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Relative to a nondegenerate metric tensor, any affine connection is uniquely the Levi-Civita connection plus a contorsion tensor contribution and a disformation tensor contribution. In derivative-last notation, with $T_{ab}{}^c={\Gamma }^c{}_{ab}-{\Gamma }^c{}_{ba}$ and $N_{abc}={\nabla }_c{g}_{ab}$, the lowered difference is $A_{abc}=\frac{1}{2}(T_{abc}+T_{bca}-T_{cab}+N_{abc}-N_{bca}-N_{cab})$. This follows by solving $T_{abc}=A_{abc}-A_{bac}$ and $N_{abc}=-A_{acb}-A_{bca}$. The difference of affine connections is a tensor.

For an affine connection, let ${\gamma }_X$ be the affinely parametrized geodesic starting at $p$ with velocity $X$. Define the exponential map by the displayed formula for initial velocities whose geodesics exist until parameter one. It is defined near zero, has differential equal to the identity there, and obeys ${\exp }_p(tX)={\gamma }_X(t)$ wherever defined. This construction requires no metric and extends the metric exponential map.

For an affine connection, define geometric torsion tensor by $\mathcal{T}(X,Y)={\nabla }_{X}Y-{\nabla }_{Y}X-[X,Y]$. The covariant Hessian of a scalar field is ${\nabla }^{2}f(X,Y)=X(Yf)-({\nabla }_{X}Y)f$. Antisymmetrizing gives $[X,Y]f-({\nabla }_{X}Y-{\nabla }_{Y}X)f=-\mathcal{T}(X,Y)f$. Thus all scalar Hessians are symmetric exactly when the connection is torsion-free. In derivative-first coordinates, ${\nabla }_{\mu }{\nabla }_{\nu }f={\partial }_{\mu }{\partial }_{\nu }f-{\Gamma }_{\mu \nu }{}^{\lambda }{\partial }_{\lambda }f$, which makes the sign explicit. Differentials of local coordinate functions span the cotangent space, so vanishing of this antisymmetric part for every $f$ forces the torsion itself to vanish.

Specify the displayed rule on left-invariant vector fields and extend by the Leibniz rule to an affine connection on the Lie group. In a left-invariant vector field frame $L_i$, the extension is ${\nabla }_{L_i}(z^j{L}_j)=L_i(z^j)L_j+\lambda z^j[L_i,L_j]$. Applying the bare bracket formula to arbitrary vector fields instead would violate the Leibniz rule unless $\lambda =1$. With $R(X,Y)Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z$, the Jacobi identity givesfor left-invariant arguments, where $B$ is the Killing form and $\mathrm{Ric}(X,Y)=\mathrm{tr}(Z\mapsto R(Z,X)Y)$. These identities determine the tensor fields everywhere. The endpoints $\lambda =0,1$ have zero curvature, with respectively left- and right-invariant parallel frames, and opposite torsion tensors. The midpoint $\lambda =1/2$ is torsion-free but need not have zero curvature.

For a local basis $e_a$ of the tangent bundle, an affine connection has components defined by ${\nabla }_{e_c}{e}_b={\Gamma }^a{}_{bc}{e}_a$. If $V=V^b{e}_b$, then $({\nabla }_{e_c}V{)}^a=e_c(V^a)+{\Gamma }^a{}_{bc}{V}^b$. The connection components depend on the chosen basis and have an inhomogeneous transformation law; they are not themselves a tensor field. In a coordinate basis, a torsion-free connection has symmetry in the two lower indices.

For a metric-compatible affine connection, its difference from the Levi-Civita connection is the contorsion tensor. It is determined by the torsion tensor. In derivative-last notation let $T_{ab}{}^c={\Gamma }^c{}_{ab}-{\Gamma }^c{}_{ba}$, the negative of geometric torsion, and lower the last slot with the metric tensor. Then $C_{abc}=\frac{1}{2}(T_{abc}+T_{bca}-T_{cab})$. This is the actual correction added to the Levi-Civita connection; a convention writing $\Gamma =S-K$ has $K=-C$.

For every contravariant slot of a tensor field, the tensor Lie derivative contributes a negative derivative of its generating vector field; every covariant slot contributes a positive one. The remaining term differentiates the tensor's components along the generator. This follows from the flow definition of the Lie derivative of a tensor field and its product and contraction rules, without choosing a affine connection.

A covariant derivative differentiates a tensor field while accounting for the changing local basis. For a vector field, an affine connection gives ${\nabla }_{\mu }{V}^{\nu }={\partial }_{\mu }{V}^{\nu }+{\Gamma }^{\nu }{}_{\mu \rho }{V}^{\rho }$. Each additional contravariant index contributes a plus connection term, and each covariant index contributes a minus term.

The correction makes the result a tensor, even under nonlinear changes of coordinates. For example, a constant Cartesian vector on a flat plane has varying components in polar coordinates; the connection terms cancel that coordinate variation. In general relativity, the Levi-Civita connection supplies the metric-compatible derivative. Parallel transport sets the derivative along a curve to zero, while the geodesic equation sets a freely falling body's tangent derivative to zero.^[ref]

The difference of two affine connections is a smooth tensor field of type $(1,2)$. Linearity over smooth functions holds in the first slot by the connection axioms. In the second slot, the two extra terms $X(f)Y$ cancel, so $\Delta (X,fY)=f\Delta (X,Y)$. This tensoriality makes the value depend only on $X_p,Y_p$. In coordinates its components are the differences of the Christoffel symbols, although either collection of connection coefficients individually is not a tensor.

On a contractible Cartesian patch of a flat background, zero linearized Riemann curvature operator says that the connection 1-forms ${\Gamma }^{(1)a}{}_{bc}d{x}^c$ are closed. The Poincaré lemma writes them as derivatives of a matrix-valued potential. Symmetry of the two lower affine connection indices then supplies a second potential, giving ${\Gamma }^{(1)a}{}_{bc}={\partial }_b{\partial }_c{\xi }^a$. Linearized metric compatibility implies that $h_{ab}-2{\partial }_{(a}{\xi }_{b)}$ is constant. Its symmetric constant part is absorbed by an affine vector field. Thus the metric perturbation is locally a coordinate change. Global topology and imposed boundary conditions can obstruct the corresponding global gauge choice.

Declare a global left-invariant, respectively right-invariant, frame parallel to obtain two flat affine connections on a Lie group. For left-invariant vector fields, their derivatives are as displayed and their torsion tensors are $-[X,Y]$ and $+[X,Y]$. The tensors vanish on Abelian groups; two genuinely torsionful connections are not asserted there.

A geometric background for Newtonian mechanics consisting of a four-dimensional affine space, an absolute time covector $t_a$, a rank-three positive spatial inverse metric tensor $h^{ab}$ with $t_a{h}^{ab}=0$, and a flat torsion-free affine connection preserving both. A trajectory parametrized by absolute time has tangent $u$ with $t_a{u}^a=1$. Its absolute acceleration is ${\nabla }_{u}u$; an unforced particle obeys ${\nabla }_{u}u=0$. The structure distinguishes inertial motion without selecting an absolute state of uniform rest. Additional force laws, including a Newtonian gravitational force when wanted, specify a dynamical theory on this background.

Motion with zero acceleration relative to the theory's inertial affine connection. In an inertial frame, its spatial trajectory is uniform and straight. The geometric statement refers to the connection, not to a chosen coordinate system; force-free motion supplies its physical interpretation. In general relativity the corresponding freely falling trajectories are timelike geodesics, with local inertial coordinates rather than a global inertial frame in a curved region.

For vector fields $X,Y$, the Lie derivative is the Lie bracket of vector fields ${\mathcal{L}}_{X}Y=[X,Y]$. Pulling $Y$ back by the local flow of $X$ and differentiating gives $dY(X)-dX(Y)$. No affine connection is needed.

In Cartesian spatial coordinates and absolute time $t$, the gravitational potential $\Phi (t,x)$ determines an affine connection by the displayed coefficients, all others zero. The geodesic equation with affine time is $d^2{x}^i/d{t}^2=-{\partial }_i\Phi$. It preserves $dt$ and the spatial inverse metric $h^{ij}={\delta }^{ij}$, reproducing freely falling Newtonian mechanics in a curved spacetime connection.

Newton–Cartan theory describes Newtonian mechanics geometrically using absolute time, a degenerate spatial inverse metric, and a compatible torsion-free affine connection. It is not a theory with a nondegenerate Lorentzian metric. Gravitationally accelerated particles are geodesics of that connection, whose curvature measures tidal gravity.

Given a metric tensor and an affine connection, $N_{abc}={\nabla }_c{g}_{ab}$ measures failure of metric compatibility. It is symmetric in $a,b$. Some authors instead define nonmetricity as $-\nabla g$, so formulas must specify the sign. Together with the torsion tensor, it determines the difference from the Levi-Civita connection.

Two affine connections have exactly the same geodesics with the same parameters if and only if their difference tensor vanishes on every diagonal pair $(v,v)$. Indeed, their covariant accelerations differ by $\Delta (\dot{\gamma },\dot{\gamma })$; starting a geodesic at each arbitrary initial vector proves necessity. Polarization then says that the symmetric part of the difference is zero. This condition is stronger than agreement of unparametrized geodesic equations.

A Killing vector field $K$ generates local isometries: its flow of a vector field ${\Phi }_s$ satisfies ${\Phi }_s^{\ast }g=g$. Equivalently, the Lie derivative of the metric tensor vanishes. For the Levi-Civita connection, metric compatibility converts the coordinate expression into the Killing equation:The Levi-Civita connection is an implicit extra assumption in this question. The printed assumption of vanishing torsion tensor alone does not imply metric compatibility. For example, on Minkowski spacetime, take the affine connection whose only nonzero Christoffel symbol is ${\Gamma }^0{}_{00}=1$. The vector field $K={\partial }_0$ preserves the Minkowski metric, but $K_0=-1$ gives ${\nabla }_0{K}_0=1$, contradicting the Killing equation. The following calculation, and the covariant formulas in part (b), use the Levi-Civita connection; this is the role of metric compatibility in Killing identities.

For the stated curvature sign convention, the Ricci identity on a covector has a minus sign:Write $A_{\mu \nu \rho }={\nabla }_{\mu }{\nabla }_{\nu }{K}_{\rho }$. Differentiating the Killing equation gives $A_{\mu \nu \rho }=-A_{\mu \rho \nu }$. Alternately using this relation and the Ricci identity givesThe first Bianchi identity and antisymmetry of the last two indices of the Riemann curvature tensor reduce the three curvature terms to $2R^{\sigma }{}_{\mu \nu \rho }{K}_{\sigma }$. Consequently the second covariant derivative of a Killing vector is

An affine connection on the tangent bundle of a smooth manifold assigns a vector field ${\nabla }_{X}Y$ to two vector fields, is $\mathbb{R}$-bilinear, and satisfiesThus its first slot is tensorial and its second obeys the Leibniz rule. Its action on covectors is defined by differentiating the pairing, and this extends it to tensor fields. A torsion-free connection has ${\nabla }_{X}Y-{\nabla }_{Y}X=[X,Y]$.

Assume that $g$ is a smooth nondegenerate metric tensor. Metric compatibility and a vanishing torsion tensor imply the Koszul formula,The right-hand side is determined by $g$ and the Lie bracket of vector fields. Nondegeneracy determines ${\nabla }_{X}Y$ uniquely; conversely this formula defines an affine connection with both required properties, proving existence as well as uniqueness. In a coordinate frame the brackets vanish. Combining the three differentiated metric tensor identities givesThis is the Levi-Civita connection. Positive definiteness is unnecessary: the argument also works for a pseudo-Riemannian metric.

For the other affine connection define $S(X,Y)={\bar{\nabla }}_{X}Y-{\nabla }_{X}Y$. The derivative of $f$ cancels between the two Leibniz rules, so $S$ is $C^{\infty }$-linear in both slots. The difference of affine connections is a tensor: here $S$ is a section of $TM\otimes T^{\ast }M\otimes T^{\ast }M$ with components ${\bar{\Gamma }}^a{}_{bc}-{\Gamma }^a{}_{bc}$. Both affine connections are torsion-free, so $S(X,Y)=S(Y,X)$.

Agreement of geodesics means agreement of their unparametrized curves; different affine parameters are allowed. For every nonzero tangent vector $v$, existence of a geodesic with that initial velocity implies that $S(v,v)$ is parallel to $v$. The symmetric bilinear diagonal-parallel lemma now determines $S$. Choose a basis $e_i$ and write $S(e_i,e_i)={\alpha }_i{e}_i$. Applying the parallelism condition to $e_i+e_j$ and $e_i-e_j$ givesWith $V(e_i)={\alpha }_i/2$, bilinearity yieldsThis is projective equivalence of affine connections. It also covers dimension one. For two Levi-Civita connections, the projective covector from metric volume densities gives the explicit answerIndeed ${\Gamma }^a{}_{ac}={\partial }_c\log \sqrt{|\det g|}$. The determinant ratio is a scalar, so this expression is a genuine covector, although either determinant alone is coordinate-dependent. This determinant expression requires nondegeneracy of $\bar{g}$; the difference-tensor and earlier trace formula require only the two torsion-free affine connections.

Conversely, the displayed tensor gives ${\bar{\nabla }}_{\dot{\gamma }}\dot{\gamma }=2V(\dot{\gamma })\dot{\gamma }$ along an affinely parametrized $\nabla$-geodesic. Choosing a new parameter $t(\tau )$ satisfying $t^{\prime \prime }/t^{\prime }=2V(\dot{\gamma })$ removes that tangential acceleration. This geodesic reparametrization under projective equivalence proves sufficiency. If agreement were required with the very same affine parameter, polarization would instead force $S=0$ and $V=0$.