When the subspaces of interest are manifolds of dimension m, the Whitney
embedding theorem tells us that the euclidean covering manifold need not be of
dimension larger than 2m+1.
The usual implicit surface is defined typically as the zero set of some function. Given the function, construct its gradient as a representation of the fiber normal to the surface. The idea is that there exist algebraic constructions of m-1 vectors that are transverse to the fiber. These m-1 vectors are defined as the associated or tangent vectors to the surface. The concept of m-1 associated vectors transversal (orthogonal) to a fiber can be developed even though the fiber is not defined as a gradient of a function.
Given an arbitrary 1-form on a space of dimension m it is always possible to
construct algebraically m-1 vectors transversal to the 1-form by means of the
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