Regular Level Sets of Averages of Nemytskiı Operators Are Contractible

LetH1(S1) be the space of periodic real functions with derivative inL2andf:R→R be a smooth function with no double roots. Then there is a diffeomorphism ofH1(S1) taking the setZ={v∈H1(S1)∣∫S1 f(v(t)) dt=0} to a hyperplane. In this paper we state and prove a general version of this example. We consider a Banach spaceVof functions from some manifoldMtoRn′and a functionf:M×Rn′→Rn: under suitable hypothesis, there is a homeomorphism ofVtakingZ={v∈V∣∫M f(m, v(m)) dm=0} to a closed subspace of codimensionn.