Riemann space

[Anville Azote]

On 11/28/06, Ya Sam <takoitov at hotmail.com> wrote:
> Another appeal to the Renaissance men (and women) on the list, who dig both
> sciences and humanities. I read a dozen of articles trying to get an
> approximate layman's notion of what it is and I have to confess that I don't
> get it. Is there a simple or even simplistic, but nonetheless digestible way
> to explain it using some easily perceptible images? I'm helpless with
> maths...
>

First, start with the notion of "space".  What does "space" mean to a
mathematician?  Fundamentally, in order to behave in a reasonable way,
a space must incorporate some notion of distance.  I can think of at
least two ways to assign distances on the Earth's surface, for
example:  for any points A and B, the distance between them can be
either the straight-line separation through the body of the planet or
the "great circle" course along the surface.  (A great circle is a
circle which divides a sphere into two equal halves; segments of great
circles are the shortest possible paths between points on the sphere
if you don't leave the sphere.)

For any two points A and B, we'd like a rule for calculating the
distance between them.  Think of it as a machine, call it "g", which
takes two points and spits out a number.  We write this process as
"g(A,B)" -- feed points A and B to g and get out a number which
measures the separation between them.  Mathematicians like to require
such rules, called "metrics", to have a few key properties:

First, the value of a metric must always be greater than or equal to
zero.  (Negative distance?  Outside of Negativland, what could that
possibly be. . . .)  Second, the metric function must only work out to
zero if the two points are identical.  In symbols, one would say that
g(A,B) = 0 if and only if A = B.  Also, it shouldn't matter in which
order we feed the points to the metric machine (the distance from
Chicago to Tunguska is the same as that from Tunguska to Chicago).  We
write this as g(A,B) = g(B,A).

Finally, we require that the metric g obey a rule called the "triangle
inequality".  Given any three points A, B and C anywhere in our space,
g must satisfy the relationship

g(A,B) + g(B,C) >= g(A,C).

This says that the distance directly between A and C can only be less
than or equal to the total distance you travel if you go from A to
some other point B and then from B to C.  In our ordinary, familiar
Euclidean space, we can see this apply:  the only way the journey from
A to B and thence to C can be the same length as the straight shot
from A to C is if B is on the line between A and C.  Otherwise, g(A,B)
+ g(B,C) will be bigger than g(A,C).  Imposing the triangle inequality
is a way to get some of the properties of familiar space to apply to
much more esoteric mathematical constructions.

It might be a fun exercise to show that the "Manhattan distance"
qualifies as a valid metric.  Consider a two-dimensional plane with x
and y coordinates, x measuring east-west (say) and y measuring
north-south.  The Manhattan distance between A and B is

g(A,B) = |Ax - Bx| + |Ay - By|

(where "Ax" means the x-coordinate of point A, etc.).  Also known as
the "taxicab metric", this represents the distance you have to travel
to get from A to B if you can only move along the Manhattan street
grid, with no diagonals.  Another, equally valid metric is the
Pythagorean,

g(A,B) = sqrt((Ax - Bx)^2 + (Ay - By)^2),

which we can visualize by making the line segment AB the hypotenuse of
a right triangle and applying the Pythagorean theorem.

A "manifold" is, loosely speaking, an object made of pieces which look
on the small scale like Euclidean space.  Imagine stitching together
several sheets of rubber, which you can flex and twist however you
like.  If we can define coordinates (latitude and longitude, say) on
each piece and have the coordinate grids on adjoining pieces mesh
together in a nice way (which technically has to do with derivatives
being well-behaved), then we have a manifold.

Assorted readings:

http://scienceblogs.com/goodmath/2006/08/metric_spaces.php

http://scienceblogs.com/goodmath/2006/10/meet_the_manifolds.php

And in fact, the rest of Mark Chu-Carroll's topology series:

http://scienceblogs.com/goodmath/goodmath/topology/