Geodesic

A geodesic is a curve whose tangent vector is parallel along the curve. Equivalently, it has zero covariant acceleration; on an embedded surface, a unit-speed geodesic has acceleration normal to the surface.

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 surface homeomorphic to a cylinder with strictly negative Gaussian curvature has at most one closed geodesic image. Geodesic disks, monogons and bigons are excluded by Gauss-Bonnet theorem; distinct essential simple geodesics would bound an annulus whose curvature integral must vanish. Repeated traversals count as the same image.

A pregeodesic is a curve whose tangent obeys ${\nabla }_{U}U=\kappa U$ for some scalar $\kappa$. A change of parameter makes it a geodesic with an affine parameter. A varying speed along a geodesic changes its parameter, not its image.

A null pregeodesic has a null tangent and can be parametrized as a null geodesic. In a two-dimensional Lorentzian metric, the acceleration of a regular null curve is orthogonal to its tangent and therefore proportional to it.

A subset of a Riemannian manifold is geodesically convex when any two of its points can be joined by a minimizing geodesic segment lying in the subset. In the hyperbolic plane the joining geodesic segment is unique. Thus in the Beltrami-Klein model, geodesically convex subsets correspond to Euclidean convex sets.

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.

On the unit sphere with one point deleted, a distinct nonantipodal pair has a length minimizer exactly when its shorter great circle arc avoids the deleted point. Antipodal pairs admit an avoiding semicircle. If the unique shorter arc crosses the puncture, smooth detours approach its length without attaining it.

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 geodesic variation $F(s,\tau )$, the deviation vector $S=F_{\ast }{\partial }_s$ describes infinitesimal separation at equal parameter values. With $T=F_{\ast }{\partial }_{\tau }$, commuting parameter derivatives give $[S,T]=0$. A torsion-free connection then gives ${\nabla }_{T}S={\nabla }_{S}T$. For an affinely parametrized variation, geodesic deviation states ${\nabla }_T^{2}S=R(T,S)T$ with the convention $R(X,Y)Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z$.

Given a Jacobi field with initial values $J(0)=a$ and $D_{t}J(0)=b$, vary the initial point along a curve with tangent $a$ and the initial velocity with covariant derivative $b$. The resulting geodesic variation has the same Jacobi initial values and therefore the same field by uniqueness of the linear equation. Smooth dependence on initial conditions gives a common parameter interval around any fixed compact geodesic segment, even on an incomplete manifold.

A geodesic triangle is bounded by three geodesic segments. On a surface its angle excess is the integral of the Gaussian curvature over the triangle.

For a tangent vector field $V$ along a curve on an embedded surface, the surface covariant derivative is the tangential projection of its ordinary derivative:It is the covariant derivative induced by the surface metric.

The orthogonal projector onto the tangent plane of a surface with unit normal vector $\mathbf{n}$. It removes the normal component of a vector and defines the surface gradient ${\nabla }_s=P\nabla$.

On a sphere of radius $a$, ${\nabla }_s\mathbf{n}=P/a$, ${\nabla }_s\cdot \mathbf{n}=2/a$, and ${\Delta }_s\mathbf{n}=-2\mathbf{n}/a^2$. For a constant vector $\mathbf{U}$, ${\nabla }_s\cdot (P\mathbf{U})=-2\mathbf{U}\cdot \mathbf{n}/a$. These identities simplify dipolar surface diffusion and interfacial transport.

For an embedded surface with unit normal vector $\mathbf{n}$, the surface gradient is ${\nabla }_{s}f=(\mathbf{I}-\mathbf{n}{\mathbf{n}}^T)\nabla f$. It projects the ambient gradient onto the tangent space and is independent of how $f$ is extended normally off the surface.

The surface divergence of an ambient vector field is the tangential trace ${\nabla }_s\cdot \mathbf{v}=({\delta }_{ij}-n_i{n}_j){\partial }_j{v}_i$. For a tangent vector field it is the intrinsic divergence. For $\mathbf{v}=\mathbf{n}$ it gives the sum of the two principal curvatures; a sphere with outward normal has ${\nabla }_s\cdot \mathbf{n}=2/a$.

For $p$ in a Riemannian manifold and $v$ sufficiently close to zero in $T_{p}M$, let ${\gamma }_v$ be the geodesic with ${\gamma }_v(0)=p$ and ${\dot{\gamma }}_v(0)=v$. The exponential map isEquivalently, ${\exp }_p(tv)={\gamma }_v(t)$ wherever both sides are defined.

For the Riemannian exponential map ${\exp }_p(v)={\gamma }_v(1)$, uniqueness of the geodesic equation gives ${\exp }_p(tv)={\gamma }_v(t)$. Differentiation at $t=0$ yields $(d{\exp }_p{)}_{0}v=v$. Smooth ODE dependence and the inverse function theorem therefore make ${\exp }_p$ a diffeomorphism on a neighborhood of zero, producing geodesic normal coordinates. This is a local assertion and requires no geodesic completeness.

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.

The geodesic flow sends a tangent vector $v$ to the velocity at time $t$ of the unique geodesic with initial velocity $v$. Smooth dependence of solutions of the geodesic equation makes it a smooth flow on the tangent bundle wherever defined.

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.

The domain of ${\exp }_p$ is the star-shaped open setSmooth dependence for ordinary differential equations makes $(p,v)\mapsto {\exp }_p(v)$ smooth on its open domain in $TM$.

For $S=({\mathbb{R}}^2\setminus \{0\})\times \{0\}\subset {\mathbb{R}}^3$ and $p=(1,0,0)$, geodesics are straight lines until they hit the missing origin. Thus $(-2,0,0)\notin {\mathcal{D}}_p$, and $(-1,0,0)$ is not in the image of ${\exp }_p$ because its unique straight segment from $p$ passes through the origin.

A normal neighbourhood of $p$ is the diffeomorphic image under ${\exp }_p$ of a star-shaped neighbourhood of $0$ in $T_{p}M$. In a sufficiently small normal ball, radial geodesics from $p$ uniquely minimize length.

The injectivity radius at $p$ is the largest radius on which ${\exp }_p$ is a diffeomorphism from the tangent-space ball onto its image. The injectivity radius of a compact Riemannian manifold is the positive infimum of these pointwise radii.

Choose an orthonormal basis of $T_{p}M$ and use its linear coordinates on a neighborhood of zero on which ${\exp }_p$ is a diffeomorphism. Transporting these coordinates through ${\exp }_p$ gives geodesic normal coordinates centered at $p$.

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 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.

A chart $x$ centered at $p$ on a star-shaped coordinate domain is a geodesic normal chart exactly when $g_{ij}(0)={\delta }_{ij}$ andfor every $x$ and $k$. This is the geodesic equation for every radial curve $t\mapsto tx$.

Choose an orthonormal basis of $T_{p}M$ and put $e(\theta )=(\cos \theta ,\sin \theta )$. Away from the centre and cut locus,defines geodesic polar coordinates centred at $p$.

The radial direction is orthogonal to the angular directions under the exponential map. On a surface this makes the first fundamental form in geodesic polar coordinateswith $\sqrt{G(r,\theta )}/r\to 1$ as $r\to 0$.

For $h(r,\theta )=\sqrt{G(r,\theta )}$, the angular Jacobi field equation on a surface becomesThe Riemannian area element is $h(r,\theta )drd\theta$.

If $h^{\prime \prime }+Kh=0$, $h(0)=0$, $h^{\prime }(0)=1$, $h>0$, and $K\le C$ for $C>0$, thenSubtract a smaller multiple of the sine solution and use the monotonicity of its Wronskian with $h$.

If $K\le 0$, then $h_{rr}=-Kh\ge 0$, so $h\ge r$. Every geodesic polar ball of radius $\epsilon$ therefore satisfies

If $K\le C$ with $C>0$, scalar comparison givesbefore $\pi /\sqrt{C}$. Consequently

A geodesic is complete when its maximal affine parameter interval is all of $\mathbb{R}$.

A geodesic $\gamma :I\to M$ is maximal when it has no geodesic extension to a strictly larger interval. Every initial position and velocity determine a unique maximal geodesic through the geodesic equation.

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.

A Riemannian manifold is geodesically complete when every maximal geodesic is defined for every real affine parameter, equivalently when every exponential map is defined on its entire tangent space.

Pointwise domination of Riemannian metrics gives domination of their distances. A Cauchy sequence for the larger distance is Cauchy for the complete smaller one. Smooth positive metrics induce the same manifold topology, so its limit is also a limit in the larger distance. The Hopf-Rinow theorem turns metric completeness into geodesic completeness. Mere positivity of the second metric, without domination, is insufficient.

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 Riemannian manifold is geodesically incomplete when some maximal geodesic has a finite endpoint in its affine-parameter interval.

Let $S\subset {\mathbb{R}}^3$ be a smooth surface. Suppose its Gaussian curvature tends to infinity in absolute value along every maximal geodesic with a finite affine-parameter endpoint. Then $S$ is an inextendible embedded surface: a smooth extension would let some geodesic reach the extension boundary in finite time while its curvature remained locally bounded.

The surface of revolutionis geodesically incomplete: a meridian reaches the missing origin in finite length. Its Gaussian curvature, computed using the curvatures of a parametrized surface of revolution, iswhich tends to infinity as $r\downarrow 0$. Every finite-time endpoint of a constant-speed geodesic must approach the origin, the only finite point in the ambient closure missing from $S$, so this surface satisfies the curvature-blowup criterion.

For a connected finite-dimensional Riemannian manifold, completeness of the Riemannian distance, geodesic completeness, and compactness of every closed bounded subset are equivalent. These conditions imply that any two points are joined by a minimizing geodesic. Pairwise existence of minimizing geodesics alone does not imply completeness: an open Euclidean ball is incomplete, yet the line segment between any two of its points stays in the ball and minimizes length.

A line is a unit-speed geodesic $\gamma :\mathbb{R}\to M$ that minimizes between every two of its points: $d_g(\gamma (s),\gamma (t))=|s-t|$.

A connected noncompact manifold is disconnected at infinity when the complement of some compact subset has at least two unbounded connected components. Equivalently, it has at least two ends.

If two embedded surfaces are tangent along a curve, their tangent planes agree there. The curve's ambient acceleration is normal to one surface exactly when it is normal to the other, so it is a geodesic of one exactly when it is a geodesic of the other.

A circular helix on a circular cylinder has radial ambient acceleration and is therefore a geodesic. Any surface tangent to the cylinder along that helix has the same curve as a geodesic.