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For a complete connected $n$-dimensional Riemannian manifold with nonnegative Ricci curvature, the displayed ratio is nonincreasing, tends to one at zero, and is at most one. In particular $\mathrm{Vol}B(p,r)\le {\omega }_n{r}^n$. In geodesic polar coordinates, the radial Jacobi field determinant is at most the Euclidean density $r^{n-1}$ until the cut time. The trace of the radial Riccati equation for distance spheres, with Cauchy-Schwarz inequality, proves this comparison. Equality for every radius forces the Euclidean polar metric and absence of a cut locus.

A geodesic is a curve whose tangent vector is parallel along itself under the Levi-Civita connection. With an affine parameter it has zero covariant acceleration:This is the Riemannian manifold version of moving straight ahead with constant speed. Its coordinate form is the geodesic equation. In Euclidean space it becomes $\ddot{\gamma }=0$, giving straight lines. On an embedded surface with its induced metric, the ordinary acceleration of such a curve is normal to the surface: all acceleration is used to follow the surface, rather than to turn within it.

On a unit round sphere, the great circle $\gamma (t)=(\cos t,\sin t,0)$ has $\ddot{\gamma }(t)=-\gamma (t)$, which is normal to the sphere. Two points at $t=0$ and $t=\pi /3$ can be joined along that circle by the short arc of length $\pi /3$ or the long arc of length $5\pi /3$. Each admits a constant-speed geodesic parametrization, yet only the short arc realizes their great-circle distance.

Every geodesic minimizes arc length on sufficiently short subarcs. A long segment can cease to be globally minimizing, as the sphere example shows; the first loss from a starting point is described by the cut locus. Thus “locally straight” and “shortest route between these endpoints” are distinct assertions. On a circular cylinder, different winding gives the same distinction, made explicit by Riemannian distance on a circular cylinder.

This provides a useful division of work. The geodesic equation predicts a path from an initial point and tangent vector; comparing arc lengths decides which candidate solves a global shortest-route problem. Variations of energy of a curve give the equation as an Euler-Lagrange equation, offering a proof outline and a connection with classical mechanics. ^[ref].

A geodesic sphere is a level set of Riemannian distance from a point. For sufficiently small positive $r$, it is the exponential map image of the radius-$r$ tangent sphere, a smooth compact hypersurface. The Gauss lemma makes its tangent space orthogonal to the radial direction. Larger distance spheres need not be smooth at the cut locus.

Before the cut locus, define $A(X)={\nabla }_X{\partial }_r$ on the tangent space of a distance sphere. This is the negative of the shape operator with outward normal under the convention $S=-\nabla \nu$. A nonsingular matrix $Y$ of radial Jacobi fields satisfies $Y^{\prime \prime }+{\mathcal{R}}_{{\partial }_r}Y=0$ and $A=Y^{\prime }{Y}^{-1}$, yielding the displayed matrix Riccati equation. Taking its trace gives $h^{\prime }+\mathrm{tr}(A^2)+\mathrm{Ric}({\partial }_r,{\partial }_r)=0$, where $h=\mathrm{tr}A={\partial }_r\log \det Y$. This is the local differential identity behind the Bishop-Gromov inequality.

For a unit-speed geodesic $\gamma (t)={\exp }_p(tv)$, a finite cut time $c(v)$ is the last time to which its segment from $p$ minimizes Riemannian distance. Its endpoint is the Riemannian cut point along that geodesic. It lies in the cut locus; it need not be a cut point in the topological sense. The segment up to the cut time still minimizes by continuity of distance.