The Gauss map for surfaces. I. The affine case

Weiner, Joel L.

doi:10.1090/S0002-9947-1986-0816302-8

The Gauss map for surfaces. I. The affine case
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by Joel L. Weiner PDF

Trans. Amer. Math. Soc. 293 (1986), 431-446 Request permission

Abstract:

Let $M$ be a connected oriented surface and let $G_2^c$ be the Grassmannian of oriented $2$-planes in Euclidean $(2 + c)$-space, ${{\mathbf {E}}^{2 + c}}$. Smooth maps $t:M \to G_2^c$ are studied to determine whether or not they are Gauss maps. Both local and global results are obtained. If $t$ is a Gauss map of an immersion $X:\;M \to {{\mathbf {E}}^{2 + c}}$, we study the extent to which $t$ uniquely determines $X$ under certain circumstances.

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First integrals of direction fields on simply connected plane domains