We have seen that space can be non-Euclidean in General Relativity. But exactly what does it mean to say that space or, for that matter, space-time is curved? A few examples can help to illustrate things. At first, we will consider only two-dimensional surfaces. Imagine that we have a flat sheet. If we draw a right triangle with sides denoted by dx and dy, and hypotenuse by dl, then by the Pythagorean theorem dl²=dx²+dy² [dx² is a shorthand notation for (dx)²]. The fact that we can write dl² in this form is a statement that the space is Euclidean or "flat". Thus, a flat sheet is flat and Euclidean – nothing very profound here. Although it may seem a bit stupid, we can rewrite dl² as dl²=g₁₁dx²+ g₂₂dy²+g₁₂dxdy+g₂₁dydx, where g₁₁= g₂₂=1 and g₁₂= g₂₁=0.

Now consider the surface of a cylinder. We can draw a right triangle, one of whose sides is a circular arc of the cylinder and the other side is parallel to the axis of the cylinder. The hypotenuse is the shortest distance that connects the sides. Is this a flat space? YES! How do you know? It is possible to cut the cylinder parallel to its axis, open it up and fold it out flat. Although the surface of a cylinder is curved, it is topologically equivalent to a flat surface. Again we can write dl²=dx²+dy², where dx=rdf , r is the radius of the cylinder, f is an angle around the cylinder, and y is the direction of the axis of the cylinder. This can also be written as dl²=g₁₁dx²+ g₂₂dy²+g₁₂dxdy+g₂₁dydx, where g₁₁= g₂₂=1 and g₁₂= g₂₁=0.

The situation changes dramatically when we look at a spherical surface. There is no way to cut the surface and lay it flat without lumps. This is why it is impossible to represent the spherical Earth on a flat projection without distortion. A spherical surface has an absolute curvature. We can look at triangles on the surface of a sphere. "Straight Lines" on a sphere are defined as the shortest distance between two points on the sphere. They are arcs of great circles (a great circle is a circle on a sphere whose diameter is equal to a diameter of the sphere -–lines of longitude are great circles, but the only line of latitude that is a great circle is the Equator). In this way, a triangle can be formed by two lines of longitude and part of the Equator. The sum of the angles of this triangle is greater than 180 degrees – the geometry is non-Euclidean. We can also draw a small right triangle on the surface of the sphere. If the triangle is sufficiently small, one side can be an arc of longitude dx=rdq (r is the radius of the sphere and q is an angle of latitude) and the other side an arc of latitude dy=rsin(q)df. In this case dl²=sin²(q)(rdf )²+(rdq)². In this case, we find dl²=g₁₁dx²+ g₂₂dy²+g₁₂dxdy+g₂₁dydx, where g₁₁= sin²(q), g₂₂=1 and g₁₂= g₂₁=0. Note that g₁₁ is now a function of theta and is no longer a constant equal to unity.

The examples illustrate the criterion necessary for a two-dimensional space to obey the laws of Euclidean geometry. If a system of coordinates can be found in which the line element (hypotenuse of a small right triangle) can be written as dl²=g₁₁dx²+ g₂₂dy²+g₁₂dxdy+g₂₁dydx, with g₁₁= g₂₂=1 and g₁₂= g₂₁=0, the geometry is Euclidean. If not the geometry is not Euclidean. This is easily generalized to three-dimensional space. . If a system of coordinates can be found in which the line element (hypotenuse of a small right triangle) can be written as dl²=g₁₁dx²+ g₂₂dy²+ g₃₃dz²+g₁₂dxdy+g₂₁dydx g₂₃dydz+g₃₂dzdy+g₃₁dzdx+g₁₃dxdz, with g₁₁= g₂₂=g₃₃ =1 and g₁₂= g₂₁= g₁₃= g₃₁ =g₃₂= g₂₃=0, the geometry is Euclidean. If not the geometry is not Euclidean. It is convenient to introduce a notation in which dx=dx₁, dy=dx₂, and dz=dx₃. Then the line element can be written as dl² = .

The extension to four dimensional space-time is straightforward. We have already seen that the space-time interval in special relativity (where space-time is flat and Euclidean) is equal to ds²=-dl²=(cdt)²-dx²+dy²+dz². If we set (cdt)=dx₄, then this can be rewritten as –ds²=dl²= with g₁₁= g₂₂=g₃₃ =1, g₄₄=-1, and all the other g_ij’s equal to zero. It then follows that four dimensional space-time is Euclidean if a coordinate system can be found in which –ds²=dl²= with g₁₁= g₂₂=g₃₃ =1, g₄₄=-1, and all the other g_ij’s equal to zero. If no such coordinate system can be found, space-time is said to be curved. Space-time in a rotating coordinate system is not curved since one can transform into an inertial frame where the g’s will have the Euclidean values. However, whenever mass is present, it is impossible to find a coordinate system in which g₁₁= g₂₂=g₃₃ =1, g₄₄=-1, and all the other g_ij’s equal to zero – the presence of mass creates curved space-time. In fact, it is the mass distribution that determines the g’s, which, in turn, determine the geometrical properties of space-time! One final note. It is possible for space to be non-Euclidean, but for space-time to be Euclidean. This is true in accelerating reference frames in the absence of mass. In different reference frames, the curvature of space can be different, but the curvature of space-time is absolute.