🚀 go-pugleaf

                                         10-31-92
Dear Mr. PRATT,

   Please, bear with me, read carefully the following example and please
observe that before finishing the example, I am not going to use the
words "Differentiable Manifold" or "Diffeomorphism" contentwise - just 
let us put aside the Cosmetics for a moment.

   In the  x,y-plane, let us  denote by  L  the closed line segment
joining the two points p  and  q  where  (with  pi = 3.14....)
 
  
(23)     the coordinates of  p  are    x =  pi/6   and  y =  pi/6
and
(24)     the coordinates of  q  are    x =  pi/3   and   y = pi/3   

  So,  pq  is the closed line segment  on the diagonal of the first
  quadrant of the  x,y-plane



  y                                   v
  |        /q                         |    Q.-.
  |      /                            |        -.
  |    /                              |        P -.
  |   p                               |
  |___________x                       |______________ u      
                      Fig.1

  
In the u,v-plane, let us denote by   C  the closed arc segment of the unit
circle  (a circle with  (0,0)  for center and with radius  1)  joining the
two points  P  and  Q  where   

(25)    the coordinates of  P  are  u = cos pi/6    and   v = sin pi/6
and
(26)    the coordinates of  Q  are  u = cos pi/3     and  v = sin pi/3

  The closed line segment  L  is shown  on the left side of Fig.1  and
the closed arc  C  is shown on the right side of Fig.1.

  I will show that    L  is diffeomorphic  with  C   (nevermind the word
diffeomorphic),  i.e.,  I will show that

(27)  there exists a  one-to-one mapping  F  from  L  onto  C  which
      is differentiable (nevermind this word), i.e., F  has partial
      derivatives of all orders and such that the inverse map of  F
      is also differentiable.

 I propose the mapping  F  given by:  

(28)       F(x,y)  =  (u(x,y), v(x,y)  =  (cos x + 0y ,   0x +  sin y) 
                                   
                                       =  (u, v)   for short

 Thus,    F  is defined by:

(29)      u  =  cos x  +  0y             with    x  in  [pi/6, pi/3]
          
          v  =    0 x  +  sin y          and     y  in  [pi/6, pi/3] 
          
Clearly, F  as given by (28)  or (29)  is differentiable since  cos x  and
sin y   have partial derivatives  (in this special case just, derivatives)
of all orders.  Also, it is trivial to verify that  F  is an onto mapping
and that, in particular  F(p) = P   and  F(q) = Q.
 
It remains to show that  the inverse of  F  is also differentiable ON 
THE ARC  C .
 
  To this end, as explained in my previous articles (to Mr. Baez),
I consider the transformation of the differentials.  From (29) it
follows: 
    
         du  =  -sinx dx + 0 dy            | -sinx   0    | dx
(30)                                   =   |              | 
         dv  =      0 dx + cos y dy        |    0    cosy | dy

 
The Jacobian DF of F, i.e., the determinant of the matrix in  (30) is


(31)     DF  = - sinx cos y   IS NONVANISHING FOR EVERY POINT ON L,        
                  
           i.e., for ever  x in [pi/6, pi/3]  and  y in [pi/6, pi/3]  
      
 Thus,  Indeed the inverse function  F  is also differentiable  on every
point of  C and  (27) is established.

   People in differential topology, REJOICE when a configuration, say  L 
is diffeomorphic with another, say  C.  Because in that case from
differential topology point of view   L  and  C  become INDISTINGUISH-
ABLE AND ONE CAN STUDY ONLY ONE OF  L  AND  C,  WHICHEVER IS MORE CON-
VENIENT.  
   
  Since, in our above example (see Fig.1), AS WE SHOWED the closed line
segment  L  is diffeomorphic to the closed arc segment  C, from differen-
tial topology point of view, we may as well abandon the arc segment  C  
and study only the line segment  L.


   Now, Mr. Pratt, contrary to what you said (if I understood you correctly)
I have given in the last part of my DIFF.part 3 the Definitions of a Dif-
ferentiable manifold and the definition of Diffeomorphism from one mani-
fold onto another. Of course a manifold  L  is called  DIFFEOMORPHIC to
another manifold  C  if and only if  there is a diffeomorphism from  L
onto  C  (in which case  C  and  L  become indistinguishable for diffe-
rential topologists an they rejoice, by studying  only one of them).
   
  Now, I repeat again, L as given in my example is a one dimensional sub-
manifol of the 2-dimensional manifold R2. Similarly, C  is a differen-
tiable submanifold of  R2. Do not worry about the cosmetics. A differentiable 
manifold is a set from every point of which an infinitesimal Euclidean 
coordinate system dx, dy, dz, ...., dt   emanates and you can deal with it 
as I have dealt.
I have also several times mentioned when a mapping is called a Diffeo-
morphism,

   Now, if you are still dissatisfied  (or ripped off by me as you warned
all the readers)  I would suggest that you take my  3 lectures and this one to
a specialist at STANFORD  (surely there must be some specialists at
STANFORD) and asked them to go over my  3 lectures and ESPECIALLY THE 
PRESENT ARTICLE  and explain them to you. Also, you mentioned at least 5
books (there are at least 30 -50 such books) and you recommended one of
them - why don't you consult the one that you recommended to the readers. 
  I recommend to the readers to read my 3 lectures and the present example 
written especially for you.

  There may be some typos in this article.  There were some typos in my
Diff. part 1.  A couple of days ago I posted their corrections.
     
    
     With best wishes and regards,

                                            Alexander ABIAN

















-- 
   The tendency of maintaining the status-quo, Reaction to provocation and
                The tendency of maintaining again a status-quo.  
    TIME HAS INERTIA  and some energy is lost to move Time forward  
  E = mcc  (Einstein)    must be replaced by    E = m(0) exp(-At) (Abian)