Summary

Below, we review the intuitive meanings of the various relations we have defined on a Riemannian manifold.

Relation Meaning
\({\mathrm{div}(u)\mathrm{d}V=L_{u}\mathrm{d}V}\) \({\mathrm{div}(u)}\) is the fraction by which a unit volume changes when transported by the flow of \({u}\).
\begin{aligned}\int_{V}\mathrm{div}(u)\mathrm{d}V & =\int_{\partial V}i_{u}\mathrm{d}V\\
& =\int_{\partial V}\left\langle u,\hat{n}\right\rangle \mathrm{d}S
\end{aligned}
The change in a volume due to transport by the flow of \({u}\) is equal to the net flow of \({u}\) across that volume’s boundary.
\({\mathrm{div}(u)=0}\) \({u}\) having zero divergence means the flow of \({u}\) leaves volumes unchanged, or the net flow of \({u}\) across the boundary of a volume is zero.
\({j\equiv\rho u}\), \({\rho}\) is the density of \({Q}\) The current vector \({j}\) is the vector whose length is the amount of \({Q}\) per unit time crossing a unit area perpendicular to \({j}\)
\begin{aligned}\frac{\mathrm{d}q}{\mathrm{d}t} & =\Sigma-\int_{\partial V}\left\langle j,\hat{n}\right\rangle \mathrm{d}S\end{aligned}
The change in \({q}\) (the amount of \({Q}\) within \({V}\)) equals the amount generated less the amount which passes through \({\partial V}\).
\begin{aligned}\frac{\partial\rho}{\partial t} & =\sigma-\mathrm{div}(j)\end{aligned}
The change in the density of \({Q}\) at a point equals the amount generated less the amount that moves away.
\({R\equiv g^{ab}R_{ab}}\) The Ricci scalar is \({n}\) times the average of the Ricci function on the set of unit tangent vectors.
\begin{aligned}\mathrm{Ric}(e_{\mu})=\underset{i\neq\mu}{\sum}g_{\mu\mu}K(e_{i},e_{\mu})\end{aligned}
The Ricci function of a unit vector is \({(n-1)}\) times the average of the sectional curvatures of the planes that include the vector.
\begin{aligned}R & =\underset{j}{\sum}g_{jj}\mathrm{Ric}(e_{j})\end{aligned}
The Ricci scalar is \({n}\) times the average of all the Ricci functions.
\begin{aligned}R & =2\sum_{i<j}K(e_{i},e_{j})\end{aligned}
The Ricci scalar is \({n(n-1)}\) times the average of all sectional curvatures.
\begin{aligned}G(e_{\mu},e_{\mu}) & =-\sum_{\begin{subarray}{c}
i<j\\
i,j\neq\mu\end{subarray}}K(e_{i},e_{j})\end{aligned}
The Einstein tensor applied to a unit vector twice is \({-(n-1)(n-2)/2}\) times the average of the sectional curvatures of the planes orthogonal to the vector.
\({d\left(\mathrm{exp}(\delta\hat{w}),\mathrm{exp}(\delta\parallel_{\varepsilon\hat{v}}\hat{w})\right)=\varepsilon\left(1-\frac{\delta^{2}}{2}K(\hat{v},\hat{w})\right)}\) \({K(\hat{v},\hat{w})/2}\) is the fraction by which the geodesic parallel to \({\hat{w}}\) starting \({\hat{v}}\) away bends towards \({\hat{w}}\).
\({\ddot{L}(t)=-L(t)K(\hat{v},\hat{w})}\) \({K(\hat{v},\hat{w})}\) is the acceleration of two parallel geodesics in the \({\hat{w}}\) direction with initial separation \({\hat{v}}\) towards each other as a fraction of the initial gap.
\({\ddot{A}(t)=-A(t)\mathrm{Ric}(v)}\) \({\mathrm{Ric}(v)/2}\) is the fraction by which the area defined by the geodesics emanating from the \({(n-1)}\)-surface perpendicular to \({v}\) changes in the direction of \({v}\).

\({\mathrm{Ric}(v)}\) is the acceleration of the parallel geodesics emanating from the \({(n-1)}\)-surface perpendicular to \({v}\) towards each other as a fraction of the initial surface.

\begin{aligned}\frac{\partial B_{\varepsilon}(M^{n})}{\partial B_{\varepsilon}(\mathbb{R}^{n})} & =1-\frac{\varepsilon^{2}}{6n}R\end{aligned}
\({\varepsilon^{2}R/6n}\) is the fraction by which the surface area of a geodesic \({n}\)-ball of radius \({\varepsilon}\) is smaller than it would be under a flat metric.
\begin{aligned}\frac{B_{\varepsilon}(M^{n})}{B_{\varepsilon}(\mathbb{R}^{n})} & =1-\frac{\varepsilon^{2}}{6(n+2)}R\end{aligned}
\({\varepsilon^{2}R/6(n+2)}\) is the fraction by which the volume of a geodesic \({n}\)-ball of radius \({\varepsilon}\) is smaller than it would be under a flat metric.

An Illustrated Handbook