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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 change $\lambda \mapsto A\lambda +B$ rescales a geodesic tangent and all its linear Killing constants by $1/A$. A null norm remains zero and cannot fix $A$, unlike proper-time normalization of a material particle. Ratios of the constants, such as the photon impact parameter $b=|h|/(ck)$ in Schwarzschild spacetime, are independent of the affine scale. A physically normalized four-momentum fixes the photon energy and hence selects a tangent scale.

A connected complete Riemannian manifold that is disconnected at infinity contains a line in a Riemannian manifold. Choose points escaping in two components outside a fixed compact set and join them by minimizing geodesics. Each segment crosses the compact set. Parametrize from a crossing point; both parameter endpoints tend to infinity in opposite directions. Compactness of the unit tangent vectors over the crossing set gives a subsequence converging to an entire geodesic. Every finite subsegment is minimizing by continuity of distance, so the limit is a line.

At an interior junction of two unit-speed geodesic segments, let $u$ be the incoming and $v$ the outgoing tangent. Choose points close to the junction in a common convex normal neighbourhood. Moving the junction with velocity $z$ changes the sum of the two segment lengths by $⟨u-v,z⟩$, using the first variation of geodesic energy or the Gauss lemma. If $u\ne v$, choose $z=v-u$ to reduce length. Hence a length-minimizing broken geodesic has matching tangents and is one smooth geodesic.

// Target: riemannian-geometry.bigb

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.

Embed radius-$L$ Anti-de Sitter spacetime as $X\cdot X=-L^2$ in signature with two timelike directions. Initial tangent $V_P$ satisfies $X_P\cdot V_P=0$ and $V_P\cdot V_P=-1$. The displayed curve remains on the hyperboloid and has unit timelike tangent; its ambient acceleration is normal, proving it is a geodesic. Every such curve reaches $-X_P$ after proper time $\pi L$. On the universal cover, this is a later antipodal event, not a closed causal loop. Global time advances by $2\pi$ over a complete ambient period.

The arc length of a piecewise smooth curve $\gamma :[a,b]\to M$ on a Riemannian manifold is the integral of its speed:An odometer measures this accumulated travel rather than displacement between the endpoints. In Euclidean space, the Riemannian metric is the ordinary inner product, so the formula becomes ${\int }_a^b\sqrt{{\sum }_i{\dot{\gamma }}_i(t{)}^2}dt$.

For a circle of radius $R$ traversed through an angle $\alpha$, use $\gamma (t)=(R\cos (\alpha t),R\sin (\alpha t))$, $0\le t\le 1$. Its speed is $R|\alpha |$, hence its arc length is $R|\alpha |$. Traversing a full circle twice gives $4\pi R$, even though the starting and ending points coincide. The same formula measures the two great circle arcs pictured in geodesic.

The curve parameter controls how fast the journey happens, not its arc length. A regular increasing or decreasing reparametrization preserves the integral by change of variables; reversing direction leaves the speed norm unchanged. Retracing part of a route does add length, because it changes the actual journey.

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.

Measure $\theta \in [-\pi /2,\pi /2]$ from the vertically downward direction on the lower half of a cylinder of radius $R$, and let $z$ be axial distance. From rest at an edge, conservation of energy gives speed $\sqrt{2gR\cos \theta }$. The travel time is $T=(2gR{)}^{-1/2}\int \sqrt{R^2+z^{\prime 2}}/\sqrt{\cos \theta }d\theta$. Since $z$ is a cyclic coordinate, its conjugate quantity $z^{\prime }/[\sqrt{\cos \theta }\sqrt{R^2+z^{\prime 2}}]=K$ is constant, giving the displayed formula. For finite positive axial separation, $0<K<1$ is fixed uniquely by integrating $z^{\prime }$ between the edges: the separation grows strictly from zero to infinity as $K$ increases from zero to one. The time integrand is strictly convex in $z^{\prime }$, so its supporting-line inequality proves global minimality for the fixed endpoints. As the starting edge is approached, $z^{\prime }\to 0$: the initial direction is directly down the curved surface, with no axial component. This differs from the constant-pitch shortest geodesic on the unrolled cylinder.

A linear isometry between tangent spaces of complete simply connected Riemannian manifolds extends to a global isometry when parallel transport along corresponding broken geodesics identifies their Riemann curvature tensors. In particular, complete simply connected manifolds of the same dimension and constant sectional curvature are isometric: their curvature tensors are determined by that common constant and their Riemannian metrics. This is the constant-curvature form often called Cartan's theorem.

A Cayley graph turns multiplication in a group into travel: the chosen generators of a group are the allowed one-step moves, and a word over an alphabet becomes a route. This lets us study a finitely generated group using graph distances, closed balls, and geodesics, rather than only equations between products.

Let $G$ be a group with a finite generating set of a group $S$ satisfying $S=S^{-1}$ and $1\notin S$. Such a symmetric subset of a group includes the inverse element of every allowed move. The Cayley graph ${\mathrm{Cay}}_S(G)$ has vertex set $G$ and joins $g$ to $gs$ for each $s\in S$. We choose right multiplication for these moves. Reversing the move uses $s^{-1}$, so for graph distance we regard $g\to gs$ and $gs\to g$ as the two orientations of one undirected edge. An involution therefore gives one edge, not two parallel edges. Each edge has length one, regardless of its length in a drawing.^[ref]

A sequence $s_1,\dots ,s_n$ labels the walkIt ends at $h$ exactly when $g^{-1}h=s_1\cdots s_n$. Thus the Cayley graph is a connected graph precisely because $S$ generates $G$, and its graph distance on vertices is the word metric. Left multiplication by $a$ sends the edge from $g$ to $gs$ to the edge from $ag$ to $(ag)s$, preserving its generator label: this is a group action by graph automorphisms and isometries. Viewing groups as symmetries of geometric objects is the motivation of Clay and Margalit's introduction.^[ref]

The useful geometric object includes its allowed moves, not just its drawing. Cayley graphs of the integers show how changing the moves adds shortcuts; word metric balls in the integer lattice show how the same group can acquire visibly different closed balls. Meier's book develops this passage from group operations to graphs as its central motivation.^[ref]

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 $(2/\ell )\arctan (\mathrm{csch}(\ell /2))\sim \pi /\ell$.

Let the scale factor grow by $f$ from an initially thermal photon distribution. geodesic redshift gives ${\nu }_i=f\nu$, and conserving photons in corresponding comoving frequency bins gives $n_{\nu }(t)=f^{-2}{n}_{f\nu }(t_i)$. Inserting Planck's law leaves the same spectral form at temperature $T_i/f$. Total photon-number conservation alone would not prove that the entire radiation spectrum stays Planckian; frequency redshift and collisionless propagation are also required. Absorption, injection or spectral mixing can invalidate this one-temperature argument.

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.

A (pseudo-)Riemannian manifold is geodesically complete when every inextendible affinely parametrized geodesic is defined for all real parameter values. In the positive-definite case, the Hopf-Rinow theorem identifies this with metric completeness. Null-only completeness in spacetime is a separate weaker condition.

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.