Geodesic · aiwiki.page

A geodesic is a curve that generalizes the notion of a straight line to curved spaces. In differential geometry, its defining property is that its tangent vector remains parallel to itself along the curve. On a Riemannian manifold, every sufficiently short geodesic segment minimizes length between its endpoints, although a longer segment need not do so. The concept also applies to spaces equipped with a connection and to the curved spacetime of relativity, where “shortest path” is not an adequate general definition. (damtp.cam.ac.uk)

Geometric definition

A Riemannian structure equips a smooth manifold with a positive-definite metric tensor $g$ , specifying lengths and angles in each tangent space. Its Levi-Civita connection provides the corresponding rule for parallel transport. A smooth curve $\gamma(t)$ is an affinely parameterized geodesic when

\nabla_{\dot\gamma}\dot\gamma=0.

Here $\dot\gamma$ is the tangent vector and $\nabla$ denotes the covariant derivative. Thus the curve has zero intrinsic acceleration. This does not mean that its acceleration vanishes when viewed in an ambient Euclidean space: a great circle bends in three-dimensional space while remaining intrinsically straight on a sphere. (damtp.cam.ac.uk)

For a nonconstant Riemannian geodesic, an affine parameter gives constant speed; rescaling it can give unit speed. Affine changes $t\mapsto at+b$ , with $a\ne0$ , preserve the equation. A general smooth reparameterization preserves the geometric path but usually introduces an acceleration term proportional to the tangent. (damtp.cam.ac.uk)

Coordinate equation

In local coordinates $x^1,\ldots,x^n$ , the defining condition becomes

\frac{d^2x^k}{dt^2} +\Gamma^k_{ij}(x(t)) \frac{dx^i}{dt}\frac{dx^j}{dt}=0,

where $\Gamma^k_{ij}$ are the Christoffel symbols. Repeated indices are summed according to the Einstein summation convention. This is a system of second-order differential equations; prescribing a starting point and tangent vector determines a unique local solution. (damtp.cam.ac.uk)

For the Levi-Civita connection,

\Gamma^k_{ij} =\frac12g^{k\ell} \left(\partial_i g_{j\ell} +\partial_j g_{i\ell} -\partial_\ell g_{ij}\right),

where $g^{k\ell}$ denotes the inverse metric and $\partial_i$ a coordinate partial derivative. In Euclidean Cartesian coordinates these coefficients vanish, so geodesics are straight lines. Nonzero coefficients alone do not establish curvature: they may also result from using curvilinear coordinates in flat space. (damtp.cam.ac.uk)

Length and variational characterization

For a piecewise smooth curve on a Riemannian manifold, length is

L(\gamma)=\int_a^b \sqrt{g(\dot\gamma,\dot\gamma)}\,dt.

The Riemannian distance $d(p,q)$ is the infimum of these lengths over curves joining $p$ to $q$ . A minimizing geodesic realizes that distance. The distinction between a geodesic and a minimizing geodesic is essential: the differential equation expresses local straightness, not global optimality. (graphics.stanford.edu)

The calculus of variations supplies another characterization through the energy functional

E(\gamma)=\frac12\int_a^b g(\dot\gamma,\dot\gamma)\,dt.

With endpoints fixed, its critical curves are precisely affinely parameterized geodesics. Applying the Euler–Lagrange equations to the associated Lagrangian yields the coordinate geodesic equation. Energy depends on parameterization, whereas length does not. A critical point need not be a minimum; sufficiently long geodesics can fail to minimize either functional. (claymath.org)

Elementary examples

On a round sphere, nonconstant geodesics follow great circles, formed by intersecting the sphere with planes through its center. Between distinct, non-antipodal points, the shorter great-circle arc minimizes length; the longer arc is also a geodesic but is not minimizing. Antipodal points admit infinitely many minimizing semicircles. Thus uniqueness for prescribed initial data does not imply uniqueness for prescribed endpoints. (sites.science.oregonstate.edu)

These examples also distinguish intrinsic distance from Euclidean distance in an embedding space. The chord between two sphere points is shorter than a surface arc, but it is not an admissible surface path. Geodesic minimization considers curves within the specified manifold. (graphics.stanford.edu)

Exponential map and completeness

The exponential map at $p$ packages geodesics with a common starting point:

\exp_p(v)=\gamma_v(1), \qquad \gamma_v(0)=p,\quad \dot\gamma_v(0)=v,

whenever the solution exists to time $1$ . Near the zero vector it provides normal coordinates, in which radial coordinate lines are geodesics. This connects local geometry to initial-value differential equations. (metaphor.ethz.ch)

A manifold is geodesically complete when every geodesic extends to all real parameter values. For a connected, finite-dimensional Riemannian manifold without boundary, the Hopf–Rinow theorem states that this is equivalent to being a complete metric space under Riemannian distance. Completeness also guarantees a minimizing geodesic between every pair of points and compactness of every closed bounded subset. It does not guarantee uniqueness of minimizers. (mat.univie.ac.at)

Curvature and nearby geodesics

A variation through geodesics produces a Jacobi field $J$ , satisfying

\frac{D^2J}{dt^2} +R(J,\dot\gamma)\dot\gamma=0

for the corresponding curvature-sign convention. The Riemann curvature tensor therefore governs how neighboring geodesics separate or converge. Conjugate points occur when a nonzero Jacobi field vanishes at two points; their occurrence constrains how far a geodesic can remain minimizing. (claymath.org)

Relativity and metric spaces

In general relativity, spacetime has a Lorentzian rather than positive-definite metric. Ideal freely falling massive test particles follow timelike geodesics, while light rays in vacuum follow null geodesics in the geometric-optics approximation. Proper time is an affine parameter for timelike motion, but not for null motion, whose proper time vanishes. Sufficiently short timelike geodesic segments locally maximize proper time among nearby timelike curves with the same endpoints, rather than minimize spatial distance. (damtp.cam.ac.uk)

In a general metric space, a minimizing geodesic can instead be defined by $d(\gamma(s),\gamma(t))=c|s-t|$ , with constant $c\ge0$ . A geodesic metric space admits such a connecting curve between every pair of points. This definition requires neither smooth coordinates nor a connection, extending geodesic geometry beyond smooth manifolds. (metaphor.ethz.ch)