 7→ (t, e 1 2 ) 7→ sin(t)e−iθ2 cos(t)e−iθ1

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

• The map (t, eiθ1 ,eiθ2 ) 7→ (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 2 2 2 2 2 dt + ϕ (t)dθ1 + ψ (t)dθ2 and the target has the rotationally symmetric metric

(ϕ(t) · ψ(t))2 dr2 + 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

2 2 2 2 2 dt + ϕ (t)ds2n+1 + ψ (t)dθ .

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) 7

can be made into a Riemannian submersion by choosing suitable metric on the quotient space. — To find the metric, we split the canonical metric 2 ds2n+1 = h + g, where h corresponds to the metric along the Hopf fiber and g the orthogonal complement. 2n+1 2n+1 V – In other words, if π : TpS → (TpS ) is the orthogonal projection (with 2 b respect to ds2n+1) whose image is the distribution generated by the Hopf action, then 2 h(v, w)=ds2n+1(π∗v, π∗w) b b and 2 g(v, w)=ds2n+1(v − π∗v, w − π∗w). b b — We can then define 2 2 2 2 2 2 2 2 2 2 dt + ϕ (t)ds2n+1 + ψ (t)dθ = dt + ϕ (t)g + ϕ (t)h + ψ (t)dθ . 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 (ϕ(t) · ψ(t))2 ds2 + ϕ2(t)g + h. ϕ2(t)+ψ2(t) (i) In the case when n = 0, we recapture the previous case, as g does not appear. 2 (ii) When n = 1, the decomposition ds3 = h + g can also be written 2 1 2 2 2 3 2 ds3 =(σ ) +(σ ) +(σ ) , 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)   y 2 × 3 × 1 2 2 2 2 3 2 (ϕ(t)·ψ(t)) 1 2 (I S S ,dt + ϕ (t)[(σ ) +(σ ) ]+ ϕ2(t)+ψ2(t) (σ ) ). (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 π π (0, ) × (S2n+1 × S1) → (0, ) × (S2n+1 × S1/S1) 2 2 as a Riemannian submersion, and the Fubini-Study metric on CPn+1 can be represented as dt2 + sin2(t)(g + cos2(t)h).