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2.4. Flows on Graßmann manifolds 33
a neighborhood of t0. Then the solution W(t,W0,t0) is holomorphic in some neighborhood of t0.
Since we shall mostly deal with constant coefficients then, by Theorem 2.3.1 and Remark 2.3.2, we know that the solutions are meromorphic in C. Clearly, as it was pointed out in the proof to Theorem 2.2.1, this also holds true in case of holomorphic in C coefficient matrices Mij (t), 1 ≤ i, j ≤ 2. From Theorem 2.2.1, iii) we get that the representation (2.9) still remains valid in case of a complex t, hence
W (t; W0, t0) = P (t)Q−1(t)
as long as Q−1(t) exists, where P and Q are defined as in Theorem 2.2.1. Using
this formula, we have with W = P Q−1 000
Thus,
ψ(W (t; W0, t0))
= Im
In P (t)Q−1(t)
= Im
Q(t) P (t)
In = Im(Φ(t,t0) P ) = Φ(t,t0)Im W .
ψ(W (t; W0, t0)) = S(t, ψ(W0), t0).
(2.28)
Equation (2.28) is valid as long as Q−1(t) exists; i.e., as long as the solution W (t; W0 , t0 ) continues to exist. This is equivalent to the tra jectory S(t, ψ(W0), t0) remaining in the subset Gn0 (Cm+n) of Gn(Cm+n), the standard chart.
Here we make the following important observation: If a solution of RDE is leaving the standard chart, it can be continued continuously to the whole (compact) Graßmann manifold by the flow S(t, S0, t0). Since, for instance, the transition matrix Φ(t,t0) associated with a constant matrix M in the system L is holomorphic in C, it makes sense to investigate W(t;W0,t0) as t → ∞ even if W (t; W0, t0) has a finite escape time. If limt→∞ W (t; W0, t0) exists in Gn(Cm+n), i.e., limt→∞ S(t; ψ(W0), t0) exists, then it can be either in the standard chart Gn0 (Cm+n) or in its complement. If limt→∞ S(t; ψ(W0), t0) ∈ Gn0 (Cm+n), then we speak of the extended limit of the solution W (t; W0, t0) of RDE.
Equation (2.28) now can be viewed as a geometric version of Radon’s for- mula (2.9). The content of (2.28) is that the embedding ψ (which identifies Cm×n with Gn0 (Cm+n)) maps the trajectories of RDE onto the trajectories of the restriction of the flow S(t, ψ(W0), t0) to Gn0 (Cm+n). Thus, modulo the iden- tification ψ, the solutions of the Riccati differential equation can be regarded as the flow S(t, ψ(W0), t0) restricted to the open and dense subset Gn0 (Cm+n). Actually, as already seen above, the complement of Gn0 (Cm+n) in Gn(Cm+n) is
Q0 00