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The notion of a ''geodesic line'' (also: ''geodesic'')
is a geometric concept which is a generalization of the concept of a
straight line (or a segment of a straight line) in Euclidean geometry
to spaces of a more general type. The definitions of geodesic lines in
various spaces depend on the particular structure (metric, line
element, linear connection) on which the geometry of the particular
space is based. In the geometry of spaces in which the metric is
considered to be specified in advance, geodesic lines are defined as
locally shortest. In spaces with a connection a geodesic line is
defined as a curve for which the tangent vector field is parallel
along this curve. In Riemannian and Finsler geometries, where the line
element is given in advance (in other words, a metric in the tangent
space at each point of the considered manifold is given), while the
lengths of lines are obtained by subsequent integration, geodesic
lines are defined as extremals of the length functional.

Geodesic lines were first studied by J. Bernoulli and L. Euler, who
attempted to find the shortest lines on regular surfaces in Euclidean
space. On such lines the
[[Geodesic curvature|geodesic curvature]] vanishes, and the principal
normal of such curves is parallel to the normal to the
surface. Geodesic lines are preserved under isometric deformation. The
motion of a conservative mechanical system with a finite number of
degrees of freedom is described by a geodesic line in a
suitably-chosen Riemannian space.

Geodesic lines in Riemannian spaces have been studied most
thoroughly. Let $M^n$ be an $n$-dimensional Riemannian space with metric
tensor $g_{ij}$ of class $C^k$, $k\ge 2$. The definition of a geodesic line as an
extremal makes it possible to write down its differential equation in
arbitrary local coordinates $x^i$, $i=1,\dots,n$, for any parametrization $\def\g{\gamma} \g = (x^i(t))$:

$$\def\d{\partial}\frac{d}{dt}\Big(\frac{\d F}{\d x^i}\Big) - \frac{\d F}{\d x^i} = 0,$$
where 
$$\def\dx{\dot{x}} F = \sqrt{g_{ik} \dx^i\dx^k}, \quad \dx^i=\frac{dx^i}{dt}.$$
Another, equivalent, form of the equations of
geodesic lines is derived from the postulate of parallelity of the
tangent vector $\dot{\gamma}=(\dx^i)$ along $\gamma$. If $t$ is the arc length $s$ along a
geodesic, or a linear function of $s$, then 
$$\frac{D}{dt}(\dot{\gamma})=0,\quad {\textrm or }\quad \ddot{x}^i + \Gamma_{jk}^i \dx^j\dx^k = 0.\label{1}$$
The definition of a
geodesic line by equation (1) also involves a canonical selection of a
parameter. In such a definition a geodesic line $\gamma=x(t,\xi)$, with initial
tangent vector $\xi$, $x(0,\xi) = x_0$, $\dx(0,\xi) = \xi$, passes through a given point $x_0$. The
mapping $\xi\mapsto x(1,\xi)$ of the tangent space at $x_0$ into the space under
consideration is the exponential mapping with pole $x_0$. Near the pole
$x_0$ it is a diffeomorphism, which introduces Riemannian coordinates
into the space under consideration.

A number of properties of geodesic lines are preserved for curves
represented by second-order equations $\ddot{x}=F(x,\dot{x})$ if, like in (1), the
function $F$ is a homogeneous function of the second degree in
$(\dx^i)$. The search for such equations in terms of tangent bundles yields
the concepts of a
[[Spray|spray]] and their integral curves. Geodesic lines are a
special case of such curves
{{Cite|La}}.

The local behaviour of geodesic curves is similar to that of straight
lines in Euclidean space. A sufficiently short arc of a geodesic line
is the shortest among all rectifiable curves with the same ends. Only
one geodesic line passes through any point in a given direction. Each
point has a neighbourhood $U$ in which any two points can be connected
by a unique geodesic line lying in $U$
{{Cite|He}}.

The question of the distance to which an arc of a geodesic line
issuing from a point $x_0$ may be extended so that it remains the
shortest among all curves in a neighbourhood of $x_0$ is one of the
problems of the calculus of variations. A comparison of a geodesic
line with nearby curves is based on the study of the second variation
of the length by considering particular fields of velocities (the
[[Jacobi vector field|Jacobi vector field]]) along the geodesic line
$\gamma(s)$ belonging to particular variations $\gamma(s,t)$. For any fixed $t$ the
curve $\gamma(s,t)$ remains a geodesic, while the parameter $s$ on it remains
canonical. If at the origin of $\gamma$ the velocity is zero, then the
points on $\gamma$ where some non-zero Jacobi vector field is zero are
called conjugate points. Geodesic lines remain the shortest of all
nearby curves up to the first conjugate point. For a geodesic arc
extended beyond the conjugate point there exists a shorter curve with
the same ends, which may be arbitrary near. A Jacobi vector field $\eta(s)$
satisfies the equation 
$$\frac{D^2\eta}{ds^2} + R(\dot{\gamma},\eta)\dot{\gamma} = 0,$$
where $\dot{\gamma}$ is a tangent vector to the
geodesic $\gamma(s)$, while $R(\dot{\gamma},\eta)$ is the
[[Curvature transformation|curvature transformation]] or, in Fermi
coordinates $x^i$, $x^1 = s$: 
$$\frac{d^2\eta^l}{ds^2} + R_{1k,1}^i\eta^k = 0,\label{2}$$
where $R^l_{ij,k}$ is the curvature tensor. The
connection between the Jacobi vector field and the curvature
determines the dependence of properties of the geodesics on the
curvature of the space. For instance, in a space of negative curvature
there are no conjugate points; if, in addition, the space is simply
connected, then any geodesic arc is shortest, and geodesic lines
issuing from a point diverge exponentially. These properties are of
importance in the theory of dynamical systems (cf.
[[Geodesic flow|Geodesic flow]]). The monotone nature of the effect of
the curvature forms the subject of several so-called comparison
theorems. In particular, the distance to the first conjugate point and
the lengths of the vectors of the Jacobi vector field on this segment
(normalized by the condition $\eta(0)=0$, $|D\eta/ds| = 1$) decrease as the curvature of
the space increases. Here, under a comparison of two geodesic lines it
is understood that all the curvatures of the second space majorize any
curvature of the first space at points corresponding to the same
lengths
{{Cite|GrKlMe}}.

In general relativity theory equation (2) is the source of physical
interpretation of the space-time curvature by the behaviour of
geodesic lines
{{Cite|Sy}}.

If the comparison is not restricted to nearby curves, the arc of a
geodesic line may cease to be shortest before it has passed the
conjugate point. This is possible even in a simply-connected space,
i.e. the reasons for it may be both topological and metric.

The question of the effect of the curvature on the extension of the
arc between conjugate points on which the geodesic line remains
shortest is of importance in the study of the connection between the
curvature and the topological structure of the space. The dependence
of the number of closed geodesic lines or the number of different
geodesic lines connecting two points on the topological structure of
the space forms a subject of
[[Variational calculus in the large|variational calculus in the
large]]{{Cite|LySc}},
{{Cite|GrKlMe}},
{{Cite|Mi}}.

Families of geodesic lines, considered as possible trajectories of
motion, form a subject of the theory of dynamical systems and ergodic
theory.

In spaces with an affine connection geodesic lines are defined by
equation (1). Local theorems on the existence and uniqueness of
geodesic lines connecting two points, and on the existence of a convex
neighbourhood, are preserved for such spaces.

Geodesic lines with similar properties are also defined in spaces with
a projective connection, and also in cases of more general connections
on manifolds.

The geometrization of the problems of variational calculus for
functionals other than the length functional generated the concept of
a
[[Finsler space|Finsler space]] and the geodesic lines in that
space. Separation of basic geometrical properties of such spaces led
to the concept of
[[Geodesic geometry|geodesic geometry]], defined by the presence and
the extendability of geodesic lines.

The most intensively studied geodesic lines in metric spaces with an
irregular metric are those on a convex surface and in a
[[Two-dimensional manifold of bounded curvature|two-dimensional
manifold of bounded curvature]]. Here, a geodesic line is not
necessarily a smooth curve; it need have no extension or — in a
two-dimensional manifold of bounded curvature — it may have a
non-unique extension. A geodesic line on a convex surface invariably
has a semi-tangent; if it can be extended, the extension is unique;
geodesic lines issue from a point in almost-all directions. It was
found that in such spaces the class of quasi-geodesic lines (cf.
[[Quasi-geodesic line|Quasi-geodesic line]]), which is the closure of
the class of geodesic lines, is more natural than the class of
geodesic lines itself
{{Cite|Al}}.

====Comments====
Riemannian coordinates are usually called normal
coordinates or geodesic polar coordinates, cf.
[[Geodesic coordinates|Geodesic coordinates]].

A convex neighbourhood, also called a normal neighbourhood, is a
neighbourhood $U$ in which any two points can be connected by a unique
geodesic in $U$.


====References====
{|
|-
|valign="top"|{{Ref|Al}}||valign="top"| A.D. Aleksandrov, "Die innere Geometrie der konvexen Flächen", Akademie Verlag (1955) (Translated from Russian)         
|-
|valign="top"|{{Ref|BeGo}}||valign="top"| M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces", Springer (1988) (Translated from French)  {{MR|0917479}}  {{ZBL|0629.53001}}       
|-
|valign="top"|{{Ref|Bu}}||valign="top"| H. Busemann, "The geometry of geodesics", Acad. Press (1955)  {{MR|0075623}}  {{ZBL|0112.37002}}       
|-
|valign="top"|{{Ref|GrKlMe}}||valign="top"| D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen", Springer (1968)  {{MR|0229177}}  {{ZBL|0155.30701}}       
|-
|valign="top"|{{Ref|He}}||valign="top"| S. Helgason, "Differential geometry, Lie groups, and symmetric spaces", Acad. Press (1978)     {{MR|0514561}}  {{ZBL|0451.53038}}    
|-
|valign="top"|{{Ref|Kl}}||valign="top"| W. Klingenberg, "Lectures on closed geodesics", Springer (1978)  {{MR|0478069}}  {{ZBL|0397.58018}}       
|-
|valign="top"|{{Ref|Kl2}}||valign="top"| W. Klingenberg, "Riemannian geometry", de Gruyter (1982) (Translated from German)  {{MR|0666697}}  {{ZBL|0495.53036}}       
|-
|valign="top"|{{Ref|La}}||valign="top"| S. Lang, "Introduction to differentiable manifolds", Interscience (1962) pp. App. III        {{MR|0155257}}  {{ZBL|0103.15101}} 
|-
|valign="top"|{{Ref|LySc}}||valign="top"| L.A. Lyusternik, L.G. [L.G. Shnirel'man] Schnirelmann, "Méthode topologiques dans les problèmes variationelles", Hermann (1934) (Translated from Russian)         
|-
|valign="top"|{{Ref|Mi}}||valign="top"| J.W. Milnor, "Morse theory", Princeton Univ. Press (1963)  {{MR|0163331}}  {{ZBL|0108.10401}}       
|-
|valign="top"|{{Ref|Ra}}||valign="top"| P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse", Deutsch. Verlag Wissenschaft.  (1959) (Translated from Russian)         
|-
|valign="top"|{{Ref|Sy}}||valign="top"| J.L. Synge, "Relativity: the general theory", North-Holland &amp; Interscience (1960)  {{MR|0118457}}  {{ZBL|0090.18504}}       
|-
|}