Riemann space

[the Robot Vegetable]

Ok, I'll try.  I studied this a billion years ago, my brain's
gone soft, and this is simplistic. 

I need to assume some things, basically the idea of the cartesian
plane.  An plane, points described by X and Y coordinates, where
certain propeties hold.  They're basically what makes sense in
our world, and the most important things is that there are no
shortcuts. (and things like isf two points are distinct, then
the distance between them is nonzero)

If you have 3 points, and they do not lie in a line, then the distance
between any two points is shorter than if you went through the
third point first.   This is (sort of) the triangle inequality.

The cartesian plane is a metric space is a Riemann Space.

A manifold is just a fancy name for some space, like the plane.  You
got your spheres, and saddle shapes, and potato shapes and whatever.
As long as you can lay a coordinate system on in, and the points
described adhere to metric space properties, then it's a Riemann
Space.

Really, the idea itself is very easy, it's a generalization of
space, where you'll plopped down a set of rules so as to do some 
calculation.  

This is very sloppy, I have a math degree, but all I do day to
day is arithmetic...