Positive vs Negative Curved Space

Suppose that we are two dimensional beings living on this surface. As far as the surface can tell, this space is flat so long they don’t go all the way around the tip. If the two dimensional beings want to determine that their space is not flat, they can do this by going all the way around the tip of the cone and returning to where they started. This is because their vector is now no longer pointing in its original direction. The angle by which their vector rotates is exactly equal to the angle of the missing section of this circle. If we double the circle, the angle by which the vector rotates is also doubled. If the two dimensional beings go all the way around the tip of the cone in the "counterclockwise" direction, their vector is rotated "counterclockwise. " Because the vector rotates in the same direction in which they travel, we say that the area has a positive intrinsic curvature. Suppose we increase the angle of the missing section of the circle to 180 degrees. The amount by which the vector is rotated has now increased to 180 degrees. Let’s consider a new example. In this new example, we will add an extra section to the circle. The angle of the extra section that we add to the circle is exactly equal to the angle by which the vector rotates when it travels all the way around the surface. This time, the vector rotates opposite to the direction of travel. When travel, we say that the area has a negative intrinsic curvature.