> Let?s look at spacetime. According to GR, it is the curvature of > spacetime that causes gravity. So, spacetime is curved around a > gravitating mass. In free space, the Einstein tensor vanishes which > means the Ricci tensor also vanishes which mean the Riemann tensor > also vanishes. So, you have vanished Riemann tensor in curved > spacetime. That means the curvature tensors really do not address the > curvature thing. The field equations are merely differential > equations that allow you solve the local geometry and nothing more. > <shrug>
Also ,don't confuse space with space-time . Space is only a 'slice' of space-time . In itself it can be flat, while still being in a curved space-time . The metric of space is three-dimensional , and embedded in the four dimensional metric of space-time. Even factoring out the gravitational effects (which predominantly affect the time-time component of the metric tensor, not it's space components ) ,we still have to consider the expansion of the universe .
Space can be predominantly flat (provided it's free from strong gravitational effects) , while still being embedded in a curved space- time .
> Write down the metric for n-sphere please. <shrug>
If the earth were perfectly round , you would return to your point of origin when going around it . Generalize . Or better yet, use Google to figure out the metric .
> At least, you admit your own version of cosmology is purely > speculation. The so-called experts believe in their speculated > ?reality? more whole heartedly.
First, don't underestimate the value of rational speculation .I speculate that the sun will rise tomorrow . All science has origin in rational speculation. Second, the most commonly accepted model of the universe is flat (in space) and infinite.Observations confirm the universe 'flat with only a 0.4% margin of error' . http://en.wikipedia.org/wiki/Shape_of_the_Universe#cite_note-2