The existence of non-degenerate functions on a compact differentiablem-manifoldM
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Let M be a compact differentiable m-manifold of class Cm in En, n=2m+1. Let x=(x1, ..., xn) represent a point in En. The union of the direction c on the direction sphere Sn−1 in En such that the scalar product c · x defines a non-degenerate fonction on M is an open subset of Sn−1 whose complement θ has a Lebesgue measure zero on Sn−1. When M is non-compact θ can be everywhere dense on Sn−1, but still has Lebesgue measure zero.