Regular Level Sets of Averages of Nemytskiı Operators Are Contractible
✍ Scribed by Iaci Malta; Nicolau C. Saldanha; Carlos Tomei
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 886 KB
- Volume
- 143
- Category
- Article
- ISSN
- 0022-1236
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✦ Synopsis
Let H 1 (S 1 ) be the space of periodic real functions with derivative in L 2 and f : R Ä R be a smooth function with no double roots. Then there is a diffeomorphism of H 1 (S 1 ) taking the set Z=[v # H 1 (S 1 ) | S 1 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 space V of functions from some manifold M to R n$ and a function f: M_R n$ Ä R n : under suitable hypothesis, there is a homeomorphism of V taking
Let H 1 (S 1 ) be the space of periodic real functions with derivative in L 2 . The set Z=[v # H 1 (S 1 ) | S 1 v 2 (t) dt=1], at first sight, looks like a sphere: infinite dimensional topology (for which a good reference is [4]) tells us that the unit sphere is diffeomorphic to a hyperplane in Hilbert space. Actually, there is a diffeomorphism of H 1 (S 1 ) taking Z to a hyperplane. In this paper, we present a generalization of this example.
Let M be a compact manifold with a smooth Riemannian metric inducing a measure + with +(M)=1. Let C (M) be the Fre chet ring of smooth real valued functions on M. Set V to be a separable Banach space continuously article no. FU962987