Gauss's lemma (differential geometry)

Gauss's lemma states that, under the Riemannian exponential map, radial directions are orthogonal to images of tangent directions to spheres centered at the origin in a tangent space.

Connect

Riemannian exponential map: The map sending a tangent vector to the endpoint of the geodesic that starts with that initial velocity. The lemma describes the metric behavior of this map along radial and spherical directions.
Normal coordinates: Local coordinates obtained by expressing nearby points as tangent vectors through the Riemannian exponential map. The lemma identifies radial coordinate lines as orthogonal to coordinate spheres.
Tangent space: A vector space of infinitesimal directions at a point of a differentiable manifold. The lemma begins with radial vectors and sphere-tangent vectors in this vector space.
Gauss–Bonnet theorem: A theorem equating integrated Gaussian curvature and boundary geodesic curvature with a topological invariant of a surface. Unlike Gauss's lemma, it relates curvature integrals to topology rather than radial and angular directions.
Geodesic: A curve whose velocity is parallel transported along itself, locally extremizing length between nearby points. Radial curves under the exponential map are geodesics, and their constant speed drives the lemma.
Geodesic polar coordinates: Coordinates that describe points near a center by geodesic distance and an initial direction. The lemma ensures the radial coordinate is orthogonal to angular coordinates.
Tangent sphere: A sphere centered at the origin of a tangent space, defined using an inner product on that space. Its tangent directions are precisely the angular directions whose images the lemma makes orthogonal to radial ones.
Riemannian curvature tensor: A tensor measuring the failure of covariant derivatives to commute and encoding intrinsic curvature. Curvature affects angular behavior of geodesic spheres, while Gauss's lemma's radial orthogonality holds independently of curvature.
Geodesic variation: A smooth family of geodesics depending on one or more parameters. Varying initial directions on a sphere produces the variation fields whose inner product with the radial velocity vanishes.
Gauss lemma for surfaces: The surface-geometric form of Gauss's lemma, relating radial geodesics to curves on geodesic circles. It gives the classical surface version of the same radial orthogonality property.