Page 51 - Textos de Matemática Vol. 36

2.4. Flows on Graßmann manifolds 41 Theorem 2.4.4. If
i) #J(ak) = 0, then Γk = ∅, or if
ii) #J(ak) = 1, then Γk = {Wk}, where Wk is given by (2.34), or if
iii) #J(ak) = κ > 1, then Γk is uncountable and several possibilities for the structure of Γk result.
α) If there exists C ∈ G0ζ(ak) with J0(C) = J(ak), then Γk is a κ − 1 parametric family of solutions of ARE of the form
v (ak)+ v (aj)D(aj,C)
wk = lα j∈J0(C)\{k} lα D(ak,C) , (2.35)
lα v˜(ak)+ v˜(aj)D(aj,C) j∈J0(C)\{k} D(ak,C)
with 1≤l≤m, 1≤α≤n.
β) Otherwise, we can split Γk into nk ≥ 2 subsets
where Γkj
nk
Γk = Γkj
j=1
nk
is a pkj -parametric set and pkj ≤ κ − 1.
j=1
Proof. (i) is clear by definition and mentioned here just for completeness. If J(ak) contains exactly one index it must be k and from (2.33) we infer (2.34) as before. If J(ak) contains two or more indices in case α) we obtain (2.35). The pa-
rameters appearing in (2.35) are the κ−1 determinants D(aj ,C) , j ∈ J (C)\{k}. D(ak,C) 0
To make clear that Γk depends on exactly κ − 1 parameters we need to know additionally that for each j ∈ J0(C) \ {k} at least one of the determinants v˜(aj ) or vlα(aj),1 ≤ l ≤ m,1 ≤ α ≤ n does not vanish. But, if all of them were zero, then the eigenvectors vν1j , vν2j , . . . , vνnj would be linearly dependent.
In case β) we start with a matrix C1 ∈ G0ζ(ak) and assume that #J0(C1) = pk1 + 1 ≥ 1 leading in the same way as before to a pk1 -parametric set of solutions Γk1. These solutions are connected with the set of C-matrices
1 k −1 In
Gζ(a ) = {C|C = V W ,W ∈ Γk1} by their initial data via (2.12) at
−1 In
t = 0, i.e., via C1 = V W . Then, we proceed inductively by choosing
C2 ∈ G0ζ(ak)\G1ζ(ak) and so on until step nk. The procedure terminates in step
nk if
G1 (ak ) ∪ G2 (ak ) ∪ · · · ∪ Gnk (ak ) = G0 (ak ), ζζζζ