Oz knocks on the wizard's door after pondering the definition of Riemann curvature. He says "It would be really nice to have some simple examples of a Riemann tensor for a suitable space. I think 2d would do. Say of a sphere or other straightforward object so one can get an idea of what a real one would look like. At some point a few simple concrete numbers is helpful for clarity, if they are appropriately chosen."
The man in a sorcerer's cap hems and haws for a minute and then speaks:
"Two dimensions is good for some purposes, boring for others. In 2 dimensions it only takes one number to describe the Riemann curvature at each point, so there is the same amount of information in the Riemann curvature tensor, the Ricci tensor, and the Ricci scalar. So we can't understand the differences between these concepts very well in 2d.
Let me describe the Ricci scalar, R, in 2d. This is positive at a given point if the surface looks locally like a sphere or ellipsoid there, and negative if it looks like a hyperboloid - or "saddle". If the R is positive at a point, the angles of a small triangle there made out of geodesics add up to a bit more than 180 degrees. If R is negative, they add up to a bit less.
For example, a round sphere of radius r has Ricci scalar curvature R = 2/r2 at every point. [With a click of his fingers, a sphere of radius r appears, with a small triangle drawn on it, edges bulging slightly.]
But how about the Riemann tensor on a round sphere? Well, for this we need some coordinates. Let's use the usual spherical coordinates theta, phi. Since there is actually disagreement at times on which is which, I remind you that for me theta is the longitude, running round from 0 to 2pi, while phi is the angle from the north pole, going from 0 to pi. [Coordinate lines appear on the sphere, which floats back and forth playfully.]
When we write something like gab or Rabcd, the indices will go from 1 to 2, with "1" corresponding to theta and "2" corresponding to phi. The components of the metric g are then:
g11 g12 = r2 sin2(phi) 0 g21 g22 0 r2
What does this mean? Remember this matrix lets us calculate dot products of vectors via
g(v,w) = gab va vb
So if I take the vector v to be, say,
1 0so that its theta component is 1 and its phi component is zero (v1 = 1, v2 = 0), we get
g(v,v) = r2 sin2(phi)
This means that its length squared is r2 sin2(phi), or in other words, its length is r sin(phi). That's supposed to make sense. You do remember your spherical coordinates, right?" The wizard flashes a pained, worried look at Oz. A helpful review of trig formulas flashes by on the sphere.
Oz replies "Hey, spherical geometry at this level is kid's stuff. Even I remember it. I have only regressed to age 17ish, this stuff you learn at 14!! "
The wizard smiles and says "Okay. But you wanted to know the Riemann tensor of the sphere, not the metric or the Ricci scalar! Well, okay, so we take the metric and feed it into this machine..." He scurries behind a curtain; loud banging noises ensue, followed by a deafening explosion and a puff of smoke; he returns somewhat blackened but smiling. "...and it computes the Riemann tensor for us. Don't worry about that machine in the other room just yet, someday you too may learn to use it... or maybe not."
So, the Riemann tensor has lots of components, namely 2 x 2 x 2 x 2 of them, but it also has lots of symmetries, so let me tell just tell you one:
R2121 = sin2(phi)/r2
Remember what this means!" He shuffles through a vast stack of old papers, plucks one out, and reads:
"Say we take a tangent vector pointing in the d direction and carry it around a little square in the b-c plane. We go in the b direction until the b coordinate has changed by epsilon, then we go in the c direction until the c coordinate has changed by epsilon, then we go back in the b direction until the b coordinate is what it started out as, and then we go back in the c direction until the c coordinate is what it started out as. Our tangent vector may have rotated a little bit since space is curved. Its component in the a direction has changed a bit, say
-epsilon2 Rabcd
So R2121 tells us how much a vector pointing in the theta direction swings over towards the phi direction when we parallel transport it around a little square. If you visualize it... [a little vector pointing east appears near the equator on the hovering globe; moves east, then south, then west, and north back to where it was, but has rotated a wee bit southwards in the process] ...you'll see this makes sense. Unless I got a sign wrong...." [The north pole on globe flips over to the south pole, the east flips over to the west, the globe lets out a snicker and disappears in a puff of smoke.]