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A local trivialization of a smooth vector bundle gives a local connection by differentiating its component functions. Weight these connections by a subordinate locally finite partition of unity and extend the weighted terms by zero. The weights sum to one, so the connection Leibniz rule survives. This proves existence on a paracompact smooth manifold. Averaging connections is valid with weights summing to one because the collection is the affine space of vector-bundle connections.

A connection on a vector bundle specifies how to compare nearby fibers without identifying them globally. Several equivalent descriptions expose its algebra, local calculations and geometry.

The first description is a vector bundle covariant derivative, a real-linear mapEquivalently ${\nabla }_{X}s$ is linear over smooth functions in the vector field $X$, and obeys ${\nabla }_X(fs)=X(f)s+f{\nabla }_{X}s$ in the section. The derivative term distinguishes a connection from a bundle homomorphism. The Leibniz rule also makes it local: use a smooth cutoff equal to one near a point to see that its value depends only on the section near that point. Thus a connection restricts consistently to all open subsets.

The second description is a family of connection matrices in local frames. Write a frame as a row $e=(e_1,\dots ,e_m)$, so a section is $s=eu$ with coefficient column $u$. The connection is thenwhere $A$ is a matrix of one-forms determined by $\nabla e_j={\sum }_i{e}_i{A}^i{}_j$. If $e^{\prime }=eg$, the same section has $u=g{u}^{\prime }$. Expanding the Leibniz rule givesHence the change of frame of a vector-bundle connection isConversely, matrices of one-forms satisfying this transformation rule define $\nabla (eu)=e(du+Au)$ in each frame. The displayed calculation proves that these local formulas agree on overlaps; they therefore glue to a connection and obey its Leibniz rule. This proves equivalence of the first two descriptions in both directions.

A third description is parallel transport. Along a smooth curve $\gamma :[a,b]\to B$, a parallel section $s(t)=e(\gamma (t))u(t)$ satisfiesExistence and uniqueness for linear ordinary differential equations supply a unique solution for each initial fiber vector, smoothly dependent on that vector and on smooth families of curves. The frame transformation law makes these local solutions glue. The resulting map $P_{\gamma ,a,b}:E_{\gamma (a)}\to E_{\gamma (b)}$ is linear and invertible; reversing the curve gives its inverse, and concatenating curves composes the maps. It is generally dependent on the path, not just its endpoints.

One can recover the covariant derivative from this transport. For $\gamma (0)=b$ and $\dot{\gamma }(0)=X$, let $P_{0,t}$ be its transport from $0$ to $t$. ThenTo verify the formula rather than assume it, use a local frame and a fundamental solution matrix $P(t)$ with $P(0)=I$ and $\dot{P}=-A(\dot{\gamma })P$. Differentiation gives $(P^{-1}u{)}^{\prime }{|}_0=\dot{u}(0)+A_b(X)u(0)$, exactly the local covariant-derivative formula. Conversely the connection reconstructed from transport defined by this equation has the same differential equation and, by uniqueness, the same transport. Thus this viewpoint is also equivalent for the transport rules arising from a connection.

A fourth description is a linear horizontal distribution on a vector bundle. In a trivialization $E{|}_U=U\times {\mathbb{R}}^m$, putProjection maps this subspace isomorphically to $T_{b}B$, and it complements the vertical tangent space. Under the coordinate change $v=g{v}^{\prime }$, differentiation gives $\dot{v}=(dg)(X)v^{\prime }+g{\dot{v}}^{\prime }$. Thus the horizontal equation transforms into ${\dot{v}}^{\prime }=-A^{\prime }(X)v^{\prime }$, with exactly the same frame-change law. The subspaces consequently glue independently of trivializations. Their fiberwise linear dependence on $v$ is essential: an arbitrary horizontal complement can describe a nonlinear connection on the total space and need not define a vector-bundle connection. Conversely a horizontal complement with this linearity has a unique local expression of the displayed form, recovering the matrices $A$. Its horizontal lifts are precisely the parallel sections just described.

One can also express the geometry on the frame bundle as a principal connection. In local frame-bundle coordinates $(b,g)$, its Lie-algebra-valued form isIt is equivariant under right translation, $R_h^{\ast }\omega ={\mathrm{Ad}}_{h^{-1}}\omega$, and equals $\xi$ on the vertical vector generated by $\xi$. These properties define a principal connection; its kernel is the horizontal distribution on the frame bundle. Pulling back by a local frame section gives $A$, and its frame-change rule is once again the same formula. This identifies the principal viewpoint with the earlier descriptions.

Connections are not a vector space with a distinguished origin. The difference of two connections is linear over smooth functions in both $X$ and $s$, because their Leibniz terms cancel. It is a one-form with values in $\mathrm{End}E$. Conversely adding any such form to a connection gives another connection. Thus the collection is an affine space of vector-bundle connections modeled on ${\Omega }^1(B;\mathrm{End}E)$. The local curvature in the usual vector-bundle convention is $F=dA+A\wedge A$, representing ${\nabla }_X{\nabla }_Y-{\nabla }_Y{\nabla }_X-{\nabla }_{[X,Y]}$. It records the infinitesimal failure of path-independent parallel transport. Question 6 uses the opposite curvature sign, which will be stated explicitly there.

A connection on a vector bundle $E\to M$ is an $\mathbb{R}$-linear mapsatisfying $\nabla (fs)=df\otimes s+f\nabla s$. Equivalently its directional version is real-bilinear and satisfiesFor a complex bundle it is complex-linear in the section variable and has the same Leibniz property.

Every smooth vector bundle over an ordinary Hausdorff, second-countable smooth manifold has such a connection. Choose trivializing open sets $U_a$, local frames $e_{a,i}$ and a subordinate locally finite partition of unity ${\rho }_a$. In each frame define the flat local connectionThen set $\nabla s={\sum }_a{\rho }_a{\nabla }^{(a)}(s{|}_{U_a})$, extending each weighted term by zero outside $U_a$. The supports lie inside the corresponding trivializing sets, making the extensions smooth, and local finiteness makes the sum smooth. Since ${\sum }_a{\rho }_a=1$, its Leibniz rule isThis is the construction of a vector bundle connection by a partition of unity.

The difference $A={\nabla }^{\prime }-\nabla$ of two connections is $C^{\infty }(M)$-linear in both its vector-field argument and its section argument. The derivative terms cancel, so $A_{X}s$ depends only on $X_p,s_p$ at each point. Smooth local coefficients therefore identify it withConversely, any such $A$ added to a connection preserves its axioms. Thus connections form an affine space modeled on ${\Omega }^1(M;\mathrm{End}E)$. Choosing a base connection ${\nabla }^0$ gives the noncanonical bijection $A\mapsto {\nabla }^0+A$; there is no distinguished zero connection and hence no canonical vector-space identification. This is the affine space of vector-bundle connections.

For a connection on $TM$, the torsion tensor is the alternating operation $\tau (X,Y)={\nabla }_{X}Y-{\nabla }_{Y}X-[X,Y]$. The connection and bracket Leibniz rules giveAlternation then gives $\tau (X,fY)=f\tau (X,Y)$ as well. Hence it is tensorial, of type one contravariant and two covariant indices:It is equivalently a $TM$-valued two-form, a torsion form.

For the local frame $e_i$ with dual coframe ${\varphi }_j$, defineThe connection is $C^{\infty }$-linear in $X$, so these are smooth differential one-forms. Expansion in the frame gives ${\nabla }_X{e}_i={\sum }_j{\omega }_{ij}(X)e_j$, and uniqueness follows by applying each ${\varphi }_j$. Here the first index is the input frame vector and the second the output coefficient; this fixes the connection one-form convention.

Expand $Y={\sum }_i{\varphi }_i(Y)e_i$ and apply the connection Leibniz rule:Subtract the corresponding expression with $X,Y$ exchanged, and then subtract ${\varphi }_j([X,Y])$. By the exterior derivative of a one-form evaluated on vector fields,Since all pairs of tangent vectors can be tested, equality of the two-forms follows:This is the Cartan first structure equation with input-first indices. Its plus sign is correct with the stated index order and wedge order; the common output-first convention writes the same identity with a minus sign and reversed wedge factors.