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A geodesic congruence is a family of nonintersecting geodesics filling an open spacetime region. Its derivative tensor describes the relative motion of neighboring geodesics.

The metric defines this globally smooth Hamiltonian on the cotangent bundle. With $\omega =d{x}^i\wedge d{p}_i$ and ${\iota }_{X_H}\omega =dH$, its Hamiltonian vector field projects to affinely parametrized geodesics. The momentum is $p_i=g_{ij}{\dot{x}}^j$, and its conserved value is half the squared speed.

For a geodesic $\gamma$ with initial velocity $v$, $(f\circ \gamma {)}^{\prime \prime }(0)=\mathrm{Hess}f(v,v)$, because the covariant acceleration vanishes. Summing over an orthonormal basis of the tangent space gives $\Delta f(p)=-{\sum }_i(f\circ {\gamma }_i{)}^{\prime \prime }(0)$ for the nonnegative Laplace-Beltrami operator.

Let a compact smooth surface of revolution have exactly two poles, and let $c=f(v{)}^2\dot{u}=f(v)\cos \theta$ be the Clairaut first integral for a surface of revolution of a complete unit-speed geodesic. If $c\ne 0$, then $f(v)\ge |c|$, so the geodesic avoids open neighbourhoods of both poles. If $c=0$, then $u$ is constant away from the poles and the geodesic lies in a meridian of a surface of revolution, whose complement contains a nonempty open set. Thus no complete geodesic has dense image.

Rotate any geodesic into an equatorial plane of the magnetically charged dilaton black hole. Conserved energy and angular momentum are $E=V\dot{t}$ and $L=R^2\dot{\varphi }$. Normalizing its tangent to $g(\dot{x},\dot{x})=-ϵ$, with $ϵ=1,0,-1$ for timelike, null and spacelike geodesics, gives ${\dot{r}}^2=E^2-V(ϵ+L^2/R^2)$. Thus its radial equation is ${\dot{r}}^2/2+\tilde{V}=0$. Radial null rays have $\dot{r}=\pm E$ and use the same Schwarzschild tortoise coordinate with $2M$ replaced by $r_{+}$.

A geodesic segment is the restriction of a geodesic to a compact parameter interval. In the hyperbolic plane, two points determine a unique geodesic segment, which minimizes the hyperbolic distance between them.

Integral curves through the identity of left-invariant vector fields are one-parameter subgroups, by flow uniqueness and left translation. They extend for all time because a fixed local existence interval translates to every point. Under a bi-invariant Riemannian metric they are geodesics by the Levi-Civita connection of a bi-invariant metric. Geodesic uniqueness proves that these are all geodesics through the identity.

For a circular cone with intrinsic metric $d{\rho }^2+{\lambda }^2{\rho }^{2}d{\theta }^2$, $0<\lambda \le 1$, local development uses plane angle $\varphi =\lambda \theta$. A plane straight segment that avoids the origin lifts to a geodesic. Choosing an angular difference at most $\pi$ gives plane difference at most $\pi \lambda$, so for $\lambda <1$ this constructs a joining geodesic between any two distinct points. Different permitted angle lifts can yield distinct joining geodesics; the missing apex cannot be used to join straight segments through the plane origin.

A geodesic variation of a given geodesic is a smooth family $F(s,t)$ with $F(0,t)=\gamma (t)$, whose curves at fixed $s$ are affinely parametrized geodesics. Differentiating its geodesic equation and using the torsion-free Levi-Civita connection gives the Jacobi field equation $D_t^{2}J+R(J,\dot{\gamma })\dot{\gamma }=0$ for $J={\partial }_{s}F{|}_{s=0}$.

For a left-invariant metric on a Lie group, the one-parameter subgroup ${\gamma }_{\xi }(t)=\exp (t\xi )$ is a geodesic exactly whenfor every $\eta$ in the Lie algebra. The Koszul formula gives $⟨{\nabla }_{\xi }\xi ,\eta ⟩=-⟨\xi ,[\xi ,\eta ]⟩$.

A great circle is the intersection of a sphere with a plane through its centre. Its shorter arcs are geodesics of the round sphere; celestial meridians and the equator are examples.

A collar around a simple closed geodesic of length $\ell$ is an embedded annulus with metric $d{r}^2+{\ell }^2{\cosh }^{2}rd{t}^2$, where $t\in \mathbb{R}/\mathbb{Z}$. The collar lemma gives half-width $w=\mathrm{arcsinh}(1/\sinh (\ell /2))$. As $\ell$ tends to zero the collar gets wide, forcing crossing geodesics to become long.

A real Möbius transformation represented by $A\in {\mathrm{SL}}_2(\mathbb{R})$ with $|\mathrm{tr}A|>2$. It has two fixed ideal boundary points and translates along the geodesic joining them; its hyperbolic translation length is $2\mathrm{arcosh}(|\mathrm{tr}A|/2)$. Conjugating by a real Möbius transformation puts it in the form $z\mapsto e^{\ell }z$ with $\ell >0$; its axis is then the imaginary axis.

The infimum of nonconstant closed geodesic lengths in curvature $-1$, with value $+\infty$ if there are none. On a closed surface with a hyperbolic metric this is a positive attained minimum, and a minimizing geodesic is simple. For fixed closed genus $g\ge 2$, a common positive lower bound gives relative compactness in the unmarked moduli space of Riemann surfaces by Mumford's compactness theorem.

For a hyperbolic Möbius transformation represented by $A\in {\mathrm{SL}}_2(\mathbb{R})$, the infimum of $d(z,Az)$ is attained on its axis and equals $2\mathrm{arcosh}(|\mathrm{tr}A|/2)$. It is the length of the corresponding closed geodesic in a quotient.

For a geodesic tangent $u$ and a stationary Killing vector field $K$, the Killing energy is $E=-g(K,u)$. It is conserved along the geodesic. When $K$ is future timelike and $u$ is future causal, $E>0$ for a nonzero tangent; it need not be positive where $K$ becomes spacelike.

A Killing tensor is a symmetric covariant tensor satisfying the displayed equation for the Levi-Civita connection. Its contraction with $r$ copies of a geodesic velocity is constant along every affinely parametrized geodesic. The converse follows because the resulting symmetric derivative is determined by its homogeneous polynomial on tangent vectors. Rank one recovers the Killing vector field after metric duality.