Page 40 - Textos de Matemática Vol. 36

30 Chapter 2. Global aspects with (2.21)
0=m21 +(m22 −m11)w−m12w2.
In Example 2.3.6 we did not see such a behavior, but, if we interpret the equation in the complex domain, then, by asymptotic considerations, we obtain exactly the same situation. Therefore, in order to get the full information, it is sometimes necessary to deal also with this complex extension.
Taking from (2.24) the complex solutions we obtain (for also complex t) the meromorphic function
p(t) c1eibt + c2e−ibt
w(t) = q(t) = ic1eibt − c2e−ibt (2.27)
which is the representation (2.14) of the solutions of the Riccati differential equation (2.26) with, in case b > 0,
w1 = lim w(t)=−i, w2 = lim w(t)=i. I(t)→∞ I(t)→−∞
If b < 0, then w1 and w2 interchange their roles. These are again the solutions of the algebraic Riccati equation
0 = 1 + w2
and both w1,w2 are also contained in the representation (2.27) by choosing c1 =0orc2 =0.
One more comment on the limits is necessary: As meromorphic functions can also be considered as continuous (in the chordal metric) functions on the Riemann sphere, we also can interpret the solutions w(t) of the Riccati differen- tial equation (2.26) by their continuous extension to the Riemann sphere. The limits above then have to be considered in the chordal metric.
But, according to the real case we also may investigate the flow on the complex projective space CP(1) which is the space of points
[q,p]={(z1,z2)∈C2 |qz2 =pz1, q,p∈C, |p|2 +|q|2 ̸=0} where C once again may be embedded via
ψ : C→CP(1)
w → ψ(w) = [1, w].
Hereanalogouslyψ(C)=CP(1)\[0,1]andψ−1([q,p])=ψ−1([1,p])=p iden- qq
tifies C with CP(1) \ [0, 1]. It can be shown that CP(1) is homeomorphic to the Riemann sphere.