🚀 go-pugleaf

Messrs AUSTERN, BAEZ, CHOW, FLOWER, HALL, MOORE, PRATT, RUSIN, SMITH       
VIDUGIRIS, WEISS, WIENER

  I am in my office and on my bookshelf I have  the following books:

  Differential Manifolds by G. de Rham, 1984
  Differential Manifolds by Y. Matsushima, 1982
  Differential Forms in Algebraic Topology by R.Bott and L.W. Tu, 1982
  Complex Cobordism and Stable Homotopy Groups, by D.l Ravenel, 1986
  An Introduction to Algebraic Topology by J.Rotman 1988
  Linear Spaces and Differentiation by A. Froelicher and A. Kriegl,1988
  Everyone of the above books has a Definition of a Differentiable Manifold. 

I also read your contributions, views, and discussions concerning various
approaches in connection with a definition of a Differentiable Manifold.
  
 So, I have been exposed (in fact rather technically) to the notion
of a Differentiable Manifold and can recite its definition as described by
some of you and by the abovementioned books (in fact I have known them
for many, many years!).  I know, that it is desirable to have an intrinsic
definition of a Differentiable manifold  without imbeddings, etc, etc, I
know all of that!  HOWEVER, HOWEVER, HOWEVER, HOWEVER, HOWEVER,
  
 HOWEVER, if you wake me up from a deep sleep around 4 a.m. and ask me
     
 WHAT IS A 2-DIMENSIONAL DIFFERENTIABLE MANIFOLD OF A  EUCL. 3-SPACE R^3
I will answer you as followws which is stored in my subconscious and there-
fore which has touched me deeply and convincingly  and which is all content
with minimum amount of noise ! 
 
 ANSWER:  A 2-dimensional Differentiable manifold of R^3 space  is the
         set of all  3-tuples   (x,y,z)  which satisfy some system of
         3-equations in  2 parameters  p, q  of the following form:


      x =  x(p,q)        with parameters  p, q ranging over some pairs of   
                         real numbers such that  x(p,q), y(p,q), z(p,q)   
 (1)  y =  y(p,q)        have part. deriv. of all orders and the Jacobian  
         
      z =  z(p,q)        |  dx/dp  dx/dq |    has rank 2 for  p,q
                         |  dy/dp  dy/dq |    ranging over the above 
                         |  dz/dp  dz/dq |    pairs.


In the Jacobian determinant all derivatives are partial.  I don't have>a "round d" on my keybord !

     Here is an example of a 2-diff. manifold of R^3
 

      x = cos p + cos q              |  -sin p     -sin q |    where this
                                     |                    |    Jacobian has  
(2)   y = sin p + sin q              |   cos p      cos q |    rank 2 if
                                     |                    |    sin(q-p) is
      z = (-1-2cos(u-v)^(1/2)        |   dz/du      dz/dv |    not zero

 (it is too late and I left the third line in the Jacobian in implicit 
  form.  Believe me I can partial differentiate!)
eliminating  p, q  in the above  3 equations we obtain
      
      z = (-x^2 - y^2 + 1)^(1/2)   which shows that (2) is the upper shell

of the unit sphere which is a 2- diff. manifold in R^3


    Now, any 2-diff  manifold in  R^4 space  will be the set of all  4-tuples 

(x,y,z,t)   which satisfy some system of 4 equations in 2-parameters p,q

of the following form:

       x = x(p,q)         with parameters p,q range over pairs of real num-
                          bers such that the functions x(p,q), y(p,q),   
(3)    y = y(p,q)         z(p,q), t(p,q) have part. deriv. of all orders 
                          and the Jacobian 
       z = z(p,q)        
                          | dx/dp    dx/dq |     has rank 2 for  p,q
       t = t(p,q)         | dy/dp    dy/dq |     ranging over the
                          | dz/dp    dz/dq |     above pairs
                          | dt/dp    dt/dq |  


 It has been shown, that for suitable  x(p,q), y(p,q), z(p,q), t(p,q)  
(3) may describe the Klein Bottle.  
 
 Thus, both the 2-sphere  and  KLEIN bottle are 2-dimensional diff. 
manifolds.  

  Hovever  whereas  2-sphere can be described by 3-equations (2)  with 
2 parameters ,the Klein bottle cannot be described by 3-equations in 2 
parameters ,but (as mentioned above) , the Klein bottle can be described
by 4 equations in 2-parameters of the form  (3).

  So, the above is the essential difference between  2-sphere and Klein
bottle.  One has to mentione that 2-sphere can also be described by  4
equations in 2-parameters, in fact by 10 equations in 2-parameters. But
this does not make Klein bottle and 2-spehre diffeomorphic, sincve 2
sphere can by described by 3 equations with 2 parameters whereas Klein
bottle cannot be so described. Indeed, Kline bottle is not diffeomorphic
with 2-spehere.

  In general, if you wake me up at 4 a.m. I will give you the following
general definition.


  DEFINITION.  An m-dimensional Differentiable Manifold of  R^n   (of
course with  m  less than or equal to  n>1) is a set of  n-tuples
(x(1), x(2), x(3), ..., x(n))  which satisfy some  n equations in  m
parameters p(1), p(2), p(3), ..., p(m)  of the form:


           x(1)  = x(1) ( p(1), p(2), p(3), ..., p(m) )     
     
           x(2)  = x(2) ( p(1), p(2), p(3), ..., p(m) )
(4)        x(3)  = x(3) ( p(1), p(2), p(3),  ..., p(m) )
    
            ............................................
            ............................................

           x(n)  = x(n) ( p(1), p(2), p(3),  ..., p(m) ) 

with parameters  p(1), ..., p(m)  ranging over the set of m-tuples of real
numbers such that the functions x(1), ..., x(n)  have part. deriv. of all
orders and such that that  n  by  m  Jacobian given by (5) has rank  m
when parameters range over the m-tuples.      


             |  dx(1)/dp(1)    dx(1)/dp(2)  ...   dx(1)/dp(m) |
             |                                                |
             |  dx(2)/dp(1)    dx(2)/dp(2)  ...   dx(2)/dp(m) | 
(5)          |                                                |
             |  dx(3)/dp(1)    dx(3)/dp(2)  ...   dx(3)/dp(m) |
             |                                                |
             |       ..................................       |
             |       ..................................       |
             |                                                |
             |  dx(n)/dp(1)    dx(n)/dp(2)   ...   dx(n)/dp(m)|  



  Please observe  that an m-diff.manifold of R^n  is  described by
                  n equations in  m parameters such that ....



I only  talk about  m-DIFF.MANIFOLDS IN R^n  (my subconscious stores
only this setup.  But it touches me deeply).

     It is around 1a.m., please do not assume that I do not know all the
other Definitions of an  m-diff. manifiold -  At  4.am.  (in fact , even 
now  at  around  1 a.m)  I LIKE THE ONE THAT I GAVE ABOVE.
 
    Of course, the Jacobian pops up automatically when in (4) one replaces
the column   of  x(1), x(2), ..., x(n)   on the left of  =  by the 
column of  dx(1), dx(2), ..., dx(n)  and performs the taking of the dif-
ferentials and rewriting the whole thing as a matrix.  I have explained
this before.
    It is 1:15 a.m.  I am exhaustd , but enjoyed typing this.  However, 
there may be some typos.  I am in no position to reread.

 
     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)