Friday, April 15, 2022

An Introduction to Differential Geometry (Part 2)

 From a general point of view in developing the basis of differential geometry  it is useful to consider the concept of a group and its connection to geometry. 

It can be shown that by assuming the mappings: 

T:  x' i   F i  (x1 ,  x2 , x3)

are one-to-one and defined on the whole space   i.e. mappings of the whole space 3   onto itself we can expect:

x' i   F i  (x1 ,  x2 , x3),      x' j   F' i  (x'1 ,  x'2 , x'3)    (i, j = 1,2,3........)

To be effected successively, thereby enabling us to obtain a composite mapping (or product) of the mappings such that: 

x' j  {F j  (x1 ,  x2 , x3), F2  (x1 ,  x2 , x3), F3  (x1 ,  x2 , x3)  }

Thus a set of mappings:   T:  x' i   F i  (x1 ,  x2 , x3) 

Is called a group of mappings or a transformation group if  has the following properties:

i) The identical mapping  x'i =    x i   (i= 1,2,3......) is contained in G

ii)  The inverse  -1   mapping of any mapping T  contained in G  is also an element of  G

iii) For any arbitrary pair of mappings contained in  G   the product of these mappings is also an element of  G

Then the direct congruent transformations form a group.  Related to the above there are three axioms of equivalence:  

1) Every geometric configuration is equivalent to itself

2) If a configuration A is equivalent to a configuration B then B is also equivalent to A.

3) If a configuration A is equivalent to a configuration B and B is also equivalent to configuration C  then A is also equivalent to C.  

Of particular importance are invariants (denoted I)  corresponding to certain geometric objects which will remain unaffected by any mapping contained in G .  Then I  is said to be an invariant of the group  G .   For example the distance:  

d  =  Ö å 3 =1 ( y i  -  x i ) 2

Between two points ( y i )  and ( x i) of  3  is an invariant with respect to the group of displacements.  If related to some motion it will in general cause a change in the value of the coordinates, but d remains the same value.  From the preceding we can consider some geometry which will be identical with  a class of invariants for a certain transformation group.  For example, affine geometry corresponds to the group of affine transformations:

x i  =  å 3 =1   a ik  x k  +  b    det (ik )   

Of projective geometry to the group of projective transformations:

  x i  =  å 3 =1   b ik  x k  +  b ik  å 3 =1   ik  x k  +  b i 

 det (b ik )    0 

Now, since vector components are invariant with respect to translations, the basic transformation shown in Part 1 (see conditions (1), (2) of that post) corresponds to a transformation of the components of a vector which is of the form:

x' i  =  å 3 =1   ik  x k  ,  å 3 =1   ik  a il  =  d kl  ,     


det (ik ) =  -1   

The preceding transformation property is basic for any vectors in Euclidean space.  A vector in that space can also be defined as an ordered system of 3 numbers - called components - which behave according to the transformation property shown.  This is provided a transformation corresponding to conditions (1), (2) of Part 1, are imposed.    

In 3D Euclidean space  ( 3 )  a vector having the origin of the coordinate system (x1 ,  x2 , x3) as its initial point and a point Q as terminal point is called the position vector of Q with respect to that particular coordinate system.  



Then the coordinates of that position vector  Q will have the same numerical value as the coordinates of the point Q.   Given a fixed coordinate system, any point in space can be uniquely determined by a certain position vector. (N.B.  The origin of the coordinate system will change if the coordinates are transformed according to that prescription defined for affine transformations)

See Also:

And:



Suggested Problems:

1) A student proposes to form a group of translations by mapping points from the line y= x +1 to the line y = x + 3,  to form a parallelogram, e.g.

Explain how this might be done and briefly describe the nature of the group.

2)  For the box shown below (with sides a = 4, b = 6, and c = 2), in a Euclidean space 3  find the coordinates of the position vector Q and its length.

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