11 EXAMPLES 75

PROBLEMS

1. (a) For any two points x, y e S2n-i c C", put

«(*> y) = arc CQS(^(JC \y))

which is a real number between 0 and TT. If s, t are any two elements of the unitary
group U(X), put

d(s, t) = sup a(> • x, f • x).

xeS2n-l

Show that d is a bi-invariant distance on U(X).

(b) For s e U(ri), let ei6j (1 ^j^m) be the distinct eigenvalues of ,y, so that C" is the
Hilbert sum of the eigenspaces V/ of s (1 ^7 <^ m), the restriction of s to V/ being the
homothety with ratio et9j; we may assume that — TT < 0, <; TT for each y. Show that if
6(s) = sup 1 0/ 1 , then d(e, s) = 6(s). (Minorize 3$(x \ s • x) by using the decomposition

of * as a sum of vectors Xj e V/ .)

(c) Let ,y, / be two elements of U(ri) such that s and the commutator ($, /) = sts"1*"1
commute, or equivalently such that s and u = tst~l commute. Show that if 6(t) < JTT,
then s and t commute. (With the notation of (b), observe that if Vj is the orthogonal
supplement of V/ , and if W, = r( V/), then W/ is the direct sum of W, n V/ and W/ n Vj ,
and deduce from the hypothesis on / that W/ n V/ = {0}.)

2. When n — 2, the two charts c^ , <p2 on S2 defined in (16.2.3) are such that <pi and <p2
define on S2 the structure of complex analytic manifold defined in (16.11.12).

3. Let U be an open neighborhood of 0 in R". In the real-analytic manifold U x Pn_j(R),
let IT be the subset consisting of points (x, z) such that for some system (r1, . . . , z") of
homogeneous coordinates for z we have xjzk — xkzj = 0 for all pairs of indices j, k (in
which case these relations are satisfied for all systems of homogeneous coordinates for
z). Show that U' is a closed analytic submanifold of dimension n in U X PM_j(R)
(consider the atlas of Pn_i(R) defined in (16.11.10)). The restriction TTXJ of the projection
pri to U' is a surjection of U' onto U and is proper; ^'(O) is a submanifold of U' iso-
morphic to P^^R), and the restriction of TT^ to U7 — ^'(0) is an isomorphism of this
open set onto U — {0}. Let r be the inverse isomorphism of U — {0} onto U' — Tr^O),
and let / be any C1 -function defined on an open neighborhood I of 0 in R and with
values in U, such that/(0) = 0 and T0(/) ^ 0. Then the function /H*r(/(/)), defined on
I — {0} and with values in U', extends by continuity to a mapping/' : I -> U', such that
/'(O) is the canonical image in Pn.i(R) of the vector T0(/) • 1. Furthermore, if /is of
class Cr, then/' is of class C""1 ; and if/, g are two Cr-functions defined on I, such that
/(O) = #(0) = 0, which have contact of order >k «> 1 at the point 0, then/' and g' have
contact of order ^k ~~ 1 at the point 0 (Section 16.5, Problem 9).

Jf V is another open neighborhood of 0 in Rn and if u : U -> V is an isomorphism
of analytic manifolds (resp. a diffeomorphism), then if V and TTV are defined as above,
there exists a unique isomorphism u' : U'-* V of analytic manifolds (resp. a unique
diffeomorphism) such that TTV ° u' = u ° TTU .

Deduce that if X is a pure differential (resp. analytic) manifold of dimension n, and
x a point of X, there exists a differential (resp, analytic) manifold X' of dimension npen ball \\y — f(x0)\\ < a; and let