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Curved space often refers to a spatial geometry which is not "flat" where a flat space is described by Euclidean geometry. Curved space can generally be described by Riemannian geometry though some simple cases can be described in other ways.
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Einstein's theory of general relativity describes space as curved, with the "curved space" being the four-dimensional space-time conceived of by Minowski. Gravity is the results of curved space. This notion of curved space becomes more tangible by thinking about an astronaut and a space capsule. Then we will introducing a beam of light into the capsule. If a beam of light is given off from the top of one capsule wall to the opposite wall while the capsule is accelerating upwards in space, the light will appear curved. This is because, in the time it takes for the light beam to move across the cabin to the opposite wall, the cabin will have accelerated upwards and the beam will appear to curve across the cabin and hit below the spot directly across from where it started. The light will also appear to curve across the top of the space capsule if the capsule is at rest in Cape Canaveral. In other words, the light beam acts as if it is being pulled down by gravity. The space-time through which it moves can be understood to be curved by the presence of a massive body: in this case, the earth. In space, the curvature of space itself causes all objects, such as light or planets or spaceships, to follow the curvature. In both cases, the gravitational effect occurs because of the curvature of space.

YouTube curved space

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Curved space is the hardest thing to understand in cosmology(the discipline that deals with the nature of the universe as a whole). This is mostly because we think in the way of Euclidean geometry, and so it pretty much goes against what we were taught. The name is also a big stepping stone to go over. The property that is meant by the "curvature" of space is not curvature in the usual sense, or even in the sense of simple mathematics. C. F. Gauss created the terminology for curved space. Just to make this point explicitly, "curve" originally meant a curved line, but in Gauss' terms a 1-D space (line) cannot be curved!

Here is another example. Take a flat piece of paper, which we can think of as a 2-D space. Roll it into a cylinder. Is it curved? Of course it is... in the usual meaning of the word. But not for Gauss: technically the surface of a cylinder has zero Gaussian curvature. So what is going on here?

The trouble was, Gauss was a man with a secret. The biggest mathematical problem in Gauss' day was "the scandal of geometry". In ancient times Euclid had made geometry the archetype of a rigourous mathematical system. To deduce all the theorems of geometry, Euclid needed some definitions and some rules of reasoning. But he also needed to make some basic assumptions about geometrical entities (lines, circles etc), that he called postulates. There were five of them:
  1. It is possible to draw a straight line from any point to any other.
  2. It is possible to extend a finite straight line continuously in a straight line.
  3. It is possible to draw a circle with any centre and radius.
  4. All right angles are equal to each other.
  5. If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles.

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