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- W1980018215 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" @default.
- W1980018215 created "2016-06-24" @default.
- W1980018215 creator A5013986452 @default.
- W1980018215 date "1986-02-01" @default.
- W1980018215 modified "2023-09-24" @default.
- W1980018215 title "The Gauss map for surfaces. II. The Euclidean case" @default.
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- W1980018215 doi "https://doi.org/10.1090/s0002-9947-1986-0816303-x" @default.
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