Connected components of the space of proper gradient vector fields

Abstract

We show that there exist two proper gradient vector fields on $$\mathbb {R}^n$$ R n which are homotopic in the category of proper maps but not homotopic in the category of proper gradient maps.

The Arnold conjecture in $\mathbb C\mathbb P^n$ and the Conley index

“…To do this, all we need to show is that there exists an isolating neighborhood which is common for every s ∈ [0, 1]. In virtue of [St,Prop 2.4] (see also [StWa,Prop. 2.14]), as in Lemma 7 it suffices to prove the following: if where for notational convenience we set g(z, λ) := g((0, z, 0), λ).…”

The Arnold conjecture in <inline-formula><tex-math id="M1">$ \mathbb C\mathbb P^n $</tex-math></inline-formula> and the Conley index

“…[0,1]. In virtue of[30, Prop 2.4] (see also[31, Prop. 2.14]), as in Lemma 3.2 it suffices to prove the following:if {((y m , z m , w m ), λ m , s m )} ⊂ E ×[λ 0 , λ 0 +1]×[0, 1] is a sequence such that {(T ((y m , z m , w m ), λ m ), g((s m • y m , z m , s m • w m ), λ m ))} ⊂ E × R is bounded, then {(y m , z m , w m )} is bounded in E. We start noticing that T ((y m , z m , w m ), λ m ) 2 = V λ0 y m 2 + | − 1 + 2(λ m − λ 0 )| • z m 2 + w m 2and this immediately implies that {y m } and {w m } are bounded (recall that V λ0 is homogeneous and bounded away from zero on the sphere of radius one).…”