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Definition
A constant-speed curve whose covariant acceleration vanishes on a Riemannian manifold.
Let be a Riemannian manifold with Levi–Civita connection . A smooth curve is a geodesic if along .
Its speed is constant. Given , there is a unique geodesic with and on its maximal interval of existence. If , it is unit speed.
Let be a finite-dimensional smooth manifold. A Riemannian metric on is a smooth symmetric covariant -tensor such that is a positive-definite inner product on for every . Equivalently, is a bundle metric on the tangent bundle . A Riemannian manifold is a pair . In local coordinates, , where the matrix is symmetric positive definite at every point and each coefficient is smooth.
Let be a Riemannian smooth manifold of dimension . Denote by the orthonormal frame bundle, a principal -bundle.