Nonlinear Equations - UFRJ

[SEC. 4.5: PROJECTIVE SPACE 53
The integral on the left is just
(∫
1
√ e −x dx 2π
R
) k+1
and from the case k = 1, we can infer that it is equal to 1.
proposition then follows for all k.
The
Proof of Proposition 4.10. Let S 2n+1 ⊂ C n+1 be the unit sphere
|z| = 1. The Hopf fibration is the natural projection of S 2n+1 onto
P n . The preimage of any (z 0 : · · · : z n ) is always a great circle in
S 2n+1 .
We claim that
Vol(P n ) = 1
2π Vol(S2n+1 ).
Since we know that the right-hand-term is π n /n!, this will prove
the Proposition.
The unitary group U(n + 1) acts on C n+1
≠0
by Q, x ↦→ Qx. This
induces transitive actions in P n and S 2n+1 . Moreover, if ‖x‖ = 1,
H(Qx) = Q(x 0 : · · · : x n )
so DH Qx = QDH x . It follows that the Normal Jacobian det(DHDH ∗ )
is invariant by U(n + 1)-action, and we may compute it at a single
point, say at e 0 . Recall our convention z i = x i + √ −1 y i . The tangent
space T e0 S n has coordinates y 0 , x 1 , y 1 , . . . , y n while the tangent space
T (1:0:···:0) P n has coordinates x 1 , y 1 , . . . , y n . With those coordinates,
⎡
⎤
DH(e 0 ) =
⎢
⎣
0 1
. ..
⎥
⎦
1
(white spaces are zeros). Thus DH(e 0 ) DH(e 0 ) ∗ is the identity.
The co-area formula (Theorem 4.7) now reads:
∫
VolS 2n+1 = dS 2n+1
S
∫
2n+1 ∫
= dP n (x) | det(DH(y) DH ∗ (y))| −1 dS 1 (y)
P n H −1 (x)
= 2πVol(P n )