Covariant Derivative · aiwiki.page

A covariant derivative is an operation in differential geometry that extends the directional derivative to vector fields, tensor fields, and sections of vector bundles on a smooth manifold. It uses a chosen connection to account for how local frames vary between points. Unlike ordinary differentiation of components, it produces a geometrically defined result that transforms appropriately when coordinates or frames change. A manifold alone generally does not specify a unique covariant derivative. (math.stanford.edu)

Geometric motivation and definition

For a scalar function, differentiation along a vector field $X$ is intrinsically defined as $X(f)$ . For vectors, the situation is different: vectors at distinct points belong to different tangent spaces, so their difference is not intrinsically meaningful. A connection supplies the additional structure needed for differentiation. Its associated covariant derivative is written $\nabla_XY$ , meaning the derivative of $Y$ in direction $X$ . (damtp.cam.ac.uk)

For smooth vector fields $X,Y,Z$ and smooth functions $f,g$ , the defining rules include

\nabla_{fX+gY}Z=f\nabla_XZ+g\nabla_YZ,

\nabla_X(Y+Z)=\nabla_XY+\nabla_XZ, \qquad \nabla_X(fY)=X(f)Y+f\nabla_XY.

Thus differentiation is linear over smooth functions in its direction argument, but obeys a product rule in the field being differentiated. This distinction explains why a connection is not itself a tensor: its value depends on first derivatives of the differentiated field. (damtp.cam.ac.uk)

Coordinate expressions

Choose local coordinates $x^i$ and their coordinate basis $\partial_i$ . Define the connection coefficients by

\nabla_{\partial_i}\partial_j=\Gamma^k{}_{ij}\partial_k.

For $Y=Y^j\partial_j$ ,

(\nabla_iY)^k=\partial_iY^k+\Gamma^k{}_{ij}Y^j.

Repeated upper and lower indices are summed according to the Einstein summation convention. The first term differentiates components; the second accounts for the connection’s action on the basis. The coefficients are not tensor components, although their combination with the partial derivative gives a tensorial result. (damtp.cam.ac.uk)

For a covector field $\omega_j$ , the induced derivative is

\nabla_i\omega_j=\partial_i\omega_j-\Gamma^k{}_{ij}\omega_k.

The minus sign ensures compatibility with the scalar pairing $\omega(Y)$ . Extension to tensor products gives one positive connection term for each upper index and one negative term for each lower index. For example,

\nabla_iT^a{}_b =\partial_iT^a{}_b +\Gamma^a{}_{ic}T^c{}_b -\Gamma^c{}_{ib}T^a{}_c.

Consequently, if $T$ has type $(p,q)$ , then $\nabla T$ has type $(p,q+1)$ . Covariant differentiation also respects tensor contraction. (damtp.cam.ac.uk)

The Levi-Civita connection

A metric tensor $g$ determines a distinguished covariant derivative through the Levi-Civita connection. It is the unique connection that is metric-compatible, $\nabla g=0$ , and has zero torsion. For arbitrary vector fields, torsion is

\mathcal T(X,Y)=\nabla_XY-\nabla_YX-[X,Y],

where $[X,Y]$ is their Lie bracket. Metric compatibility means that differentiation preserves the metric pairing through the product rule. (math.mit.edu)

In coordinates, the Levi-Civita coefficients are the Christoffel symbols

\Gamma^k{}_{ij} =\frac12g^{k\ell} \left(\partial_i g_{j\ell} +\partial_j g_{i\ell} -\partial_\ell g_{ij}\right),

where $g^{k\ell}$ denotes the inverse metric. The same construction applies to nondegenerate indefinite metrics, including those used in relativity. (math.mit.edu)

Nonzero coefficients do not necessarily indicate curvature. For example, applying this formula to the Euclidean plane in polar coordinates,

ds^2=dr^2+r^2d\theta^2,

gives

\Gamma^r{}_{\theta\theta}=-r,\qquad \Gamma^\theta{}_{r\theta} =\Gamma^\theta{}_{\theta r}=1/r.

These terms describe the changing coordinate basis on a flat space, rather than intrinsic curvature. (damtp.cam.ac.uk)

Parallel transport, geodesics, and curvature

Along a curve $\gamma(t)$ , the connection defines

\frac{DV^k}{dt} =\frac{dV^k}{dt} +\Gamma^k{}_{ij}\dot\gamma^iV^j.

A vector field satisfying $DV/dt=0$ is parallel along the curve. Solving this differential equation defines parallel transport between its tangent spaces. An affinely parametrized geodesic for the Levi-Civita connection satisfies

\nabla_{\dot\gamma}\dot\gamma=0:

its velocity is parallel along its own trajectory. (math.stanford.edu)

Curvature measures a failure of covariant derivatives to commute. With one common sign convention,

R(X,Y)Z =\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z.

For the Levi-Civita connection, this defines the Riemann curvature tensor. The bracket term removes the contribution caused by noncommuting direction fields. Curvature and torsion are distinct properties; vanishing torsion does not imply vanishing curvature. (math.mit.edu)

Vector bundles and physical applications

The construction extends to a vector bundle $E\to M$ . A connection differentiates a smooth section $s$ along $X$ , satisfying

\nabla_X(fs)=X(f)s+f\nabla_Xs.

In a local frame, it takes the form $\nabla=d+A$ , where $A$ is a matrix-valued one-form. Changes of frame alter $A$ , but not the underlying geometric operation. (math.stanford.edu)

In general relativity, the Levi-Civita derivative provides differentiation compatible with the geometry of spacetime. Spinor fields require a corresponding spin connection. In gauge theory, a gauge connection similarly compensates for position-dependent changes of internal frame. For instance, one convention in electrodynamics is

D_\mu\psi=(\partial_\mu+ieA_\mu)\psi.

Under $A_\mu\mapsto A_\mu+\partial_\mu\lambda$ and $\psi\mapsto e^{-ie\lambda}\psi$ , the derivative transforms as $D_\mu\psi\mapsto e^{-ie\lambda}D_\mu\psi$ , without additional derivatives of the transformation parameter. (damtp.cam.ac.uk)