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 O_{1} in S, Ψ[O_{1}] = [S] ∩ O_{2} for some open set O_{2} 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 → R^{2} 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) = s^{3} satisfies (I1) and (I3) but is not an imbedding because (Ψ_{0})_{∗} : R_{0} → R_{0} is not injective. (Here R_{0} 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 → R^{2} 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 O_{1} in R such that given any open set O_{2} in R^{2}, Ψ[O_{1}] ≠ O_{2} ∩ 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_{a}ξ^{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).