Geodesic

geodesicsgeodesic flowgeodesic equationaffine parametergeodesic spraygeodetic distancestraight linesaffineAffine geodesicaffinely parametrized
In differential geometry, a geodesic is a curve representing in some sense the shortest path between two points in a surface, or more generally in a Riemannian manifold.wikipedia
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Differential geometry of surfaces

surfaceshape operatorsmooth surface
In differential geometry, a geodesic is a curve representing in some sense the shortest path between two points in a surface, or more generally in a Riemannian manifold.
This is well illustrated by the non-linear Euler–Lagrange equations in the calculus of variations: although Euler developed the one variable equations to understand geodesics, defined independently of an embedding, one of Lagrange's main applications of the two variable equations was to minimal surfaces, a concept that can only be defined in terms of an embedding.

Line (geometry)

linestraight linelines
It is a generalization of the notion of a "straight line" to a more general setting.
Thus in differential geometry a line may be interpreted as a geodesic (shortest path between points), while in some projective geometries a line is a 2-dimensional vector space (all linear combinations of two independent vectors).

Geodesy

geodeticgeodesistgeodetic survey
The term "geodesic" comes from geodesy, the science of measuring the size and shape of Earth.
In the general case, the solution is called the geodesic for the surface considered.

Geodesic curvature

curvature vectorgeodesic curvespath curvature
In a Riemannian manifold or submanifold geodesics are characterised by the property of having vanishing geodesic curvature.
In Riemannian geometry, the geodesic curvature k_g of a curve \gamma measures how far the curve is from being a geodesic.

Affine connection

connectionaffineaffine connections
More generally, in the presence of an affine connection, a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it.
Affine connections may also be used to define (affine) geodesics on a manifold, generalizing the straight lines of Euclidean space, although the geometry of those straight lines can be very different from usual Euclidean geometry; the main differences are encapsulated in the curvature of the connection.

Calculus of variations

variationalvariational calculusvariational methods
The shortest path between two given points in a curved space, assumed to be a differential manifold, can be defined by using the equation for the length of a curve (a function f from an open interval of R to the space), and then minimizing this length between the points using the calculus of variations.
Such solutions are known as geodesics.

Great circle

Great Circle Routegreat circlesarcs of great circle
For a spherical Earth, it is a segment of a great circle. On a sphere, the images of geodesics are the great circles.
These great circles are the geodesics of the sphere.

General relativity

general theory of relativitygeneral relativity theoryrelativity
Geodesics are of particular importance in general relativity.
In modern parlance, their paths are geodesics, straight world lines in curved spacetime.

Orbit

orbitsorbital motionplanetary motion
In particular, the path taken by a falling rock, an orbiting satellite, or the shape of a planetary orbit are all geodesics in curved spacetime.
However, Albert Einstein's general theory of relativity, which accounts for gravity as due to curvature of spacetime, with orbits following geodesics, provides a more accurate calculation and understanding of the exact mechanics of orbital motion.

Riemannian geometry

Riemannianlocal to global theoremsRiemann geometry
Geodesics are commonly seen in the study of Riemannian geometry and more generally metric geometry.
Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions.

Metric tensor

metricmetricsround metric
In a Riemannian manifold M with metric tensor g, the length L of a continuously differentiable curve γ : [a,b] → M is defined by
On a Riemannian manifold, the curve connecting two points that (locally) has the smallest length is called a geodesic, and its length is the distance that a passenger in the manifold needs to traverse to go from one point to the other.

Parallel transport

parallelparallel-transporttransported
More generally, in the presence of an affine connection, a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it.
Then a connection on the tangent bundle of M, called an affine connection, distinguishes a class of curves called (affine) geodesics.

Riemannian manifold

Riemannian metricRiemannianRiemannian manifolds
In differential geometry, a geodesic is a curve representing in some sense the shortest path between two points in a surface, or more generally in a Riemannian manifold. In a Riemannian manifold M with metric tensor g, the length L of a continuously differentiable curve γ : [a,b] → M is defined by Applying this to the Levi-Civita connection of a Riemannian metric recovers the previous notion.
Even though Riemannian manifolds are usually "curved", there is still a notion of "straight line" on them: the geodesics.

Intrinsic metric

length spaceintrinsicinduced length metric
At the other extreme, any two points in a length metric space are joined by a minimizing sequence of rectifiable paths, although this minimizing sequence need not converge to a geodesic.
If the space has the stronger property that there always exists a path that achieves the infimum of length (a geodesic) then it may be called a geodesic metric space or geodesic space.

Geodesics as Hamiltonian flows

geodesic Hamiltoniangeodesics
By applying variational techniques from classical mechanics, one can also regard geodesics as Hamiltonian flows.
In mathematics, the geodesic equations are second-order non-linear differential equations, and are commonly presented in the form of Euler–Lagrange equations of motion.

Jacobi field

Jacobi equation
In an appropriate sense, zeros of the second variation along a geodesic γ arise along Jacobi fields.
In Riemannian geometry, a Jacobi field is a vector field along a geodesic \gamma in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic.

Levi-Civita connection

Christoffel symbolconnectionsLevi-Civita
Applying this to the Levi-Civita connection of a Riemannian metric recovers the previous notion.
If the covariant derivative is the Levi-Civita connection of a certain metric, then the geodesics for the connection are precisely those geodesics of the metric that are parametrised proportionally to their arc length.

Christoffel symbols

Christoffel symbolChristoffel coefficientsChristoffel connection
:where are the Christoffel symbols of the metric.
In differential geometry, an affine connection can be defined without reference to a metric, and many additional concepts follow: parallel transport, covariant derivatives, geodesics, etc. also do not require the concept of a metric.

Free fall

free-fallfreefallfree-falling
Timelike geodesics in general relativity describe the motion of free falling test particles.
In general relativity, an object in free fall is subject to no force and is an inertial body moving along a geodesic.

Sphere

sphericalhemisphereglobose
On a sphere, the images of geodesics are the great circles.
The analogue of the "line" is the geodesic, which is a great circle; the defining characteristic of a great circle is that the plane containing all its points also passes through the center of the sphere.

Geodesics in general relativity

geodesicgeodesicsnull geodesic
Timelike geodesics in general relativity describe the motion of free falling test particles.

Geodesic manifold

geodesically completegeodesic completeness
Any γ extends to all of ℝ if and only if M is geodesically complete.
Equivalently, consider a maximal geodesic

Closed geodesic

closed
A closed orbit of the geodesic flow corresponds to a closed geodesic on M.
In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold is a geodesic that returns to its starting point with the same tangent direction.

Flow (mathematics)

flowflowsLocal flow
Geodesic flow is a local R-action on the tangent bundle TM of a manifold M defined in the following way
Specific examples of vector flows include the geodesic flow, the Hamiltonian flow, the Ricci flow, the mean curvature flow, and the Anosov flow.

Hamilton–Jacobi equation

Hamilton–Jacobi theoryHamilton's principal functionHamilton-Jacobi theory
They are solutions of the associated Hamilton equations, with (pseudo-)Riemannian metric taken as Hamiltonian.
For example, the Hamilton–Jacobi equations can be used to determine the geodesics on a Riemannian manifold, an important variational problem in Riemannian geometry.