Let F : ℝ³ → ℝ be defined by F(x,y,z) = x² + y² + z² - 1. By the regular value theorem, F⁻¹(0) = S² is a smooth submanifold of ℝ³ because...
AF is a smooth function and 0 is in its image
BThe derivative dF is surjective (has rank 1) at every point of F⁻¹(0)
CS² is compact and Hausdorff
DF⁻¹(0) is a closed subset of ℝ³
The regular value theorem (preimage theorem) states that if c is a regular value of F — meaning dF_p is surjective for every p ∈ F⁻¹(c) — then F⁻¹(c) is a smooth submanifold of codimension equal to the dimension of the codomain. Here dF = (2x, 2y, 2z), which is nonzero (hence surjective as a map to ℝ) at every point of S² (where x²+y²+z² = 1). So 0 is a regular value and S² is a smooth 2-dimensional submanifold of ℝ³.
Question 2 True / False
An immersion is always an embedding.
TTrue
FFalse
Answer: False
An immersion is a smooth map whose derivative is injective at every point, but it need not be an embedding. An embedding additionally requires the map to be a homeomorphism onto its image (in the subspace topology). The figure-eight curve t ↦ (sin 2t, sin t) is an immersion of ℝ into ℝ² that is not an embedding because it crosses itself. A more subtle example: the irrational-slope line on a torus is an injective immersion that is not an embedding because its image is dense in the torus.
Question 3 Short Answer
What is the relationship between the tangent space of a submanifold S ⊂ M at a point p and the tangent space of the ambient manifold M at p?
Think about your answer, then reveal below.
Model answer: TpS is naturally a linear subspace of TpM. If S has dimension k and M has dimension n, then TpS is a k-dimensional subspace of the n-dimensional space TpM. When S is defined as the level set F⁻¹(c) of a smooth map F : M → N, then TpS = ker(dFp) — the tangent space of S is the kernel of the derivative of the defining map. The normal directions to S at p correspond to the quotient TpM/TpS (or, with a metric, to the orthogonal complement).
This is the infinitesimal version of the inclusion S ↪ M. The inclusion map i : S → M has derivative di_p : TpS → TpM which is injective, so TpS embeds as a subspace. For level sets, the tangent space consists of all tangent vectors that are 'tangent to the constraint surface' — vectors along which the constraint function F does not change to first order.
Question 4 True / False
The dimension of a submanifold defined as the level set of a smooth map F : M → ℝᵏ at a regular value is dim(M) - k.
TTrue
FFalse
Answer: True
When c is a regular value of F : Mⁿ → ℝᵏ (so dFp has rank k at every point of F⁻¹(c)), the preimage theorem says F⁻¹(c) is a smooth submanifold of dimension n - k. Each component of F imposes one independent constraint, removing one dimension. For example, one equation F = 0 in ℝ³ gives a surface (3-1=2), two independent equations give a curve (3-2=1), and three independent equations give isolated points (3-3=0).