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The coordinate transformation law for a affine connection has a homogeneous term with one upper and two lower index factors, and an inhomogeneous term involving second coordinate derivatives. The latter is the same for both Levi-Civita connections, so it cancels in their difference. Consequentlywhich is exactly the tensor component transformation law. Equivalently, $(\tilde{\nabla }-\nabla {)}_{X}Y$ is linear over functions in both $X$ and $Y$.

A smooth vector field along a map $\gamma :[a,b]\to M$ is a smooth section of the pullback tangent bundle ${\gamma }^{\ast }TM$: it assigns $V(t)\in T_{\gamma (t)}M$, with smooth coefficients in each local frame. It need not be the restriction of one ambient vector field, since a curve may return to the same point with different values of $V$.

An affine connection, also called a Koszul connection, induces a pullback connection on this bundle. In coordinates, write $V=V^i(t){\partial }_i$ and ${\nabla }_{{\partial }_j}{\partial }_k={\Gamma }^i{}_{jk}{\partial }_i$. Its covariant derivative along a curve isThe connection transformation law, or the Leibniz rule applied to a change of frame, makes this expression independent of the frame. In particular it is defined even when $\dot{\gamma }=0$; it differentiates the section's coefficients as well as the frame.

The field is parallel precisely when $D_{t}V=0$. In a frame along a coordinate segment this is the linear ordinary differential equationThe Picard-Lindelöf theorem gives a unique solution with prescribed initial value. Smooth coefficients on compact subintervals are bounded, and the usual norm estimate for a linear ordinary differential equation prevents finite-time blowup there. A finite sequence of coordinate segments covers the image of the compact interval $[a,b]$; solve successively and use uniqueness on overlaps. This gives a unique parallel field on the whole interval for every $V_a\in T_{\gamma (a)}M$.

Define parallel transport by ${\tau }_t(V_a)=V(t)$. Linearity of the equation and uniqueness give ${\tau }_t(c{V}_a+d{W}_a)=c{\tau }_t(V_a)+d{\tau }_t(W_a)$. Solving along the reversed curve supplies its inverse. Therefore each ${\tau }_t:T_{\gamma (a)}M\to T_{\gamma (t)}M$ is a linear isomorphism.

For two affine connections, put $\Delta (X,Y)={\nabla }_{X}Y-{\tilde{\nabla }}_{X}Y$. Their rules givesince the two $X(f)Y$ terms cancel. The tensoriality of this operation proves that the difference of affine connections is a tensor, of type $(1,2)$, with smooth coordinate coefficients ${\Gamma }^i{}_{jk}-{\tilde{\Gamma }}^i{}_{jk}$.

A parametrized geodesic satisfies $D_t\dot{\gamma }=0$, or the geodesic equationWith $v^i={\dot{\gamma }}^i$, this becomes the first-order system ${\dot{\gamma }}^i=v^i$, ${\dot{v}}^i=-{\Gamma }^i{}_{jk}(\gamma )v^j{v}^k$. Its right side is smooth, so the Picard-Lindelöf theorem gives a unique local solution for each initial point and tangent vector. This establishes uniqueness with the parametrization fixed.

The two accelerations differ byIf $\Delta (v,v)=0$ for every tangent vector, either acceleration vanishes exactly when the other does. Conversely, start the $\nabla$-geodesic with arbitrary initial vector $v$ at an arbitrary point. If it is also a $\tilde{\nabla }$-geodesic with the same parameter, evaluating this identity initially gives $\Delta (v,v)=0$. HencePolarization makes the latter condition equivalent to $\Delta (u,v)+\Delta (v,u)=0$: parametrized geodesics determine the symmetric part of an affine connection. In particular torsion-free connections are determined by their parametrized geodesics, since their difference is also symmetric and must then vanish.

An affine connection on $M$ is an operation $(Y,X)\mapsto {\nabla }_{Y}X$ on smooth vector fields that is real-bilinear, linear over $C^{\infty }(M)$ in the differentiating field, and satisfiesIt is a connection on a vector bundle for the bundle $TM$, also called a linear connection on the manifold. Its curvature form of a connection has the operator formThis convention fixes its sign. Expanding the two connection product rules shows that the extra derivatives of a scalar cancel, so $R$ is linear over smooth functions in all three fields. For a Levi-Civita connection, this is the Riemann curvature tensor.

The torsion form is $T(Y,Z)={\nabla }_{Y}Z-{\nabla }_{Z}Y-[Y,Z]$. For a Riemannian metric $g$, its Levi-Civita connection is the unique torsion-free connection satisfying metric compatibilityExistence and uniqueness follow from the Koszul formula. In coordinates its coefficients areSymmetry in $b,c$ gives zero torsion, and substitution verifies metric compatibility.

Use spatial indices $i,j=1,2,3$, set ${\varphi }_i={\partial }_i\varphi$, and assume $\varphi$ is smooth. The inverse metric has $g^{tt}=-c^{-2}{e}^{-2\varphi /c^2}$ and $g^{ij}={\delta }^{ij}$. The Christoffel symbols of the Levi-Civita connection follow from ${\Gamma }^{\alpha }{}_{\beta \gamma }=g^{\alpha \delta }({\partial }_{\beta }{g}_{\gamma \delta }+{\partial }_{\gamma }{g}_{\beta \delta }-{\partial }_{\delta }{g}_{\beta \gamma })/2$:All other components vanish. Although $g_{tt}=-c^2-2\varphi +O(c^{-2})$ diverges, its affine connection has a smooth limit on compact subsets:This is the Newtonian connection from an exponential lapse. It is torsion-free, and its geodesic equation, using $t$ as an affine parameter when $\dot{t}\ne 0$, is $d^2{x}^i/d{t}^2=-{\partial }_i\varphi$. Thus $\varphi$ has the role of a Newtonian gravitational potential. There is no finite limiting nondegenerate Lorentzian metric in these fixed coordinates; $-g(c)/c^2$ tends to $d{t}^2$, a rank-one tensor. The limiting connection preserves $dt$ and the contravariant spatial tensor with components $h^{ij}={\delta }^{ij}$, $h^{t\mu }=0$, which describe the degenerate temporal/spatial structures of this limit.

Fix the Riemann curvature tensor conventionFor the limiting affine connection, every product of two nonzero connection coefficients is zero, because its only nonzero coefficients have spatial upper index and both lower indices $t$. Its only possibly nonzero curvature components, apart from antisymmetry in the last two indices, areThe Ricci tensor is thereforeAll other components vanish, so the limiting connection is a connection with vanishing Ricci tensor exactly when $\varphi$ satisfies the Laplace equation on spatial Euclidean space. This is Ricci-flatness, not necessarily vanishing curvature: $\varphi =x^2-y^2$ is a harmonic function but has nonzero Hessian matrix and hence a nonflat limiting connection. Reversing the curvature sign convention reverses the displayed Ricci sign but leaves the equivalence unchanged.

Use $x^0=t$ and the Levi-Civita symbol ${ϵ}^1{}_{23}=1$. Define a smooth affine connection byThe lower indices are symmetric, so it is a torsion-free connection. With $t$ itself as parameter, the time component of the geodesic equation is identically satisfied, while its spatial components becomeThe last term is $2(\mathbf{B}\times \dot{\mathbf{x}}{)}^i$, with the wedge in the source interpreted as the three-dimensional cross product. Thus the spacetime lifts $(t,\mathbf{x}(t))$ are actually affinely parametrized geodesics, which proves the requested unparametrized claim. Conversely a geodesic with nonzero constant $dt/ds$ can be parametrized by $t$ and obeys this particle equation; purely spatial geodesics with $dt/ds=0$ are not such trajectories. No field equations or metric compatibility are needed. This construction is the affine connection for a velocity-linear force.

An inertial frame is a nonrotating frame in which all force-free test bodies follow straight trajectories with constant velocity. The qualification “all” matters: a specially chosen coordinate system can make one accelerated trajectory look straight without making the frame inertial. In practice, approximately isolated test bodies and gyroscopes help identify inertial motion; identifying the relevant forces and calibrating instruments are part of testing the theory. The definition does not select a privileged state of uniform rest.

In its usual coordinate formulation, Newtonian mechanics states $m\ddot{\mathbf{x}}=\mathbf{F}$ in an inertial frame. A Galilean transformation, $t^{\prime }=t+b$, ${\mathbf{x}}^{\prime }=R\mathbf{x}+\mathbf{V}t+\mathbf{a}$ with constant orthogonal $R$ and constant $\mathbf{V}$, maps the class of inertial frames to itself. It preserves zero acceleration; a time-dependent translation generally does not. In the translating frame ${\mathbf{x}}^{\prime }=\mathbf{x}-\mathbf{a}(t)$, the same physical law reads $m{\ddot{\mathbf{x}}}^{\prime }=\mathbf{F}-m\ddot{\mathbf{a}}$. The extra fictitious force records the frame's acceleration, rather than a new interaction.

It is possible to state Newtonian mechanics without mentioning an inertial frame explicitly. On Galilean spacetime, let $u$ be a trajectory tangent normalized by the absolute time covector, $t_a{u}^a=1$. Thenuses an affine connection to define acceleration. The free-particle equation is ${\nabla }_{u}u=0$. This formulation removes a chosen coordinate frame but retains the geometric distinction between inertial motion and accelerated motion. Omitting both the inertial frame language and all equivalent inertial structure would leave the law unable to specify which trajectories count as unforced.

The same distinction applies to special relativity. Einstein's original presentation uses a class of inertial frames and calibrated rods, clocks and light signals.^[ref] A geometric presentation instead uses Minkowski spacetime, its Lorentzian metric and its Levi-Civita connection. A free massive particle obeys ${\nabla }_{u}u=0$, now with proper time parametrization and $g(u,u)=-c^2$. There is no preferred inertial rest frame, but there is a distinguished family of inertial trajectories and a physical Minkowski interval.

Accelerating coordinates are allowed in special relativity. They produce nonzero connection coefficients even though the Riemann curvature tensor remains zero. General relativity is needed when spacetime curvature and dynamical gravitation matter, not merely because the chosen observer accelerates. Thus neither theory requires a particular inertial frame in its coordinate-free statement; each still needs the structure that determines inertial motion.

For a Riemannian metric on an ambient manifold and an embedded submanifold, let $\pi :TN{|}_M\to TM$ be orthogonal projection. Project the restriction of a connection to an embedded submanifold to obtain $D_{X}Y=\pi {\nabla }_{X}Y$. Since $\pi Y=Y$, the Leibniz rule and linearity over smooth functions hold, making $D$ an affine connection. If the ambient connection is the Levi-Civita connection, pairing with tangent vectors shows that $D$ is a metric connection; projecting the zero-torsion identity shows it is a torsion-free connection. By uniqueness, $D$ is the induced Levi-Civita connection.

For an embedded submanifold $i:M\hookrightarrow N$, the ambient affine connection induces a pullback connection on $i^{\ast }TN=TN{|}_M$. Extend a local section off $M$ and differentiate in a direction tangent to $M$. If two extensions agree on $M$, their difference has zero coefficients on $M$, whose tangential derivatives are zero. Thus the restricted derivative is independent of the extension. Its value may have a normal component, so obtaining a connection on $TM$ requires a further projection.

Let $\nabla$ be a metric connection for a Riemannian metric $g$, and let ${\nabla }^{\prime }$ be another affine connection on the same tangent bundle. The difference of affine connections is a tensor $A=\nabla -{\nabla }^{\prime }$. Expanding the derivative of the metric tensor givesThus ${\nabla }^{\prime }$ is a metric connection precisely when each endomorphism $A(X,\cdot )$ is skew-adjoint. This imposes no antisymmetry in $X,Y$; that separate condition characterizes agreement of parametrized geodesics. If both conditions hold, $g(A(X,Y),Z)$ is alternating in all three arguments.

At a fixed point $p$ of an $n$-dimensional smooth manifold $M$ without boundary, the tangent space $T_{p}M$ is the real vector space of all tangent vectors at $p$. In manifold charts $x=(x^1,\dots ,x^n)$ and $y=(y^1,\dots ,y^n)$, a tangent vector has componentsThe coordinate basis transforms inversely, so the tangent vector is unchanged. This invertible Jacobian matrix also preserves vector addition and scalar multiplication, giving a vector space of dimension $n$ independently of coordinates. ^[ref]

For example, at $(x,y)=(1,0)$ in ${\mathbb{R}}^2$, use the alternative manifold chart $u=x$, $w=y+x^2$. The Jacobian matrix sends $(v_x,v_y)=(2,3)$ to $(v_u,v_w)=(2,7)$. Indeed, the smooth curve $(x(t),y(t))=(1+2t,3t)$ gives $w(t)=1+7t+4t^2$. Conversely, ${\partial }_u{|}_p={\partial }_x{|}_p-2{\partial }_y{|}_p$ and ${\partial }_w{|}_p={\partial }_y{|}_p$, so $2{\partial }_u{|}_p+7{\partial }_w{|}_p=2{\partial }_x{|}_p+3{\partial }_y{|}_p$. The numbers change; their action on every smooth function does not.

For an embedded submanifold $M\subseteq {\mathbb{R}}^N$, the tangent space can be identified with the vector subspaceA proof outline explains why every coordinate vector is possible: in a manifold chart $x$, lift the short straight path $x(p)+ta$ using $x^{-1}$. Its tangent vector has components $a$. For a regular local parametrization $X:U\subseteq {\mathbb{R}}^n\to M$, the same chain rule gives $T_{p}M=\mathrm{im}D{X}_q$, where $X(q)=p$. This is the tangent space from a local parametrization.

For the sphere $S_R^2=\{p\in {\mathbb{R}}^3:p\cdot p=R^2\}$, differentiating that equation gives $p\cdot v=0$. Conversely, for any nonzero $v$ with $p\cdot v=0$, the smooth curvelies on the sphere and has ${\gamma }_v^{\prime }(0)=v$; use a constant smooth curve for $v=0$. Thus $T_p{S}_R^2=\{v:p\cdot v=0\}$, not merely a subset of that plane. At $p=(0,0,R)$ this vector subspace is $v_z=0$, while the affine tangent plane drawn through $p$ is $z=R$.

This also explains a mechanical constraint without requiring a force calculation. In the ideal mathematical model of a particle confined to a stationary sphere, its allowed velocities at $p$ are exactly $T_p{S}_R^2$: the radial component must vanish, but either independent tangential direction is available. For a general stationary holonomic constraint $F(q)=0$ with independent equations, the regular level set theorem gives $T_{q}M=\ker D{F}_q$. These are instantaneous kinematic possibilities; they do not determine the actual subsequent motion. On a general smooth manifold, tangent spaces at different base points cannot be identified canonically; an affine connection supplies parallel transport when such comparisons are needed.

A multilinear operation on vector fields is tensorial when it is linear over smooth functions in every argument. Its value at a point then depends only on the argument vectors at that point. To see this, use a smooth cutoff function to reduce to local fields, expand them in a local frame, and apply linearity over smooth functions to their coefficients. A smoothly valued tensorial operation therefore defines a tensor field. The curvature of an affine connection is tensorial, whereas a covariant derivative differentiates a scalar coefficient in its second argument and is not tensorial there.

Declare a global frame of left-invariant vector fields parallel on a semisimple Lie group. This affine connection is flat and preserves a left-invariant metric tensor because its frame components are constant. If $[e_a,e_c]=C_a{}^b{}_c{e}_b$, its torsion tensor is $T_a{}^b{}_c=-C_a{}^b{}_c$. For fixed $c$, the matrix with entries $T_a{}^b{}_c$ is $\mathrm{ad}{e}_c$. Its trace vanishes because a semisimple Lie algebra is perfect and adjoints of brackets are commutators. The trace of the product for $c,d$ is the Killing form $K_{cd}$, proving both contractions. This flat metric connection differs from the curved torsion-free Levi-Civita connection of the Killing metric.

The torsion of an affine connection is the displayed antisymmetric tensor. It measures the failure of the antisymmetrized covariant derivative to agree with the Lie bracket of vector fields. With derivative-last coefficients ${\nabla }_{\mu }{Y}^{\rho }={\partial }_{\mu }{Y}^{\rho }+{\Gamma }^{\rho }{}_{\nu \mu }{Y}^{\nu }$, its components are ${\mathcal{T}}^{\rho }{}_{\mu \nu }={\Gamma }^{\rho }{}_{\nu \mu }-{\Gamma }^{\rho }{}_{\mu \nu }$. Some conventions use the negative component tensor. Geodesic equations depend on the symmetric connection coefficients, so they alone cannot detect torsion.

A regular curve with tangent $v=dx/du$ is an unparametrized geodesic of an affine connection if ${\nabla }_{v}v=f(u)v$. Choose a new parameter $s$ with $ds/du=r(u)>0$ and $r^{\prime }/r=f$. Then $w=dx/ds=v/r$ obeys ${\nabla }_{w}w=r^{-2}({\nabla }_{v}v-(r^{\prime }/r)v)=0$. Conversely a reparametrized affine geodesic has acceleration proportional to its tangent. This criterion concerns the geometric curve, not a fixed parametrization.