#This file was created by <dlj0> Sat Sep 21 00:35:36 1996
#LyX 0.10 (C) 1995 1996 Matthias Ettrich and the LyX Team
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\language default
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\epsfig dvips
\papersize usletter 
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\baselinestretch 1.00 
\secnumdepth 3 
\tocdepth 3 
\paragraph_separation skip 
\quotes_language english 
\quotes_times 2 
\paperorientation portrait 
\papercolumns 4 
\papersides 2 
\paperpagestyle headings 

\layout LaTeX Title

Math 423, Fall, 1996
\layout Date


\latex latex 

\backslash 
today
\layout Address

Lehigh University
\layout Email
\labelwidthstring E-mail address: 

dlj0@lehigh.
edu
\layout Standard


\latex latex 

\backslash 
maketitle
\layout Section*

Inverse Function Theorems and Coordinates 
\layout Standard

The classical Inverse and Implicit Function Theorems are two of the best
 examples, once interpreted, of the relationship between 
\shape italic 
algebraic
\shape default 
 properties of the differential 
\begin_inset Formula \( \phi _{*} \)
\end_inset 

 and 
\shape italic 
analytic
\shape default 
 properties of  
\begin_inset Formula \( \phi  \)
\end_inset 

 itself.
  
\layout Theorem*


\series bold 

\shape up 
[Inverse Function Theorem (Classical)]
\series default 

\shape default 
 Let 
\begin_inset Formula \( U\subset \R ^{n} \)
\end_inset 

 be open, and let 
\begin_inset Formula \( f:U\rightarrow \R ^{n} \)
\end_inset 

 be 
\begin_inset Formula \( C^{\infty } \)
\end_inset 

.
 If the Jacobian matrix 
\begin_inset Formula 
\[
\left[ \frac{\partial f^{i}}{\partial x^{j}}\right] \]

\end_inset 

 is nonsingular at 
\begin_inset Formula \( x_{0}\in U \)
\end_inset 

, then there is a neighborhood 
\begin_inset Formula \( V\subset U \)
\end_inset 

 of 
\begin_inset Formula \( x_{0} \)
\end_inset 

 (and 
\begin_inset Formula \( W\subset \R ^{n} \)
\end_inset 

 of 
\begin_inset Formula \( f(x_{0}) \)
\end_inset 

), so that  
\begin_inset Formula \( f:V\rightarrow W \)
\end_inset 

 is a bijection, and so that 
\begin_inset Formula \( f:V\rightarrow W \)
\end_inset 

 and 
\begin_inset Formula \( f^{-1}:W\rightarrow V \)
\end_inset 

 are both 
\begin_inset Formula \( C^{\infty } \)
\end_inset 

.
 
\layout Proof

Consult the Math 220 text.

\layout Definition*

The following terminology is standard, and will make it easier to discuss
 this material.
 Let 
\begin_inset Formula \( \phi :M\rightarrow N \)
\end_inset 

 be smooth.
 
\layout Itemize


\begin_inset Formula \( \phi  \)
\end_inset 

 is an 
\shape italic 
immersion
\shape default 
 if 
\begin_inset Formula \( \phi _{*} \)
\end_inset 

 is everywhere injective.
 
\layout Itemize


\begin_inset Formula \( \phi  \)
\end_inset 

 is an 
\shape italic 
submersion
\shape default 
 if 
\begin_inset Formula \( \phi _{*} \)
\end_inset 

 is everywhere surjective.
 
\layout Itemize
\cursor 24 

\begin_inset Formula \( \phi :M\rightarrow N \)
\end_inset 

 is 
\shape italic 
submanifold
\shape default 
 if 
\begin_inset Formula \( \phi  \)
\end_inset 

 i
\begin_inset Formula \( \supset \sqsubseteq \supseteq  \)
\end_inset 

s an injective immersion.
 
\layout Itemize


\begin_inset Formula \( \phi :M\rightarrow N \)
\end_inset 

 is an 
\shape italic 
embedding
\shape default 
 if it is a submanifold which is a  homeomorphism onto 
\begin_inset Formula \( \phi (M)\subset N \)
\end_inset 

 as a topological subspace.
 
\layout Definition*

A set of smooth functions 
\begin_inset Formula \( \{y^{1},\dots ,y^{k}\} \)
\end_inset 

 defined on some neighborhood of 
\begin_inset Formula \( x\in M \)
\end_inset 

 is called 
\shape italic 
independent
\shape default 
 at 
\begin_inset Formula \( x \)
\end_inset 

 if 
\begin_inset Formula \( \{dy^{1},\dots ,dy^{k}\} \)
\end_inset 

 are linearly independent in 
\begin_inset Formula \( T_{*}(M,x) \)
\end_inset 

.
 
\layout Corollary*


\series bold 

\shape up 
(a)
\series default 

\shape default 
 If 
\begin_inset Formula \( \psi :M\rightarrow N \)
\end_inset 

 is such that 
\begin_inset Formula \( \psi _{*}:T_{*}(M,x)\rightarrow T_{*}(N,\psi (x)) \)
\end_inset 

 is an isomorphism, then there is a neighborhood 
\begin_inset Formula \( U \)
\end_inset 

 of 
\begin_inset Formula \( x \)
\end_inset 

 (and 
\begin_inset Formula \( W \)
\end_inset 

 of 
\begin_inset Formula \( \psi (x) \)
\end_inset 

) so that 
\begin_inset Formula \( \psi :U\rightarrow W \)
\end_inset 

 is a diffeomorphism.
 
\layout Proof

This is nothing but a re-statement of the Inverse Function Theorem, couched
 in manifold clothing.
 
\layout Corollary*


\series bold 

\shape up 
(b) 
\series default 

\shape default 
If 
\begin_inset Formula \( \{y^{1},\dots ,y^{n}\} \)
\end_inset 

 is an independent set of functions on an 
\begin_inset Formula \( n \)
\end_inset 

-manifold 
\begin_inset Formula \( M \)
\end_inset 

 at 
\begin_inset Formula \( x \)
\end_inset 

, they form a coordinate system in a  neighborhood of 
\begin_inset Formula \( x \)
\end_inset 

.
 
\layout Proof

This is also a trivial consequence.
 Such a set of functions  really ought to be called a 
\shape italic 
basis
\shape default 
 of functions at 
\begin_inset Formula \( x \)
\end_inset 

.
 
\layout Corollary*


\series bold 

\shape up 
(c)
\series default 

\shape default 
 If 
\begin_inset Formula \( \{y^{1},\dots ,y^{k}\} \)
\end_inset 

 is an independent set of functions on an 
\begin_inset Formula \( n \)
\end_inset 

-manifold 
\begin_inset Formula \( M \)
\end_inset 

 at 
\begin_inset Formula \( x \)
\end_inset 

, where 
\begin_inset Formula \( k<n \)
\end_inset 

.
 Then they form part of a coordinate system in a neighborhood of 
\begin_inset Formula \( x \)
\end_inset 

, in that there are functions 
\begin_inset Formula \( \{y^{k+1},\dots ,y^{n}\} \)
\end_inset 

 so that 
\begin_inset Formula \( \{y^{1},\dots ,y^{n}\} \)
\end_inset 

 form a coordinate system in a neighborhood of 
\begin_inset Formula \( x \)
\end_inset 

.
 
\layout Proof

This is perhaps less trivial.
 However, it can be reduced to the previous case.
 Let 
\begin_inset Formula \( \{x^{1},\dots ,x^{n}\} \)
\end_inset 

 be a coordinate system in a  neighborhood of 
\begin_inset Formula \( x \)
\end_inset 

.
 For some 
\begin_inset Formula \( j \)
\end_inset 

, 
\begin_inset Formula \( \{y^{1},\dots ,y^{k},x^{j}\} \)
\end_inset 

 will still be independent at 
\begin_inset Formula \( x \)
\end_inset 

, since 
\begin_inset Formula \( \{dx^{1},\dots ,x^{n}\} \)
\end_inset 

 is a basis of 
\begin_inset Formula \( T^{*}(M,x) \)
\end_inset 

,  while 
\begin_inset Formula \( \{dy^{1},\dots ,y^{k}\} \)
\end_inset 

 is an independent set which does 
\shape italic 
not
\shape default 
 span.
 Continue adding until the resulting set spans at 
\begin_inset Formula \( x \)
\end_inset 

, which by Corollary (b) will imply that the new set is a coordinate system.

\layout Corollary*


\series bold 

\shape up 
(d)
\series default 

\shape default 
 If 
\begin_inset Formula \( \psi :M^{n}\rightarrow N^{p} \)
\end_inset 

 is such that  
\begin_inset Formula \( \psi _{*}:T_{*}(M,x)\rightarrow T_{*}(N,\psi (x)) \)
\end_inset 

 is surjective.
 Then the pull-back 
\begin_inset Formula \( \{y^{1}\circ \psi ,\dots ,y^{p}\circ \psi \} \)
\end_inset 

 of a coordinate system on a neighborhood of 
\begin_inset Formula \( \psi (x) \)
\end_inset 

 in 
\begin_inset Formula \( N \)
\end_inset 

 will be part of a coordinate system on 
\begin_inset Formula \( M \)
\end_inset 

 near 
\begin_inset Formula \( x \)
\end_inset 

.
 
\layout Proof

Since 
\begin_inset Formula \( \psi _{*} \)
\end_inset 

is surjective, 
\begin_inset Formula \( \psi ^{*} \)
\end_inset 

 is injective.
 Then, since 
\begin_inset Formula \( \{dy^{1},\dots ,dy^{p}\} \)
\end_inset 

 is independent, so is  
\begin_inset Formula \( \{\psi _{*}(dy^{1}),\dots ,\psi _{*}(dy^{p})\} \)
\end_inset 

.
 This brings us back to (c).

\layout Corollary*


\series bold 

\shape up 
(e)
\series default 

\shape default 
 If 
\begin_inset Formula \( \{y^{1},\dots ,y^{q}\} \)
\end_inset 

 is an set of  functions on an 
\begin_inset Formula \( n \)
\end_inset 

-manifold 
\begin_inset Formula \( M \)
\end_inset 

 at 
\begin_inset Formula \( x \)
\end_inset 

 so that 
\begin_inset Formula \( \{dy^{q}\} \)
\end_inset 

 spans 
\begin_inset Formula \( T^{*}(M,x) \)
\end_inset 

.
 Then, a subset form a coordinate system in a  neighborhood of 
\begin_inset Formula \( x \)
\end_inset 

.
 
\layout Corollary*


\series bold 

\shape up 
(f) 
\series default 

\shape default 
If 
\begin_inset Formula \( \psi :M^{n}\rightarrow N^{p} \)
\end_inset 

 is such that  
\begin_inset Formula \( \psi _{*}:T_{*}(M,x)\rightarrow T_{*}(N,\psi (x)) \)
\end_inset 

 is injective.
 Then a subset of  the pull-back  
\begin_inset Formula \( \{y^{1}\circ \psi ,\dots ,y^{p}\circ \psi \} \)
\end_inset 

 of a coordinate system  on a neighborhood of 
\begin_inset Formula \( \psi (x) \)
\end_inset 

 in 
\begin_inset Formula \( N \)
\end_inset 

 will form a coordinate system on 
\begin_inset Formula \( M \)
\end_inset 

 near 
\begin_inset Formula \( x \)
\end_inset 

.
 
\layout Subsection*

Slices 
\layout Standard

Let 
\begin_inset Formula \( \{x^{1},\dots x^{n}\} \)
\end_inset 

 be a coordinate system on a neighborhood 
\begin_inset Formula \( U \)
\end_inset 

 of 
\begin_inset Formula \( M \)
\end_inset 

 and let 
\begin_inset Formula \( c\leq n \)
\end_inset 

.
 Let  
\begin_inset Formula \( S_{a^{c+1},\dots ,a^{n}}:=\{x\in U\, |\, x^{i}=a^{i},c<i\leq n\}. \)
\end_inset 


\layout Standard

Then, 
\begin_inset Formula \( S \)
\end_inset 

 is a submanifold of 
\begin_inset Formula \( M \)
\end_inset 

, called a 
\shape italic 
slice
\shape default 
 of the coordinate system.
 In these coordinates, the slice 
\begin_inset Formula \( S \)
\end_inset 

 is a coordinate 
\begin_inset Formula \( (c-1) \)
\end_inset 

--plane.

\layout Theorem*


\series bold 

\shape up 
[Slice Theorem]
\series default 

\shape default 
 Let 
\begin_inset Formula \( \psi :M^{n}\rightarrow N^{p} \)
\end_inset 

 be an immersion, and let 
\begin_inset Formula \( x\in M \)
\end_inset 

.
 Then there is a neighborhood 
\begin_inset Formula \( U \)
\end_inset 

 of 
\begin_inset Formula \( x \)
\end_inset 

 and a coordinate system on 
\begin_inset Formula \( N \)
\end_inset 

 for which 
\begin_inset Formula \( \psi (U) \)
\end_inset 

 is a slice of the coordinate system in a neighborhood of 
\begin_inset Formula \( \psi (x) \)
\end_inset 

.
 
\layout Proof

The trick is to set up correctly the coordinates.
 Let 
\begin_inset Formula \( \tau  \)
\end_inset 

 be a coordinate chart on 
\begin_inset Formula \( N \)
\end_inset 

 in a neighborhood 
\begin_inset Formula \( W \)
\end_inset 

 of 
\begin_inset Formula \( \psi (x) \)
\end_inset 

, with coordinates  
\begin_inset Formula \( \{y^{1},\dots ,y^{p}\} \)
\end_inset 

.
 By (f), re-number the coordinates so that the first 
\begin_inset Formula \( n \)
\end_inset 

 pull back to coordinates on 
\begin_inset Formula \( M \)
\end_inset 

 near 
\begin_inset Formula \( x \)
\end_inset 

, defining a coordinate system 
\begin_inset Formula \( \widetilde{\tau } \)
\end_inset 

.
 Define new functions 
\begin_inset Formula \( \{x^{i}\} \)
\end_inset 

 by 
\begin_inset Formula 
\[
x^{i}\, :=\left\{ \begin{array}{ll}
y^{i}, & \mathrm{if}\, i\leq n\\
y^{i}-y^{i}\circ \psi \circ \widetilde{\tau }^{-1}\circ \pi _{n}\circ \tau , & \mathrm{if}\, i>n
\end{array}
\right. \]

\end_inset 

 where 
\begin_inset Formula \( \pi _{n} \)
\end_inset 

 is the projection 
\begin_inset Formula \( \R ^{p}\rightarrow \R ^{n} \)
\end_inset 

 onto the first 
\begin_inset Formula \( n \)
\end_inset 

 coordinates.
 
\newline 
These are independent near 
\begin_inset Formula \( \psi (x) \)
\end_inset 

, and 
\begin_inset Formula \( \psi (M) \)
\end_inset 

 is clearly given as the slice 
\begin_inset Formula \( x^{n+1}=0,\dots ,x^{p}=0 \)
\end_inset 

.
 
\layout Corollary*

If 
\begin_inset Formula \( \psi :M\rightarrow N \)
\end_inset 

 is an embedding, and 
\begin_inset Formula \( \psi (M) \)
\end_inset 

 is closed, then each 
\begin_inset Formula \( g\in C^{\infty }(M) \)
\end_inset 

 has a smooth extension into  
\begin_inset Formula \( C^{\infty }(N) \)
\end_inset 

.
 
\layout Proof

This is a simple partition of unity argument applied to  neighborhoods generated
 from coordinate systems as in the Slice Theorem.
    
