An Introduction to Differential Geometry (1)

The basic core of differential geometry entails the application of differential calculus to what we call parametric curves, e.g.  y= x(t).  Consider the curve in 3-space (x,y,z) below:

Here the  position of P becomes a function if the arc length  s  traverses from P ₀  to P on the curve.  Then the vector:

R =  ix  + j y + k z

is also a function of s.   Of particular interest will be the derivative:

dR/ ds =  i(dx/ds) + j(dy/ds) + k(dz/ds)

By taking the limit:

dR/ ds =    lim  D _{s   -> 0}   (D R/ D s)

It can be shown one obtains a unit vector tangent to the curve at P, say, and pointing in the direction in which arc length s increases along the curve. One thereby arrives at the vector T:

T = dR/ ds

But before one can run he must learn to walk and in the same way we need to first become familiar with the basic laws of vector calculus as they apply to Euclidean space.   Thus, consider the scalar product of two vectors  A  and B  is  also called the dot product because of the dot symbol used to denote it.   Thus, in general if  A,  B  are two arbitrary vectors where Θ  denotes the angle between A  and  B  as shown below:

Then we can write:

 A ·B    =  ‖A‖ ‖B‖   cos Θ,

hence also:

cos(Θ) = (A ·B)/ ‖A‖ ‖B‖ 

Note that as well:

cos(Θ) =   A ·B   / Ö A ·A  Ö B ·B 

This is important in context of differential geometry   

And we have in the transformed system   A ·B    = A1 B1   = ‖A‖ ‖B‖   cos Θ,                  

Thereby we see that two non-null vectors are orthogonal to each other (i.e. at 90 degrees) if and only if their scalar product vanishes.  This means we have: 

 cos(Θ) = (A ·B)/ [A][B]   =  0   

Meanwhile the vector product of two vectors, i.e.  V  =  A x B   is defined by:

V=  A x B  =

   [ e ₁      e  ₂    e₃ ]

   [  a ₁      a  ₂    a₃]

   [  b ₁     b  ₂    b ₃] 

Where e _i  denotes a unit vector having the positive direction of the ith coordinate axis of the Cartesian coordinate system in space R ₃.

This is a prelude to getting a handle on right-handed and left -handed orthogonal systems of coordinates, which we also need to proceed further. Such a right-handed system (in 3 dimensions) is shown below:

With orthogonal parallel coordinates (x₁ ,  x₂ , x₃) correspond to points: (1,0,0),  (0,1,0) and (0, 0, 1), respectively, so each have the distance '1' from the origin.  In general, a coordinate system is called right-handed if the axes assume the same sort of orientation as the thumb, index finger and middle finger of the right hand. A system is said to be left-handed if the axes assume the same sort of orientation - in their natural configuration - as the thumb, index finger and middle finger of the left hand.  Thus we will have:

The use of the notation (x₁ ,  x₂ , x₃)  for the orthogonal coordinates is more convenient  than the familiar (x, y, z) since it enables the use of the more compact form x _i for the coordinates   of a point).   Then any other similar Cartesian coordinate system  e.g.  (x'₁ ,  x'₂ , x'₃)  will be related to the given one by a particular linear transformation of the form:

x' _i  =  å ³ _{b =1}   a _ik  x _k  +  b _i

Whose coefficients satisfy the conditions: 

1)  å ³ _b =1   a _ik  a _il  =  d _kl   =   

                     {0    for  k ≠  1    

                            {1  for   k  =  1

2)   d _ik = a _ik   =

(1….0…..0)

(0….1…..0)

(0….0… ..1)

Where  d _ik   is the Kronecker  delta

.

The transition from one Cartesian coordinate system to another can always be effected by an appropriate rigid motion of the axis of the original system. Usually this entails a suitable translation and a suitable rotation. Any rigid motion which carries a Cartesian coordinate system into another Cartesian system is called a direct congruent transformation. 

 From the preceding we can also write:

A  =   a _ik     =  

   [ a ₁₁      a ₁₂   a₁₃]

   [ a ₁₂₁    a ₂₂   a₂₃]

   [ a ₃₁     a  ₃₂   a₃₃]

Which is a quadratic matrix. In general we note that a system of   m · n   quantities arranged in a rectangular array of m horizontal rows and n vertical columns is called a matrix, and the quantities are the elements of the matrix.   If m equals n the matrix is called 'square'  and the number n is the order of the matrix. Thus, the coefficients  a _ik  for the preceding linear transformation equation form a quadratic matrix.  The corresponding determinant is then:

det (A)  =  det (a _ik )  

 If in particular A  equals the Kronecker delta (unit matrix) then the new Cartesian coordinate system is given by:

x'_{i =    x i  +   b i}             i=  1, 2, 3.....

Which is a translation of the coordinate system.  If, however,   _b i =  0  then we find: 

x'_{i =    x i} 

And the transformed coordinates are the same as the original ones.  Such a special transformation is called the identical transformation.  If _b i =  0  and the coordinates  a _ik   are arbitrary, but such that the two conditions (1) and (2)  are satisfied, then:

  

x' _i  =  å ³ _b =1   a _ik  x _k  +  b _i  

 Corresponds to a rotation of the coordinate system with the origin (0,0,0 ) as center.  Such a rotation is called a direct orthogonal transformation.   Finally,  note that a transformation 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   

Can be geometrically interpreted as a motion composed of a rotation about the origin and a reflection in the plane.  This type of transformation is called an opposite orthogonal transformation because it transforms a right-handed coordinate system into a left-handed one and vice versa.

Suggested Problems:

1) Consider two vectors A, B spanning a subspace of   R ⁴   where:

A = (1, 2, 1, 0) and B = (1, 2, 3, 1)  

Find:  A/ ‖A ‖    and:  B/ ‖A‖

2)  Given  x'_{i =    x i  +   b i}

And:  x _i   =

(2….1…..0)

(3….-5.….6)

(-7….0… .4)

Find the new coordinates x'_i   if :

  _b i  =

(1….2…..1)

(2….1.….3)

(3….2… .1)

Write out the matrix elements for  a _ik .

Next: Incorporating group theory into differential geometry.