# Summary

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

RelationMeaning
$${\mathrm{div}(u)\mathrm{d}V=L_{u}\mathrm{d}V}$$$${\mathrm{div}(u)}$$ 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}) & =-g_{\mu\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 $${\hat{v}}$$ twice is $${-\left\langle \hat{v},\hat{v}\right\rangle (n-1)(n-2)/2}$$ 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 $${\hat{w}}$$ with separation direction $${\hat{v}}$$ bends towards $${\hat{w}}$$.
$${\left.\frac{\ddot{L}}{L}\right|_{t=0}=-K(\hat{v},\hat{w})}$$$${K(\hat{v},\hat{w})}$$ is the acceleration of two parallel geodesics in the $${\hat{w}}$$ direction with initial separation direction $${\hat{v}}$$ towards each other as a fraction of the initial gap.
$${\left.\frac{\ddot{A}}{A}\right|_{t=0}=-\mathrm{Ric}(v)}$$$${\mathrm{Ric}(v)/2}$$ 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.