🚀 go-pugleaf

I have defined and given examples of DIFFERENTIABLE MANIFOLDS.
But, if you really want to know the GIST , the naked truth without 
cosmetic paraphernalia and if for once you want to get to the soul of
Manifold, then read the following:


 DEFINITION (Abian)
 
 A manifold is a set of  n-tuples  (x, y, ..., t) satisfying n-equations 
 x = ...,  y = ..., ... ,  t = ..., in  m parameters p, q, ..., v.
   
 PERIOD.
    p, q, ..., v  can range over  Reals, Complexes, Quaternions, 
    or  Matrices, or any abstract Ring, Field, etc, etc

   The corresponding functions depending on the cases can be Real valued,
Complex valued, Quaternion valued, Matrix valued, etc, etc.

 EXAMPLE.
         x = 3p+5q+6v     (say over Reals)  is a Manifold
    
 EXAMPLE.
         x = sinp + e^q
         y = 5 log p - cos 3v^2      (say over ...)    is a Manifold
 
 EXAMPLE.
         
         x = 3u
         y = 6v                 (say over ...)  is a manifold
         z = 7v

 EXAMPLE
 
         x = F(p,q,..., v)
         y = G(p,q,..., v)           I don't care how many  n
                                     equations and how many m
          ...............            parameters.  All I care
                                     is that  n > 0 and  m > 0          
          ...............

         t = H(p,q,..., v)               

  NOW, if you impose conditions on the equations then you will have 
      "RESPECTABLE", NOBLE", "SMOOTH",  "NICE", "GORGEOUS",..  manifolds.

      IF equations are polynomials  THE MANIFOLD IS CALLED ALGEBRAIC.
      IF EQUATIONS ARE  (PARTIALLY) DIFFERENTIABLE (you can have nuances  
      up to some order) OF ANY ORDER, AND THE  RANK OF THE  JACOBIAN 
      n by m  Matrix  is  min {n,m}  THEN YOU HAVE  A DIFFERENTIABLE 
      MANIFOLD. 
   
      So, depending on the conditions that you impose on the equations
      you will have various manifolds and their corresponding Theories.
     
      You can create over night hundreds of (sometimes quite interest-
      ing) Manifolds and call them after your NAME.

      You can  be very abstract - your equations can be MAPPINGS  and
      Defined by English sentences with or without math. formulas.
      Your equations may have integrals in them and ANYTHING YOU LIKE.
    
      SO,  A MANIFOLD  (Algebraic, Differentiable, or .....)  is
      the set of  n tuples  (x,y,z, ..., t) described by
      n equations  x = ..., y =...,  z = ..., ..., t = ...  in
      m  parameters  p, q, ..., v.
                  
        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)