Translation (geometry) of the object. The translation vector formula_8 common to all points of the object describes a particular type of displacement of the object, usually called a "linear" displacement to distinguish it from displacements involving rotation, called "angular" displacements. When considering spacetime, a change of time coordinate is considered to be a translation. For example, the Galilean group and the Poincaré group include translations with respect to time. Translation (geometry) In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same distance in a given direction. In Euclidean geometry a
Gauss's lemma (Riemannian geometry) a linear map. This will therefore prove the lemma. Since the maps formula_75 are geodesics, the function formula_78 is constant. Thus, Gauss's lemma (Riemannian geometry) In Riemannian geometry, Gauss's lemma asserts that any sufficiently small sphere centered at a point in a Riemannian manifold is perpendicular to every geodesic through the point. More formally, let "M" be a Riemannian manifold, equipped with its Levi-Civita connection, and "p" a point of "M". The exponential map is a mapping from the tangent space at "p" to "M": which is a diffeomorphism in a neighborhood of zero. Gauss' lemma asserts that the image