Let $M$ be a manifold and $\phi_t$ be a smooth flow associated to a smooth vector field on $M$.
Are the following 3 conditions equivalent?
1)For every fixed $t$ the diffeomorphism $\phi_t$ is a hyperbolic diffeomorphism.(No continuity assumption on stable and unstable distributions as a grassman valued map in $t$)
2)The classical definition of hyperbolicityThe splitting of the tangent bundle to stable and unstable subspace(The splitting independent of t)
3)The condition $2$ with extra assumption that stable and unstable distributions are invariant under $D\phi_t$
