Riemannian geometry

Killing vector field

In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold. More simply, the flow generates a symmetry, in the sense that moving each point of an object the same distance in the direction of the Killing vector will not distort distances on the object. (Wikipedia).

Killing vector field
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Cartan–Ambrose–Hicks theorem | Poincaré group | Group action | Covariant derivative | Lie group | Upper half-plane | Schwarzschild metric | Anti-de Sitter space | Sectional curvature | Lie bracket of vector fields | Symmetry | Special conformal transformation | Conformal map | Lorentz group | Conformal Killing vector field | Kronecker delta | Isometry group | Hodge theory | Poincaré metric | Divergence | Symmetric space | De Sitter space | Riemann curvature tensor | Levi-Civita connection | Homomorphism | Geodesics as Hamiltonian flows | Minkowski space | Indefinite orthogonal group | Structure constants | Pseudo-Euclidean space | Mathematics | Lie derivative | Killing form | Abstract index notation | Isometry | Riemannian manifold | Euclidean space | Involution (mathematics) | Killing tensor | Lie algebra | Local coordinates | Tensor | Harmonic function | Integral curve | Compact space | Killing spinor | Manifold | Metric tensor | Ricci curvature | Spacetime symmetries | Pseudo-Riemannian manifold | Poincaré half-plane model | Kerr metric | Vector field