Metric connection

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

On a closed oriented Riemannian four-manifold, a sequence of $SU(2)$ anti-self-dual connections of second Chern number one has, modulo gauge and subsequence, either smooth convergence with charge one or smooth convergence away from one point to a flat charge-zero connection. In the second case the curvature-energy measures converge to $8{\pi }^2{\delta }_x$. Small-energy gauges give convergence away from finitely many points; removable singularities extend the limit; Chern-Simons clutching integers quantize each positive defect. Total charge one leaves only the two stated cases. A flat limit need not have trivial holonomy on a nonsimply connected base.

// Destination: geometry-and-topology.bigb

A smooth anti-self-dual connection with compact structure group on a punctured four-ball and finite $L^2$ curvature extends smoothly over the puncture after changing gauge and extending the bundle. The gauge on the punctured ball can carry nontrivial clutching degree; the extended bundle need not agree globally with the original one in a bubbling sequence.

// Destination: geometry-and-topology.bigb

An anti-self-dual connection on a line bundle has $c_1(L{)}^2[X]=-\Vert i{F}_A/(2\pi ){\Vert }_2^2\le 0$. A line bundle with positive square therefore admits no such connection for any metric of the chosen orientation. The splitting $L\oplus L^{-1}$ has $c_2=-c_1(L{)}^2$, giving the equivalent nonnegative-instanton-charge obstruction.

// Destination: geometry-and-topology.bigb