×

A note on the admissible Gauss map. (English) Zbl 0597.53002

Let \(G(n,n+k)\) be a Grassmann manifold. For an n-dimensional Riemannian manifold M, isometrically immersed in Euclidean \((n+k)\)-space E, the Gauss map \(\Gamma_ E: M \to G(n,n+k)\) is well-known. A deformation \(I\times M \to E\) is said to be an admissible one (with respect to \(\Gamma_ E)\) if the Gauss image of M(t), \(t\in I\), is fixed against t for each point of M [the reviewer, Proc. Am. Math. Soc. 76, 140-144 (1979; Zbl 0419.53029)].
The present author defines the Gauss map \(\Gamma_{-p}: M\to G(n,n+k)\) where M is an n-dimensional submanifold of the unit hypersphere S of \(E^{n+k+1}\) and the point \(-p\in S\) is such that the antipodal point p is not a point of M, with the use of parallel displacement along geodesics through -p. Then an admissible deformation \(I\times M \to S\) is also defined with respect to \(\Gamma_{-p}\). The author studies the relation of these two deformations and a stereographic projection.
Reviewer: Y.Muto

MSC:

53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces

Citations:

Zbl 0419.53029
PDFBibTeX XMLCite