A-Equivalence and Local Normal Forms of Smooth Maps

Definition (Smooth Map-Germ).

Let $M$ and $N$ be smooth manifolds, $p\in M$, and $q\in N$. An equivalence relation is defined on the set of smooth maps $f:U\to N$, where $U$ is any open neighborhood of $p$. Two maps $f_1:U_1\to N$ and $f_2:U_2\to N$ are equivalent if there exists an open neighborhood $W\subseteq U_1\cap U_2$ of $p$ such that $f_1{|}_W=f_2{|}_W$. An equivalence class $[f]$ is a smooth map-germ from $(M,p)$ to $(N,q)$, where $f(p)=q$.

Intuition: A map-germ captures the behavior of a function in an infinitesimally small neighborhood of a point, discarding any information about the function’s behavior far from that point.

Definition (A-Equivalence).

Let $f,g:(M,p)\to (N,q)$ be two smooth map-germs. They are A-equivalent (or right-left equivalent) if there exist diffeomorphism-germs $\varphi :(M,p)\to (M,p)$ and $\psi :(N,q)\to (N,q)$ such that the relation $\psi \circ f=g\circ \varphi$ holds for representatives in a neighborhood of $p$.

Intuition: A-equivalence formalizes the idea of two functions having the same “local structure” up to a smooth change of coordinates in both the source (domain) and the target (codomain). The diffeomorphisms $\varphi$ and ψ act as these local coordinate changes.

Definition (Standard Normal Forms).

Let $m,n,r$ be non-negative integers.

1. The standard rank-r map is the map ${\rho }_r:{\mathbb{R}}^m\to {\mathbb{R}}^n$ given by $${\rho }_r(x_1,\dots ,x_m)=(x_1,\dots ,x_r,0,\dots ,0).$$
2. The standard projection is the case where $r=n\le m$, denoted $\pi ={\rho }_n$.
3. The standard inclusion is the case where $r=m\le n$, denoted $\iota ={\rho }_m$.

Definition (Immersion and Submersion via A-Equivalence).

Let $f:(M,p)\to (N,q)$ be a smooth map-germ. Let $m=\dim M$ and $n=\dim N$.

1. $f$ is a submersion at $p$ if $m\ge n$ and $f$ is A-equivalent to the standard projection germ $\pi :({\mathbb{R}}^m,0)\to ({\mathbb{R}}^n,0)$.
2. $f$ is an immersion at $p$ if $m\le n$ and $f$ is A-equivalent to the standard inclusion germ $\iota :({\mathbb{R}}^m,0)\to ({\mathbb{R}}^n,0)$.

Intuition: Instead of defining these concepts with derivatives, we define them by their “shape.” A submersion is any map that, after a suitable change of coordinates, looks exactly like a projection. An immersion is any map that looks like a clean, non-self-intersecting inclusion of a lower-dimensional space into a higher-dimensional one.

Theorem (Equivalence to Classical Definitions).

Let $f:M\to N$ be a smooth map between manifolds with $f(p)=q$, and let $d{f}_p:T_{p}M\to T_{q}N$ be its differential at $p$.

1. The map-germ of $f$ at $p$ is a submersion if and only if its differential $d{f}_p$ is surjective.
2. The map-germ of $f$ at $p$ is an immersion if and only if its differential $d{f}_p$ is injective.

Proof. This proof establishes the crucial link between the geometric definition (A-equivalence) and the analytic definition (rank of the differential). The core argument relies on the Inverse Function Theorem.

(A-Equivalence $⟹$ Rank Condition) If $f$ is a submersion, then $\psi \circ f\circ {\varphi }^{-1}=\pi$ for some local diffeomorphisms $\varphi ,\psi$. The chain rule gives $d{\psi }_q\circ d{f}_p\circ d({\varphi }^{-1}{)}_p=d{\pi }_0$. As $d\psi$ and $d\varphi$ are isomorphisms and $d{\pi }_0$ is surjective, $d{f}_p$ must be surjective. An identical argument holds for immersions, using the injectivity of $d{\iota }_0$.

(Rank Condition $⟹$ A-Equivalence) This direction constructs the coordinate changes using the Inverse Function Theorem. We prove it for submersions; the proof for immersions is analogous. Assume $d{f}_p$ is surjective. In local coordinates, let $f:{\mathbb{R}}^m\to {\mathbb{R}}^n$ with $p=0,q=0$. The $n\times m$ Jacobian matrix of $f$ at $0$ has rank $n$. After reordering coordinates in ${\mathbb{R}}^m$, we can assume the first $n$ columns are linearly independent. Define a new map $F:{\mathbb{R}}^m\to {\mathbb{R}}^m$ by

$$F(x)=(f_1(x),\dots ,f_n(x),x_{n+1},\dots ,x_m).$$

The Jacobian of $F$ at $0$ is invertible, so by the Inverse Function Theorem, $F$ is a local diffeomorphism. Let $\varphi =F^{-1}$ be the desired source coordinate change. In the new coordinates $u=F(x)$, the map $f$ becomes $f\circ \varphi (u)=f\circ F^{-1}(u)$. By construction of $F$, the first $n$ components of $u$ are exactly $f_1(x),\dots ,f_n(x)$. So, $f(x)$ is just the first $n$ components of $u$. Therefore,

$$f\circ \varphi (u)=(u_1,\dots ,u_n)=\pi (u).$$

Taking $\psi =\mathrm{id}$, we have $\psi \circ f\circ \varphi =\pi$, so $f$ is A-equivalent to the projection. $\square$

Definition (Constant Rank Map-Germ).

A smooth map-germ $f:(M,p)\to (N,q)$ has constant rank r if it is A-equivalent to the standard rank-r map ${\rho }_r:({\mathbb{R}}^m,0)\to ({\mathbb{R}}^n,0)$.

Theorem (Constant Rank Theorem).

A smooth map-germ $f:(M,p)\to (N,q)$ has constant rank $r$ in the sense of A-equivalence if and only if there is a neighborhood $U$ of $p$ such that the rank of the differential $d{f}_x$ is equal to $r$ for all $x\in U$.

Proof. (A-Equivalence $⟹$ Constant Rank of Differential) If $f$ is A-equivalent to ${\rho }_r$, there exist local diffeomorphisms $\varphi ,\psi$ such that $\psi \circ f\circ {\varphi }^{-1}={\rho }_r$ on some neighborhood. By the chain rule, $d\psi \circ d{f}_x\circ d({\varphi }^{-1})=d({\rho }_r)$. Since $d\psi$ and $d\varphi$ are isomorphisms, they preserve the rank. The rank of $d({\rho }_r)$ is constant and equal to $r$ everywhere. Therefore, the rank of $d{f}_x$ must be constant and equal to $r$ in the neighborhood where the diffeomorphisms are defined.

(Constant Rank of Differential $⟹$ A-Equivalence) This is the constructive part of the theorem. Assume $\mathrm{rank}(d{f}_x)=r$ for all $x$ in a neighborhood of $p$. We work in local coordinates where $f:{\mathbb{R}}^m\to {\mathbb{R}}^n$, with $p=0,q=0$.

1. Construct the source diffeomorphism $\varphi$: Since $\mathrm{rank}(d{f}_0)=r$, we can reorder coordinates such that the top-left $r\times r$ submatrix of the Jacobian $J_f(0)$ is invertible. Define a map $F:{\mathbb{R}}^m\to {\mathbb{R}}^m$ by $$F(x_1,\dots ,x_m)=(f_1(x),\dots ,f_r(x),x_{r+1},\dots ,x_m).$$ The Jacobian of $F$ at $0$ is invertible, so by the Inverse Function Theorem, $F$ is a local diffeomorphism near $0$. Let $\varphi =F$. We will work in the new source coordinates $u=F(x)$. The map becomes $g=f\circ F^{-1}$.
2. Analyze the simplified map $g$: Let $u=(u_1,\dots ,u_m)$. By construction of $F$, the first $r$ components of $u$ are $(f_1(x),\dots ,f_r(x))$. Thus, $g(u)=f(F^{-1}(u))$ has its first $r$ components as $(u_1,\dots ,u_r)$. So $g(u)$ has the form $$g(u_1,\dots ,u_m)=(u_1,\dots ,u_r,h_{r+1}(u),\dots ,h_n(u))$$ for some functions $h_j$.
3. Use the rank condition: The Jacobian of $g$ has the form $$J_g(u)=\left(\begin{array}{cc}{I}_r& 0\\ \frac{\partial h}{\partial u^{\prime }}& \frac{\partial h}{\partial u^{\prime \prime }}\end{array}\right)$$ where $u^{\prime }=(u_1,\dots ,u_r)$ and $u^{\prime \prime }=(u_{r+1},\dots ,u_m)$. Since $\mathrm{rank}(d{g}_u)=\mathrm{rank}(d{f}_{F^{-1}(u)})=r$ everywhere locally, the bottom-right block must be zero, i.e., $\frac{\partial h}{\partial u^{\prime \prime }}=0$. This implies that the functions $h_j$ depend only on $u_1,\dots ,u_r$. So, $g$ has the form $$g(u_1,\dots ,u_m)=(u_1,\dots ,u_r,h_{r+1}(u_1,\dots ,u_r),\dots ,h_n(u_1,\dots ,u_r)).$$
4. Construct the target diffeomorphism $\psi$: We define a map to “straighten” the target space. Let $\Psi :{\mathbb{R}}^n\to {\mathbb{R}}^n$ be defined by $$\Psi (v_1,\dots ,v_n)=(v_1,\dots ,v_r,v_{r+1}-h_{r+1}(v_1,\dots ,v_r),\dots ,v_n-h_n(v_1,\dots ,v_r)).$$ The Jacobian of $\Psi$ is lower triangular with 1s on the diagonal, so it is invertible. By the Inverse Function Theorem, $\Psi$ is a local diffeomorphism. Let $\psi =\Psi$.
5. Final Composition: We compute $\psi \circ g(u)=\Psi (g(u))$. This gives $$\Psi (u_1,\dots ,u_r,h_{r+1}(u^{\prime }),\dots ,h_n(u^{\prime }))=(u_1,\dots ,u_r,h_{r+1}(u^{\prime })-h_{r+1}(u^{\prime }),\dots ,h_n(u^{\prime })-h_n(u^{\prime }))$$ which simplifies to $$(u_1,\dots ,u_r,0,\dots ,0)={\rho }_r(u).$$ We have found $\psi$ and $\varphi$ such that $\psi \circ f\circ {\varphi }^{-1}={\rho }_r$. Thus, $f$ is A-equivalent to the standard rank-r map. $\square$

Theorem (Coordinate Subspaces as Submanifolds).

Any affine subspace of ${\mathbb{R}}^n$ is a submanifold of ${\mathbb{R}}^n$. For instance, the $k$-dimensional subspace $S=\{(x_1,\dots ,x_k,c_{k+1},\dots ,c_n)\in {\mathbb{R}}^n\}$ for constants $c_i$ is a submanifold.

Proof. A subset $S\subset N$ is a submanifold if, for every point $p\in S$, there is a coordinate chart $(U,\mathbf{x})$ on the ambient manifold $N$ around $p$ such that $\mathbf{x}(U\cap S)$ is a “flat slice” of the coordinate space ${\mathbb{R}}^n$. For $N={\mathbb{R}}^n$ and $S$ an affine subspace, the identity chart $({\mathbb{R}}^n,\mathrm{id})$ trivially satisfies this condition for all points in $S$, as $S$ is already a flat slice in its own coordinate space. $\square$

Theorem (A-Equivalence and Local Submanifold Structure).

Let $S\subset N$ be a subset and $q\in S$. Let $\psi :(N,q)\to (N,q)$ be a diffeomorphism-germ. Then $S$ is a submanifold-germ at $q$ if and only if the image $\psi (S)$ is a submanifold-germ at $\psi (q)=q$.

Proof. Diffeomorphisms are smooth, invertible maps with smooth inverses, so they preserve all local geometric properties. Suppose $S$ is a submanifold at $q$. This means there exists a local chart $(U,\mathbf{x})$ on $N$ around $q$ that “straightens out” $S$ into a flat coordinate subspace. We can construct a new chart for $N$ around $q$, $\mathbf{y}=\mathbf{x}\circ {\psi }^{-1}$. This new chart is well-defined because it is a composition of a diffeomorphism and a chart map. This chart straightens out the transformed set $\psi (S)$, since $\mathbf{y}(\psi (S))=(\mathbf{x}\circ {\psi }^{-1})(\psi (S))=\mathbf{x}(S)$, which is a flat coordinate subspace. The converse follows by applying the same logic to the inverse diffeomorphism ${\psi }^{-1}$. $\square$

Theorem (Regular Value Theorem).

Let $f:M\to N$ be a smooth map and let $y\in N$ be a regular value of $f$. The preimage $f^{-1}(y)$ is a properly embedded submanifold of $M$ with dimension $\dim M-\dim N$.

Proof. Let $x$ be any point in the preimage $f^{-1}(y)$.

1. By definition, $y$ being a regular value means that for every $x\in f^{-1}(y)$, the differential $d{f}_x$ is surjective.
2. From our Equivalence Theorem, a surjective differential $d{f}_x$ is equivalent to the map-germ of $f$ at $x$ being a submersion.
3. By the definition of a submersion, this means $f$ is locally A-equivalent to the standard projection $\pi$. That is, there exist local diffeomorphisms ${\varphi }_x$ near $x$ and ${\psi }_x$ near $y$ such that ${\psi }_x\circ f\circ {\varphi }_x^{-1}=\pi$.
4. The preimage of a point under the projection $\pi$ is a coordinate subspace, which is a submanifold.
5. The original preimage, $f^{-1}(y)$, is related to this simple coordinate subspace by local diffeomorphisms. Since diffeomorphisms preserve the property of being a submanifold, $f^{-1}(y)$ must be a submanifold locally at each point $x$.
6. This argument holds for every point $x\in f^{-1}(y)$. The collection of these local submanifold charts forms an atlas for $f^{-1}(y)$, proving it is a submanifold of dimension $\dim M-\dim N$. $\square$