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 m1 vectors that are transverse to the fiber. These m1 vectors are defined as the associated or tangent vectors to the surface. The concept of m1 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 1form on a space of dimension m it is always possible to
construct algebraically m1 vectors transversal to the 1form by means of the
interior product.
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