What is called the curvature of space?

Moving along the spaces that are slightly not flat…

At the beginning of the XIX century, it became clear that, in addition to the usual-for-eyes Euclidean space, there may be spaces with other fundamental properties. First, Lobachevsky has studied hyperbolic spaces, a two-dimensional example of which is a saddle; and soon Riemann has described elliptical ones, an example of which is a sphere. The sum of angles of a triangle drawn on a saddle is less than on a plane. Furthermore, it is possible to draw two intersecting lines on a saddle, one of which is bent outward, and another — inward. In this regard, it is credited with negative curvature. On a sphere, on the contrary, the sum of angles of a triangle is always greater than 180 degrees, and two intersecting arcs are bent in the same direction. Sphere’s curvature is considered to be positive and equal to its inverse radius. Saddles as well as spheres are two-dimensional spaces immersed in a three-dimensional space. Similarly, there are curved three-dimensional spaces immersed in a four-dimensional one, etc.

A two-dimensional curved space.

Only in a flat space a straight road is the shortest one.

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