Some lifting and approximation properties for maps in $W^{1,2}({\mathbb{B}}^3;{\mathbb{S}}^2)$

Abstract

Smooth maps $u:{\mathbb{B}}^3\to {\mathbb{S}}^2$ can be lifted to $\hat{u}:{\mathbb{B}}^3\to {\mathbb{S}}^3$ using the Hopf fibration $h:{\mathbb{S}}^3\to {\mathbb{S}}^2$ via the factorization $u=h\circ \hat{u}$. In this note we characterize the $W^{1,2}$-maps which have this lifting property in terms of exactness of the pullback form $u^{\ast }\ddot{I}{‰}_{{\mathbb{S}}^2}$, and deduce a smooth approximation property preserving the constraint $u^{\ast }\ddot{I}{‰}_{{\mathbb{S}}^2}=d\hat{I}\cdot$.

7 pages