Covariant derivative

Definition

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component and the intrinsic covariant derivative component.

Related concepts

Abstract index notation Adjoint bundle Affine bundle Affine connection Albert Einstein Almost-contact manifold Almost complex manifold Almost flat manifold Almost symplectic manifold Angular momentum Antisymmetric tensor Arc length Associated bundle Atiyah–Singer index theorem Atlas (topology)Augustin-Louis Cauchy Banach manifold Basis (linear algebra)Basis (mathematics)Bernhard Riemann Calabi–Yau manifold Carl Friedrich Gauss Cartan connection Cartan formalism (physics)Cauchy stress tensor Change of basis Christoffel symbols Classification of manifolds Closed and exact differential forms Closed manifold Cofibration Collapsing manifold Comma Complete manifold Complex differential form Complex manifold Computer vision Connection (algebraic framework)Connection (fibred manifold)Connection (mathematics)Connection (principal bundle)Connection (vector bundle)Connection form Connection on a vector bundle Contact manifold Continuum mechanics Coordinate system Coordinates (elementary mathematics)Cotangent bundle Cotangent space Covariance and contravariance of vectors Covariant transformation Crelle's Journal Curvature of Riemannian manifolds Darboux's theorem De Rham cohomology Derivative Diffeology Diffeomorphism Differentiable curve Differentiable manifold Differential calculus over commutative algebras Differential form Differential geometry Differential of a function Differential operator Differential structure Diffiety Dimension Directional derivative Distribution (differential geometry)Dual bundle Dyadics Ehresmann connection Einstein manifold Einstein notation Einstein summation convention Einstein tensor Electromagnetic tensor Electromagnetism Elwin Bruno Christoffel Encyclopedia of Mathematics Engineering Euclidean geometry Euclidean space European Mathematical Society Exponential map (Lie theory)Exponential map (Riemannian geometry)Exterior calculus Exterior covariant derivative Exterior derivative Exterior form Exterior product Fiber bundle Fibered manifold Fibration Finsler manifold Flat manifold Foliation Foundations of Differential Geometry Frame bundle Frobenius theorem (differential topology)Fréchet manifold Fundamental theorem of Riemannian geometry G-structure on a manifold Gauge covariant derivative Gauge theory (mathematics)Gauss's lemma (Riemannian geometry)Gauss–Bonnet theorem General covariance General relativity Generalized Stokes theorem Geodesic Geodesics Geometrization conjecture Glossary of Riemannian and metric geometry Glossary of differential geometry and topology Glossary of tensor theory Gluon field strength tensor Gregorio Ricci-Curbastro Hadamard manifold Hermann Grassmann Hermann Weyl Hermitian manifold Hilbert manifold History of manifolds and varieties Hodge star operator Hopf–Rinow theorem Hyperbolic manifold Hypercomplex manifold Hyperkähler manifold Immersion (mathematics)Index notation Inner product space Integral curve Interior product Introduction to the mathematics of general relativity Isometry Jacobian matrix and determinant Jan Arnoldus Schouten Jean-Louis Koszul Jet bundle K-theory Katsumi Nomizu Kenmotsu manifold Koszul connection Kronecker delta Kähler manifold Leonhard Euler Levi-Civita connection Levi-Civita symbol Lie algebra bundle Lie algebra cohomology Lie bracket of vector fields Lie derivative Lie group Lie group–Lie algebra correspondence Linear map List of formulas in Riemannian geometry List of manifolds Local diffeomorphism Manifold Manifold with boundary Mathematician Mathematics Matrix (mathematics)Metric signature Metric tensor Metric tensor (general relativity)Michael Spivak Mixed tensor Moment of inertia Morse theory Moving frame Multi-index notation Multilinear algebra Multilinear map Multivector Musical isomorphism Nash embedding theorem Neighborhood (mathematics)Nilmanifold Noether's theorem Non-Hausdorff manifold Nonmetricity tensor Normal bundle One-form Operation (mathematics)Orbifold Orientability Orthogonal projection Orthonormal Parallel transport Parallelizable manifold Partial derivative Penrose graphical notation Physics Poincaré conjecture Poisson manifold Prime manifold Principal bundle Principal part Product rule Pseudo-Riemannian manifold Pseudotensor Pullback (differential geometry)Pushforward (differential)Quaternionic manifold Raising and lowering indices Ricci-flat manifold Ricci calculus Ricci curvature Ricci flow Riemann curvature tensor Riemann tensor Riemannian geometry Riemannian manifold Riemannian volume form Rizza manifold Sard's theorem Sasakian manifold Scalar curvature Schur's lemma (Riemannian geometry)Secondary calculus and cohomological physics Section (fiber bundle)Sectional curvature Semicolon Sheaf (mathematics)Shlomo Sternberg Shoshichi Kobayashi Singularity theory Smooth function Smooth structure Smoothness Spin tensor Spinor Spinor bundle Stable normal bundle Stein manifold Stratified space Stratifold Stress–energy tensor Sub-Riemannian manifold Subbundle Submanifold Submersion (mathematics)Supermanifold Symmetric tensor Symmetrization Symplectic manifold Tame manifold Tangent bundle Tangent space Tangent vector Tangential component Tensor Tensor (intrinsic definition)Tensor algebra Tensor analysis Tensor bundle Tensor calculus Tensor contraction Tensor density Tensor derivative (continuum mechanics)Tensor field Tensor operator Tensor product Tensors in curvilinear coordinates Tetrad (index notation)Topological manifold Torsion of connection Torsion tensor Translation (geometry)Transport phenomena Transpose Tullio Levi-Civita Two-point tensor Uniformization theorem Van der Waerden notation Vector-valued differential form Vector (mathematics and physics)Vector bundle Vector calculus Vector field Vector fields Vector flow Vector space Voigt notation Volume form Wedge product Weyl tensor Whitney embedding theorem Wiley Interscience Woldemar Voigt Élie Cartan

42 concepts already in your glossary