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

(1)
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.
Other articles related to "geodesic, geodesics":
... In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold is the projection of a closed orbit of the geodesic flow on the manifold ...
... It is one of a family of theorems that quantify the assertion that a pair of geodesics emanating from a point p spread apart more slowly in a region of high curvature ... Let pqr be a geodesic triangle in M, such that the geodesic pq is minimal and if δ ≥ 0, the length of the side pr is less than ... Let p′q′r′ be a geodesic triangle in the space form Mδ such that the length of sides p′q′ and p′r′is equal to that of pq and pr respectively and the angle at p ...
... notion of "straight line" generalizes to that of a geodesic ... coordinate system), any two points can be joined by a geodesic ... neighborhood, the steps to producing a LeviCivita parallelogram are Start with a geodesic AB and another geodesic AA′ ...
... Let (X, d) be a geodesic metric space, i.e ... a metric space for which every two points x, y ∈ X can be joined by a geodesic segment, an arc length parametrized continuous curve γ → X, γ(a) = x, γ(b) = y, whose length is precisely d(x ... Let Δ be a triangle in X with geodesic segments as its sides ...
... The French Geodesic Mission (also called the Geodesic Mission to Peru, Geodesic Mission to the Equator and the SpanishFrench Geodesic Mission) was an 18th ... The mission was one of the first geodesic (or geodetic) missions carried out under modern scientific principles, and the first major international scientific expedition ...