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1 The Complex Projective Space Definition. Complex projective n-space, denoted by CP n, is defined to be the set of 1-dimensional complex-linear subspaces of C n+1, with the quotient topology inherited from the natural projection π : C n+1 \{0} CP n. Definition*. A complex linear subspace of C n+1 of complex dimension one is called line. Define the complex projective space CP n as the space of all lines in C n+1. Thus, CP n is the quotient of C n+1 \{0} by the equivalence relation z w. λ C \{0} w = λz. Namely, two points of C n+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 =(z 0,,z n ) C n+1 and define U i = {[z] :z i 0} CP n, i.e. the space of all lines not contained in the complex hyperplane {z i =0}. We then obtain a bijection ϕ i : U i C n via ( ) z ϕ i ([z 0,,z n 0 zi 1 ]) :=,, zi z i, zi+1 z i,, zn z i. Thus CP n becomes a smooth manifold, because, assuming w.l.o.g. i<j, the transition maps ϕ j ϕ 1 i : ϕ(u i U j )={z =(z 1,,z n ) C n : z j 0} ϕ(u i U j ) ϕ j ϕ 1 i (z 1,,z n )=ϕ([z 1,,z i, 1,z i+1,,z n ]) ( z 1 zi =,, zj z j, 1 ) z j, zi+1 z j,, zj 1 z j, zj+1 z j,, zn z j are diffeomorphisms. The vector space structure of C n+1 induce an analogous structure on CP n by homogenization: Each linear inclusion C m+1 C n+1 induces an inclusion CP m CP n. The image of such an inclusion is called linear subspace. The image of a hyperplane in C n+1 is again called hyperplane, and the image of a two-dimensional space C 2 is called line. Typeset by AMS-TEX 1

2 2 Instead of considering CP n as a quotient of C n+1 \{0}, we may also view it as a compactification of C n. One says that the hyperplane H at infinity is added to C n ; this means the following: the inclusion C n CP n is given by (z 1,,z n ) [1,z 1,,z n ]. Then CP n \ C n = {[z] =[0,z 1,,z n ]} =: H, where H is a hyperplane CP n 1. It follows that (1) CP n = C n CP n 1 = C n C n 1 C 0. Proposition. CP 1 is diffeomorphic to S 2. Proof. It follows from (1) that the two spaces are homeomorphic. In order to see that they are diffeomorphic, we recall that S 2 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 ( x ϕ 1 (x 1,x 2,x 3 1 )= 1 x 3, x 2 ) 1 x 3 ( x ϕ 1 (x 1,x 2,x 3 1 )= 1+x 3, x 2 ) 1+x 3, and the transition map z 1 z. This, however, is nothing but the transition map [1,z] [ 1 z, 1] of CP1. Proposition. The quotient map π : C n+1 \{0} CP n is smooth. The restriction of π to S 2n+1 is a surjective submersion. Define an action of S 1 on S n+1 by z (w 1,,w n+1 )=(zw 1,,zw n+1 ). This action is smooth, free and proper. Thus, we have the following. Proposition. CP n = S 2n+1 /S 1. Each line in C n+1 intersects S 2n+1 in a circle S 1, and we obtain the point of CP n defined by this line by identifying all points on S 1. Proposition. CP n can be uniquely given the structure of smooth, compact, real 2n-dimensional manifold on which the Lie group U(n +1) acts smoothly and transitively. In other words, CP n is a homogeneous U(n +1)-space. Proof. The unitary group U(n+1) acts on C n+1 and transforms complex subspaces into complex subspaces, in particular lines to lines. Therefore, U(n + 1) acts on CP n.

3 Proposition. The round metric on S 2n+1 decends to a homogeneous and isotropic Riemannian metric on CP n+1, called the Fubini-Study metric. The projection π : S 2n+1 CP n is called Hopf map. In particular, since CP 1 = S 2, we obtain a map with fiber S 1. Hopf Fibration We have the smooth map H :(u, v) π : S 3 S 2 H : C 2 \{0} S 2 ( v 2 u 2 ) u 2 + v 2, 2uv u 2 + v 2. 3 On S 3 (1), write the metric as dt 2 + sin 2 (t)dθ cos 2 (t)dθ 2 2, t [0,π/2], and use the complex notation, (t, e iθ1,e iθ2 ) (sin(t)e iθ1, cos(t)e iθ2 ) to describe the isometric embedding (0, π 2 ) S1 S 1 S 3 (1) C 2. Since the Hopf fibers come from complex scalar multiplication, we see that they are of the form θ (t, e i(θ1+θ),e i(θ1+θ) ). On S 2 ( 1 2 ) use the metric dr 2 + sin2 (2r) dθ 2, r [0, π 4 2 ], with coordinates ( 1 (r, e iθ ) 2 cos(2r), 1 ) 2 sin(2r)eiθ. The Hopf fibration in these coordinates, therefore, looks like (t, e iθ1,e iθ2 ) (t, e i(θ1 θ2) ).

4 4 Now on S 3 (1) we have an orthognal frame { θ1 + θ2, t, cos 2 (t) θ1 sin 2 (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 }, cos 2 (t) θ1 sin 2 (t) θ2 cos(t) sin(t) t r, cos2 (r) θ + sin 2 (r) θ cos(r) sin(r) = 2 sin(2r) θ, thus showing that it is an isometry on vectors perpendicular to the fiber. Note that the map ( ) (t, e iθ1,e iθ2 ) (t, e i(θ1 θ2) cos(t)e iθ 1 sin(t)e ) iθ2 sin(t)e iθ2 cos(t)e iθ1 gives us the isometry from S 3 (1) to SU(2). The map (t, e iθ1,e iθ2 ) (t, e i(θ1 θ2) ) from I S 1 S 1 to I S 1 is actually always a Riemannian submersion when the domain is endowed with the doubly warped product metric dt 2 + ϕ 2 (t)dθ ψ2 (t)dθ 2 2 and the target has the rotationally symmetric metric dr 2 (ϕ(t) ψ(t))2 + ϕ 2 (t)+ψ 2 (t) dθ2.. This submersion can be generalized to higher dimensions as follows. On I S 2n+2 S 1 consider the doubly warped product metric dt 2 + ϕ 2 (t)ds 2 2n+1 + ψ 2 (t)dθ 2. The unit circle acts by complex scalar multiplication on both S 2n+1 and S 1, and consequently induces a free isometric action on the space: if λ S 1 and (z,w) S 2n+1 S 1, then λ (z,w) =(λz, λw). The quotient map I S 2n+1 S 1 I ((S 2n+1 S 1 )/S 1 )

5 can be made into a Riemannian submersion by choosing suitable metric on the quotient space. To find the metric, we split the canonical metric ds 2 2n+1 = h + g, where h corresponds to the metric along the Hopf fiber and g the orthogonal complement. In other words, if π : T p S 2n+1 (T p S 2n+1 ) V is the orthogonal projection (with respect to ds 2 2n+1 ) whose image is the distribution generated by the Hopf action, then h(v, w) =ds 2 2n+1( π v, π w) and g(v, w) =ds 2 2n+1(v π v, w π w). We can then define dt 2 + ϕ 2 (t)ds 2 2n+1 + ψ 2 (t)dθ 2 = dt 2 + ϕ 2 (t)g + ϕ 2 (t)h + ψ 2 (t)dθ 2. Now notice that (S 2n+1 S 1 )/S 1 = S 2n+1 and that S 1 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 S 2n+1 can be written ds 2 + ϕ 2 (ϕ(t) ψ(t))2 (t)g + ϕ 2 (t)+ψ 2 (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 ds 2 3 = h + g can also be written ds 2 3 =(σ 1 ) 2 +(σ 2 ) 2 +(σ 3 ) 2, where (σ 1 ) 2 = h, (σ 2 ) 2 +(σ 3 ) 2 = g, and {σ 1,σ 2,σ 3 } is the coframing coming from the identification S 3 = SU(2). The Riemannian submersion in this case can therefore be written (I S 3 S 1,dt 2 + ϕ 2 (t)[(σ 1 ) 2 +(σ 2 ) 2 +(σ 3 ) 2 ]+ψ 2 (t)dθ 2 ) (I S 3 S 1,dt 2 + ϕ 2 (t)[(σ 2 ) 2 +(σ 3 ) 2 ]+ (ϕ(t) ψ(t))2 ϕ 2 (t)+ψ 2 (t) (σ1 ) 2 ). (iii) If we let ϕ = sin(t), ψ = cos(t) and t I = [0,π/2], then we obtain the generalized Hopf fibration S 2n+3 CP n+1 defined by (0, π 2 ) (S2n+1 S 1 ) (0, π 2 ) (S2n+1 S 1 /S 1 ) as a Riemannian submersion, and the Fubini-Study metric on CP n+1 can be represented as dt 2 + sin 2 (t)(g + cos 2 (t)h). 5