## Friday, August 5, 2022

### An Introduction To Differential Geometry - Part 6

Overview Of Tensors & Operatioins:

Since this part of differential geometry will initiate the use of tensors we begin by going through some preliminaries.  Those readers who've not yet been exposed to tensor algebra can access some examples here:

We have already seen, experienced, developed multiple occasions to use vectors, i.e.

But all such vectors are just tensors of rank 1.

By extension, a tensor of rank 2 (e.g. moment of inertia tensor I i j ):

I i j  =  m [x 2   d i j     x j  x j  ]

The most basic tensor of all is the unit tensor, defined:

1  =  i^ i^ + j^ j^ + k^ k^  =

(1......0........0)

(0........1.......0)

(0........0.......1)

Also:

× C  =  i^ Cx + j^ Cy + k^ Cz

Further properties:

A tensor is symmetric if:  i j   j i

A tensor is anti- symmetric if:  i j   = -  ji

The latter will look something like this:

(0 ……a12………a 13)

(-a12…..0……….a23)

(- a13……-a23……0)

The fundamental  metric tensor : mn  or    ab  *   is of special importance in differential geometry. Thus

mn  =  ab =

(11      12      13           14 )
(
21     - g 22     23         24)
(
31       32      33        34 )
(
41        42       43      44)

A metric which is defined by a quadratic differential form, e.g.

ds2   =  å n m =1   å n =1 mn du m du n

Or:

ds2   =  å n a =1   å n b =1 ab du a du b

Is called a Riemannian metric.  This references a general case in which we have introduced n real dimensions, u1,  2   … n   in order to obtain a Riemannian space.  Going forward we will also use an abbreviated summation convention, e.g.

ds2   ab du a du b

or:  ds2   mn du m du n

Fundamental forms:

F(x,y) =  11 x 2 +  2  12  xy +  22 y 2

Of two real variables x and y is called positive definite if it has a positive value for any pair (x, y)     (0, 0)  At regular points on a surface if these are consistently positive definite we expect a 'first fundamental form'

A curve on some surface  S,: xu1 ,  )   can be determined by a parametric representation, i.e.  u1 =   u1(t),   2  =   u2(t)

Where the parameter t is a real variable.

Example: Any circular helix;  x(t) = (r cos t, r sin t, ct)

is a curve on a surface S and can be represented by:

u1 = t,    2  = ct    or as:  2  = cu1

Of particular importance in differential geometry are the coordinate curves:  u1 = const.,   2  = const.  on a surface S: x ( u1 ,  ).  These correspond to parallels to the coordinate axes in the  u1  2 -plane.  (Refer back to Part 5).  A set of curves on a surface S which depend continuously on a real parameter is said to be a one parameter family of curves on S.  For example a family of parabolas on a Cartesian surface,

A family of curves can also be determined by a suitable linear differential equation of first order, e.g.

dy/ dx = f(x,y)

Or:   d2/ du1     =  F  (u1 , )

The direction of the tangent to a curve on a surface S: x ( u1 ,  )   is determined by the vector:

x' =  dx/dt =  ( x / u1 )   du1 /  dt  +  ( x / u2 )   du2 /  dt =

x 1 u1'  +  x 2  / u2

Which in general depends on t.  Note the vector x is a linear combination of the vectors x 1  and x 2 .

More generally, the first fundamental form is always associated with a Riemannian geometry.  We already saw:

u1 =   u1(t),   2  =   u2(t)

It is instructive here to determine the element of arc of such a curve which leads in a practical application to the first fundamental form.  We have seen above the summation form:

ds2   =  å n a =1   å n b =1 ab du a du b

From which we can also write:  x a · x b  ab

x 1  ·  x 1 g11             x 1  ·  x 2 g12           x 2  ·  x 1 g21

And:    x 2 ·  x 2 22

And the resulting full quadratic form:

ds2   g11 (du1)2   + 2 g12 du1 du2  + g22 (du2)2

Is called the first fundamental form.

Example Problem:  Let a sphere be represented by:

x ( u1 ,  ) =  (r cos 2  cos u1 , r cos 2 sin u1 , r sin  2)

Find the first fundamental form corresponding to this representation:

Soln.

The first partial derivatives are;

x 1    = (- r cos 2  sin u1 , r cos 2 sin u1 , 0 )

x 2   = (- r sin 2  cos u1 , - r sin 2 sin u1 , r cos 2 )

This yields:  x 1  ·  x 1 =  rcos  2 2

x 1  ·  x 2 g12 = 0   And:   x 2 ·  x 2 22  r 2

Then the first fundamental form corresponding to the sphere as represented is:

ds2   rcos  2 2 (du1)r 2 (du2)2

Suggested Problems:

1)  Find the first fundamental form of the cylinder:

x ( u1 ,  ) =  ( h 1 (u1)h 2 (u1) 2)

2) Find the first fundamental forms corresponding to Cartesian and spherical coordinates in space.

3) What differential equation would give rise to the family of curves shown below?