Consider F : ℝ⁴ → ℝ² defined by F(x,y,z,w) = (x² + y² - 1, z² + w² - 1). What is the dimension of the submanifold F⁻¹(0,0), and what familiar manifold is it?
ADimension 1 — it is a circle S¹
BDimension 2 — it is the torus S¹ × S¹
CDimension 3 — it is the 3-sphere S³
DDimension 2 — it is the 2-sphere S²
The Jacobian of F is a 2×4 matrix. At any point of F⁻¹(0,0), the two rows (2x, 2y, 0, 0) and (0, 0, 2z, 2w) are linearly independent (since x²+y²=1 and z²+w²=1 ensure the rows are nonzero). So dF has rank 2 everywhere on the preimage, making (0,0) a regular value. The dimension is 4-2=2. The preimage is {(x,y,z,w) : x²+y²=1 and z²+w²=1} = S¹ × S¹, the flat torus embedded in ℝ⁴.
Question 2 Short Answer
The value 1 is a regular value of F(x,y) = x² + y², but the value 0 is not. Why?
Think about your answer, then reveal below.
Model answer: At every point (x,y) with x² + y² = 1, the derivative dF = (2x, 2y) is nonzero (since x and y cannot both be zero on the unit circle), hence surjective as a map to ℝ. So 1 is a regular value and F⁻¹(1) = S¹ is a smooth 1-manifold. At the only point of F⁻¹(0) = {(0,0)}, dF = (0,0) is the zero map, which is not surjective. So 0 is a critical value. The preimage F⁻¹(0) is a single point — still a manifold, but the theorem does not apply (the conclusion happens to hold by coincidence, not by the theorem).
This example illustrates that the regular value theorem gives a sufficient condition, not a necessary one. The preimage of a critical value might or might not be a manifold — you need to analyze it by other means. The preimage of a regular value is guaranteed to be a manifold.
Question 3 Short Answer
A smooth map F : M → N is called a submersion at p if dFp : TpM → TF(p)N is surjective. What does the submersion theorem guarantee?
Think about your answer, then reveal below.
Model answer: The submersion theorem (also called the local submersion theorem or canonical form for submersions) says that near a point where F is a submersion, there exist local coordinates on M and N such that F looks like the standard projection (x¹,...,xⁿ) ↦ (x¹,...,xᵏ) where k = dim(N). In particular, every fiber F⁻¹(c) near that point is a smooth submanifold of dimension dim(M) - dim(N).
This is the manifold version of the implicit function theorem. It says that submersions are locally as simple as possible — they are locally equivalent to linear projections. The rank theorem generalizes further: if dF has constant rank r near p, then in suitable coordinates F looks like (x¹,...,xⁿ) ↦ (x¹,...,xʳ, 0,...,0). The regular value theorem is the special case where F is a submersion along an entire fiber.
Question 4 Short Answer
Why is the regularity condition (dF surjective) necessary? What goes wrong at a critical value?
Think about your answer, then reveal below.
Model answer: At a critical value c, the preimage F⁻¹(c) can fail to be a manifold — it can have singularities such as cusps, corners, self-intersections, or dimension changes. For example, the level set of F(x,y) = x² - y² at c=0 is two crossing lines (an X shape), which is not a manifold at the origin. The regularity condition ensures the derivative has enough rank to apply the implicit function theorem, which provides local coordinate charts making F⁻¹(c) a smooth manifold.
The implicit function theorem requires the derivative to have maximal rank to solve for some variables in terms of others. When the rank drops, you cannot solve and the level set can develop singularities. The study of what happens at critical values is the subject of singularity theory and Morse theory.