The failure of second derivatives to commute
In Guide 2 you built the Levi-Civita connection nabla, the one way to differentiate vector fields that is both metric-compatible and torsion-free. It let you write nabla_X Y, the rate at which Y changes as you move along X. The natural next question is whether the order matters: is differentiating first along X then along Y the same as the reverse? For ordinary partial derivatives on R^n the answer is the boring yes — mixed partials commute. On a curved manifold the answer is a resounding no, and that defect is the whole subject of this guide.
Define the Riemann curvature tensor as exactly the obstruction to commuting: R(X,Y)Z = nabla_X nabla_Y Z - nabla_Y nabla_X Z - nabla_[X,Y] Z. The first two terms compare differentiating in the two orders; the last term, with the Lie bracket [X, Y], is a correction that subtracts the part of the discrepancy coming merely from X and Y not commuting as flows. What is left over is honest curvature — the bending of the manifold itself, not of your chosen coordinate grid.
The same defect, seen as holonomy around a loop
There is a picture that makes R unforgettable. Take a vector at p and parallel-transport it around a small closed loop spanned by X and Y. On flat space it returns unchanged. On a curved manifold it returns rotated, and the infinitesimal rotation per unit area of the loop is exactly R(X,Y). This is the holonomy reading: curvature measures how much parallel transport, which Guide 2 promised was path-dependent, actually depends on the path. The commutator definition and this loop picture are two faces of one fact.
Make it concrete on the sphere S^2 of radius 1. Stand at the north pole holding an arrow. Walk down to the equator, slide a quarter of the way around, and walk back to the pole, parallel-transporting the whole time so the arrow never turns relative to your path. You arrive home with the arrow rotated by 90 degrees — exactly the area, pi/2, of the spherical triangle you traced, because the sphere has constant curvature 1. That visible, unfakeable rotation IS the curvature tensor doing its job; nothing about your route was special, the deficit angle equals enclosed curvature.
Lowering the upper index with the metric gives the fully covariant tensor R_ijkl = g(R(e_i, e_j) e_k, e_l), and in this form the symmetries become clean: R is antisymmetric in the first pair (i,j), antisymmetric in the second pair (k,l), symmetric under swapping the two pairs, and satisfies the first Bianchi identity (the cyclic sum over the last three indices vanishes). These symmetries cut the number of independent components of R from n^4 down to n^2(n^2 - 1)/12 — which is 1 in dimension 2, and 20 in dimension 4, the count that matters for general relativity.
Rung one: sectional curvature, the most geometric piece
The raw tensor R has many components and little intuition. The cure is to feed it geometric inputs. Pick a 2-dimensional plane sigma inside the tangent space T_p M, spanned by two vectors u and v. The sectional curvature of that plane is the number K(sigma) = g(R(u,v)v, u) / (g(u,u) g(v,v) - g(u,v)^2). The denominator is just the squared area of the parallelogram on u and v, included so the answer depends only on the plane sigma, not on which basis you chose for it.
What makes K worth its name is the meaning, not the formula. Exponentiate the plane sigma — sweep out the little surface in M traced by geodesics leaving p in every direction inside sigma. That surface is a genuine 2-dimensional submanifold, and its Gaussian curvature at p, the bending you knew from the curves-and-surfaces rung, is exactly K(sigma). So sectional curvature is the intrinsic Gauss curvature of the geodesic slice in a chosen 2-plane direction. Positive K means slices curl up like a sphere; negative K means they flare out like a saddle; zero means locally flat.
Rungs two and three: Ricci and scalar by averaging
Sectional curvature is rich but unwieldy — it is a function on the bundle of 2-planes. We get more tractable invariants by averaging. The Ricci curvature in a unit direction v is, up to a constant, the average of the sectional curvatures K(sigma) over all planes sigma containing v. Formally it is a trace of R: Ric(v,v) = sum over i of g(R(e_i, v)v, e_i) for an orthonormal basis e_i. It is a symmetric 2-tensor, the same type as the metric g itself, which is why it can be compared to g.
Ricci has a vivid meaning: Ric(v,v) controls how the volume of a thin cone of geodesics shooting out in direction v shrinks or spreads compared to flat space. Positive Ricci means nearby geodesics reconverge and volumes lag behind Euclidean growth — that focusing is the engine behind the Bonnet-Myers theorem, which says a complete manifold with Ricci bounded below by a positive constant must be compact with finite diameter. State the hypothesis honestly: it is a positive lower bound on Ricci, not on sectional curvature, and dropping it (flat R^n has Ric = 0) lets the manifold run off to infinity.
Average one more time and you reach the bottom rung. The scalar curvature S is the trace of Ricci, S = sum over i of Ric(e_i, e_i) — a single number at each point, the most compressed curvature invariant there is. It governs how the volume of a small geodesic ball deviates from the Euclidean ball of the same radius: vol(B_r) = omega_n r^n (1 - S r^2 / (6(n+2)) + ...), so positive S means small balls hold less volume than they would in flat space. Compressing R all the way to one number loses a lot — S cannot see directional focusing — but it is the right object for the simplest global statements.
R_ijkl (1,3)-tensor, full curvature ~ n^2(n^2-1)/12 components
| fix a 2-plane sigma = span(u,v), normalize by area
v
K(sigma) = g(R(u,v)v,u) / area(u,v)^2 one number per 2-plane
| average over all planes through v (trace once)
v
Ric(v,v) = sum_i g(R(e_i,v)v, e_i) symmetric 2-tensor, same type as g
| average over all directions v (trace again)
v
S = sum_i Ric(e_i,e_i) a single scalar at each point
dim 2: R, Ric, S all carry the SAME info (R_1212 = K = S/2)Why three flavors, and where they take you
It is fair to ask why we keep three curvatures instead of one. The honest answer is that they carry genuinely different amounts of information, and the right one depends on the theorem. In dimension 2 the distinction collapses — R, Ricci, and scalar all reduce to the single Gauss curvature, with S = 2K. From dimension 3 up they separate: there exist manifolds with positive scalar curvature but Ricci negative in some direction, and manifolds with positive Ricci but some sectional curvatures negative. The hierarchy sectional > Ricci > scalar is a strict loss of information at each averaging step.
Each flavor anchors its own world. Sectional bounds drive the comparison geometry of the next guide-rung — the Rauch and Toponogov theorems and the celebrated sphere theorem (pinched positive sectional curvature forces a sphere). Ricci is the hinge of Bishop-Gromov volume comparison and of the splitting theorem, and it is the quantity that flows under Ricci flow, dg/dt = -2 Ric, the heat-like evolution Perelman used in proving geometrization. Scalar curvature governs the positive mass theorem and the existence questions of the Yamabe problem.
Two honest cautions before you climb on. First, sign conventions differ across books: many texts define R with the opposite sign, and some put the sectional-curvature numerator as g(R(u,v)u, v); always check whether your source makes the round sphere have positive or negative K before trusting any formula. We use the convention that the unit sphere has K = +1. Second, an Einstein manifold (Ric = lambda g) is NOT the same as constant sectional curvature in dimension 4 or higher — Einstein is a weaker, Ricci-level condition, and conflating the two is a common beginner error. With the tensor and its three distillations in hand, Guide 5 turns outward to submanifolds, where this intrinsic curvature meets the extrinsic bending of the second fundamental form.