Space-time curvature: spheres & donuts

We can define the curvature of space-time in terms of how much the angle of a vector changes when it returns to its original position. When the vector travels around this area, it is rotated by 90 degrees. If we double the area, we also double the amount by which the vector is rotated. We can therefore define the curvature in terms of the amount by which the vector is rotated divided by the size of the area around which the vector traveled. Note that when the vector travelled around the area in the clockwise direction, the vector rotated clockwise. If the vector rotates in the same direction in which it is travelling, we will refer to this as positive curvature. If the vector rotates in the opposite direction in which it is travelling, we will refer to it as negative curvature. An example of negative curvature is a saddle point, such as the region colored in green. If a vector travels "counter-clockwise" around this area, the vector is rotated "clockwise. " This other region has positive curvature. If a vector travels "counter-clockwise" around this area, the vector is rotated "counter-clockwise. " The vector rotates the same way as the direction of travel, as expected for a region of positive curvature. If a vector travels around an area composed of the sum of two smaller areas, then the amount by which the vector will rotate is the sum of the rotations associated with each individual area. In this example, the vector rotates 90 degrees plus 90 degrees, which is equal to 180 degrees. If the vector rotates around the region composed of this area with positive curvature and this area with negative curvature, the effects of the two areas will cancel each other out. The vector rotates by 90 degrees plus negative 90 degrees, meaning that the vector returns pointing in its original direction. As it turns out, if we add up all the positive and negative curvatures everywhere on a donut, we get a total curvature of zero.