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In mathematics, particularly differential geometry, a geodesic (/ˌdʒiːɵˈdiːzɨk/ JEEoDEEzik or /ˌdʒiːɵˈdɛsɨk/ JEEoDESik) is a generalization of the notion of a "straight line" to "curved spaces". 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. If this connection is the LeviCivita connection induced by a Riemannian metric, then the geodesics are (locally) the shortest path between points in the space.
The term "geodesic" comes from geodesy, the science of measuring the size and shape of Earth; in the original sense, a geodesic was the shortest route between two points on the Earth's surface, namely, a segment of a great circle. The term has been generalized to include measurements in much more general mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph.
Geodesics are of particular importance in general relativity. Geodesics in general relativity describe the motion of inertial test particles.
The shortest path between two points in a curved space can be found by writing the equation for the length of a curve (a function f from an open interval of R to the manifold), and then minimizing this length using the calculus of variations. This has some minor technical problems, because there is an infinite dimensional space of different ways to parameterize the shortest path. It is simpler to demand not only that the curve locally minimize length but also that it is parameterized "with constant velocity", meaning that the distance from f(s) to f(t) along the geodesic is proportional to s−t. Equivalently, a different quantity may be defined, termed the energy of the curve; minimizing the energy leads to the same equations for a geodesic (here "constant velocity" is a consequence of minimisation). Intuitively, one can understand this second formulation by noting that an elastic band stretched between two points will contract its length, and in so doing will minimize its energy. The resulting shape of the band is a geodesic.
In Riemannian geometry geodesics are not the same as "shortest curves" between two points, though the two concepts are closely related. The difference is that geodesics are only locally the shortest distance between points, and are parameterized with "constant velocity". Going the "long way round" on a great circle between two points on a sphere is a geodesic but not the shortest path between the points. The map t → t^{2} from the unit interval to itself gives the shortest path between 0 and 1, but is not a geodesic because the velocity of the corresponding motion of a point is not constant.
Geodesics are commonly seen in the study of Riemannian geometry and more generally metric geometry. In general relativity, geodesics describe the motion of point particles under the influence of gravity alone. 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. More generally, the topic of subRiemannian geometry deals with the paths that objects may take when they are not free, and their movement is constrained in various ways.
This article presents the mathematical formalism involved in defining, finding, and proving the existence of geodesics, in the case of Riemannian and pseudoRiemannian manifolds. The article geodesic (general relativity) discusses the special case of general relativity in greater detail.
The most familiar examples are the straight lines in Euclidean geometry. On a sphere, the images of geodesics are the great circles. The shortest path from point A to point B on a sphere is given by the shorter arc of the great circle passing through A and B. If A and B are antipodal points (like the North Pole and the South Pole), then there are infinitely many shortest paths between them. Geodesics on an ellipsoid behave in a more complicated way than on a sphere; in particular, they are not closed in general (see figure).
In metric geometry, a geodesic is a curve which is everywhere locally a distance minimizer. More precisely, a curve γ: I → M from an interval I of the reals to the metric space M is a geodesic if there is a constant v ≥ 0 such that for any t ∈ I there is a neighborhood J of t in I such that for any t_{1}, t_{2} ∈ J we have
This generalizes the notion of geodesic for Riemannian manifolds. However, in metric geometry the geodesic considered is often equipped with natural parameterization, i.e. in the above identity v = 1 and
If the last equality is satisfied for all t_{1}, t_{2} ∈I, the geodesic is called a minimizing geodesic or shortest path.
In general, a metric space may have no geodesics, except constant curves. 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.
In a Riemannian manifold M with metric tensor g, the length of a continuously differentiable curve γ : [a,b] → M is defined by
The distance d(p, q) between two points p and q of M is defined as the infimum of the length taken over all continuous, piecewise continuously differentiable curves γ : [a,b] → M such that γ(a) = p and γ(b) = q. With this definition of distance, geodesics in a Riemannian manifold are then the locally distanceminimizing paths.
The minimizing curves of L in a small enough open set of M can be obtained by techniques of calculus of variations. Typically, one introduces the following action or energy functional
It is then enough to minimize the functional E, owing to the Cauchy–Schwarz inequality
with equality if and only if dγ/dt is constant.
The Euler–Lagrange equations of motion for the functional E are then given in local coordinates by
where are the Christoffel symbols of the metric. This is the geodesic equation, discussed below.
Techniques of the classical calculus of variations can be applied to examine the energy functional E. The first variation of energy is defined in local coordinates by
The critical points of the first variation are precisely the geodesics. The second variation is defined by
In an appropriate sense, zeros of the second variation along a geodesic γ arise along Jacobi fields. Jacobi fields are thus regarded as variations through geodesics.
By applying variational techniques from classical mechanics, one can also regard geodesics as Hamiltonian flows. They are solutions of the associated Hamilton–Jacobi equations, with (pseudo)Riemannian metric taken as Hamiltonian.
A geodesic on a smooth manifold M with an affine connection ∇ is defined as a curve γ(t) such that parallel transport along the curve preserves the tangent vector to the curve, so
 ( 
at each point along the curve, where is the derivative with respect to . More precisely, in order to define the covariant derivative of it is necessary first to extend to a continuously differentiable vector field in an open set. However, the resulting value of (1) is independent of the choice of extension.
Using local coordinates on M, we can write the geodesic equation (using the summation convention) as
where are the coordinates of the curve γ(t) and are the Christoffel symbols of the connection ∇. This is just an ordinary differential equation for the coordinates. It has a unique solution, given an initial position and an initial velocity. Therefore, from the point of view of classical mechanics, geodesics can be thought of as trajectories of free particles in a manifold. Indeed, the equation means that the acceleration of the curve has no components in the direction of the surface (and therefore it is perpendicular to the tangent plane of the surface at each point of the curve). So, the motion is completely determined by the bending of the surface. This is also the idea of general relativity where particles move on geodesics and the bending is caused by the gravity.
The local existence and uniqueness theorem for geodesics states that geodesics on a smooth manifold with an affine connection exist, and are unique. More precisely:
In general, I may not be all of R as for example for an open disc in R^{2}. The proof of this theorem follows from the theory of ordinary differential equations, by noticing that the geodesic equation is a secondorder ODE. Existence and uniqueness then follow from the Picard–Lindelöf theorem for the solutions of ODEs with prescribed initial conditions. γ depends smoothly on both p and V.
Geodesic flow is a local Raction on tangent bundle TM of a manifold M defined in the following way
where t ∈ R, V ∈ TM and denotes the geodesic with initial data . Thus, G_{t}(V) = exp(tV) is the exponential map of the vector tV. A closed orbit of the geodesic flow corresponds to a closed geodesic on M.
On a (pseudo)Riemannian manifold, the geodesic flow is identified with a Hamiltonian flow on the cotangent bundle. The Hamiltonian is then given by the inverse of the (pseudo)Riemannian metric, evaluated against the canonical oneform. In particular the flow preserves the (pseudo)Riemannian metric , i.e.
In particular, when V is a unit vector, remains unit speed throughout, so the geodesic flow is tangent to the unit tangent bundle. Liouville's theorem implies invariance of a kinematic measure on the unit tangent bundle.
The geodesic flow defines a family of curves in the tangent bundle. The derivatives of these curves define a vector field on the total space of the tangent bundle, known as the geodesic spray.
More precisely, an affine connection gives rise to a splitting of the double tangent bundle TTM into horizontal and vertical bundles:
The geodesic spray is the unique horizontal vector field W satisfying
at each point v ∈ TM; here π_{∗} : TTM → TM denotes the pushforward (differential) along the projection π : TM → M associated to the tangent bundle.
More generally, the same construction allows one to construct a vector field for any Ehresmann connection on the tangent bundle. For the resulting vector field to be a spray (on the deleted tangent bundle TM \ {0}) it is enough that the connection be equivariant under positive rescalings: it need not be linear. That is, (cf. Ehresmann connection#Vector bundles and covariant derivatives) it is enough that the horizontal distribution satisfy
for every X ∈ TM \ {0} and λ > 0. Here d(S_{λ}) is the pushforward along the scalar homothety A particular case of a nonlinear connection arising in this manner is that associated to a Finsler manifold.
Equation (1) is invariant under affine reparameterizations; that is, parameterizations of the form
where a and b are constant real numbers. Thus apart from specifying a certain class of embedded curves, the geodesic equation also determines a preferred class of parameterizations on each of the curves. Accordingly, solutions of (1) are called geodesics with affine parameter.
An affine connection is determined by its family of affinely parameterized geodesics, up to torsion (Spivak 1999, Chapter 6, Addendum I). The torsion itself does not, in fact, affect the family of geodesics, since the geodesic equation depends only on the symmetric part of the connection. More precisely, if are two connections such that the difference tensor
is skewsymmetric, then and have the same geodesics, with the same affine parameterizations. Furthermore, there is a unique connection having the same geodesics as , but with vanishing torsion.
Geodesics without a particular parameterization are described by a projective connection.
