Suppose f : N → M. Then d_{p}f : T_{p}(N) → T_{p}(M). If df_{p} is 1 − 1, for all p ∈ N, then f is called an immersion.

I.e., f is an immersion iff f has rank n

If df_{p} is onto for all p ∈ N, then f is called a submersion.

I.e., f is an submersion iff f has rank m

Defn. Suppose f : M → N is smooth. p ∈ M is a critical point and f(p) is a critical value if rank df_{p} < n. If p ∈ M is not a critical point, then it is a regular point. If q ∈ N is not a critical value, then it is a regular value. Note: q ∈ N is a regular value iff f^{−1}(q) = ∅ or ∀p ∈ f^{−1}(q), df_{p} = n. Defn: K is a m-submanifold of N if ∀q ∈ K ⊂ N, ∃g^{smooth} : V^{open} ⊂ N → R^{n−m}, q ∈ V such that 1.) g is smooth 2.) K ∩ V = g^{−1}(0) and 3.) rank d_{p}g = n − m Defn: Suppose f: M → N is a 1 − 1 immersion, and suppose f: M → f(M) is a homeomorphism, where f(M) ⊂ N has the relative topology. Then f is an embedding, and f(M) is an embedded submanifold. Thm. f : M → N embedding implies f(M) is a submanifold of N. Thm 2.3.13: Let q be a regular value of f: M → N. Then either f^{−1}(q) = ∅ or f^{−1}(q) is an (m − n)-submanifold of M.