Question: Consider a smooth vector bundle $\pi:V\to B$ and the space $\mathcal{F}$ of $C^k$ linear flows $\Bbb{R}\times V \to V$ endowed with the strong $C^k$ Whitney topology. Is it true that for all linear flows $\Phi$ in an open-dense (or perhaps residual) subset of $\mathcal{F}$, $V$ will split as a $\Phi^t$-invariant Whitney sum $$V = E^1 \oplus \cdots \oplus E^m$$

where each of the $E^i$ is a rank one or two vector subbundle? (And if true, is smoothness necessary -- e.g., can we set $k = 0$?)

Motivation: The constant matrix $A$ defining the linear ODE on $\Bbb{R}^n$:

$\dot{x} = Ax$

generically has distinct eigenvalues, and therefore $\mathbb{R}^n$ splits as a direct sum of one and two dimensional linear subspaces which are invariant under the induced flow.

Similarly, by Floquet theory it follows for a generic $2\pi$-periodic $A(t)$, the trivial bundle $S^1\times \Bbb{R}^n$ over $S^1 = [0,2\pi]/\{0,2\pi\}$ has a $\Phi^t$-invariant splitting into rank one or two subbundles, where $\Phi:\Bbb{R}\times (S^1\times \Bbb{R}^n)$ is the linear flow induced by

$\dot{x} = A(t)x.$


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