Can an inhabitant of a spacetime region measure its curvature tensor? So, lets say that I am an ant living on a 2-D spherical surface that is stretching to the equator...like half a sphere. I can not describe this surface in terms of the outside coordinates only someone living in the outside world can do this. So, can I really determine that I am in fact living on a spherical and not flat surface? If I try to measure the phi coordinate change as I move over the R coordinate (phi is the angle that runs from 0 to 2pi and R runs from 0 to rpi where r is radius of a sphere) at two separate places as I move to the north I can measure that this distance is getting smaller. I hope that I am clear enough about what I mean. But, would not the measuring stick also get smaller? Making it imposible for the inhabitant of this world to determine its geometry. So how can someone living in this world determine its geometry while his measuring equpment distorts acording to this? 
 A: You could construct a circle of radius $r$ and measure the circumference $c$. In a flat space, $c=2\pi r$. However, in a positively (negatively) curved space, $c<2\pi r$ ($c>2\pi r$).
Alternatively, you could pick three non-collinear points and measure the three angles that they form. In a flat space, the sum of angles will be 180$^\circ$. However, in a positively (negatively) curved space it will be $>180^\circ$ ($<180^\circ$).
A: If an inhabitant of a spacetime couldn't measure its curvature, then there would be no general relativity, because general relativity is the study of how the curvature of our spacetime affects us...
Your intuition is correct that some geometric aspects of curvature cannot be measured from "inside" the curved manifold.  These aspects make up its "extrinsic" curvature.  But other aspects can be measured from inside the manifold - these aspects make up its "intrinsic" curvature.  General relativity is only concerned with intrinsic curvature, because what would be the point of discussing a "curvature" that we could never measure?
The triangle example in the previous answer is a good one.  The thing to keep in mind is that if you consider a small enough patch, everything looks locally flat so you don't need to worry about the effect of curvature on your measuring devices.  So if you measure the three angles right at the "corners" and you find they add up to more or less than 180 degrees, you can trust that the spacetime really is curved because your measurements were made over small enough regions that you can be sure the curvature didn't distort them.
Here's another example: send two light rays out in parallel.  If the rays are initially very close together, you can trust that they're parallel because at such small differences the curvature won't distort your measurements.  If the light rays eventually cross or diverge, then your know your spacetime must be curved.  You may find angles more intuitive then lengths because angles are inherently local ("located" in an infinitesimally small region).
A: So as I was going through smart stuff of general relativity, still confused with this problem of detecting the curvature I realised how not very smart I was. In my question I am confused because I think that the stick shoul shrink as unit distance of a coordinate shrinks but of course it wont....sphere does not care if you and how you draw your coordinates. According to my incorrect thinkink on this matter behaviour of a stick would depend on a way in which I drew my coordinates...so if I chose that north pole is where the equator is stick would sudenly shrink because of that choice and that is just nonsense! So stick would be curved but that is it...
