Submanifold theory determines when equations or parametrizations define smooth spaces. The constant-rank theorem supplies the local normal forms from which regular level sets, immersed images, and embeddings follow.
Immersions, submersions, and embeddings
Definition. A smooth map \(F:M^m\to N^n\) is an immersion at \(p\) if \(\mathrm{d}F_p\) is injective, and a submersion if it is surjective. It is an embedding if it is an immersion and a homeomorphism onto its image with the subspace topology.
Theorem (constant rank). If \(\operatorname{rank}\mathrm{d}F=r\) near \(p\), there are coordinates about \(p\) and \(F(p)\) in which \[ F(x^1,\ldots,x^m)=(x^1,\ldots,x^r,0,\ldots,0). \]
The proof is the inverse function theorem applied after selecting a nonzero \(r\times r\) Jacobian minor. It simultaneously generalizes the local normal forms for immersions and submersions.
Regular level sets
Theorem (regular value). If \(F:M^m\to N^n\) and \(q\in N\) is a regular value, meaning \(\mathrm{d}F_p\) is surjective for every \(p\in F^{-1}(q)\), then \(F^{-1}(q)\) is an embedded submanifold of dimension \(m-n\), with \[ T_pF^{-1}(q)=\ker\mathrm{d}F_p. \]
Proof. The submersion normal form gives local coordinates in which \(F(x)=(x^1,\ldots,x^n)\). The level set is then cut out by \(x^1=\cdots=x^n=\text{constant}\), so the remaining \(m-n\) coordinates parametrize it. Differentiating the level-set identity gives the tangent-space formula.
Example (sphere and orthogonal group). For \(F(x)=|x|^2\), every \(r^2>0\) is regular and \[ T_xS_r^{n-1}=\{v:x\cdot v=0\}. \] For \(G(A)=A^\top A\) on \(M_n(\mathbb R)\), the level \(G^{-1}(I)=O(n)\) has tangent space at \(I\) \[ T_IO(n)=\{X:X^\top+X=0\}, \] the skew-symmetric matrices.
Immersed versus embedded images
The map \(t\mapsto(\sin2t,\sin3t)\) can be immersive at a crossing, but its image is not locally Euclidean there with the subspace topology. An injective immersion need not be an embedding if its inverse onto the image is not continuous. A proper injective immersion, however, is an embedding.
Proposition. If \(F:M\to N\) is a proper injective immersion, then \(F\) is an embedding.
Proof. A proper map between locally compact Hausdorff spaces is closed. Thus the continuous bijection \(F:M\to F(M)\) is closed and has continuous inverse. Together with the immersion property, this is exactly an embedding.
Transversality
Submanifolds \(A,B\subset M\) meet transversely at \(p\) if \[ T_pA+T_pB=T_pM. \] Their intersection is then a submanifold of dimension \(\dim A+\dim B-\dim M\). Transversality is stable under small perturbations and makes the expected dimension count geometrically valid.