The Gauss map for surfaces. II. The Euclidean case

Abstract:

We study smooth maps $t:\;M \to G_2^c$ of a Riemann surface $M$ into the Grassmannian $G_2^c$ of oriented $2$-planes in ${{\mathbf {E}}^{2 + c}}$ and determine necessary and sufficient conditons on $t$ in order that it be the Gauss map of a conformal immersion $X:\;M \to {{\mathbf {E}}^{2 + c}}$. We sometimes view $t$ as an oriented riemannian vector bundle; it is a subbundle of ${\mathbf {E}}_M^{2 + c}$, the trivial bundle over $M$ with fibre ${{\mathbf {E}}^{2 + c}}$. The necessary and sufficient conditions obtained for simply connected $M$ involve the curvatures of $t$ and ${t^ \bot }$, the orthogonal complement of $t$ in ${\mathbf {E}}_M^{2 + c}$, as well as certain components of the tension of $t$ viewed as a map $t:\;M \to {S^C}(1)$, where ${S^C}(1)$ is a unit sphere of dimension $C$ that contains $G_2^c$ as a submanifold in a natural fashion. If $t$ satisfies a particular necessary condition, then the results take two different forms depending on whether or not $t$ is the Gauss map of a conformal minimal immersion. The case $t:\;M \to G_2^2$ is also studied in some additional detail.