Affine Connections and Covariant Derivatives

Vectors at different points live in different tangent spaces, so ordinary differentiation of their coordinate components has no invariant meaning on its own. An affine connection supplies the missing comparison rule. Its coefficients transform inhomogeneously, and that extra term cancels the non-tensorial part of a component derivative. Torsion, nonmetricity, and curvature are additional properties of a connection; none is set to zero in the definition.

Required background. Tensor Fields and Natural Operations supplies vector fields, covectors, tensor products, and coordinate transformations; Pseudo-Riemannian Metrics and Signature supplies index raising and lowering while making clear that a connection need not be metric compatible.

Helpful background. Calculus and Linear Algebra for Relativity supplies directional derivatives and first-order linear systems.

A connection as a differentiation rule

An affine connection on $TM$ is a map

$$\nabla:\Gamma(TM)\times\Gamma(TM)\longrightarrow\Gamma(TM), \qquad (X,Y)\longmapsto\nabla_XY,$$

that is smooth-function linear in $X$, real linear in $Y$, and satisfies

$$\nabla_X(fY)=X(f)Y+f\nabla_XY.$$

Thus $\nabla_{fX+hZ}Y=f\nabla_XY+h\nabla_ZY$, but $\nabla_X(fY)$ contains a derivative. These asymmetric rules are exactly what a directional derivative should satisfy. No metric appears in them.

The connection extends uniquely to all tensor fields by setting $\nabla_Xf=X(f)$, imposing the product rule, and requiring commutation with contractions. For a covector $\alpha$,

$$(\nabla_X\alpha)(Y) =X\!\left(\alpha(Y)\right)-\alpha(\nabla_XY).$$

This minus sign ensures $\nabla_X(\alpha(Y))=(\nabla_X\alpha)(Y)+\alpha(\nabla_XY)$. Write $\nabla_{\partial_\mu}\partial_\nu =\Gamma^\rho{}_{\mu\nu}\partial_\rho$ to define the connection coefficients; the first lower index is the differentiation direction. For a tensor with one upper and one lower index,

$$\nabla_\mu T^\rho{}_\sigma =\partial_\mu T^\rho{}_\sigma +\Gamma^\rho{}_{\mu\lambda}T^\lambda{}_\sigma -\Gamma^\lambda{}_{\mu\sigma}T^\rho{}_\lambda.$$

Every upper index receives a plus term and every lower index a minus term. Calegari gives the axiomatic construction and tensor extension in Calegari 2022, §§4.1–4.3, pp. 31–33, PDF.

Why partial derivatives fail

Choose coordinates and define coefficients by

$$\nabla_{\partial_\mu}\partial_\nu =\Gamma^\rho{}_{\mu\nu}\partial_\rho.$$

For $V=V^\nu\partial_\nu$,

$$\nabla_\mu V^\rho =\partial_\mu V^\rho+\Gamma^\rho{}_{\mu\nu}V^\nu.$$

Under a nonlinear coordinate change, $\partial_{\mu'}V^{\rho'}$ contains a term with a second derivative of the coordinate map. It therefore fails the tensor transformation law. The coefficients acquire the opposite inhomogeneous term, so the sum $\nabla_{\mu'}V^{\rho'}$ does transform as a $(1,1)$ tensor.

This explains both sides of the construction: $\partial_\mu V^\rho$ is not usually tensorial, and $\Gamma^\rho{}_{\mu\nu}$ is not a tensor. Their particular combination is. Tong derives the component formula and its coordinate covariance in Tong 2019, §3.2.1, pp. 101–104, PDF.

Flat polar-coordinate check

On the Euclidean plane, Cartesian coordinates have vanishing Levi-Civita coefficients. In polar coordinates,

$$ds^2=dr^2+r^2d\phi^2,$$

the nonzero coefficients are

$$\Gamma^r{}_{\phi\phi}=-r, \qquad \Gamma^\phi{}_{r\phi} =\Gamma^\phi{}_{\phi r}=\frac1r.$$

The Cartesian constant vector $\partial_x$ becomes

$$\partial_x =\cos\phi\,\partial_r-\frac{\sin\phi}{r}\,\partial_\phi.$$

Its partial derivatives are nonzero. The covariant derivatives cancel:

$$\begin{aligned} \nabla_\phi V^r &=-\sin\phi+(-r)\left(-\frac{\sin\phi}{r}\right)=0,\\ \nabla_\phi V^\phi &=-\frac{\cos\phi}{r}+\frac1r\cos\phi=0, \end{aligned}$$

and the two $r$ derivatives vanish similarly. A geometrically constant field therefore has nonconstant polar components; the connection removes the coordinate-basis variation.

Properties not built into a connection

The torsion and nonmetricity of a connection are

$$T(X,Y)=\nabla_XY-\nabla_YX-[X,Y], \qquad Q(X;Y,Z)=(\nabla_Xg)(Y,Z).$$

Both are tensors. An arbitrary affine connection may have $T\ne0$ and $Q\ne0$. Metric compatibility means $Q=0$; the Levi-Civita connection is the unique connection satisfying both $T=0$ and $Q=0$. Until those conditions are stated, lowering an index before differentiating need not agree with lowering it afterward.

The difference of two connections is tensorial, but a single connection is not itself a tensor. Its local coefficients may vanish in a chosen basis at a point while its derivatives, transport around loops, or torsion remain nontrivial. Curvature measures one obstruction to extending such a simplification over a neighborhood; its classification lies beyond this chapter.

Second derivatives and torsion

Scalar fields illustrate where the connection first becomes visible. The first covariant derivative is just the ordinary differential,

$$\nabla_\mu f=\partial_\mu f,$$

because a scalar has no tensor index. But $\nabla_\nu f$ is then a covector, so its second derivative is

$$\nabla_\mu\nabla_\nu f =\partial_\mu\partial_\nu f -\Gamma^\rho{}_{\mu\nu}\partial_\rho f.$$

The ordinary Hessian is not tensorial under nonlinear coordinate changes; the coefficient term repairs it. Its antisymmetric part is

$$\nabla_\mu\nabla_\nu f-\nabla_\nu\nabla_\mu f =-T^\rho{}_{\mu\nu}\partial_\rho f.$$

Here the second derivative means differentiating the covector $df$, including its lower index. It must be distinguished from merely composing two directional operators on scalar functions: $\partial_\mu\partial_\nu f -\partial_\nu\partial_\mu f=0$ in a coordinate frame, regardless of torsion.

The same distinction applies to vector fields. Define the tensor second derivative by

$$(\nabla^2V)(X,Y) =\nabla_X(\nabla_YV)-\nabla_{\nabla_XY}V.$$

With $R(X,Y)V=\nabla_X\nabla_YV-\nabla_Y\nabla_XV -\nabla_{[X,Y]}V$, antisymmetrizing gives

$$(\nabla^2V)(X,Y)-(\nabla^2V)(Y,X) =R(X,Y)V-\nabla_{T(X,Y)}V.$$

In a coordinate frame the pure operator commutator is still $[\nabla_{\partial_\mu},\nabla_{\partial_\nu}]V =R(\partial_\mu,\partial_\nu)V$, even with torsion. The additional torsion term belongs to the tensor second derivative. This distinction fixes what is meant whenever compact double-derivative notation is used.

Differentiation along a curve

The connection also differentiates a smooth vector field $V(\lambda)$ along a curve $\gamma$, including when it is not the restriction of one vector field on a neighborhood. Define

$$\frac{DV^\rho}{d\lambda} =\frac{dV^\rho}{d\lambda} +\Gamma^\rho{}_{\mu\nu}(\gamma(\lambda)) \frac{d\gamma^\mu}{d\lambda}V^\nu.$$

The connection transformation law makes this coordinate independent. If a local extension of $V$ exists, the formula agrees with $\nabla_{\dot\gamma}V$ and is independent of the extension. At a self-intersection, however, the vector may take different values on the two visits, so such an extension need not exist; the displayed definition still works. It becomes the parallel-transport ODE when $DV/d\lambda=0$.

Finally, index motion commutes with $\nabla$ only under metric compatibility:

$$\nabla_\mu V_\nu =g_{\nu\rho}\nabla_\mu V^\rho +(\nabla_\mu g_{\nu\rho})V^\rho.$$

Dropping the last term is equivalent to assuming zero nonmetricity. Keeping it visible prevents Levi-Civita formulas from being attributed to every affine connection.

Exercises

Starting from $(\nabla_\mu\alpha)_\nu=\partial_\mu\alpha_\nu- \Gamma^\rho{}_{\mu\nu}\alpha_\rho$, show directly that the covariant derivative of the contraction $\alpha_\nu V^\nu$ equals its ordinary derivative.

Solution

Using the vector and covector rules,

$$\begin{aligned} (\nabla_\mu\alpha)_\nu V^\nu +\alpha_\nu\nabla_\mu V^\nu &=(\partial_\mu\alpha_\nu)V^\nu -\Gamma^\rho{}_{\mu\nu}\alpha_\rho V^\nu\\ &\quad+\alpha_\nu\partial_\mu V^\nu +\alpha_\nu\Gamma^\nu{}_{\mu\rho}V^\rho. \end{aligned}$$

Relabeling dummy indices shows that the two connection terms cancel. The remaining terms are $\partial_\mu(\alpha_\nu V^\nu)$, as required for a scalar.

References

• Calegari, Danny. Introduction to Differential Geometry. Preliminary version, May 26, 2022, University of Chicago, §§4.1–4.3, pp. 31–33. Course PDF.
• Tong, David. General Relativity. University of Cambridge lecture notes, 2019, §3.2.1, pp. 101–104. Official course PDF.