The Gauss map for surfaces. I. The affine case
HTML articles powered by AMS MathViewer
- 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.References
- Kinetsu Abe and Joseph Erbacher, Isometric immersions with the same Gauss map, Math. Ann. 215 (1975), 197–201. MR 372789, DOI 10.1007/BF01343889
- Yu. A. Aminov, Determination of a surface in a four-dimensional Euclidean space from its image in a Grassmannian, Mat. Sb. (N.S.) 117(159) (1982), no. 2, 147–160, 287 (Russian). MR 644766
- D. Bleecker and L. Wilson, Stability of Gauss maps, Illinois J. Math. 22 (1978), no. 2, 279–289. MR 487812, DOI 10.1215/ijm/1256048737
- David A. Hoffman and Robert Osserman, The Gauss map of surfaces in $\textbf {R}^{n}$, J. Differential Geom. 18 (1983), no. 4, 733–754 (1984). MR 730925
- David A. Hoffman and Robert Osserman, The Gauss map of surfaces in $\textbf {R}^3$ and $\textbf {R}^4$, Proc. London Math. Soc. (3) 50 (1985), no. 1, 27–56. MR 765367, DOI 10.1112/plms/s3-50.1.27
- Kurt Leichtweiss, Zur Riemannschen Geometrie in Grassmannschen Mannigfaltigkeiten, Math. Z. 76 (1961), 334–366 (German). MR 126808, DOI 10.1007/BF01210982
- John A. Little, On singularities of submanifolds of higher dimensional Euclidean spaces, Ann. Mat. Pura Appl. (4) 83 (1969), 261–335. MR 271970, DOI 10.1007/BF02411172
- Joel L. Weiner, A uniqueness theorem for submanifolds of Euclidean space, Illinois J. Math. 25 (1981), no. 1, 16–26. MR 602891
- Joel L. Weiner, The Gauss map for surfaces in $4$-space, Math. Ann. 269 (1984), no. 4, 541–560. MR 766013, DOI 10.1007/BF01450764 —, First integrals of direction fields on simply connected plane domains (preprint).
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 293 (1986), 431-446
- MSC: Primary 53A07; Secondary 53A05
- DOI: https://doi.org/10.1090/S0002-9947-1986-0816302-8
- MathSciNet review: 816302