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Then the bijection to Uα is defined by
 b1   . 
 bi1−1
 e1  bi1+1
 .  ψα(Λα) = Im . ,
bin −1   en 
bin +1  .
 .  bm+n
Chapter 2.
Global aspects
where e1, ..., en are the unit (row) vectors in Cn. If m = n = 1, then clearly we
may identify G1(C2) or G1(R2) with CP(1) or RP(1), respectively. The so-called
standard chart is U ,ψ−1 where α = {1,2,...,n} and ψ is the embedding α0α0 0 α0
ψα0 =ψ:Cm×n −→Uα0 =:Gn0(Cm+n) In
where clearly by definition of ψ we have ψ(W ) = Im W .
The image Gn0 (Cm+n) = ψα0 (Cm×n) consists of those n-dimensional subspaces
0
which are complementary to the m-dimensional subspace Im I . Notice that
m
Gn0 (Cm+n) is an open and dense subset of Gn(Cm+n) since its complement is the finite union of sets of dimension less than mn. It is also natural to identify Cm×n with Gn0 (Cm×n) and to regard Gn(Cm+n) \ Gn0 (Cm+n) as a hyper plane
of points at infinity which have been added to Cm×n to compactify the space. Define a flow on Gn(Cm+n) given by
S(t, S0, t0) := Φ(t, t0)(S0) where Φ(t,t0) is the transition matrix associated with
M11 (t) M12 (t) M(t) = M21(t) M22(t)
and Φ(t,t0)(S0) denotes the image of the n-dimensional subspace S0 under the non–singular linear transformation Φ(t,t0). Let W(t;W0,t0) denote the flow of the Riccati differential equation RDE
W˙ =M21(t)+M22(t)W−WM11(t)−WM12(t)W
with initial value W(t0;W0,t0) = W0. If we consider t also to be a complex parameter, then the coefficient matrices in RDE have to be holomorphic in