Projecting (n-1)-cycles to zero on hyperplanes in Rn+1
Bruce Solomon
Abstract
The projection of a compact oriented submanifold Mn-1 in Rn+1 on a hyperplane Pn can fail to bound any region in P. We call this ``projecting to zero.'' Example: The equatorial S1 in S2 projects to zero in any plane containing the x3-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 Rn+1. We also show, using examples, that all the hypotheses in this statement are sharp.
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