Curvature measures the failure of covariant derivatives to commute and the failure of parallel transport to be path independent. Its contractions control volume, geodesic deviation, and the geometry of the metric itself.
Riemann curvature
Definition. With the convention used here, \[ R(X,Y)Z =\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z. \] The \((0,4)\)-tensor is \[ \operatorname{Rm}(X,Y,Z,W)=g(R(X,Y)Z,W). \]
Proposition. \(R\) is \(C^\infty(M)\)-linear in all three arguments and hence is a tensor.
Proof. Expand, for example, \(R(fX,Y)Z\). Derivatives of \(f\) from the first two terms cancel those from \([fX,Y]=f[X,Y]-Y(f)X\). The other slots are similar. This cancellation is why the bracket term is required.
In coordinates, \[ R^\ell{}_{kij} =\partial_i\Gamma^\ell_{jk}-\partial_j\Gamma^\ell_{ik} +\Gamma^m_{jk}\Gamma^\ell_{im} -\Gamma^m_{ik}\Gamma^\ell_{jm}. \]
Symmetries and Bianchi identities
Theorem. For the Levi-Civita connection, \[ \begin{aligned} \operatorname{Rm}(X,Y,Z,W)&=-\operatorname{Rm}(Y,X,Z,W),\\ \operatorname{Rm}(X,Y,Z,W)&=-\operatorname{Rm}(X,Y,W,Z),\\ \operatorname{Rm}(X,Y,Z,W)&=\operatorname{Rm}(Z,W,X,Y), \end{aligned} \] and \[ R(X,Y)Z+R(Y,Z)X+R(Z,X)Y=0. \] The covariant derivative satisfies the second Bianchi identity \[ (\nabla_XR)(Y,Z)+(\nabla_YR)(Z,X)+(\nabla_ZR)(X,Y)=0. \]
Sectional, Ricci, and scalar curvature
Definition. For a two-plane \(\sigma=\operatorname{span}(u,v)\), \[ K(\sigma)= \frac{\operatorname{Rm}(u,v,v,u)} {|u|^2|v|^2-g(u,v)^2}. \] For an orthonormal frame \((e_i)\), \[ \operatorname{Ric}(X,Y)=\sum_i\operatorname{Rm}(e_i,X,Y,e_i), \qquad \operatorname{Scal}=\sum_i\operatorname{Ric}(e_i,e_i). \]
Sectional curvature determines the full Riemann tensor. Ricci curvature is its trace and governs infinitesimal volume distortion; scalar curvature is a second trace.
Example (constant curvature). A manifold has constant sectional curvature \(\kappa\) exactly when \[ \operatorname{Rm}(X,Y,Z,W) =\kappa\bigl(g(X,W)g(Y,Z)-g(X,Z)g(Y,W)\bigr). \] Then \(\operatorname{Ric}=(n-1)\kappa g\) and \(\operatorname{Scal}=n(n-1)\kappa\). Euclidean space, the round sphere, and hyperbolic space realize \(\kappa=0,1,-1\) after scaling.
Geodesic deviation
Theorem (Jacobi equation). The variational field \(J\) of a variation through geodesics satisfies \[ \frac{D^2J}{\mathrm{d}t^2}+R(J,\dot\gamma)\dot\gamma=0. \]
Positive sectional curvature focuses nearby geodesics; negative curvature separates them. Jacobi fields also identify the differential of the exponential map and its singular points, called conjugate points.
Contracted Bianchi identity
Contracting the second Bianchi identity gives \[ \operatorname{div}\!\left( \operatorname{Ric}-\tfrac12\operatorname{Scal}\,g \right)=0. \] The divergence-free Einstein tensor is therefore forced by Riemannian geometry rather than introduced ad hoc.