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For an affine connection, let ${\gamma }_X$ be the affinely parametrized geodesic starting at $p$ with velocity $X$. Define the exponential map by the displayed formula for initial velocities whose geodesics exist until parameter one. It is defined near zero, has differential equal to the identity there, and obeys ${\exp }_p(tX)={\gamma }_X(t)$ wherever defined. This construction requires no metric and extends the metric exponential map.

Identify $T_{p}M$ with coordinate vectors using a basis and invert the affine exponential map near zero. Radial geodesics then have coordinates $sX$. Their equations imply ${\Gamma }^{\lambda }{}_{\mu \nu }(p)X^{\mu }{X}^{\nu }=0$ for every $X$, hence the displayed vanishing of the symmetric part. The antisymmetric part may survive when the torsion tensor is nonzero. For a general connection the Levi-Civita connection coefficients and first metric derivatives need not vanish in these coordinates.

A smooth nondegenerate component of the fixed-point set of an isometry is a totally geodesic submanifold. If a geodesic starts tangent to the fixed component, its image under the isometry has identical position and tangent. Uniqueness of the geodesic equation makes the two geodesics coincide throughout their common domain, so the original geodesic stays in the fixed set.

The Beltrami-Klein model realizes the hyperbolic plane inside an open Euclidean disk, with geodesics given by straight chords. Its metric differs from the Euclidean metric, and angles are generally distorted. The straight chords make geodesic convexity especially easy to see.

In geodesic normal coordinates at $p$, radial geodesics have coordinates $x(t)=tv$. The geodesic equation at $t=0$ gives ${\Gamma }_{jk}^i(p)v^j{v}^k=0$ for every $v$. The Levi-Civita connection is torsion free, making these coefficients symmetric in $j,k$; the polarization identity then gives ${\Gamma }_{jk}^i(p)=0$. This does not imply vanishing curvature, which depends on derivatives of the coefficients.

A nonconstant geodesic whose position and tangent velocity repeat after a positive time. On a flat torus it is a straight line modulo the lattice, closing when its displacement is a Euclidean lattice vector. A primitive closed geodesic traverses its image without being an iterate of a shorter periodic geodesic.

A simple closed curve that is a geodesic of length $\ell$ on a hyperbolic surface has an embedded collar of half-width $w$ with $\sinh w\sinh (\ell /2)=1$. In circumference-one coordinates its metric is $d{r}^2+{\ell }^2{\cosh }^{2}rd{t}^2$, and its conformal modulus of an annulus is $(2/\ell )\arctan (\mathrm{csch}(\ell /2))\sim \pi /\ell$.

Suppose these sets exhaust a manifold, and its Riemannian metric is Euclidean beyond some radius in the end coordinates. A finite-time geodesic has fixed speed and cannot increase its exterior radius faster than that speed. It remains in one compact core enlargement, with velocity in a compact disk bundle. The geodesic equation therefore continues beyond every finite parameter endpoint. The compactness hypothesis is substantive: the open exterior of a closed Euclidean ball, with empty core, is incomplete at its inner boundary despite having a Euclidean metric at infinity.

A local isometry from a nonempty complete Riemannian manifold to a connected Riemannian manifold is a surjective covering map. Lift radial geodesics in a normal ball using completeness; the exponential map at each inverse-image centre maps a tangent ball diffeomorphically onto a sheet. Uniqueness of the geodesic initial-value problem makes these sheets disjoint and exhaustive. Surjectivity follows by lifting piecewise geodesic paths. Conversely a Riemannian covering of a complete target is complete, by lifting complete target geodesics and using the Hopf-Rinow theorem. If the target is disconnected, the first assertion gives a covering only of the union of components met by the map.

In a homogeneous Riemannian manifold, choose $r>0$ so that one closed metric ball $\overline{B}(o,r)$ is compact. Such a ball exists from local compactness and the metric topology. Every radius-$r$ ball is isometric to it. A Cauchy sequence eventually lies in one such compact ball, has a convergent subsequence, and therefore converges. This proves metric completeness; the Hopf-Rinow theorem gives geodesic completeness and minimizing geodesics between points.

A conjugate point to a spacelike surface occurs where normal geodesics develop a focal degeneracy: the transverse Jacobi field map from initial points on the surface ceases to be invertible. Collapse of the transverse area makes the null expansion diverge. Beyond a first such point, the normal null geodesic cannot continue as a generator of the achronal boundary of the surface's future.

A convex normal neighbourhood is an open set whose points have unique smoothly varying joining geodesics, with the joining geodesic contained in the set. The corresponding star-shaped restriction of each exponential map is a diffeomorphism onto the neighbourhood. Hence its differential is invertible and no such joining geodesic admits a nonzero Jacobi field vanishing at both endpoints. Every point of a Riemannian manifold has arbitrarily small such neighbourhoods. This is stronger than merely requiring the existence of a minimizing geodesic in the set.

A mechanical Lagrangian quadratic in velocities can be represented by geodesics of a Lorentzian metric with an extra null coordinate. For $L=\frac{1}{2}|\dot{\mathbf{x}}{|}^2-2\mathbf{A}\cdot \dot{\mathbf{x}}-U$, use $g=d{\mathbf{x}}^2+2dt(du-2\mathbf{A}\cdot d\mathbf{x}-Udt)$. The null Killing vector field ${\partial }_u$ gives the conserved momentum $p_u=t\prime$. Fixing $p_u=1$ projects the geodesic equation to the mechanical Euler-Lagrange equations. This is useful for relating mechanical forces to spacetime geometry.

In dimension $n\ge 2$, along a nonconstant geodesic, such a Jacobi field is determined injectively by $D_{t}J(0)$, which lies in the $(n-1)$-dimensional normal space. On the round unit sphere the geodesic from a point to its antipode, parametrized with speed $\pi$ on $[0,1]$, has fields $J(t)=\sin (\pi t)E(t)$ for all parallel normal fields $E$, attaining the bound. For a constant geodesic the endpoint-vanishing Jacobi field space is zero.

An extendible geodesic is a geodesic segment that can be continued beyond at least one endpoint within the same manifold after retaining an affine parameter. An artificial truncation at a regular event is extendible even when its continuation eventually reaches a singularity.

For $E(\gamma )=\frac{1}{2}{\int }_0^1|\dot{\gamma }{|}^{2}dt$ and variation vector field $V$, the first variation is $[⟨V,\dot{\gamma }⟩{]}_0^1-{\int }_0^1⟨V,D_t\dot{\gamma }⟩dt$. For fixed endpoints the boundary term vanishes, so the critical points of the energy are exactly affinely parametrized geodesics.