Analysis Theorems

9 Geometry of Mappings

Theorem 9.1 (Inverse Mapping Theorem). If \(f:\FF ^n\to \FF ^n\) is \(\cC ^1\) near a point \(a\) and \(f'(a)\) is invertible near the origin, then \(f\) is a \(\cC ^1\) diffeomorphism near \(a\).

Proof. WLOG, \(a=f(a) = 0, f'(a) = I\) via an affine transformation. Now as \(r(x) = f(x)-x\) is \(\cC ^1\) near the origin and \(r'(0) = 0\), we have \(\Vert r'(x) \Vert _o\leq 1/2\) near the origin. Then by Corollary 4.7 we have \(\Vert r(b)-r(a)\Vert _2\leq \frac {1}{2}\Vert b-a\Vert _2\) near the origin which gives

\[\Vert f(b)-f(a)\Vert = \Vert f(b)-f(a)\Vert +\frac {1}{2}\Vert b-a\Vert -\frac {1} 2 \Vert b-a\Vert \]

\[ \geq \Vert f(b)-f(a)\Vert + \Vert r(b)-r(a)\Vert -\frac {1}{2} \Vert b-a\Vert \geq \frac {1}{2}\Vert b-a\Vert \]

Hence the map is injective, and by Theorem 3.13, this is a local homeomorphism near the origin. Now let \(g\) be the local inverse of \(f\) near \(0\), we would like to show \(g\) is differentiable at \(0\). If \(f(x)=y\) and \(f(x+h) = y+k\), then as \(f\) is differentiable,

\[ f(x+h) - f(x) = f'(x)h + \ee _f(h) \implies k = f'(x)(g(y+k)-g(y)) + \ee _f(h) \]

\[\implies g(y+k)-g(y) = f'(x)^{-1}k-f'(x)^{-1}\ee _f(h)\]

and as \(\Vert f'(x)^{-1}\Vert _o\) is bounded near \(0\), it suffices to show \(\frac {\Vert \ee _f(h)\Vert }{\Vert k\Vert } \to 0\) as \(k \to 0\).

Indeed, we have

\[\frac {\Vert \ee _f(h)\Vert }{\Vert k\Vert }=\frac {\Vert \ee _f(h)\Vert }{\Vert h\Vert }\frac {\Vert h\Vert }{\Vert k \Vert } = \frac {\Vert \ee _f(h)\Vert }{\Vert h\Vert }\frac {\Vert h\Vert }{\Vert f(x+h)-f(x) \Vert }\leq 2\frac {\Vert \ee _f(h)\Vert }{\Vert h\Vert }\frac {\Vert h\Vert }{\Vert h \Vert } \]

which goes to \(0\) as \(k \to 0\). Thus \(g\) is differentiable and since its derivative is the inverse of \(f'\), \(g\) is \(\cC ^1\). □

Theorem 9.2 (Decomposition Theorem). If a map \(f:\RR ^n \to \RR ^n\) is \(\cC ^1\) near a point \(a\) and \(\det (f'(a)) \neq 0\), then near \(a\), \(f\) is the composite of \(\cC ^1\) diffeomorphisms \(\phi _i\) that only change one variable.

Proof. WLOG, \(a = f(a) = 0\). We say that \(f\) is of type \(r\) if \(f\) doesn’t change at least \(r-1\) coordinates. By induction it suffices to show that if \(f\) is type \(r\), then there is a \(\cC ^1\) diffeomorphism \(\phi \) so that \(f \circ \phi \) is type \(r+1\), so we assume \(f\) fixes the first \(r-1\) coordinates, denoting these \(x_I\), and denoting the last \(n-r\) coordinates \(x_{II}\). Then after some relabeling \(f'\) looks like

\[\begin {pmatrix} I_{r-1}&0&0\\ \partial _If_r&\partial _rf_r&\partial _{II}f_r\\ \partial _If_{II}&\partial _rf_{II}&\partial _{II}f_{II} \end {pmatrix}\]

Since \(f'(x)\) is invertible near 0, we can assume \(\partial _rf_r \neq 0\) near by relabeling the \(f_i\). Then we can define \(\psi \) near \(0\) as the function that is \(f_r\) on the \(x_r\) coordinate and the identity on all other coordinates. Its derivative looks like

\[\begin {pmatrix} I_{r-1}&0&0\\ \partial _If_r&\partial _rf_r&\partial _{II}f_r\\ 0&0&I_{n-r} \end {pmatrix}\]

so is invertible and by Theorem 9.1 has a local inverse \(\phi \), which is what we want. □

Note that the \(m =n\) case of the next theorem is the Inverse Mapping Theorem.

Theorem 9.3. If \(f:\RR ^m\to \RR ^n, m\geq n\) is \(\cC ^1\) near a point \(a\) with \(f'(a)\) rank \(n\), then there is a \(\cC ^1\) diffeomorphism \(\phi :\RR ^m \to \RR ^m\) near \(a\) so that \(f\circ \phi -f(a)\) is a linear map near \(a\).

Proof. WLOG, \(a = 0, f(a) = 0\). We label the first \(n\) coordinates of \(\RR ^m\) \(x_I\) and the last \(n-m\) \(x_{II}\). Then \(f'\) looks like

\[\begin {pmatrix} \partial _If_I&\partial _{II}f_I \end {pmatrix}\]

and we define \(\psi :\RR ^m\to \RR ^m\) near \(0\) as \(f_I\) on the first \(n\) coordinates and \(x_{II}\) on the rest. Its derivative looks like

\[\begin {pmatrix} \partial _If_I&\partial _{II}f_I\\ 0&I_{m-n} \end {pmatrix}\]

so by possibly reordering coordinates we can assume it is invertible and we can use Theorem 9.1 to locally make an inverse \(\phi :\RR ^m\to \RR ^m\) that fixes the coordinates \(x_{II}\). Now we have

\[\begin {pmatrix}x_I \\ x_{II}\end {pmatrix} = (\psi \circ \phi )\begin {pmatrix}x_I\\x_{II}\end {pmatrix} = \begin {pmatrix} (f\circ \phi )(x)\\ x_{II}\end {pmatrix}\]

so indeed \(f\circ \phi \) is locally linear. □

Theorem 9.4 (Implicit Function Theorem). If \(f:\RR ^m\to \RR ^n,m>n\) is \(\cC ^1\) near \(0\) and \(\det (\partial _{I}f(0))\neq 0\), then for a small cell \(I^m = I_I\times I_{II}\) near \(0\), there is a \(\cC ^1\) function \(h:I_{II}\to I_{I}\) so that \(f(x_I,x_{II}) = f(0)\) iff \(x_I = h(x_{II})\).

Proof. By Theorem 9.3 we locally have a \(\cC ^1\) map \(\phi :\RR ^m\to \RR ^n\) that is a function \(g\) on the first \(n\) coordinates and the identity on the last \(m-n\) coordinates such that \(f\circ \phi \) is the linear map \((x_I,x_{II})\mapsto (x_I)\). Then we define \(h(x_{II}) = g(0,x_{II})\). Now locally \(f(x) = 0\) iff \(x = \phi (y)\) and \((f\circ \phi )(y)=0\) iff \(x = \phi (0,y_{II})\) iff \(x_I = h(y_{II})=h(x_{II})\). □

Corollary 9.5. Locally a \(\cC ^1\) \(k\)-submanifold of \(\RR ^n\) is given by implicit \(\cC ^1\) functions in \(k\) variables.

Proof. If \(V\) is such a submanifold, after a local change of coordinates at a point \(a\), it is a linear subspace of \(\RR ^n\). We can collapse this subspace, which is \(\cC ^1\) and by Theorem 9.4 the kernel composite of this with the local change of coordinates is given by implicit functions in \(k\) variables. Conversely if \(V\) is given locally by implicit functions in \(k\) variables, those implicit functions are a \(\cC ^1\) diffeomorphism to a linear embedding of \(\RR ^k\) in \(\RR ^n\). □

Note that the \(n=m=r\) case of the Rank Theorem below is the Inverse Mapping Theorem.

Theorem 9.6 (Rank Theorem). If \(f:\RR ^m \to \RR ^n\) is \(\cC ^1\) near \(a\) and \(f'(x)\) is rank \(r\) near \(a\), then there are \(\cC ^1\) maps \(\phi :\RR ^m\to \RR ^m\) defined near \(a\) and \(\theta :\RR ^n\to \RR ^n\) defined near \(f(a)\) such that \(\theta \circ f\circ \phi \) is a linear map of rank \(r\).

Proof. WLOG, \(a =0,f(a) =0\). By relabeling coordinates in the domain and range, we may assume that the principle \(r\times r\) submatrix of \(f'(0)\) is invertible. Let us label the first \(r\) coordinates of \(\RR ^m\) \(x_I\) and the last \(m-r\) \(x_{II}\). Then we can consider the map \(\psi :\RR ^m \to \RR ^m\) that is \(f\) on \(x_I\), and the identity on \(x_{II}\). By hypothesis, \(\psi '(0)\) is invertible, so by the Inverse Mapping Theorem we can let \(\phi \) be its local inverse, which is \(g\) on \(x_I\) and the identity on \(x_{II}\). Now we can call the first \(r\) coordinates of \(\RR ^n\) \(y_I\) and the last \(n-r\) \(y_{II}\). Now \(h = f \circ \phi \) is the map that is \(x_I\) on the first \(r\) coordinates, and \(h_{II}\) on the last \(n-r\). \(h'\) looks like

\[\begin {pmatrix} I_r&0\\\partial _Ih_{II}&\partial _{II}h_{II} \end {pmatrix}\]

but since it is rank \(r\) near \(a\) by the chain rule, we must have \(\partial _{II}h_{II} = 0\), so \(h\) only depends on \(x_I\). Now we can define \(\theta :\RR ^n\to \RR ^n\) near \(0\) as the identity on the first \(r\) coordinates, and \(y_{II} - h_{II}(y_I)\) on the last \(n-r\). Then \(\theta '(0)\) is rank \(n\), and \(\theta \circ h\) is \(x_I\) on the first \(r\) coordinates and \(0\) on the last \(n-r\), which is linear. □

Corollary 9.7. If \(f:\RR ^m\to \RR ^n\) is \(\cC ^1\) near a point \(a\), and \(f'(x)\) is rank \(r\) near \(a\), then the image of a small neighborhood around \(a\) is a \(\cC ^1\) \(r\)-submanifold of \(\RR ^n\), and the preimage of \(f(a)\) is a \(\cC ^1\) \(m-r\)-submanifold if \(f\) is restricted close enough to \(a\).

Proof. Use Theorem 9.6 to obtain \(\phi \) and \(\theta \). Now after applying these \(\cC ^1\) diffeomorphisms, the preimage of \(f(a)\) is the kernel of a linear map, so is an open subset of \(\RR ^{m-r}\). Similarly, the image of a neighborhood of \(a\) is an open subset of \(\RR ^r\). □

Corollary 9.8 (Lagrange Multipliers). If \(g:\RR ^n\to \RR \) is \(\cC ^1\) in an open set \(U\), where \(\cC ^1\) \(r\)-submanifold \(V\) that is the locus of \(f_i,1 \leq n-r\) in \(U\), any extremum \(c\) of \(g\) in \(V\) must satisfy \(\nabla g(c) = \sum _1^{n-r}\lambda _i\nabla f_i(c)\).

Proof. Note that for any parameterized curve \(\phi \) on \(V\) that sends \(0\) to \(c\), we have \(f_i(\phi (c))=0\), so by the chain rule, \(\nabla f_i(c)\cdot \phi '(0) = 0\), so the tangent space is exactly the space perpendicular to the \(\nabla f_i\). Now in order to have an extremum of \(g\) at \(c\) on \(V\), we need \(g'(c) = 0\), but then for any parameterized curve \(\phi \) on \(V\) sending \(0\) to \(x\), again we have \(g(\phi (c)) = 0\), so by the chain rule, \(\nabla g(c)\) lies in the tangent space, so is a linear combination of the \(\nabla f_i(c)\). □