# Curvature and geodesics

Geometrically, the Ricci function $${\mathrm{Ric}(v)}$$ at a point $${p\in M^{n}}$$ can be seen to measure the extent to which the area defined by the geodesics emanating from the $${(n-1)}$$-surface perpendicular to $${v}$$ changes in the direction of $${v}$$. Considering the three dimensional case in an orthonormal frame (and again dropping the hats in $${\hat{e}_{i}}$$ to avoid clutter), we have

\displaystyle \begin{aligned}\mathrm{Ric}(e_{2}) & =\left\langle \check{R}(e_{1},e_{2})\vec{e}_{2},e_{1}\right\rangle +\left\langle \check{R}(e_{3},e_{2})\vec{e}_{2},e_{3}\right\rangle \\ & =K(e_{1},e_{2})+K(e_{3},e_{2}). \end{aligned}

If we form a cube made from parallel transported vectors as we did for the first Bianchi identity, we can see that each sectional curvature term in $${\mathrm{Ric}(e_{2})}$$ takes an edge of the cube and measures the length of the difference between the cube-aligned component of its parallel transport in the $${e_{2}}$$ direction and the edge of the cube at a point parallel transported in the $${e_{2}}$$ direction.

The above depicts how each sectional curvature measures the convergence of geodesics, while their sum forms the Ricci curvature function, which measures the change in the area of the $${(n-1)}$$-surface formed by geodesics perpendicular to its argument. In the figure we assume without loss of generality (see below) that $${\check{R}(e_{1},e_{2})\vec{e}_{2}}$$ is parallel to $${e_{1}}$$.

The figure above details the sectional curvature $${K(e_{1},e_{2})=\beta^{1}\check{R}(e_{1},e_{2})\vec{e}_{2}}$$ assuming that $${\check{R}(e_{1},e_{2})\vec{e}_{2}}$$ is parallel to $${e_{1}}$$, so that $${\left\langle \check{R}(e_{1},e_{2})\vec{e}_{2},e_{1}\right\rangle =\left\Vert \check{R}(e_{1},e_{2})\vec{e}_{2}\right\Vert }$$. The parallel transport of $${e_{2}}$$ along itself is depicted as parallel, so that the geodesic parametrized by arclength $${\phi(t)}$$ is a straight line in the figure. The vector $${\parallel_{\delta e_{2}}\parallel_{\varepsilon e_{1}}\delta e_{2}}$$ is the parallel transport of $${\parallel_{\varepsilon e_{1}}\delta e_{2}}$$ by $${\delta}$$ in the direction parallel to $${e_{2}}$$, and therefore the geodesic $${\phi_{\varepsilon}(t)}$$ tangent to $${\parallel_{\varepsilon e_{1}}\delta e_{2}}$$ at $${q}$$ has tangent $${\parallel_{\delta e_{2}}\parallel_{\varepsilon e_{1}}\delta e_{2}}$$ after moving a distance $${\delta}$$. If we consider the function $${f(t)}$$ whose value at $${t=\delta}$$ is the quantity $${(L-\varepsilon)}$$ in the figure (i.e. $${f(t)}$$ measures the offset of the geodesic from the right edge of the stack of parallel cubes), its derivative is the slope of the tangent, so that to lowest order in $${t}$$ we have

\displaystyle \begin{aligned}\dot{f}(t) & =-\varepsilon t^{2}K(e_{1},e_{2})/t\\ & =-\varepsilon tK(e_{1},e_{2})\\ \Rightarrow f(t) & =-\varepsilon t^{2}K(e_{1},e_{2})/2. \end{aligned}

We can generalize this logic to arbitrary unit vectors $${\hat{v}}$$ and $${\hat{w}}$$ to conclude that $${K(\hat{v},\hat{w})/2}$$ is the “fraction by which the geodesic parallel to $${\hat{w}}$$ with separation direction $${\hat{v}}$$ bends towards $${\hat{w}}$$.” More precisely, in terms of the distance function and the exponential map, to order $${\varepsilon}$$ and $${\delta^{2}}$$ we have

$$\displaystyle d\left(\mathrm{exp}(\delta\hat{w}),\mathrm{exp}(\delta\parallel_{\varepsilon\hat{v}}\hat{w})\right)=\varepsilon\left(1-\frac{\delta^{2}}{2}K(\hat{v},\hat{w})\right).$$

In the general case $${L}$$ in the figure is the distance between two geodesics infinitesimally separated in the $${\hat{v}}$$ direction, so if we define $${L(t)}$$ as this distance at any point along the parametrized geodesic tangent to $${\hat{w}}$$, the above becomes

\displaystyle \begin{aligned}L(t) & =L(0)\left(1-\frac{t^{2}}{2}K(\hat{v},\hat{w})\right)\\ \Rightarrow\left.\frac{\ddot{L}}{L}\right|_{t=0} & =-K(\hat{v},\hat{w}), \end{aligned}

where the double dots indicate the second derivative with respect to $${t}$$. Thus $${K(\hat{v},\hat{w})}$$ 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.”

Now, the distance $${\left|\varepsilon-L\right|=\varepsilon\delta^{2}K(e_{1},e_{2})/2}$$ defines a strip $${S}$$ bordering the surface orthogonal to $${e_{2}}$$ a distance $${\delta}$$ in the $${e_{2}}$$ direction. This strip thus has an area $${\varepsilon^{2}\delta^{2}K(e_{1},e_{2})/2}$$. If we sum this with the other strip of area $${\varepsilon^{2}\delta^{2}K(e_{3},e_{2})/2}$$, to order $${\varepsilon^{2}}$$ and $${\delta^{2}}$$ we measure the extent to which the area $${A}$$ defined by the geodesics emanating from the surface perpendicular to $${e_{2}}$$ changes in the direction of $${e_{2}}$$. But the sum of sectional curvatures is just the Ricci function, so that in general $${\mathrm{Ric}(v)/2}$$ 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 $${v}$$.” More precisely, we can follow the same logic as above, defining the “infinitesimal geodesic $${(n-1)}$$-area” $${A(t)}$$ along a parametrized geodesic tangent to $${v}$$, so that to order $${\varepsilon^{2}}$$ and $${t^{2}}$$ we have

\displaystyle \begin{aligned}A(t) & =\varepsilon^{2}\left(1-\frac{t^{2}}{2}\mathrm{Ric}(v)\right)\\ \Rightarrow\left.\frac{\ddot{A}}{A}\right|_{t=0} & =-\mathrm{Ric}(v). \end{aligned}

Thus $${\mathrm{Ric}(v)}$$ 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.” Note that if our previous assumption that $${\check{R}(e_{1},e_{2})\vec{e}_{2}}$$ is parallel to $${e_{1}}$$ is dropped, the only impact is that of an $${e_{3}}$$ component on the area calculation; to address this, a more accurate picture would be to extend the area to include all four quadrants defined by both negative and positive values of $${e_{1}}$$ and $${e_{3}}$$, in which case any change in area due to an $${e_{3}}$$ component cancels. In the case of a pseudo-Riemannian manifold, “areas” and “volumes” become less geometric concepts; however, we have a clear picture in the case of a Lorentzian manifold that the Ricci function applied to a time-like vector $${v\equiv\partial/\partial x^{0}=\partial/\partial t}$$ tells us how the infinitesimal space-like volume $${V}$$ of free-falling particles (i.e. following geodesics) changes over time according to $${\ddot{V}/V=-\mathrm{Ric}(v)=-R_{00}=-R^{\mu}{}_{0\mu0}}$$.