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On Non-Orientable Closed Surfaces in Euclidean Spaces

Published online by Cambridge University Press:  20 November 2018

C. T. Yang
C. T. Yang*
Affiliation:
University of Pennsylvania
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Let us begin with a simple result.

Proposition. Let X be a non-orientable closed surface differentiably imbedded into the euclidean 4-space R4. Then there is a line in R4 which intersects X at more than two points.

Proof. Let c be any point of R4 and let r be the smallest number such that X is contained in the closed 4-spheroid W of centre c and radius r. Clearly the boundary 3-sphere S of W intersects X.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 14 , 1962 , pp. 660 - 668
Copyright
Copyright © Canadian Mathematical Society 1962

References

1. Borsuk, K., On the k-independent subsets of the euclidean space and of the Hilbert space, Bull. Acad. Polon Sci. CI. III, 5 (1957), 351–356.Google Scholar
2. Yang, C. T., On k-independent sets (unpublished).Google Scholar