We returned to our consideration of the parallel transport of a vector around a closed loop on the manifold. We calculated the rotation for an infinitesimal loop, by calculating how the inner product of the parallel transported vector $v^i$ with a reference unaltered covector $w_l$ changes. This calculation was essentially a generalized derivation of Stokes' theorem, and gave as a result a change in dot product which was $\Delta = \int\ ds\, dt \ T^k\ S^j\ R_{kji}^{\ \ \ \ \ l}\ v^i\ w_l$. Here $ds\, dt$ is the infinitesimal area of the loop; $T^k$ and $S^j$ are the tangent vectors pointing along the orthogonal segments of the loop; and $R_{kji}^{\ \ \ \ \ l}$ characterizes the manifold itself; specifically, it is a rank $(1,3)$ tensor which encodes the failure of covariant derivatives to commute:

$$(\ \nabla_i \ \nabla_j - \nabla_j \ \nabla_i\ )\ w_k \equiv R_{ijk}^{\ \ \ \ \ l}\ w_l \ \ \ \ .$$

We noted why covariant derivatives fail to commute. There are basically two reasons: (1), since covariant derivatives includes terms with Christoffel symbols, whether particular Christoffel symbols are differentiated or undifferentiated depends on the order $\nabla_i \ \nabla_j$, as only the Christoffel symbol associated with the first applied $\nabla_j$ experiences differentiation when the second $\nabla_i$ acts. (2), more subtly, $w_k$ is a covector while $\nabla_i w_k$ is a rank (0,2) tensor, and the covariant derivative acts differently on tensors of different rank (it includes Christoffel contractions on each tensor index). So which covariant derivative acts first, on a covector, and which acts second and differently on the tensor, determines a distinct result.

Calculating this noncommutation of covariant derivatives determines the Riemann tensor, the prime characterization of curvature for a manifold.

We showed why -- since $R_{ijk}^{\ \ \ \ \ l}$ transforms as a tensor -- $R_{ijk}^{\ \ \ \ \ l}$ must have all its components vanish on a flat manifold. Thus it is a true measure of curvature, unlike Christoffel symbols which can become nonzero on a flat manifold, due just to choice of coordinates.

We introduced the simpler measures of curvature obtained by tracing over -- or ``contracting'' -- the Riemann tensor, to obtain covariant tensors of lower rank (and hence simpler transformation laws). These were the Ricci tensor and the Ricci scalar. We mentioned some of the useful symmetry properties of these tensors and counted their independent components, showing that for 2-dimensional manifolds (surfaces) only one number, a scalar, is necessary to describe the curvature; for 3-dimensional manifolds, only the Ricci tensor is necessary; and for 4-dimensional manifolds -- and higher -- the full Riemann tensor is necessary.

We also introduced the Einstein tensor, with vanishing four-divergence. This feature will become important when we return to our challenge of covariantizing the laws of gravity, next time.

--

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