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  R ³  i.e. mappings of the whole space R 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 G  of mappings:   T:  x' _i  =  F _i  (x₁ ,  x₂ , x₃) 

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

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

ii)  The inverse  T ^-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 _b =1 ( y _i  -  x _i ) ²

Between two points ( y _i )  and ( x _i) of  R 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 _k =1   a _ik  x _k  +  b _i     det (a _ik )  ≠  0 

Of projective geometry to the group of projective transformations:

  x _i  =  å ³ _b =1   b _ik  x _k  +  b _ik  / å ³ _b =1   a _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 _k =1   a _ik  x _k  ,  å ³ _i =1   a _ik  a _il  =  d _{kl  ,}     

det (a _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  ( R 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:

• Revisiting Vectors And Solid Geometry (Part 1)

And:

• Revisiting Vectors & Solid Geometry (Part 2)

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 R ³  find the coordinates of the position vector Q and its length.