Projecting (n-1)-cycles to zero on hyperplanes in R^{n+1}

Details Projecting (n-1)-cycles to zero on hyperplanes in R^{n+1} Projecting (n-1)-cycles to zero on hyperplanes in R^{n+1}

2002-09-18

arxiv.org/abs/math/0209226v1

Mathematics

Differential Geometry

14 pages, 4 figures

Scientific paper

The projection of a compact oriented submanifold M^{n-1} in R^{n+1} on a hyperplane P^{n} can fail to bound any region in P. We call this ``projecting to zero.'' Example: The equatorial S^1 in S^2 projects to zero in any plane containing the x_3-axis. Using currents to make this precise, we show: A lipschitz (homology) (n-1)-sphere embedded in a compact, strictly convex hypersurface cannot project to zero on n+1 linearly independent hyperplanes in R^{n+1}. We also show, using examples, that all the hypotheses in this statement are sharp.

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