Homage to Gauss: Theorema Egregium ("Remarkable Theorem")

The great mathematician Carl Friedrich Gauss once wrote his great theory about the curvature of spaces - the Theorema Egregium ("The Remarkable Theorem"). Gauss wrote how different curved surfaces could not "mate" with each other in simple 3D space.

Does this imply that space is not Euclidean, and is in fact hyperbolic? The implications would be that Omega-zero would be less than one and that space (and the Universe and everything in it) would begin to stop accelerating and begin retracting at some point along the space-time curvature.

It seems that this would be the case if before the (previous) Big Bang, some curved surfaces were to be embedded in the intial spacelike curve/event horizon. The forces created between these different (energy) surfaces would have provided thrust for the intial Big Bang. Dark Energy from the bulk surrounding the incompatible surfaces would have provided further negative energy - or an emerging energy vacuum thus providing the impetus for the expansion and inflation of the universe during/after the (initial/previous) Big Bang.

Fortunately the light cones of either event are beyond our observable event horizon. (Or is this unfortunate?). Traveling along some point on the space-time curvature would theoretically provide us with the vantage point from which to observe this future light cone. Which your author suspects is probably true due to the implications of the Theorema Egregium. Certain curvatures are incompatible and do not meet and match. Eventually, the light cone would become observable along this incompatible stretch of curvature in the space-time curvature and the apparent Euclidean curvature of our observable local space-time geometry.

More to come...? 8/3/06