Everything Is the Rank of df_p
In the last guide we built, for a smooth map f: M -> N and a point p in M, the differential df_p: T_p M -> T_{f(p)} N — an honest linear map between vector spaces. The single most useful number attached to it is its rank: the dimension of its image. Linear algebra already tells you a linear map between an m-dimensional and an n-dimensional space can have rank at most min(m, n); the geometry of f near p is governed almost entirely by which rank df_p actually achieves. Three special cases get names, and the whole rest of differential topology is built on them.
Say dim M = m and dim N = n. We call f an immersion at p when df_p is injective (rank m, which forces m <= n); a submersion at p when df_p is surjective (rank n, which forces m >= n); and we call df_p having the largest possible rank min(m, n) being of maximal rank. An immersion infinitesimally squeezes M into N without collapsing any direction; a submersion infinitesimally spreads M onto all of N. These are conditions at a single point, but smoothness means rank cannot drop without warning, so a point-condition propagates to a neighborhood.
The Local Model: Constant Rank Theorem
Here is the engine behind the whole guide. The constant rank theorem says: if f has constant rank r on a neighborhood, then around p and f(p) you can choose charts in which f becomes the simplest possible linear-looking map — it just keeps r coordinates and zeroes the rest. It is the manifold version of putting a matrix into reduced row-echelon form, except the row operations are now smooth changes of coordinates. Everything below is this one theorem read in three special cases.
immersion (rank m, m<=n): (x_1,...,x_m) |-> (x_1,...,x_m, 0,...,0) submersion (rank n, m>=n): (x_1,...,x_m) |-> (x_1,...,x_n) constant rank r: (x_1,...,x_m) |-> (x_1,...,x_r, 0,...,0)
Read the immersion line and you see why an immersion is locally injective: in those charts f is literally the inclusion of R^m as the first m coordinate axes of R^n, which is obviously one-to-one. Read the submersion line and you see why a submersion is locally surjective and, better, why its fibers look so tidy: f is the projection forgetting the last m - n coordinates, so each fiber is a copy of R^{m-n} sitting as a coordinate slice. This second reading is the seed of the regular value theorem.
Immersion vs Embedding: The Subtle Gap
An immersion is a local statement, so it can globally misbehave. The figure-eight is the classic warning: a line immersed in the plane that crosses itself, so the map is not injective. Worse, even an injective immersion can fail to give you a faithful copy of M inside N. The standard example is the irrational-slope line wrapping densely on the torus: it is injective and an immersion everywhere, yet its image comes arbitrarily close to itself, so the subspace topology it inherits from the torus is not the topology it had as a line.
To rule this out we demand more. An embedding is an injective immersion that is also a homeomorphism onto its image (with the subspace topology). That extra clause — topological faithfulness — is exactly what the figure-eight and the dense line lack. When you do have an embedding, the image is a genuine embedded submanifold: it sits inside N with its own smooth structure agreeing with the one N hands it. A clean theorem rescues you in the most common case: every injective immersion from a compact manifold is automatically an embedding, because a continuous injection from a compact space to a Hausdorff space is always a homeomorphism onto its image.
The Regular Value Theorem: Submanifolds for Free
Now flip from immersions to submersions and you get the most-used theorem in the subject. Call a point q in N a regular value of f if df_p is surjective at every p in the fiber f^{-1}(q) — that is, f is a submersion at each preimage of q. (If the fiber is empty, q is vacuously regular.) The regular value theorem then states: if q is a regular value, the level set f^{-1}(q) is an embedded submanifold of M of dimension m - n, and at each of its points its tangent space is exactly the kernel of df_p.
Why is it true? Read the submersion line of the constant rank theorem again: near each preimage f looks like the projection R^m -> R^n forgetting the last m - n coordinates, so the fiber f^{-1}(q) is locally the slice where the first n coordinates are fixed — a clean copy of R^{m-n}, with its tangent space being the directions that project to zero, i.e. ker df_p. The hypothesis 'q is regular' is precisely what licenses that local picture at every point of the fiber simultaneously, gluing the slices into one smooth submanifold.
- Pick the equation. To realize S^{n-1} as a submanifold of R^n, use f: R^n -> R given by f(x) = x_1^2 + ... + x_n^2.
- Compute the differential. Here df_p is the row vector (2 x_1, ..., 2 x_n), i.e. 2x^T; it is surjective onto R exactly when it is nonzero.
- Find where rank drops. df_p = 0 only at x = 0, the single critical point; its critical value is f(0) = 0.
- Choose a regular value. Any c > 0 misses the critical value, so 1 is regular and f^{-1}(1) = S^{n-1} is a smooth (n-1)-dimensional submanifold — no charts written by hand.
- Read off the tangent space. At p in S^{n-1}, T_p S^{n-1} = ker df_p = { v : x . v = 0 } — exactly the hyperplane orthogonal to the position vector, the answer you already expected.
What This Buys You, and the Honest Fine Print
The same recipe builds the orthogonal group O(n) as a submanifold of matrices (take f(A) = A^T A, regular value the identity), the special linear group, and countless others — Lie groups will arrive this way in a later rung. The regular value theorem is also the cleanest source of submanifolds-with-boundary and the natural setting for the transversality you will meet next, where instead of hitting a single value you ask two maps to meet a submanifold cleanly. And the local picture you just used — a submersion is a projection — is the structural fact that makes a submersion a locally trivial fibration over its image.
Now the honesty. The theorem needs q to be a regular value, and most values are — but not all. A critical value (one with a critical point in its fiber) can give a level set that is not a manifold at all: the cone x^2 + y^2 - z^2 = 0 has a singular vertex exactly because 0 is a critical value of that f. The natural follow-up question — are regular values plentiful? — is answered by Sard's theorem, the headline of the next guide: the critical values form a measure-zero set, so almost every value you pick is regular. Until then, never apply the regular value theorem without checking that your chosen value actually avoids the critical points.
One last reassurance about scope. Embeddings are not exotic: the Whitney embedding theorem says every smooth m-manifold embeds into R^{2m+1} (and, with more work, into R^{2m}), so the abstract manifolds you defined chart-by-chart in guide 1 really do live as honest submanifolds of some Euclidean space. Be careful with the slogan, though — it is an existence theorem about smooth, not isometric, embeddings; fitting a manifold in while preserving a given metric is the harder Nash embedding story, and a different course entirely.