Curvature in three-dimensional space
We live in three-dimensional space and we are going to consider the idea that three-dimensional space is curved. You say, “But how can you imagine it being bent in any direction?” Well, we can’t imagine space being bent in any direction because our imagination isn’t good enough. (Perhaps it’s just as well that we can’t imagine too much, so that we don’t get too free of the real world.) But we can still define a curvature without getting out of our three-dimensional world. All we have been talking about in two dimensions was simply an exercise to show how we could get a definition of curvature which didn’t require that we be able to “look in” from the outside. We can determine whether our world is curved or not in a way quite analogous to the one used by the gentlemen who live on the sphere and on the hot plate. We may not be able to distinguish between two such cases but we certainly can distinguish those cases from the flat space, the ordinary plane. How? Easy enough: We lay out a triangle and measure the angles. Or we make a great big circle and measure the circumference and the radius. Or we try to lay out some accurate squares, or try to make a cube. In each case we test whether the laws of geometry work. If they don’t work, we say that our space is curved. If we lay out a big triangle and the sum of its angles exceeds $180$ degrees, we can say our space is curved. Or if the measured radius of a circle is not equal to its circumference over $2\pi$, we can say our space is curved. You will notice that in three dimensions the situation can be much more complicated than in two. At any one place in two dimensions there is a certain amount of curvature. But in three dimensions there can be several components to the curvature. If we lay out a triangle in some plane, we may get a different answer than if we orient the plane of the triangle in a different way. Or take the example of a circle. Suppose we draw a circle and measure the radius and it doesn’t check with $C/2\pi$ so that there is some excess radius. Now we draw another circle at right angles—as in Fig. 42–15. There’s no need for the excess to be exactly the same for both circles. In fact, there might be a positive excess for a circle in one plane, and a defect (negative excess) for a circle in the other plane.  Perhaps you are thinking of a better idea: Can’t we get around all of these components by using a sphere in three dimensions? We can specify a sphere by taking all the points that are the same distance from a given point in space. Then we can measure the surface area by laying out a fine scale rectangular grid on the surface of the sphere and adding up all the bits of area. According to Euclid the total area $A$ is supposed to be $4\pi$ times the square of the radius; so we can define a “predicted radius” as $\sqrt{A/4\pi}$. But we can also measure the radius directly by digging a hole to the center and measuring the distance. Again, we can take the measured radius minus the predicted radius and call the difference the radius excess, \begin{equation*} r_{\text{excess}}=r_{\text{meas}}-\biggl( \frac{\text{measured area}}{4\pi}\biggr)^{1/2}, \end{equation*}  which would be a perfectly satisfactory measure of the curvature. It has the great advantage that it doesn’t depend upon how we orient a triangle or a circle.  But the excess radius of a sphere also has a disadvantage; it doesn’t completely characterize the space. It gives what is called the mean curvature of the three-dimensional world, since there is an averaging effect over the various curvatures. Since it is an average, however, it does not solve completely the problem of defining the geometry. If you know only this number you can’t predict all properties of the geometry of the space, because you can’t tell what would happen with circles of different orientation. The complete definition requires the specification of six “curvature numbers” at each point. Of course the mathematicians know how to write all those numbers. You can read someday in a mathematics book how to write them all in a high-class and elegant form, but it is first a good idea to know in a rough way what it is that you are trying to write about. For most of our purposes the average curvature will be enough.2