Hypersurfaces

Let (S, C_S) and (M, C_M) be manifolds of dimension k and n, respectively, with 1 ≤ k ≤ n. A smooth map : S → M is said to be an imbedding if it satisfies the following three conditions.

(I1) Ψ is injective.

(I2) At all points p in S, the associated (push-forward) linear map (Ψ_p)_∗ : S_p → M_Ψ(p) is injective.

(I3) ∀ open sets O₁ in S, Ψ[O₁] = [S] ∩ O₂ for some open set O₂ in M. (Equivalently, the inverse map Ψ⁻¹ : Ψ[S] → S is continuous with respect to the relative topology on [S].)

Several comments about the definition are in order. First, given any point p in S, (I2) implies that (Ψ_p)_∗[S_p] is a k-dimensional subspace of M_Ψ(p). So the condition cannot be satisfied unless k ≤ n. Second, the three conditions are independent of one another. For example, the smooth map Ψ : R → R² defined by (s) = (cos(s), sin(s)) satisfies (I2) and (I3) but is not injective. It wraps R round and round in a circle. On the other hand, the smooth map : R → R defined by (s) = s³ satisfies (I1) and (I3) but is not an imbedding because (Ψ₀)_∗ : R₀ → R₀ is not injective. (Here R₀ is the tangent space to the manifold R at the point 0). Finally, a smooth map : S → M can satisfy (I1) and (I2) but still have an image that “bunches up on itself.” It is precisely this possibility that is ruled out by condition (I3). Consider, for example, a map : R → R² whose image consists of part of the image of the curve y = sin(1/x) smoothly joined to the segment {(0, y) : y < 1}, as in the figure below. It satisfies conditions (I1) and (I2) but is not an imbedding because we can find an open interval O₁ in R such that given any open set O₂ in R², Ψ[O₁] ≠ O₂ ∩ Ψ[R].

Suppose(S, C_S) and (M, C_M) are manifolds with S ⊆ M. We say that (S, C_S) is an imbedded submanifold of (M, C_M) if the identity map id: S → M is an imbedding. If, in addition, k = n − 1 (where k and n are the dimensions of the two manifolds), we say that (S, C_S) is a hypersurface in (M, C_M). Let (S, C_S) be a k-dimensional imbedded submanifold of the n-dimensional manifold (M, C_M), and let p be a point in S. We need to distinguish two senses in which one can speak of “tensors at p.” There are tensors over the vector space Sp (call them S-tensors at p) and ones over the vector space M_p (call them M-tensors at p). So, for example, an S-vector ξ ̃^a at p makes assignments to maps of the form f ̃: O ̃ → R where O ̃ is a subset of S that is open in the topology induced by C_S, and f ̃ is smooth relative to C_S. In contrast, an M-vector ξ^a at p makes assignments to maps of the form f : O → R where O is a subset of M that is open in the topology induced by C_M, and f is smooth relative to C_M. Our first task is to consider the relation between S-tensors at p and M-tensors there.

Let us say that ξ^a ∈ (M_p)^a is tangent to S if ξ^a ∈ (id_p)_∗[(S_p)^a]. (This makes sense. We know that (id_p)_∗[(S_p)^a] is a k-dimensional subspace of (M_p)^a; ξ^a either belongs to that subspace or it does not.) Let us further say that η_a in (M_p)^a is normal to S if η_aξ^a =0 ∀ ξ^a ∈ (M_p)^a that are tangent to S. Each of these classes of vectors has a natural vector space structure. The space of vectors ξ^a ∈ (M_p)^a tangent to S has dimension k. The space of co-vectors η_a ∈ (M_p)^a normal to S has dimension (n − k).

Advertisement