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A linear representation $\rho :G\to GL(V)$ of a principal bundle's structure group defines the quotient of $P\times V$ by $(p,v)\sim (pg,\rho (g{)}^{-1}v)$. It is a vector bundle over the same base. A principal connection induces parallel transport and a local covariant derivative $d+{\rho }_{\ast }(A)$ on its sections.

Expand the string embedding map along target geodesics as $X={\exp }_{\bar{X}}Y$. The quadratic Euclidean fluctuation action has a covariant derivative term and a curvature term $-R_{acbd}{Y}^a{Y}^b\partial {\bar{X}}^c\partial {\bar{X}}^d$. The logarithmically divergent coincident contraction is proportional to ${\alpha }^{\prime }{g}^{ab}$; its curvature contraction is the Ricci tensor. A compensating metric counterterm gives ${\beta }_{ab}^g={\alpha }^{\prime }{R}_{ab}+O({\alpha }^{\prime 2})$. This is why the target gravitational equation arises from quantum worldsheet Weyl anomaly cancellation, rather than from the classical worldsheet metric variation. Infrared regulation and removal of equation-of-motion terms by field redefinitions specify the usual one-loop scheme.

On a closed Riemannian manifold, a Killing vector field $X$ satisfies ${\nabla }^{\ast }\nabla X={\mathrm{Ric}}^{♯}X$ for the nonnegative rough Laplacian. This follows by differentiating ${\nabla }_i{X}_j+{\nabla }_j{X}_i=0$, contracting and commuting covariant derivatives; the contraction also gives $\mathrm{div}X=0$. Taking the inner product with $X$ and using integration by parts gives the displayed identity. If Ricci curvature is nonpositive, the nonnegative left side must vanish, so $X$ is a parallel vector field. In particular every Killing vector field on a closed flat Riemannian manifold is a parallel vector field.

For a torsion-free connection, the commutator of coordinate covariant derivatives vanishes on a scalar field and acts on a vector field through the displayed Riemann curvature tensor. On a covector field it instead gives $[{\nabla }_{\alpha },{\nabla }_{\beta }]A_{\mu }=-R^{\nu }{}_{\mu \alpha \beta }{A}_{\nu }$. The curvature commutator is a tensor derivation, so each index of a general tensor contributes independently with the corresponding sign. Including the lower derivative index in the second covariant derivative is essential for this identity.

The Christoffel trace identity converts the covariant derivative of a vector field into a coordinate divergence weighted by the metric volume density. This is valid for the Levi-Civita connection. Applied to the raised gradient of a scalar field, it gives $\square \varphi =|g{|}^{-1/2}{\partial }_{\mu }(\sqrt{|g|}{g}^{\mu \nu }{\partial }_{\nu }\varphi )$ in Lorentzian signature, the curved d'Alembert operator.

For a torsion-free linear connection on a manifold, the commutator of two covariant derivatives obeys the tensor-product Leibniz rule: mixed first-derivative terms cancel after antisymmetrizing. It vanishes on scalar fields, so ${\Delta }_{ab}(fQ)=f{\Delta }_{ab}Q$; its action is therefore pointwise tensorial. Its vector action defines the Riemann curvature tensor. Applying the commutator to a scalar contraction proves that every covariant tensor index contributes minus the corresponding curvature action, while every contravariant index contributes plus. Torsion-free does not by itself imply metric compatibility.

With convention ${\nabla }^{\mu }{F}_{\mu \nu }=-{\mu }_0{J}_{\nu }$, write the electromagnetic field tensor as $F_{\mu \nu }={\nabla }_{\mu }{A}_{\nu }-{\nabla }_{\nu }{A}_{\mu }$. Commuting the contracted covariant derivatives gives ${\nabla }^{\mu }{\nabla }_{\nu }{A}_{\mu }={\nabla }_{\nu }({\nabla }^{\mu }{A}_{\mu })+R^{\alpha }{}_{\nu }{A}_{\alpha }$. The Lorenz gauge condition removes the gradient-of-divergence term, leaving the displayed wave equation for a covector field. A locally inertial coordinate system does not eliminate the Ricci tensor term.

A differential three-form is a smooth, totally antisymmetric covariant tensor of rank three. For a Killing-Yano two-form, its covariant derivative is such a tensor, because it is antisymmetric in the first two and last two slots.

For a constant orthogonal Cartesian coordinate change $x^{\prime }=Rx$, a Cartesian second-rank tensor obeys $T_{ij}^{\prime }=R_{ik}{R}_{jl}{T}_{kl}$. The chain rule gives ${\partial }_j^{\prime }=R_{jm}{\partial }_m$. Consequently ${\partial }_j^{\prime }{T}_{ij}^{\prime }=R_{ik}{R}_{jl}{R}_{jm}{\partial }_m{T}_{kl}=R_{ik}{\partial }_l{T}_{kl}$, the transformation rule for a vector. Under non-Cartesian position-dependent coordinate changes, partial derivatives must be replaced by the appropriate covariant derivative.

For the Levi-Civita connection, the second Bianchi identity gives ${\nabla }_i{R}^{ik}{}_{mn}={\nabla }_m{R}^k{}_n-{\nabla }_n{R}^k{}_m$. Applying another covariant derivative and the contracted Bianchi identity reduces its divergence to a commuting scalar Hessian plus curvature commutators on the Ricci tensor. The latter are the difference of symmetric tensors: a square of the Ricci tensor and $R^{ab}{R}_{anbm}$, which is symmetric in $m,n$ by curvature pair exchange and Ricci symmetry. Both differences vanish. This identity does not require vacuum field equations and is independent of the overall curvature sign convention.

The four-acceleration is the covariant derivative of four-velocity along a massive observer's worldline: $A^{\mu }=d{U}^{\mu }/d\tau +{\Gamma }_{\nu \rho }^{\mu }{U}^{\nu }{U}^{\rho }$. In Cartesian Minkowski spacetime coordinates the Christoffel symbols vanish, leaving the ordinary proper time derivative. Differentiating the constant velocity norm gives $U\cdot A=0$; in the instantaneous rest frame, $A=(0,\mathbf{a})$, so $A\cdot A=-a^2$ with $+---$ metric signature. The magnitude $a$ is the proper acceleration.

A gauge field is a connection associated with a local symmetry in a gauge theory. Its gauge potential defines the internal-symmetry contribution to a covariant derivative. For an Abelian gauge field, the gauge field strength is $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$.

A classical field theory is a gauge theory when its description has a local redundancy represented by gauge transformations. A gauge field defines the covariant derivative that compares matter fields at different points, and its gauge field strength measures the curvature of that comparison. Electromagnetism is the familiar Abelian example; Yang-Mills theory generalizes it to a non-Abelian internal group. Both classical equations and their quantum field theory can have this structure. Physically equivalent gauge descriptions must give the same observables, even though the gauge potential changes.

For a family of affinely parametrized geodesics with tangent $U$ and connecting field $J$, the Jacobi equation in the convention used here isIt describes the relative acceleration produced by spacetime curvature.

Here $J$ describes an infinitesimal separation and the covariant derivatives compare it in a frame transported along the reference geodesic. The Riemann curvature tensor thus becomes something measured with neighboring trajectories. The Newtonian version follows by differentiating gravitational acceleration with respect to position. In the exterior of a spherical mass, let ${\xi }_r$ be radial separation and ${\xi }_{\perp }$ separation in either transverse direction. Newton's law of universal gravitation givesThe upper particle falls less strongly than the lower one, stretching their radial separation; side-by-side particles accelerate toward converging radial directions, compressing their transverse separation. See Oregon State's derivation of tidal acceleration.

For a spherical Earth model with $r=R_{\oplus }=6.371\times 10^6\mathrm{m}$, $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$ and $GM=g{R}_{\oplus }^2$, a one-metre radial separation gives ${\ddot{\xi }}_r=2g/R_{\oplus }=3.07958\times 10^{-6}\mathrm{m}{\mathrm{s}}^{-2}$; a one-metre transverse separation gives $-1.53979\times 10^{-6}\mathrm{m}{\mathrm{s}}^{-2}$. Initially comoving particles therefore change separation after one second by approximately $\frac{1}{2}\ddot{\xi }{t}^2$: $+1.54$ micrometres radially or $-0.770$ micrometres transversely. This short-time calculation holds the reference radius fixed to leading order. Its beauty is that common falling motion drops out, leaving a measurable stretching pattern that reveals curvature.

A metric space in which every two points are joined by a metric geodesic segment. Equivalently, they admit a continuous path whose metric path length equals their distance. In this general setting no Riemannian metric or covariant derivative is required.

For smooth localized compatible wave map Cauchy data and compact Riemannian target, one-dimensional wave maps stay smooth globally. In null coordinates, the equation is $D_v{\varphi }_u=D_u{\varphi }_v=0$. Compatibility of the target metric with the covariant derivative makes $|{\varphi }_u{|}^2$ independent of $v$ and $|{\varphi }_v{|}^2$ independent of $u$. These transported derivative bounds and higher energy estimates prevent finite-time breakdown.

The Killing equation tests whether a candidate vector field generates a local isometry, without needing to solve for its flow of a vector field. For the Levi-Civita connection of a metric tensor $g$, it isParentheses mean averaging over the two index orders. The equation says the symmetric part of the covariant derivative vanishes, not that the entire covariant derivative vanishes. The Lie derivative of the metric tensor gives the coordinate formThe last two terms account for how tangent vectors move with the field; checking only $K^{\rho }{\partial }_{\rho }{g}_{\mu \nu }$ misses this. See Carroll, section 5, equations (5.33) and (5.41)-(5.42).

For the Euclidean Killing vector field $K=-y{\partial }_x+x{\partial }_y$, the Euclidean metric in Cartesian coordinates makes covariant derivatives ordinary partial derivatives. Its diagonal equations give ${\partial }_x{K}_x={\partial }_y{K}_y=0$, and its mixed equation gives ${\partial }_x{K}_y+{\partial }_y{K}_x=1-1=0$. Thus a varying vector field can preserve a constant metric tensor. In contrast, $K=x{\partial }_x+y{\partial }_y$ gives $({\mathcal{L}}_{K}g{)}_{xx}=2$ and fails the Killing equation, despite the constant metric components.

If $K={\partial }_z$ is a coordinate basis vector, its components are constant and the coordinate form reduces to ${\partial }_z{g}_{\mu \nu }=0$. Hence independence of the metric components from $z$ supplies a Killing vector field. This shortcut applies to that coordinate generator, not to an arbitrary varying vector field.

On a connected Riemannian manifold, these data determine a Killing field uniquely, without a completeness assumption. The second covariant derivative of a Killing vector makes its restriction to every geodesic a Jacobi field. The difference of two fields with identical data has zero Jacobi initial values and vanishes in a convex normal neighborhood. The set on which both its value and covariant derivative vanish is closed by continuity and open by the same local argument, so connectedness makes it the whole manifold.

A Killing spinor is a nonzero supersymmetry parameter that makes the fermionic transformations vanish on a bosonic background. In undeformed minimal four-dimensional supergravity it is parallel, ${\nabla }_{\mu }ϵ=0$; in its cosmological deformation the equation uses ${\mathcal{D}}_{\mu }={\nabla }_{\mu }+(m/2){\gamma }_{\mu }$. Its commuting Dirac current is a Killing vector, since the spinor equation makes the symmetric covariant derivative of the current vanish. General supergravity theories also impose the vanishing of matter-fermion transformations.

A Killing-Yano tensor is a differential form satisfying the displayed equation for the Levi-Civita connection. Its covariant derivative is totally antisymmetric. Contraction with the velocity of an affinely parametrized geodesic gives a form that is carried by parallel transport along that geodesic. This differs from a Killing tensor, which is symmetric.