Embedded and Immersed Submanifolds

A smooth constraint $F(x)=q$ defines an embedded submanifold when $q$ is a regular value: the differential $dF$ must be surjective at every point of the level set. Then the tangent space is exactly $\ker dF$. Immersions describe spaces that are locally embedded but may intersect themselves globally; embeddings additionally preserve the topology of the source inside the target.

Required background. Calculus and Linear Algebra for Relativity — use rank, kernels, the inverse and implicit function theorems, and local coordinate changes.

Immersions, embeddings, and inclusions

For smooth manifolds $S^k$ and $M^n$, a smooth map $i:S\to M$ is an immersion if $di_p$ is injective for every $p$. It is an embedding if it is an immersion and a homeomorphism from $S$ onto $i(S)$ with the subspace topology. An embedded submanifold is a subset equipped with this compatible smooth structure; its inclusion into $M$ is an embedding.

The topological condition is genuinely additional. Consider

$$\gamma:S^1\longrightarrow\mathbb R^2, \qquad \gamma(t)=(\sin 2t,\sin t),$$

with $t$ understood modulo $2\pi$. Its derivative $(2\cos2t,\cos t)$ never vanishes, so $\gamma$ is an immersion. But $\gamma(0)=\gamma(\pi)=(0,0)$; the image is a figure eight, and the map is not injective, hence not an embedding. Each branch looks embedded locally, while the self-intersection prevents the image from being a one-dimensional embedded submanifold near the crossing.

These distinctions, including the local normal forms behind them, are stated in Lee 2013, Chs. 4–5, pp. 77–124.

The regular-value theorem

Let $F:M^m\to N^n$ be smooth, and let $q\in N$. The point $q$ is a regular value when $dF_p:T_pM\to T_qN$ is surjective for every $p\in F^{-1}(q)$. If $q$ is a regular value, then $F^{-1}(q)$ is an embedded submanifold of dimension $m-n$, and

$$T_pF^{-1}(q)=\ker dF_p.$$

The theorem also permits an empty level set. Its proof uses the submersion normal form: near each point of the level set, suitable source and target coordinates put $F$ into the form

$$(x^1,\ldots,x^m)\longmapsto(x^1,\ldots,x^n).$$

Taking the target coordinates to vanish at $q$, the level set is locally $x^1=\cdots=x^n=0$, visibly a coordinate slice. This proves its dimension and tangent-space formula: tangent vectors have zero first $n$ components, precisely the kernel of $dF$. The local normal form and regular-level conclusion appear in Gualtieri 2017, §2.5, Theorem 2.16 and Proposition 2.20, pp. 22–23, PDF.

Sphere and singular crossing

Define $F:\mathbb R^3\to\mathbb R$ by $F(X,Y,Z)=X^2+Y^2+Z^2$. On $F^{-1}(1)$,

$$dF_p(v)=2(Xv_X+Yv_Y+Zv_Z).$$

Because $p\ne0$, this linear map is surjective, so $1$ is a regular value. The theorem gives $S^2$ as an embedded two-manifold and

$$T_pS^2=\{v\in\mathbb R^3:Xv_X+Yv_Y+Zv_Z=0\}.$$

This agrees with the velocity calculation: differentiating $F(\gamma(t))=1$ yields $dF_p(\dot\gamma)=0$.

The rank condition is sufficient, not necessary at each presentation, but its failure removes the guarantee. For $G(x,y)=x^2-y^2$, the level $G^{-1}(0)$ is the union of the lines $y=x$ and $y=-x$. Away from the origin it is locally a smooth curve; at the origin, $dG=0$ and four branches meet. No neighborhood of the crossing is homeomorphic to an interval, so the level set is not a one-manifold there. This counterexample shows exactly what the regularity hypothesis prevents.

Local defining functions and scope

Every embedded codimension-$n$ submanifold is locally a regular level set: near each $p\in S$, one can find $n$ smooth functions with independent differentials whose common zero set is $S$. This is a local assertion. A single global defining map may fail to exist because of topology, and a parametrized immersion need not determine an embedded image.

The submanifold inherits smooth structure and tangent spaces, but not automatically a metric, orientation, or preferred normal vector. Those require additional data. Likewise, this page establishes geometric constraints, not equations of motion: a physical constraint surface or hypersurface becomes dynamical only after a theory supplies evolution and compatibility conditions.

The sphere calculation supplies an independent consistency check between three descriptions: velocities of curves constrained to remain on the sphere, the kernel of the defining function’s differential, and the image of a stereographic-chart differential. All three produce the same two-dimensional tangent plane. Such agreement is more informative than a rank count alone because it tests both the theorem’s hypotheses and the resulting tangent-space formula.

Local versus global placement

An injective immersion is not automatically an embedding when the source is noncompact: its inverse onto the image may fail to be continuous in the subspace topology. For example, on the torus $\mathbb R^2/(2\pi\mathbb Z)^2$, the map

$$i:\mathbb R\longrightarrow T^2, \qquad i(t)=(t,\alpha t)\pmod{2\pi}, \qquad \alpha\notin\mathbb Q,$$

has nonzero derivative and is injective, but winds densely around the torus. There are arbitrarily large $t$ for which $i(t)$ is arbitrarily close to $i(0)$, so the inverse from the image to $\mathbb R$ is not continuous. No crossing is needed for embeddedness to fail.

Properness is a useful sufficient addition: a map is proper if inverse images of compact sets are compact, and an injective proper immersion is an embedding. For compact source and Hausdorff target, injectivity already makes a continuous map a homeomorphism onto its image, so an injective immersion is then an embedding.

This is a separate issue from a self-intersection. A self-intersecting immersion visibly fails injectivity, whereas a nonproper injective immersion can approach part of its own image without reaching it in the source topology. Both examples warn against deciding embeddedness from differential rank alone. Rank controls the local model; the topology of the map controls the global placement.

When constraints are given by several functions, independence of their differentials is the invariant condition. Counting the number of written equations is not enough: redundant constraints do not raise codimension, and dependent gradients signal that the regular-value theorem cannot be invoked as stated.

Exercises

Let $F:\mathbb R^3\to\mathbb R$ be $F(x,y,z)=x^2+y^2$. For which $c$ is $F^{-1}(c)$ a regular level set, and what are its dimension and tangent space? Analyze $c=0$ separately.

Solution

Since $dF=2x\,dx+2y\,dy$, every $c>0$ is regular: on $x^2+y^2=c$, $(x,y)$ cannot both vanish. The level is a two-dimensional cylinder, and at $p=(x,y,z)$

$$T_pF^{-1}(c)=\{(v_x,v_y,v_z):xv_x+yv_y=0\}.$$

For $c<0$ the level is empty, which is allowed by the theorem. For $c=0$, the level is the $z$-axis, which is nevertheless an embedded one-manifold, even though $dF=0$ there. Thus regularity is sufficient but not necessary for a particular level set; the same set can be defined regularly by the two-component map $(x,y)$.

References

• Gualtieri, Marco. Geometry and Topology I. University of Toronto course notes, 2017, §2.5, pp. 21–23, PDF.
• Lee, John M. Introduction to Smooth Manifolds. 2nd ed., Graduate Texts in Mathematics 218, Springer, 2013, Chs. 4–5, pp. 77–124.