Connection (vector bundle, $\nabla$)

A connection is a linear map $\nabla :\Gamma (E)\to {\Omega }^1(B;E)$ satisfying $\nabla (fs)=df\otimes s+f\nabla s$. In a local frame it has the form $d+A$ for a matrix-valued one-form $A$.

A vector field is parallel for a specified connection on a vector bundle on the tangent bundle when its covariant derivative in every direction vanishes. Its values along every smooth curve are related by parallel transport. For a Levi-Civita connection, its length is constant on each connected component, since the connection preserves the Riemannian metric.

A smooth self-adjoint projection $p:M\to M_N(\mathbb{C})$ of locally constant rank defines the Hermitian vector bundle $\mathrm{im}p$ inside the trivial bundle. Projecting the flat derivative gives the unitary connection $\nabla s=pds$ for $ps=s$. Differentiating $p^2=p$ gives $pdp=dp(1-p)$ and $pdpp=0$. For a section $s$, ${\nabla }^{2}s=pdp\wedge ds=p(dp{)}^{2}s$, proving the curvature formula. Relative to $\mathrm{im}p\oplus \ker p$, $dp$ is off-diagonal and $(dp{)}^2$ is diagonal. Therefore $d\mathrm{tr}(p(dp{)}^{2k})=\mathrm{tr}(dp{)}^{2k+1}=0$. These are explicit closed representatives of the Chern character.

The graded Leibniz rule terms of two connections cancel, making their difference $C^{\infty }$-linear in the section. It is therefore an element of $A^1(\mathrm{End}E)$. Conversely adding any such form to a connection gives another connection, making the space of connections an affine space.

With local frame written as a row $e$ and $D(eu)=e(du+Au)$, a change $e^{\prime }=eg$ gives $A^{\prime }=g^{-1}Ag+g^{-1}dg$. The vector-bundle curvature transforms by conjugation. This convention makes the inhomogeneous derivative term and every subsequent vector-bundle curvature sign explicit.

The difference of two connections on a vector bundle is a smooth one-form with values in the endomorphism bundle: the Leibniz derivative terms cancel, leaving pointwise linearity in both the tangent vector and section value. Conversely every such form added to a connection gives another connection. Thus the collection is an affine space with this model vector space. Choosing ${\nabla }^0$ supplies a noncanonical origin; there is no distinguished zero connection.

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 induces a connection on its top exterior power by differentiating each factor in turn. Its local connection one-form is $\mathrm{Tr}A$ and its curvature form of a connection is $\mathrm{Tr}{\Theta }_D$.

For a metric vector bundle connection over a Riemannian manifold, the rough Laplacian is the formally nonnegative connection operatorwhere $e_i$ is a local orthonormal frame. The correction term makes the expression independent of the orthonormal frame. On a closed manifold, integration by parts gives $⟨{\nabla }^{\ast }\nabla s,s{⟩}_{L^2}=\Vert \nabla s{\Vert }_{L^2}^2$.

An affine connection on a smooth manifold is a connection on a vector bundle for its tangent bundle. It satisfies ${\nabla }_{fX}Y=f{\nabla }_{X}Y$ and ${\nabla }_X(fY)=X(f)Y+f{\nabla }_{X}Y$. Its curvature isThe Leibniz rule makes curvature tensorial in all three slots. An affine connection is also called a linear connection on the manifold; a Levi-Civita connection is the distinguished torsion-free metric-compatible example.

Two torsion-free connections are projectively equivalent if they have the same unparametrized geodesics. Their difference of affine connections is a tensor $S$. A change of parameter adds only tangential acceleration, so equality of curves requires $S(v,v)$ to be parallel to $v$ for every tangent vector. The symmetric bilinear diagonal-parallel lemma gives exactly the displayed condition. Conversely the geodesic reparametrization under projective equivalence absorbs this tangential term. Agreement with the same affine parameters instead forces $S=0$.

Let $\tau$ be an affine parameter for a geodesic of $\nabla$ and suppose ${\bar{\nabla }}_{X}Y-{\nabla }_{X}Y=V(X)Y+V(Y)X$. Then ${\bar{\nabla }}_{\dot{\gamma }}\dot{\gamma }=2V(\dot{\gamma })\dot{\gamma }$. For the new velocity $d\gamma /dt=\dot{\gamma }/t^{\prime }$, the acceleration is $(t^{\prime }{)}^{-2}[{\bar{\nabla }}_{\dot{\gamma }}\dot{\gamma }-(t^{\prime \prime }/t^{\prime })\dot{\gamma }]$, which vanishes under the displayed differential equation. Its nonzero solution $t^{\prime }=C\exp (2\int V(\dot{\gamma })d\tau )$ changes the parameter while preserving the curve.

For two projectively equivalent connections that are Levi-Civita connections, contraction of their difference gives $(n+1)V_c={\bar{\Gamma }}^a{}_{ac}-{\Gamma }^a{}_{ac}$. The Jacobi formula gives ${\Gamma }^a{}_{ac}={\partial }_c\log \sqrt{|\det g|}$. Subtraction proves the displayed expression. The two determinant densities transform by the same coordinate Jacobian factor, so their ratio is a scalar and its derivative a covector. The formula is a necessary consequence of projective equivalence, not a test asserting that arbitrary metric tensors have identical geodesics.

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 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.

Identify $T_{p}M$ with coordinate vectors using a basis and invert the affine exponential map near zero. Radial geodesics then have coordinates $sX$. Their equations imply ${\Gamma }^{\lambda }{}_{\mu \nu }(p)X^{\mu }{X}^{\nu }=0$ for every $X$, hence the displayed vanishing of the symmetric part. The antisymmetric part may survive when the torsion tensor is nonzero. For a general connection the Levi-Civita connection coefficients and first metric derivatives need not vanish in these coordinates.

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.

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.

With derivative-last connection coefficients and $N_{abc}={\nabla }_c{g}_{ab}$, the displayed tensor is the symmetric-in-$a,b$ correction associated with the nonmetricity tensor. Raising its final slot gives the correction to connection coefficients. It vanishes for metric compatibility. Reversing the sign used to define nonmetricity reverses this formula.

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.

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 $K=-C$.

A one-form $k$ with ${\nabla }_b{k}_a=0$. If $\ell =d\varphi$ satisfies ${\nabla }_b{\ell }_a+{\ell }_a{\ell }_b=0$, then $k=e^{\varphi }\ell =d(e^{\varphi })$ is parallel.

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.

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.

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.

The difference of two torsion-free connections is symmetric, because subtracting their zero-torsion identities gives $\Delta (X,Y)=\Delta (Y,X)$. Agreement of parametrized geodesics makes this difference skew-symmetric as well, by parametrized geodesics determine the symmetric part of an affine connection. Over the real numbers it must therefore vanish. Without the torsion-free condition, an arbitrary skew-symmetric difference changes torsion without changing the parametrized geodesics.

For smooth spatial fields $E^i,C^i{}_j$, define a torsion-free connection on $\mathbb{R}\times {\mathbb{R}}^n$ by ${\Gamma }^i{}_{00}=-E^i$, ${\Gamma }^i{}_{0j}={\Gamma }^i{}_{j0}=-C^i{}_j/2$ and all other coefficients zero. The time geodesic equation is $\ddot{t}=0$. For $\dot{t}\ne 0$, use $t$ as affine parameter; the spatial geodesic equation becomes $d^2{x}^i/d{t}^2=E^i+C^i{}_{j}d{x}^j/dt$. In dimension three, $C^i{}_j=2{ϵ}^i{}_{kj}{B}^k$ realizes $\ddot{\mathbf{x}}=\mathbf{E}+2\mathbf{B}\times \dot{\mathbf{x}}$. Such a connection need not preserve a metric; a spacetime description of these trajectories only needs affine geometry.

Connections ${\nabla }^1,{\nabla }^2$ on $T{M}_1,T{M}_2$ give a connection onby taking the direct sum of their pullback connections. In product coordinates its Christoffel symbols have the two factor blocks and zero mixed blocks. On arbitrary fields, the ordinary derivatives of their coefficients are still taken in both factors. On fields lifted separately from the factors it satisfies ${\nabla }_{Y_1+Y_2}(X_1+X_2)={\nabla }_{Y_1}^1{X}_1+{\nabla }_{Y_2}^2{X}_2$.

For a product affine connection, its curvature satisfiesExpand the defining derivative commutator on lifted vector fields. Opposite-factor fields commute and have zero mixed covariant derivatives, leaving exactly the two factor curvatures. Tensoriality gives the formula for arbitrary tangent vectors. Thus the product connection is flat if the two factors are flat; for a product Riemannian metric this applies to its Levi-Civita connection.

A connection on a vector bundle pulls back along a smooth curve $\gamma$ to differentiate its time-dependent sections. For the tangent bundle, a section $V(t)$ of the pullback tangent bundle hasThe Leibniz rule under a change of frame proves independence of the coordinates. The derivative includes the time dependence of the coefficients, so it is meaningful even at a zero tangent velocity. A vector field along a map need not extend to one ambient field if the curve self-intersects. Parallel fields satisfy $D_{t}V=0$, a linear ordinary differential equation whose unique solutions define parallel transport.

In a local frame $e$, a connection on a vector bundle is written $\nabla e=Ae$, where $A$ is a matrix-valued one-form. Under a change of frame $e^{\prime }=eg$, it transforms as $A^{\prime }=g^{-1}Ag+g^{-1}dg$.

The covariant exterior derivative extends a connection to bundle-valued forms byLocally it is $d_A=d+A\wedge$.

Exterior product of differential forms combined with endomorphism composition is associative and generally noncommutative. For an endomorphism-valued one-form $a$, $(a\wedge a)(u,v)=[a(u),a(v)]$. The endomorphism bundle connection extends as a graded derivation of this algebra.

At $v\in E$, the vertical subspace is $V_{v}E=\ker (d\pi {)}_v$. A horizontal subspace is a complement $H_{v}E$ to $V_{v}E$. A linear connection is equivalently a smooth choice of such complements compatible with the vector-space structure in the fibres.

In a local trivialization, a vector-bundle connection with matrix $A$ defines the displayed complement of the vertical tangent space. The frame-change law makes these complements independent of the trivialization. Their dependence on the fiber coordinate is linear; an arbitrary horizontal complement need not satisfy this condition and need not define a connection on a vector bundle. Horizontal lifts satisfy the ordinary differential equation defining parallel transport.

In a frame of a vector bundle, write a vector bundle covariant derivative as $\nabla (ev)=e(dv+\Gamma v)$. The associated horizontal subspace at $(x,v)$ consists of tangent pairs $(\xi ,-{\Gamma }_x(\xi )v)$. Under a frame change $e^{\prime }=eg$, the connection one-form changes by ${\Gamma }^{\prime }=g^{-1}\Gamma g+g^{-1}dg$, and this horizontal equation changes to the corresponding one in the new frame. Thus the subspaces glue to a unique smooth linear connection on a vector bundle. Its vertical projection gives back the original derivative. The Leibniz rule ensures locality, allowing this construction from an operator initially defined on global sections.

If a connection on $E\to B$ has connection matrix $A=A_i(y)d{y}^i$ in a local frame and $f:M\to B$, its pullback connection has matrixIt is characterized by ${\nabla }_X^{f^{\ast }A}(f^{\ast }s)=f^{\ast }({\nabla }_{df(X)}^{A}s)$.

For a smooth map $\varphi :M^{\prime }\to M$ and a connection on a vector bundle $E\to M$, the curvature form of a connection on the pullback vector bundle satisfiesas an endomorphism of $E_{\varphi (p)}$. In a frame of a vector bundle written as a row, with section components written as columns, the connection one-form is $\Omega$ and its curvature form of a connection is $d\Omega +\Omega \wedge \Omega$. The pullback connection has matrix ${\varphi }^{\ast }\Omega$; commutation of the pullback of a differential form with the exterior derivative and wedge product of differential forms proves the identity. No pushforward to an ambient vector field is required: $d\varphi (X)$ is a vector field along a map.

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.

The dual connection is induced by a connection on a vector bundle through the evaluation pairing. A connection on $E$ induces one on $E^{\ast }$ byIts connection matrices are the negatives of the transposes of those for $E$.

Connections on $E$ and $F$ induce the connection on $E\otimes F$ determined byIn tensor-product local frames its connection matrix is $A_E\otimes I+I\otimes A_F$.

The curvature of a tensor product connection isFor line bundles, this is simply $F_{E\otimes F}=F_E+F_F$.

A connection with local matrix $A$ induces on an endomorphism-valued $r$-form $\mu$ the covariant exterior derivative

The endomorphism bundle connection is ${\tilde{D}}_X\alpha =D_X\circ \alpha -\alpha \circ D_X$ on sections. Expanding twice makes the two middle terms cancel, proving the displayed commutator identity. Subtracting the derivative along the Lie bracket of vector fields gives $\tilde{R}(X,Y)\alpha =[R(X,Y),\alpha ]$. The operators $D_X$ are differential operators, rather than fibrewise endomorphisms; the corrected curvature operator is fibrewise linear.

For the endomorphism bundle connection, $({\tilde{\nabla }}_X\varphi )(s)={\nabla }_X(\varphi s)-\varphi ({\nabla }_{X}s)$. Expanding two derivatives and subtracting ${\tilde{\nabla }}_{[X,Y]}$ givesThus its curvature form of a connection is the commutator action of the original curvature.

The endomorphism bundle connection of a rank-$r>0$ bundle is flat if and only if the original curvature satisfies $F_A=\omega \mathrm{id}$. By the curvature of an endomorphism bundle connection, flatness means each $F_A(X,Y)$ commutes with every fiber endomorphism. The centre of the full matrix algebra consists of scalar matrices, and $\omega =r^{-1}\mathrm{tr}{F}_A$ is the required smooth two-form. The original connection can still have nonzero scalar curvature.

The solder form on a smooth manifold is the identity $TX$-valued one-form $\theta (V)=V$. In a coordinate frame it is $\theta =d{x}^i\otimes {\partial }_i$.

The torsion form of a connection on $TX$ is the covariant exterior derivative of the solder form. It satisfies

If ${\nabla }_X{e}_i={\sum }_j{\omega }_{ij}(X)e_j$ and ${\varphi }_i$ is the dual coframe, the first index of $\omega$ is the input frame vector. Expand the connection derivative of $Y={\sum }_i{\varphi }_i(Y)e_i$ and antisymmetrize in $X,Y$. The formula for the exterior derivative of a one-form evaluated on vector fields gives ${\tau }_j=d{\varphi }_j-{\sum }_i{\varphi }_i\wedge {\omega }_{ij}$. This is the first structure equation with the indicated index convention. Reversing the wedge order changes the sign.

A connection on the tangent bundle is torsion-free when ${\nabla }_{X}Y-{\nabla }_{Y}X=[X,Y]$. Its Christoffel symbols are symmetric in their two lower indices.

A connection on a Riemannian vector bundle is metric-compatible whenEquivalently, its parallel transport maps are isometries.

A metric connection satisfies metric compatibility, ${\nabla }_{\alpha }{g}_{\mu \nu }=0$. This allows the metric tensor to commute with covariant derivatives and index raising or lowering.

For any torsion-free connection, expand the coordinate Lie derivative of a tensor field and cancel the symmetric connection terms to obtain the displayed identity. If the connection is metric-compatible, lowering the vector index commutes with the covariant derivative and it becomes ${\mathcal{L}}_{\xi }{g}_{ab}=2{\nabla }_{(a}{\xi }_{b)}$. Symmetry of the connection alone does not suffice: with $g=e^{2x}d{x}^2$, zero connection coefficients and $\xi ={\partial }_x$, the two expressions ${\mathcal{L}}_{\xi }g$ and $2{\nabla }_x{\xi }_x$ are $2e^{2x}$ and $4e^{2x}$, respectively.

The ambient Levi-Civita connection induces this metric connection on the normal bundle of an embedded submanifold. Its tangential counterpart appears in the Weingarten formula. Projecting the derivative preserves the connection on a vector bundle Leibniz rule and the fiber inner product.

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.

A smooth positive Hermitian bundle metric admits local orthonormal frames by the smooth Gram-Schmidt process. In such a frame, metric compatibility gives $A_{ji}(V)+\overline{A_{ij}(V)}=0$ for real vector fields $V$, so the connection matrix is skew-Hermitian. The orthonormal frame is smooth; a holomorphic orthonormal frame is not generally available.

For a metric connection on a vector bundle, the covariant derivative along a curve satisfies $(d/dt)⟨u,v⟩=⟨D_{t}u,v⟩+⟨u,D_{t}v⟩$. If both sections are parallel, the right side vanishes. Thus parallel transport preserves the fibre inner product, not merely vector lengths, and is a linear isometry between the endpoint fibres.

In a local orthonormal frame $(e_i)$, write $\nabla e_j=A^i{}_j{e}_i$. Metric compatibility and $d⟨e_i,e_j⟩=0$ giveThus the connection one-form of a real metric connection takes values in the skew-symmetric matrices.

The unique torsion-free metric connection satisfiesThis formula both proves uniqueness of the Levi-Civita connection and computes it from the metric and Lie bracket of vector fields.

The Koszul formula uniquely determines $g({\nabla }_{X}Y,Z)$ for all vector fields $X,Y,Z$, so nondegeneracy of the Riemannian metric determines ${\nabla }_{X}Y$. The resulting operation is $C^{\infty }$-linear in $X$, obeys the Leibniz rule in $Y$, is compatible with the metric, and is torsion-free. It is therefore the unique Levi-Civita connection.

A connection on a complex vector bundle with Hermitian metric $h$ is unitary whenIn a unitary local frame its connection matrix is skew-Hermitian, and so is its curvature matrix.

For a connection with compact structure group on a four-dimensional ball, sufficiently small $L^2$ curvature permits a gauge with $d^{\ast }a=0$, suitable boundary condition and $\Vert a{\Vert }_{W^{1,2}}\le C\Vert F{\Vert }_2$ after rescaling. For anti-self-dual connections, local elliptic regularity also gives scale-invariant curvature derivative estimates on smaller balls. These are local analytic inputs, not the global bubbling compactness conclusion.

// Destination: geometry-and-topology.bigb

The real curvature $i{F}_A$ of a unitary line connection has a fixed de Rham class. Let $\eta$ be its harmonic representative and write $\eta -i{F}_A=d\beta$. Then $A-i\beta$ has curvature $\eta$. Any two such connections differ by a closed imaginary one-form. If $b_1=0$, that form is exact and comes from a circle-valued gauge transformation. In general their gauge classes form a torsor for $H^1(X;\mathbb{R})/(2\pi H^1(X;\mathbb{Z}))$.

// Destination: geometry-and-topology.bigb

A determinant-preserving unitary bundle automorphism covering the identity on the base. For an $SU(2)$ bundle its values are in the fiberwise special unitary group. We use $A^u=u^{-1}Au+u^{-1}du$. Its curvature transforms by $F_{A^u}=u^{-1}{F}_{A}u$. The automorphism is a change of unitary frame, not a change of the base point.

// Destination: geometry-and-topology.bigb

The group of unitary bundle gauge transformations under composition. It acts on connections and their solution spaces. The stabilizer of a connection is the centralizer of its holonomy acting by parallel automorphisms, so a quotient can have singularities when this stabilizer increases.

// Destination: geometry-and-topology.bigb

Near a fixed connection $A$, impose $d_A^{\ast }a=0$ on perturbations. The linearized gauge-fixing equation is $d_A^{\ast }{d}_A\xi$, invertible on the orthogonal complement of its parallel kernel. In Sobolev completions above the multiplication threshold, the implicit function theorem gives a local slice; its remaining identifications are by the compact stabilizer of $A$. This reduces the gauge quotient to a finite-dimensional Kuranishi model when the curvature equation is elliptic.

// Destination: geometry-and-topology.bigb

The subgroup of unitary bundle gauge transformations equal to the identity at a chosen framed base point. Quotienting flat connections by this subgroup retains the full flat holonomy representation; quotienting by the whole group also divides by simultaneous conjugation.

// Destination: geometry-and-topology.bigb

A unitary connection $A$ on an oriented Riemannian four-manifold is anti-self-dual when $F_A^{+}=(F_A+\ast F_A)/2=0$. Its curvature is covariantly closed by the Bianchi identity and solves the Yang-Mills equations. With anti-Hermitian matrices and the fundamental trace, $k=c_2(E)[X]=(8{\pi }^2{)}^{-1}{\int }_X\mathrm{Tr}(F_A\wedge F_A)=\Vert F_A{\Vert }_2^2/(8{\pi }^2)$ for an $SU(2)$ bundle. Thus its charge is nonnegative.

// Destination: geometry-and-topology.bigb

Locally a U(1) connection whose gauge curvature has no $(2,0)$ or $(0,2)$ part can be written $A=\partial u+\bar{\partial }v$ by the Dolbeault-Poincaré lemma. Unitarity permits $v=-\bar{u}$. Writing $u=p+iq$, a gauge transformation removes $idq$ and leaves $A=\partial p-\bar{\partial }p$. Setting $f=-2p$ gives the displayed unitary form, whose gauge curvature is $\partial \bar{\partial }f$. Its remaining anti-self-duality of gauge curvature condition is the scalar Laplace equation. Alternatively a complex gauge transformation by $e^{-u}$ gives $A=\bar{\partial }(v-u)$; this second gauge generally does not preserve unitarity.

// Destination: complex-geometry.bigb

The infinitesimal gauge and curvature complex is $0\to {\Omega }^0(\mathrm{ad}E)\stackrel{d_A}{\to }{\Omega }^1(\mathrm{ad}E)\stackrel{d_A^{+}}{\to }{\Omega }^{2,+}(\mathrm{ad}E)\to 0$. Its composite on $\xi$ is $[F_A^{+},\xi ]=0$. Its cohomology describes infinitesimal stabilizers, deformations and obstructions. The gauge-fixed operator is $D_A=d_A^{\ast }\oplus d_A^{+}$.

// Destination: geometry-and-topology.bigb

On a simply connected negative-definite four-manifold, an ASD connection of full circle holonomy has $h_A^0=1$. If $h_A^2=0$, the index formula gives $h_A^1=8k-2$. The neutral diagonal deformation space vanishes because $b_1=0$, leaving a complex vector space of dimension $4k-1$ with stabilizer action $z\mapsto {\lambda }^{2}z$. Kuranishi reduction and the Coulomb slice give the local quotient ${\mathbb{C}}^{4k-1}/S^1$, equivalently the cone on ${\mathbb{CP}}^{4k-2}$.

// Destination: geometry-and-topology.bigb

For an $SU(2)$ bundle of charge $k$ on a closed oriented four-manifold, $\mathrm{ind}{D}_A=h_A^1-h_A^0-h_A^2=8k-3(1-b_1+b^{+})$. Equivalently it is $-2p_1(\mathrm{ad}E)[X]-3(\chi +\sigma )/2$, using $p_1(\mathrm{ad}E)=-4c_2(E)$. The Euler characteristic of the deformation complex has the opposite sign.

// Destination: analysis.bigb