Riemann space

[Ray Easton]

On Tuesday, Nov 28, 2006, at 13:16 US/Central, Ya Sam wrote:

> So Riemann had the idea of the curved space before Einstein?

There simply are curved spaces -- for example, the surface of a sphere. 
  Gauss was the first to find a notion of "curvature" that was intrinsic 
to surfaces (two-dimensional spaces) --  a notion did not depend upon 
the way the surface was embedded into some three-dimensional space that 
contained it.  Riemann generalized this Gaussian notion to higher 
dimensional spaces.

I know much more about the math than about the history thereof, but so 
far as I know, there is nothing in Riemann's work that says anything 
about whether or not the space of the universe is "curved".  In this 
context it might be noted that Riemann's machinery applies to "nice" 
spaces (manifolds) of arbitrary finite dimension -- it is not primarily 
a theory of three (or four) dimensional space.

> Did he have concrete geometrical figures in mind as images to describe 
> his notion of space?

Any space which is curved in the sense of Riemann and which is not 
two-dimensional cannot be embedded in three-dimensional Euclidean space 
  -- making it awfully damn difficult to produce a picture of such.  Of 
the dozens of topologists/geometers that I've asked about this, I've 
found only one who claimed he could visualize such spaces.  I'm pretty 
sure he was deluded.