Riemannian geometry

Riemannian geometry studies smooth manifolds equipped with smoothly varying inner products on tangent spaces.

The symmetric covariant two-tensor defined using the Levi-Civita connection by ${\mathrm{Hess}}_{g}f(X,Y)=X(Yf)-({\nabla }_{X}Y)f$. Its negative metric trace is the positive Laplace-Beltrami operator applied to $f$. Along a geodesic it gives the second derivative of $f$.

Spectral geometry studies which properties of a Riemannian manifold can be recovered from the spectrum of geometric differential operators, especially the Laplace-Beltrami operator. Explicit spherical harmonics, Fourier series on a flat torus, and isospectral manifolds illustrate three different aspects of the problem.

For the positive Laplace-Beltrami operator on a closed compact Riemannian manifold, the heat trace is $Z_M(t)={\sum }_j{e}^{-t{\lambda }_j}={\int }_M{K}_M(t,x,x)dV(x)$. It records every eigenvalue with multiplicity. In a finite free isometric quotient it equals $|U{|}^{-1}{\sum }_{u\in U}{\int }_N{K}_N(t,x,ux)dV(x)$. The orbital integrals form a class function of the ambient finite isometry group, so Gassmann equivalence gives equal quotient heat traces and hence Sunada theorem.

For the unit $S^d$ and the nonnegative Laplace-Beltrami operator, the eigenvalues are $\ell (\ell +d-1)$, with multiplicities $\left(\genfrac{}{}{0ex}{}{\ell +d}{d}\right)-\left(\genfrac{}{}{0ex}{}{\ell +d-2}{d}\right)$ for $\ell \ge 0$. The eigenfunctions are restrictions of degree-$\ell$ harmonic polynomials. The harmonic decomposition of homogeneous polynomials and the Stone-Weierstrass theorem prove completeness.

Two Riemannian manifolds are isospectral for a specified differential operator if their eigenvalues, counted with multiplicities and with the same boundary conditions, agree. Riemannian isometries preserve the Laplace-Beltrami operator spectrum, but the converse can fail. The transplantation theorem and the Sunada theorem give systematic constructions.

Suppose a finite group $G$ acts by Riemannian isometries on a compact Riemannian manifold $M$, and Gassmann equivalent subgroups $H_1,H_2$ act freely. The quotient manifolds $H_i\backslash M$ have equal Laplace-Beltrami operator spectra. For an eigenfunction space $E$, the quotient multiplicity is $|H_i{|}^{-1}{\sum }_{h\in H_i}\mathrm{tr}(h|E)$, and the equality follows by summing its character of a representation over conjugacy classes. Nonconjugacy inside $G$ alone does not guarantee that the quotients are nonisometric.

Choose distinct odd primes and take products of the nonisomorphic Gassmann equivalent regular subgroups, selecting one of the two groups at each prime. Product conjugacy-class counts give $2^n$ Gassmann equivalent subgroups of the same finite product of symmetric groups. Their fundamental groups are distinguished by whether the unique Sylow subgroup at each chosen prime is abelian. The closed-manifold realization of a finitely presented fundamental group and Sunada theorem therefore produce $2^n$ pairwise nonhomeomorphic, and hence nonisometric, isospectral closed four-manifolds. Gassmann equivalence alone, without a topology distinction, does not imply nonhomeomorphism.

For two assemblies of congruent Euclidean tiles, let $M_s,N_s$ encode the gluing or boundary reflection at each labelled face. An invertible constant matrix $C$ satisfying $C{M}_s=N_{s}C$ carries tile restrictions of Laplacian eigenfunctions bijectively to those on the second assembly. Boundary values satisfy $M_{s}u=u$ and outward normal derivatives satisfy $M_{s}v=-v$, so the intertwining identities preserve matching and boundary conditions. Use diagonal $-1$ for a Dirichlet boundary condition and $+1$ for a Neumann boundary condition.

Reflection across a cut splits Laplacian eigenfunctions into odd and even parts. The odd restriction has a Dirichlet boundary condition on the cut; the even restriction has a Neumann boundary condition there. The displayed matrix intertwines the two-tile swap with $\mathrm{diag}(-1,1)$. A connected uniform-Dirichlet rectangle of width $2a$ is therefore isospectral to the disjoint union of width-$a$ rectangles with all-Dirichlet and three-Dirichlet/one-Neumann conditions. This illustrates why changing boundary conditions can conceal connectedness in the spectrum.

A propeller domain consists of seven congruent reflected triangles: one central triangle and three arms of two triangles each. There are different coloured gluing patterns whose Dirichlet Laplacian and Neumann Laplacian spectra agree by the transplantation theorem. Varying the three side lengths of a scalene triangle in a suitable open region gives three parameters; the central triangle is identifiable from the three reflex corners, allowing a direct nonisometry proof.

A class of metrics is spectrally rigid if equality of their specified spectra forces the metrics to be isometric, or, in a deformation version, if continuous isospectral deformations are trivial. The spectrum of a flat torus determines every two-dimensional flat torus up to isometry. The Wolpert generic spectral rigidity theorem is a generic, rather than universal, uniqueness statement.

For closed hyperbolic surfaces of genus $g\ge 2$, there is a closed proper real-analytic exceptional subset of Teichmüller space such that a surface outside it is determined up to isometry by its unmarked length spectrum, equivalently its Laplace-Beltrami operator spectrum. The analytic part of the proof controls possible spectral matchings using finite determining length data; the geometric part recognizes persistent matchings as changes of marking, using the collar lemma and variations of Fenchel–Nielsen coordinates. Orientation reversal remains invisible to the spectrum.

A Riemannian manifold is a smooth manifold equipped with a smoothly varying positive-definite inner product on each tangent space.

A Riemannian manifold is homogeneous when its isometry group acts transitively on points. For example, a Lie group with a left-invariant metric is homogeneous under left translations. Homogeneity gives the same local metric-ball geometry at every point, which yields the completeness of homogeneous Riemannian manifolds. It does not imply two-point homogeneity: that stronger property also controls directions and pairs at equal distance.

A connected Riemannian manifold is two-point homogeneous if its isometry group is transitive on ordered pairs at each fixed Riemannian distance. The equal-distance condition is necessary because isometries preserve distance. Round spheres and Euclidean spaces are examples. The unit tangent transitivity characterizes two-point homogeneity lemma shows that this pair condition is equivalent to transitivity on the unit tangent bundle.

For a connected Riemannian manifold, two-point homogeneity is equivalent to the isometry group being transitive on the unit tangent bundle. One direction follows by taking short equal-length radial geodesic segments: the Gauss lemma identifies their distance, and injectivity of the exponential map identifies the initial directions after the endpoints are matched. Conversely, unit tangent transitivity implies point homogeneity, hence completeness. The Hopf-Rinow theorem supplies minimizing geodesics for arbitrary equal-distance pairs. Matching their initial unit tangent vectors and using uniqueness of the geodesic equation matches their other endpoints.

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 conical singularity is a failure of smoothness at the apex of a metric cone. If $\vartheta$ has period $P$, small circles have circumference-to-radius ratio $aP$. The two-dimensional apex is smooth precisely when this equals $2\pi$. Euclidean black-hole regularity condition applies this test to the imaginary-time and radial plane.

A Killing-Yano tensor is a differential form satisfying the displayed equation for the Levi-Civita connection. Its covariant derivative is totally antisymmetric. Contraction with the velocity of an affinely parametrized geodesic gives a form that is carried by parallel transport along that geodesic. This differs from a Killing tensor, which is symmetric.

For a Killing-Yano two-form, ${\nabla }_a{Y}_{bc}$ is a differential three-form. Along a geodesic with velocity $v$, the covector $w_c=Y_{bc}{v}^b$ is carried by parallel transport, since $v^a{v}^b{\nabla }_a{Y}_{bc}=0$.

The square is a rank-two Killing tensor. It is symmetric because it pairs the covectors $Y_{a\cdot }$ and $Y_{b\cdot }$ using the metric. Along every affinely parametrized geodesic, its contraction $K(v,v)$ is the squared norm of the carried by parallel transport covector $Y(v,\cdot )$. Metric compatibility makes that norm constant, proving the Killing tensor equation. On a Riemannian manifold this tensor is positive semidefinite.

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.

A symmetric two-tensor satisfying this equation gives a quadratic geodesic first integral. Examples include the Riemannian metric, symmetric products of metric-dual Killing vector fields, and the square of a Killing-Yano two-form.

A quadratic homogeneous polynomial in the fibre variables of the cotangent bundle is a first integral of the geodesic Hamiltonian exactly when its symmetric lowered coefficients form a rank-two Killing tensor. For $v=g^{-1}p$, the Poisson bracket is $\{K,H\}={\nabla }_{(a}{K}_{bc)}{v}^a{v}^b{v}^c$, so the equivalence follows by comparing cubic coefficients at every point.

A spin manifold is an oriented Riemannian manifold whose orthonormal frame bundle lifts from $SO(n)$ to its double cover $\mathrm{Spin}(n)$. Such a lift defines a spinor bundle.

A spin structure is a consistent choice of lifting oriented orthonormal frame changes to their spin double cover, permitting globally defined spinor fields. For a spatial circle in a worldsheet, the two choices give periodic and antiperiodic fermions, hence the Ramond sector and Neveu–Schwarz sector. Local field equations do not choose the global spin structure.

A spinor field is a section of a spinor bundle. Clifford multiplication by tangent vectors acts fiberwise on spinor fields.

The Dirac operator is the first-order elliptic operator obtained by composing the spin connection with Clifford multiplication. On an even-dimensional compact spin manifold it exchanges positive and negative chirality spinors.

An equivariant index theorem expresses the supertrace of a symmetry on the kernel and cokernel of an elliptic operator as an integral of local characteristic data over the symmetry's fixed-point set.

An immersion into a Riemannian manifold inherits a metric by taking ambient inner products of tangent vectors. For a parametrized curve $\mathbf{r}(u)$ in Euclidean space, the induced metric coefficient is $g={\partial }_u\mathbf{r}\cdot {\partial }_u\mathbf{r}$.

A Riemannian surface is a two-dimensional smooth manifold equipped with a smoothly varying inner product on each tangent plane.

The arc length of a regular curve $\gamma :[a,b]\to M$ is

For a piecewise smooth curve, $L(\alpha )=\int \Vert {\alpha }^{\prime }(t)\Vert dt$ in an ambient Euclidean space, or the corresponding Riemannian metric norm on a manifold. This is invariant under regular monotone reparametrization.

The distance induced by a connected Riemannian manifold is the infimum of the lengths of piecewise smooth curves joining two points:

In a connected Riemannian manifold, choose a small compact geodesic sphere around $p$ with $0<r<d(p,q)$. Minimize $d(\cdot ,q)$ on this sphere. Every path from $p$ to $q$ crosses it, so an almost-minimizing path has length at least $r+{\min }_{S_r(p)}d(\cdot ,q)$. The triangle inequality gives the reverse bound, yielding the displayed equality. No global completeness is needed.

A minimizing geodesic realizes the Riemannian distance between its endpoints. A geodesic segment is locally minimizing, but need not remain globally minimizing beyond a cut point.

On a complete connected Riemannian manifold, let $c(v)$ for a unit tangent vector $v$ be the supremum of times for which $t\mapsto {\exp }_p(tv)$ minimizes Riemannian distance from $p$. The cut locus consists of the endpoints ${\exp }_p(c(v)v)$ with $c(v)<\infty$. Radial minimization fails beyond this first cut time; distance spheres can cease to be smooth there.

For Riemannian manifolds $(M,g)$ and $(N,h)$, the product metric on $M\times N$ is $g\oplus h$. Its Levi-Civita connection and curvature split by factors, so its Ricci and scalar curvatures are the corresponding sums.

For closed manifolds with the product Riemannian metric, the positive Laplace-Beltrami operator is the sum of the two factor operators. Products of their orthonormal eigenbases form a complete basis, and eigenvalues add. Hence the product eigenspace at $\lambda$ is the displayed finite orthogonal sum of tensor products. The discrete assertion requires a discrete spectral setting.

An isometry between metric spaces preserves every distance. A Riemannian isometry preserves the Riemannian metric and therefore distances, angles, geodesics, and intrinsic curvature.

The isometry group of a metric space consists of its bijective isometries, with composition as the group operation. For a Riemannian metric, these are the diffeomorphisms preserving the metric tensor. Right translations of a right-invariant Riemannian metric supply a faithful subgroup isomorphic to the underlying Lie group.

The fixed-point set of a map $f:M\to M$ is $M^f=\{x:f(x)=x\}$. For an isometry of a Riemannian manifold, each smooth component of the fixed-point set is totally geodesic.

A Euclidean isometry has the form $\mathbf{x}\mapsto \mathbf{a}+R\mathbf{x}$, where $R$ is orthogonal. It preserves inner products of displacement vectors, distances, angles, and arc length.

A glide reflection of the plane composes reflection in a line with a nonzero translation parallel to that line. It reverses orientation and has no fixed point. It cannot be one reflection, because a reflection fixes a line, or a product of two reflections, because such a product preserves orientation. The finite reflection decomposition of a Euclidean isometry therefore shows that it requires exactly three reflections.

Every Euclidean isometry of ${\mathbb{R}}^n$ is a product of at most $n+1$ operations of reflection in a hyperplane. An isometry fixing zero is orthogonal: the polarization identity preserves inner products, and its values on an orthonormal basis determine its linear action. An orthogonal transformation is a product of at most $n$ linear reflections, by reflecting the image of the first basis vector back to that vector and inducting on its orthogonal complement. For an arbitrary isometry, first reflect its image of zero back to zero, if necessary, and apply the orthogonal result. Affine reflection hyperplanes, rather than only hyperplanes through zero, are essential here.

An isometric embedding $f:X\to Y$ between metric spaces is an injective function satisfying $d_Y(f(x),f(x^{\prime }))=d_X(x,x^{\prime })$ for all $x,x^{\prime }\in X$.

An isometric extension of a connected metric manifold is a proper embedding onto an open subset of a larger connected manifold of the same dimension, preserving the metric tensor. Smooth complete connected Riemannian manifolds admit no such proper extension; Lorentzian geodesic completeness has different subtleties.

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.

The energy of a curve is one half the integral of its squared speed.

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.

For a variation $F(s,t)$ of a curve $\gamma (t)=F(0,t)$, the variation vector field is $V(t)={\partial }_{s}F(0,t)$. A fixed-endpoint variation has $V=0$ at both endpoints.