Connections Levi Civita and Parallel Transport

Let $E\to M$ be a finite-rank smooth real vector bundle on a Hausdorff, second-countable smooth manifold; manifolds with boundary are allowed. Write $\mathrm{\Gamma }(E)$ for its smooth sections, as in Smooth sections, local sections, and support. Define ${\mathrm{\Omega }}^1(M;E)=\mathrm{\Gamma }(\mathrm{Hom}(TM,E))\mathrm{.}$ The Hom bundle is the canonical smooth bundle constructed below from the supplied smooth tangent bundle and E. Thus an element $\eta$ assigns a linear map ${\eta }_p:T_{p}M\to E_p$ to each point, smoothly in bundle charts. This is also the meaning of an $E$-valued one-form, customarily written as a section of $T^{\ast }M\otimes E$. No tangent-only tensor construction is needed for this definition.

A connection on $E$ is an $\mathbb{R}$-linear map $\mathrm{\nabla }:\mathrm{\Gamma }(E)\to {\mathrm{\Omega }}^1(M;E)$ such that, for every $f\in C^{\mathrm{\infty }}(M)$ and $s\in \mathrm{\Gamma }(E)$, $\mathrm{\nabla }(fs)=df\otimes s+f\mathrm{\nabla }s\mathrm{.}$ Here $(df\otimes s{)}_p(v)=d{f}_p(v)s(p)$, using The differential of a smooth real-valued function. In local coordinates and a local frame of $E$, this map has entries $({\mathrm{\partial }}_{i}f)s^a$, so it is a smooth Hom-bundle section. The tensor notation in the rule therefore already has a defined meaning.

This definition specifies a given operator, not its existence. The local operator rules and the Hom construction from supplied smooth bundles use no choice axiom. If $M$ is empty or $E$ has rank zero, both spaces are zero spaces and there is exactly one such operator. If $M$ has dimension zero, $TM$ is the zero bundle and the same conclusion holds, even when $E$ has positive rank. Rank one uses the same definition without a preferred trivialization.

Smooth Hom bundles from supplied smooth bundles. The following construction supplies the Hom notation used here and in the subsequent dual and tensor constructions. Let $A,B\to M$ be supplied smooth real vector bundles of ranks $r,s$, in the sense of Smooth vector bundles, rank, fibres, and trivial bundles, with Hausdorff second-countable total spaces. The same local product convention is used when $M$ has boundary. We construct the smooth structure on the fibrewise set $H={\coprod }_{p\in M}\mathrm{Hom}(A_p,B_p)$, rather than infer global topology from transition matrices alone.

If $r,s>0$, let $D\subseteq A^r\times B^r$ consist of tuples $(a_1,\dots ,a_r,b_1,\dots ,b_r)$ over a common base point $p$ such that $(a_1,\dots ,a_r)$ is a basis of $A_p$. Give $D$ the subspace topology of the finite product The product set ${\prod }_{i\in I}{X}_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space. Fix one countable base of each of the two supplied total spaces; this is a finite choice of witnesses in ZF, not a countable family of choices. Finite product boxes give a countable base of $A^r\times B^r$, because their indices lie in a finite product of countable sets, countable by repeated application of A product of two at most countable sets is at most countable. Taking traces gives a countable base of $D$ by Second countability is hereditary. No manifold structure on that ambient product or its fibre-product subspace is needed.

Define $q:D\to H$ by sending a tuple to the unique linear map $a_i\mapsto b_i$. The map is onto: each individual finite-dimensional fibre has a basis, and a specified map determines the corresponding $b_i$; this pointwise existence does not choose bases simultaneously. Give $H$ the quotient topology The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection. For any one common local trivialization of $A,B$ over $U$, the open saturated subspace $D_U$ is homeomorphic to $U\times GL(r,\mathbb{R})\times M_{s\times r}(\mathbb{R})$. Write its coordinates $(p,C,V)$, with the frame vectors the columns of $C$ and the image vectors the columns of $V$. The proposed Hom chart sends a fibre map to its matrix $T$ in the fixed local frames, and its composite with $q$ is $(p,C,V)⟼(p,V{C}^{-1})\mathrm{.}$ This map is a continuous open surjection: $(p,C,V)\mapsto (p,C,V{C}^{-1})$ is a homeomorphism with inverse $(p,C,T)\mapsto (p,C,TC)$, and projection off $GL(r,\mathbb{R})$ is open, as is seen on product boxes. Smooth matrix inversion, in particular continuity, is supplied by Matrix inversion preserves $C^k$ regularity where the determinant is nonzero. Thus the composite is quotient by A continuous open surjection, a continuous closed surjection, and a continuous surjection admitting a continuous section are all quotient maps. Since $D_U$ is open and saturated, the quotient criterion also says that $H_U$ is open and the restricted quotient topology agrees with its subspace topology. Comparing the two quotient criteria proves that the proposed Hom chart $H_U\to U\times M_{s\times r}(\mathbb{R})$ is a homeomorphism.

These local formulas also prove that $q$ is globally open: for open $O\subseteq D$, its image intersected with every $H_U$ is the open image of $O\cap D_U$ in this chart. The images of the countable base of $D$ therefore form a countable base of $H$. Indeed, if $h\in W$ with $W$ open in $H$, choose a single $d\in q^{-1}(h)$ and a basic $C_0$ with $d\in C_0\subseteq q^{-1}(W)$; then $h\in q(C_0)\subseteq W$. This is a proof for an arbitrary one point and neighbourhood, not a simultaneous choice of representatives. Distinct maps over distinct base points have disjoint neighbourhoods pulled back from $M$; distinct maps over the same base point have disjoint neighbourhoods inside one open Hom chart. Hence $H$ is Hausdorff as well as second countable.

If original bundle coordinates change by $g$ on $A$ and $h$ on $B$, the Hom matrix changes by $T\mapsto hT{g}^{-1}$. These maps and their inverses are smooth by finite matrix algebra and the smooth-inversion supplier. At a boundary point, extend the finitely many smooth matrix entries to an ambient coordinate neighbourhood and shrink it so the nonzero determinant stays nonzero; the same inverse formula restricts to the half-space. Thus these charts form a smooth bundle atlas, with total-space charts given by a base chart times a Euclidean matrix space; a boundary chart remains a half-space chart. The fibre operations are linear in these charts. Every original local trivialization participates, so the topology and maximal smooth atlas are canonical and independent of frames. If either rank is zero, $H$ is canonically the zero bundle $M$ and needs no matrix inversion; an empty base gives the empty total space. Finally $A^{\ast }=\mathrm{Hom}(A,M\times \mathbb{R})$ uses the explicit trivial line, whose product base is countable by the same finite argument. Its transition is $g^{-T}$. This proves the required dual and Hom bundle existence and smooth matrix rules from supplied bundles without a choice axiom or a selected countable family of trivializations.

For a connection as in Connection on a smooth vector bundle, a smooth vector field $X\in \mathrm{\Gamma }(TM)$ and a section $s\in \mathrm{\Gamma }(E)$, define the covariant derivative in direction $X$ by ${\mathrm{\nabla }}_{X}s=(\mathrm{\nabla }s)(X),({\mathrm{\nabla }}_{X}s)(p)=(\mathrm{\nabla }s{)}_p(X(p))\mathrm{.}$ It is a smooth section: in any local coordinates and bundle frame, if the matrix of $\mathrm{\nabla }s$ has entries $b_i^a$ and $X$ has components $X^i$, its components are the finite sums ${\sum }_i{b}_i^a{X}^i$. For a single tangent vector $v\in T_{p}M$, the notation ${\mathrm{\nabla }}_{v}s$ means $(\mathrm{\nabla }s{)}_p(v)$. In particular the direction is used only at $p$, while the section can be differentiated there.

Zero direction gives zero; zero section gives zero by real linearity of $\mathrm{\nabla }$. For a zero-dimensional base the sum is empty. Empty base and rank-zero bundle also give the unique zero section. A direction is not required to be nonzero or to extend along any prescribed curve. This is evaluation of supplied data and needs no choice.

The directional operator of a connection obeys ${\mathrm{\nabla }}_{fX+gY}s=f{\mathrm{\nabla }}_{X}s+g{\mathrm{\nabla }}_{Y}s,{\mathrm{\nabla }}_X(as+bt)=a{\mathrm{\nabla }}_{X}s+b{\mathrm{\nabla }}_{X}t,$ ${\mathrm{\nabla }}_X(fs)=X(f)s+f{\mathrm{\nabla }}_{X}s$ for smooth functions $f,g$, real numbers $a,b$, vector fields $X,Y$ and sections $s,t$. Conversely, a map $D:\mathrm{\Gamma }(TM)\times \mathrm{\Gamma }(E)\to \mathrm{\Gamma }(E)$ with these three laws comes from a unique connection by evaluation. This includes manifolds with boundary.

For a connection on $E\to M$, if two sections agree near $p$, their covariant derivatives at $p$ agree in every tangent direction. The derivative in direction $X$ depends only on $X(p)$. For every open $U\subseteq M$, there is a unique connection on $E{\mathrm{\mid }}_U$ agreeing with the original on restricted global sections; restrictions to nested open sets compose. Boundary points are included.

Let $e=(e_1,\dots ,e_r)$ be a local frame of $E$ on an open set $U$, in the sense of Local and global frames of a vector bundle. Restrict a given connection to $U$ by A bundle connection is local and restricts to open sets. Its connection one-form in the frame $e$ is the matrix $\omega =({\omega }^i{}_j)$ of scalar one-forms on $U$ characterized by ${\mathrm{\nabla }}_X{e}_j={\sum }_i{\omega }^i{}_j(X)e_i\mathrm{.}$ For each $j$, the smooth Hom section $\mathrm{\nabla }{e}_j$ has unique fibre coordinates in $e$; these coordinates are linear in the direction and smooth, hence are scalar one-forms. Smoothness may be tested in any coordinate chart and frame, using Smoothness of a section is equivalent to smooth local components.

Our convention treats $e$ as a row and the coefficients $u$ of $s=eu$ as a column. A frame change is $e^{\mathrm{\prime }}=eA$ with a smooth invertible matrix $A$, so $u=A{u}^{\mathrm{\prime }}$ for the same section. Products of matrices and one-forms use ordinary matrix multiplication with scalar functions as coefficients. In a coordinate chart, ${\omega }^i{}_j={\sum }_k{\mathrm{\Gamma }}^i{}_{kj}d{x}^k$; this notation does not assert tensorial transformation for $\mathrm{\Gamma }$.

For rank zero the frame and connection matrix are empty; for rank one $\omega$ is a scalar one-form. On a zero-dimensional base every one-form is zero. The frame is supplied locally, so this definition involves no global selection or existence theorem.

In a local frame $e$, write $s=eu$ with coefficient column $u$. Then ${\mathrm{\nabla }}_{X}s=e(X(u)+\omega (X)u),\mathrm{\nabla }s=e(du+\omega u)\mathrm{.}$ Here the last expression means an $E$-valued one-form, and $X(u)$ and $du$ act entrywise.

If $e^{\mathrm{\prime }}=eA$ are two frames on an overlap, their matrices for the same connection satisfy ${\omega }^{\mathrm{\prime }}=A^{-1}\omega A+A^{-1}dA\mathrm{.}$ All entries of $dA$ are differentials of scalar functions; the order of the matrix factors is as displayed.

Let a smooth vector bundle be covered by supplied frames $e_{\alpha }$ on $U_{\alpha }$, with $e_{\beta }=e_{\alpha }{A}_{\alpha \beta }$ on overlaps. Matrices of smooth one-forms ${\omega }_{\alpha }$ are the local matrices of a unique connection if and only if ${\omega }_{\beta }=A_{\alpha \beta }^{-1}{\omega }_{\alpha }{A}_{\alpha \beta }+A_{\alpha \beta }^{-1}d{A}_{\alpha \beta }$ on every overlap. No local finiteness or choice assumption is required for this gluing statement.

For two connections ${\mathrm{\nabla }}^0,{\mathrm{\nabla }}^1$ on the same bundle $E$, there is a unique smooth section $B$ of $\mathrm{Hom}(TM,\mathrm{End}E)$ such that ${\mathrm{\nabla }}_X^{1}s-{\mathrm{\nabla }}_X^{0}s=B(X)s\mathrm{.}$ This is called an endomorphism-valued one-form. In a frame it has matrix ${\omega }_1-{\omega }_0$.

If $\mathrm{\nabla }$ is a connection and $B\in \mathrm{\Gamma }(\mathrm{Hom}(TM,\mathrm{End}E))$, then $(\mathrm{\nabla }+B{)}_{X}s={\mathrm{\nabla }}_{X}s+B(X)s$ defines a connection. If the set of connections is nonempty, it is an affine space modeled on the real vector space of endomorphism-valued one-forms: that vector space acts freely and transitively by this addition.

Assume AC. Every finite-rank smooth real vector bundle over a Hausdorff second-countable smooth manifold, with boundary allowed, admits a connection. No canonicity or smaller sufficient choice bound is asserted.

Let $f:N\to M$ be smooth and let $\mathrm{\nabla }$ be a connection on $E\to M$. Give $f^{\ast }E=\{(q,e)\in N\times E:f(q)=\pi (e)\}$ the subspace topology. If $\mathrm{\Phi }:E{\mathrm{\mid }}_U\to U\times {\mathbb{R}}^r$ is a vector-bundle chart and $\mathrm{\Phi }(e)=(\pi (e),v)$, then $\tilde{\mathrm{\Phi }}(q,e)=(q,v),{\tilde{\mathrm{\Phi }}}^{-1}(q,v)=(q,{\mathrm{\Phi }}^{-1}(f(q),v))\mathrm{.}$ Both displayed maps are continuous in the subspace and product topologies, so this is a homeomorphism from $(f^{\ast }E){\mathrm{\mid }}_{f^{-1}U}$ to $f^{-1}U\times {\mathbb{R}}^r$. On overlaps its change of coordinates is $(q,v)⟼(q,g_{\beta \alpha }(f(q))v),$ which is smooth and fibrewise linear. These charts therefore supply the smooth rank-$r$ bundle structure directly. Its total space is Hausdorff and second countable: $N\times E$ is a smooth manifold by Products of smooth manifolds have a canonical product smooth structure, and both properties pass to the subspace $f^{\ast }E$ by $T_0$, $T_1$, and Hausdorffness are hereditary and Second countability is hereditary.

In a pulled-back frame $f^{\ast }e$ on $f^{-1}U$, the pullback connection $f^{\ast }\mathrm{\nabla }$ is specified by $(f^{\ast }\mathrm{\nabla }{)}_X((f^{\ast }e)u)=(f^{\ast }e)(X(u)+(f^{\ast }\omega )(X)u),$ where $(f^{\ast }\omega {)}_q(v)={\omega }_{f(q)}(d{f}_{q}v)$ entrywise and $u$ is any smooth coefficient column on $N$. These are local prescriptions under the gluing criterion Local connection forms glue exactly when they obey the transformation law; their compatibility and hence well-definedness are proved in the next theorem.

In particular, coefficient functions are not required to factor through $f$. For constant $f$, a constant frame of $E_{f(N)}$ gives zero pulled-back matrix and ordinary differentiation of arbitrary $u$; it does not make every varying section parallel. Empty source and rank-zero bundles use empty coefficient data. No injectivity, immersion, submersion, or choice of an extension is part of this definition.

The pullback prescription defines a unique connection $f^{\ast }\mathrm{\nabla }$, independent of frames. Under the canonical bundle isomorphisms it satisfies ${\mathrm{id}}^{\ast }\mathrm{\nabla }=\mathrm{\nabla }$ and $g^{\ast }(f^{\ast }\mathrm{\nabla })=(f\circ g{)}^{\ast }\mathrm{\nabla }$. For a local section $s$ of $E$, $(f^{\ast }\mathrm{\nabla }{)}_X(f^{\ast }s)(q)=(\mathrm{\nabla }s{)}_{f(q)}(d{f}_q{X}_q)\mathrm{.}$ The right side is interpreted in the fibre of the pullback bundle at $q$.

The dual connection on the smooth bundle $E^{\ast }$ of Connection on a smooth vector bundle is characterized by $({\mathrm{\nabla }}_X^{\ast }\alpha )(s)=X(\alpha (s))-\alpha ({\mathrm{\nabla }}_{X}s)\mathrm{.}$ The right side is function-linear in $s$: replacing $s$ by $fs$ introduces $X(f)\alpha (s)$ once with each sign, so those terms cancel.

Here is the construction, including well-definedness. In a frame $e$ write $s=eu$ and write $\alpha$ with dual coefficient column $a$, so $\alpha (s)=a^{T}u$. The coordinate formula Local coordinate formula for a bundle connection gives $X(a^{T}u)-a^T(X(u)+\omega (X)u)=(X(a)-\omega (X{)}^{T}a{)}^{T}u\mathrm{.}$ Thus the required coefficients are $X(a)-\omega (X{)}^{T}a$, and the local connection matrix in the dual frame is $-{\omega }^T$. These local rules are smooth, real-linear and obey the section Leibniz rule by the scalar product rule. On an overlap both give the intrinsic displayed pairing formula; agreement against every local frame section forces equality of the dual covectors. They therefore glue to a connection, equivalently by Local connection forms glue exactly when they obey the transformation law. Testing against a basis also proves uniqueness. This verifies the implicit characterization rather than assuming that it defines an operator.

The derivative on scalar functions is $X(f)$; duality preserves the evaluation pairing with that scalar connection. For rank one the matrix changes sign; for rank zero evaluation and all coefficients are empty. Zero sections give zero derivatives, while empty bases give the unique operator. This construction is local and canonical from the given connection and requires no choice axiom.

For finite-rank smooth real vector bundles $E_1,\dots ,E_k$ on $M$, define the fibre tensor product to be the vector space of multilinear maps $E_{1,p}^{\ast }\times \cdots \times E_{k,p}^{\ast }\to \mathbb{R}$. The elementary tensor $v_1\otimes \cdots \otimes v_k$ evaluates to ${\prod }_j{\alpha }_j(v_j)$. These spaces form a canonical smooth bundle, denoted $E_1\otimes \cdots \otimes E_k$, with product frames. Every local section is a finite sum of product-frame tensors with smooth coefficients. The empty product is the trivial real line.

Give each of finitely many bundles $E_j$ a connection. Use the tensor bundle constructed in Finite tensor products of smooth vector bundles, including its multilinear model on the dual fibres. The product connection is characterized on local elementary tensors by ${\mathrm{\nabla }}_X(s_1\otimes \cdots \otimes s_k)={\sum }_j{s}_1\otimes \cdots \otimes {\mathrm{\nabla }}_X^j{s}_j\otimes \cdots \otimes s_k\mathrm{.}$ For the empty product, the connection on scalar functions is ${\mathrm{\nabla }}_{X}f=X(f)$.

To construct it without a decomposition assumption, for a multilinear-model section $T$ and local dual sections ${\alpha }_j$ put $({\mathrm{\nabla }}_{X}T)({\alpha }_1,\dots ,{\alpha }_k)=X(T({\alpha }_1,\dots ,{\alpha }_k))-{\sum }_{j}T({\alpha }_1,\dots ,({\mathrm{\nabla }}^j{)}_X^{\ast }{\alpha }_j,\dots ,{\alpha }_k)\mathrm{.}$ Use the dual connections of Dual connection. Multiplying any argument by a smooth $f$ produces two opposite $X(f)T({\alpha }_1,\dots ,{\alpha }_k)$ terms; thus this expression is function-linear in every dual argument. In dual frames its coefficients are smooth and depend only on their fibre values, so it defines a smooth tensor section. The same intrinsic expression on overlaps gives agreement. Its real-linearity, function-linearity in $X$, and rule ${\mathrm{\nabla }}_X(fT)=X(f)T+f{\mathrm{\nabla }}_{X}T$ follow by direct expansion, establishing the connection. This also gives the gluing criterion of Local connection forms glue exactly when they obey the transformation law.

On a pure tensor, the scalar product rule and the dual pairing identity reduce this expression to the first displayed sum. Hence it respects the balanced identity $(fs)\otimes t=s\otimes (ft)$: either derivative gives the same term $X(f)s\otimes t$ and the same two terms multiplied by $f$. Every section has a finite local product-frame expansion, so that rule and the connection Leibniz identity force uniqueness.

For bundles $E,F$, identify $F\otimes E^{\ast }$ with $\mathrm{Hom}(E,F)$ by $t\otimes \alpha \mapsto (v\mapsto \alpha (v)t)$. In local bases these elementary tensors are precisely the matrix units, proving this is a smooth fibrewise isomorphism. The induced Hom connection is therefore $({\mathrm{\nabla }}_{X}A)(s)={\mathrm{\nabla }}_X^F(A(s))-A({\mathrm{\nabla }}_X^{E}s)\mathrm{.}$ The two $X(f)A(s)$ terms cancel when $s$ is replaced by $fs$, ensuring a fibrewise map. Rank-zero factors give zero bundles, whereas the empty product is the scalar line; these conventions are different. No choice beyond finitely many local frames is needed.

For the induced tensor connections, every fixed permutation $\tau$ of tensor slots intertwines covariant differentiation. So does any contraction $C$ of an $E^{\ast }$ slot against an $E$ slot equipped with dual connections: ${\mathrm{\nabla }}_X(\tau T)=\tau ({\mathrm{\nabla }}_{X}T),{\mathrm{\nabla }}_X(CT)=C({\mathrm{\nabla }}_{X}T)\mathrm{.}$ A full contraction takes values in scalar functions, with derivative $X$.

A connection on $E$ induces a connection on each ${\bigwedge }^{k}E$, including the scalar bundle ${\bigwedge }^{0}E$. For sections $S$ and $T$ of exterior degrees $p$ and $q$, ${\mathrm{\nabla }}_X(S\wedge T)=({\mathrm{\nabla }}_{X}S)\wedge T+S\wedge ({\mathrm{\nabla }}_{X}T)\mathrm{.}$ There is no graded sign: the covariant derivative in a fixed direction has degree zero.

Let $\pi :E\to M$ be a smooth vector bundle. Let $I\subseteq \mathbb{R}$ be an interval with nonempty interior and let $\gamma :I\to M$ be smooth, with smooth local extensions at any included endpoints. Give ${\gamma }^{\ast }E=\{(t,e)\in I\times E:\gamma (t)=\pi (e)\}$ the subspace topology. For a bundle chart $\mathrm{\Phi }:E{\mathrm{\mid }}_U\to U\times {\mathbb{R}}^r$, writing $\mathrm{\Phi }(e)=(\pi (e),v)$, the pulled-back chart and its inverse are $(t,e)⟼(t,v),(t,v)⟼(t,{\mathrm{\Phi }}^{-1}(\gamma (t),v))\mathrm{.}$ They are continuous for the subspace and product topologies, and the overlap maps are $(t,v)\mapsto (t,g_{\beta \alpha }(\gamma (t))v)$, hence smooth and fibrewise linear. Thus they directly define a smooth rank-$r$ bundle over $I$, with the stated smooth-up-to-endpoint convention. The interval $I$, as a subspace of $\mathbb{R}$, and the manifold $E$ are Hausdorff and second countable. Their product is Hausdorff by Arbitrary products preserve $T_0$, $T_1$, and Hausdorffness, and products of their two fixed countable bases form a countable basis. Hence both properties pass to the subspace ${\gamma }^{\ast }E$ by $T_0$, $T_1$, and Hausdorffness are hereditary and Second countability is hereditary.

A section of $E$ along $\gamma$ is a smooth section of this bundle ${\gamma }^{\ast }E$. Equivalently it is a smooth map $V:I\to E$ with $\pi (V(t))=\gamma (t)$. For $E=TM$ it is a vector field along $\gamma$. In a pulled-back frame it has the form $V(t)={\sum }_j{v}^j(t)e_j(\gamma (t))$ with arbitrary smooth coefficients $v^j$.

Values belong to the fibre over the parameter value, even if $\gamma (t)=\gamma (u)$ for $t\mathrm{\ne }u$. For example, for a constant curve at $p$ and $v\mathrm{\ne }0$ in $E_p$, the section $V(t)=tv$ is allowed and cannot be $s\circ \gamma$ for an ambient section $s$. Thus an ambient extension is not part of this definition.

For a piecewise smooth curve on a compact interval, use a finite subdivision into smooth pieces (each smooth up to its endpoints). A piecewise smooth section is continuous on the whole interval and smooth on each piece. On a singleton interval it is just a fibre vector; on an empty interval it is the empty section. These conventions define sections, without attempting differentiation on a singleton.

For a given connection on $E$ and a smooth curve on an interval with nonempty interior, the covariant derivative along the curve is $D_{t}V=({\gamma }^{\ast }\mathrm{\nabla }{)}_{\mathrm{\partial }\mathrm{/}\mathrm{\partial }t}V\mathrm{.}$ Here $V$ is a section of the pullback bundle as in Vector field and section along a smooth curve, and the pullback connection exists by Pullback connection is well defined and functorial. It is real-linear and satisfies $D_t(fV)=f^{\mathrm{\prime }}V+f{D}_{t}V$ for functions of the parameter.

At an included endpoint the smooth up-to-boundary coefficients have their one-sided derivative; equivalently any smooth local extension of the coefficients gives this value, because extensions agreeing on a one-sided interval have the same derivative there. For a piecewise smooth curve the derivative is defined on each piece, with separate one-sided derivatives at corners; those derivatives are not required to agree. A singleton interval carries no derivative operator under this convention and will have identity transport by definition. A zero section or a rank-zero bundle gives zero covariant derivative. Constant base curves can have sections with nonzero coefficient derivative.

The operator $D_t$ is independent of local frames. If $V=s\circ \gamma$ near a parameter value for an ambient local section $s$, then $D_{t}V=(\mathrm{\nabla }s{)}_{\gamma (t)}(\dot{\gamma }(t))\mathrm{.}$ Consequently any two such extensions give the same derivative. This is an agreement assertion when an ambient extension exists, not an assertion that every section along a curve extends.

If $V(t)=e(\gamma (t))v(t)$ and $B(t)={\omega }_{\gamma (t)}(\dot{\gamma }(t))$, then $D_{t}V=e(\gamma (t))(v^{\mathrm{\prime }}(t)+B(t)v(t))\mathrm{.}$ Thus $D_{t}V=0$ is the linear system $v^{\mathrm{\prime }}=-Bv$.

A section $V$ along a smooth curve is parallel if $D_{t}V=0$. By Local frame formula for covariant differentiation along a curve, in a pulled-back frame this means $v^{\mathrm{\prime }}(t)=-{\omega }_{\gamma (t)}(\dot{\gamma }(t))v(t)\mathrm{.}$ For a piecewise smooth curve, parallel means continuous on the full parameter interval and parallel on each smooth piece. One-sided derivatives are used at the endpoints of a piece; no equality of the two derivatives at a corner is required. On a singleton, every prescribed fibre vector is a parallel section by convention; on an empty interval the empty section is parallel.

The zero section is parallel, and along a constant curve the coefficients in a constant fibre frame are constant precisely for parallel sections. The definition does not assert that all local coefficients in arbitrary moving frames are constant.

Let $\gamma :I\to M$ be smooth on an interval, with smooth one-sided data at included endpoints. For every $t_0\in I$ and $v_0\in E_{\gamma (t_0)}$ there is exactly one parallel section $V$ on all of $I$ with $V(t_0)=v_0$. The same holds for a piecewise smooth curve on a compact interval, with continuity at its finitely many corners. A singleton carries its prescribed vector. No AC is required for a supplied connection and curve.