Connection on a principal bundle as splitting of the Atiyah algebroid

Intuition Beginner

A principal bundle attaches a copy of a symmetry group to every point of a base space. From the last unit you have one object that records every invariant infinitesimal move of the whole structure: a move either pushes the base point somewhere, or only twists inside the fibre, or does some of both. That object is the Atiyah algebroid, and it comes with a built-in "shadow" map that reads off the base direction a move points along.

A connection answers one question. For each direction you might walk along the base, which single invariant move should count as "go that way and add no spurious twist"? Choosing one such move for every base direction, smoothly and compatibly, is exactly a connection. The shadow of the chosen move must be the base direction you asked for, so the choice is a one-sided inverse to the shadow map.

Here is the everyday picture. Walking on a hillside while carrying a level tray, you want to move forward without tilting the tray for no reason. "Forward with no needless tilt" is the connection's chosen move for the forward direction. Different people might disagree on what counts as no-tilt; each consistent rule is a different connection.

Why does curvature appear? Take two base directions, walk a tiny loop, and compare the move you predicted with the move you actually get by combining the two chosen lifts. If they match for every loop, the connection is flat. The mismatch, packaged as a pure-twist correction, is the curvature.

Visual Beginner

Picture the base as a curved surface with one marked point, and above it the fibre drawn as a small disc of the symmetry group. The Atiyah algebroid over the marked point is the space of invariant moves: each move has a part along the surface and a part spinning in the disc. The shadow map flattens a move down to its surface part.

A connection is a section of this picture: it lifts each base direction back up to a chosen invariant move whose shadow is that very direction, and whose twist part is whatever the rule says it should be. The chosen moves form a clean copy of the base directions sitting inside the algebroid, complementary to the pure twists. Curvature measures how that chosen copy fails to be closed under combining moves.

Worked example Beginner

Take the simplest bundle: the base is a flat disc, and over every point sits the same circle of rotation angles. A point upstairs is a base point together with a chosen angle, and the whole space is the product of the disc with the circle. This is a product bundle.

The invariant moves here split with no effort. A move can push the base point in some flat direction, and it can add a fixed turn rate to the angle. Because everything is a product, "fixed turn rate" already means the same thing over the whole fibre. So an invariant move is two numbers of base push together with one number of turn rate.

Now build the connection. The natural rule is: for each base direction, lift it to the move that pushes that way and adds turn rate zero. Walking any loop on the disc with this rule and combining the lifts, you return with turn rate zero again, because zero plus zero is zero. So this connection is flat.

What this tells us: the connection is the choice "lift with zero turn rate," and that choice is a clean copy of the base directions sitting inside the algebroid. Curvature here is zero because the chosen copy is closed under combination. Every other connection on a more twisted bundle is built to imitate this picture locally; the chosen copy need not stay closed globally, and that failure is curvature.

Check your understanding Beginner

Formal definition Intermediate+

Let $\pi :P\to M$ be a principal $G$-bundle with $G$ a Lie group acting on the right, smoothly, freely, and properly 03.05.07. Recall from 03.05.22 the Atiyah algebroid $\mathrm{At}(P)=TP/G$, the vector bundle over $M$ whose sections are the $G$-invariant vector fields $\mathfrak{X}(P{)}^G$, with anchor ${\pi }_{\ast }:\mathrm{At}(P)\to TM$ descending $d\pi$ and kernel the adjoint bundle $\mathrm{ad}(P)=(P\times \mathfrak{g})/G$, where $\mathfrak{g}=\mathrm{Lie}(G)$. These fit into the Atiyah sequence, a short exact sequence of vector bundles over $M$, $$ 0 \longrightarrow \mathrm{ad}(P) \xrightarrow{\ \iota\ } \mathrm{At}(P) \xrightarrow{\ \pi_\ast\ } TM \longrightarrow 0 . $$

Definition (connection as a splitting). A connection on $P$ is a right-splitting of the Atiyah sequence: a vector-bundle morphism $\sigma :TM\to \mathrm{At}(P)$ over ${\mathrm{id}}_M$ with $$ \pi_\ast \circ \sigma = \mathrm{id}_{TM} . $$ Its image $\sigma (TM)\subseteq \mathrm{At}(P)$ is a sub-bundle complementary to $\mathrm{ad}(P)$, the horizontal sub-algebroid.

Equivalently, a connection is a left-splitting (a retraction onto the kernel): a vector-bundle morphism $\omega :\mathrm{At}(P)\to \mathrm{ad}(P)$ with $$ \omega \circ \iota = \mathrm{id}{\mathrm{ad}(P)} . $$ The two presentations determine each other by $\omega = \mathrm{id}{\mathrm{At}(P)} - \iota^{-1}|{\ker\omega},\sigma,\pi\ast$—concretely,$\sigma$and$\omega$arepairedby$\ker\omega = \mathrm{im},\sigma$and$\ker\pi_\ast = \mathrm{im},\iota$,so$\mathrm{At}(P) = \mathrm{im},\sigma \oplus \mathrm{ad}(P)$and$\omega$istheprojectionontothesecondsummandalongthefirst.Themap$\omega$,transportedupto$P$, is the connection one-form.

The notation used below: $\mathfrak{X}(P{)}^G$ are the $G$-invariant vector fields on $P$; $\mathrm{Ad}$ is the adjoint representation of $G$ on $\mathfrak{g}$ and $\mathrm{ad}$ its differential; ${\zeta }_{\xi }(p)=\frac{d}{dt}{|}_{0}p\cdot \exp (t\xi )$ is the fundamental vector field of $\xi \in \mathfrak{g}$; ${\Omega }^1(P,\mathfrak{g}{)}^G$ denotes the $\mathrm{Ad}$-equivariant $\mathfrak{g}$-valued one-forms on $P$; and $d_{\nabla }$ is the exterior covariant derivative on $\mathrm{ad}(P)$-valued forms induced by a connection.

The transported connection one-form is the $\mathfrak{g}$-valued one-form ${\omega }_P\in {\Omega }^1(P,\mathfrak{g})$ defined by ${\omega }_P(v)=\xi$ when the vertical part of $v$ at $p$ is ${\zeta }_{\xi }(p)$; the splitting $\sigma$ encodes the horizontal subspaces $H_p=\ker ({\omega }_P{)}_p$. The defining conditions on ${\omega }_P$ are ${\omega }_P({\zeta }_{\xi })=\xi$ for all $\xi \in \mathfrak{g}$ (it reproduces fundamental fields) and $R_h^{\ast }{\omega }_P={\mathrm{Ad}}_{h^{-1}}{\omega }_P$ ($G$-equivariance); these two are exactly what make ${\omega }_P$ descend to a left-splitting $\omega$ of the Atiyah sequence.

A non-example fixes the boundary. A bundle morphism $\tau :TM\to \mathrm{At}(P)$ with ${\pi }_{\ast }\circ \tau =0$ is not a connection: it lands in $\mathrm{ad}(P)$ and lifts every base direction to a pure twist, recording no horizontal direction at all. The splitting condition ${\pi }_{\ast }\circ \sigma =\mathrm{id}$, not mere linearity, is the content.

Key theorem with proof Intermediate+

Theorem (three descriptions of a connection coincide). Let $\pi :P\to M$ be a principal $G$-bundle. The following data are in natural bijection ^{[Mackenzie Ch. V §1]}.

1. Right-splittings $\sigma :TM\to \mathrm{At}(P)$ of the Atiyah sequence, ${\pi }_{\ast }\circ \sigma ={\mathrm{id}}_{TM}$.
2. $G$-invariant horizontal distributions $H\subseteq TP$ complementary to the vertical bundle $V=\ker d\pi$.
3. Connection one-forms ${\omega }_P\in {\Omega }^1(P,\mathfrak{g}{)}^G$ with ${\omega }_P({\zeta }_{\xi })=\xi$ for all $\xi \in \mathfrak{g}$ and $R_h^{\ast }{\omega }_P={\mathrm{Ad}}_{h^{-1}}{\omega }_P$.

Proof. The proof runs $\text{(2)}\to \text{(1)}\to \text{(2)}$ and $\text{(2)}\leftrightarrow \text{(3)}$.

From (2) to (1). Let $H\subseteq TP$ be a $G$-invariant complement to $V$. For $v\in T_{x}M$ and $p\in P_x$, the differential $d\pi :H_p\to T_{x}M$ is a linear isomorphism, so there is a unique horizontal lift ${\tilde{v}}_p\in H_p$ with $d\pi ({\tilde{v}}_p)=v$. Invariance $H_{p\cdot h}=(d{R}_h)H_p$ gives ${\tilde{v}}_{p\cdot h}=(d{R}_h){\tilde{v}}_p$, so $p\mapsto {\tilde{v}}_p$ is a $G$-invariant vector field and descends to ${\sigma }_H(v)\in \mathrm{At}(P{)}_x=(TP/G{)}_x$. Then ${\pi }_{\ast }{\sigma }_H(v)=d\pi ({\tilde{v}}_p)=v$, and ${\sigma }_H$ is linear in $v$ because $d\pi {|}_{H_p}$ is a linear isomorphism. So ${\sigma }_H$ is a right-splitting.

From (1) to (2). Let $\sigma :TM\to \mathrm{At}(P)$ split ${\pi }_{\ast }$. Lift each $\sigma (v)\in \mathrm{At}(P{)}_x$ to its $G$-invariant representative vector field on $P_x$, and set $$ H_p := {,\sigma(v)p : v \in T{\pi(p)}M,} \subseteq T_pP . $$ Because the representatives are $G$-invariant, $H_{p\cdot h}=(d{R}_h)H_p$. The condition ${\pi }_{\ast }\circ \sigma =\mathrm{id}$ forces $d\pi {|}_{H_p}$ to be injective, hence $H_p\cap V_p=0$; a dimension count $\dim H_p=\dim M$, $\dim V_p=\dim G$, $\dim T_{p}P=\dim M+\dim G$ gives $H_p\oplus V_p=T_{p}P$. So $H$ is a $G$-invariant horizontal complement. The two assignments are mutually inverse: ${\sigma }_{H_{\sigma }}=\sigma$ because the horizontal lift of $v$ inside $H_{\sigma }$ is the representative of $\sigma (v)$, and $H_{{\sigma }_H}=H$ because the image of ${\sigma }_H$ recovers the horizontal subspaces.

From (2) to (3) and back. Given $H$, every $v\in T_{p}P$ decomposes uniquely as $v=v^{\mathrm{hor}}+{\zeta }_{\xi }(p)$ with $v^{\mathrm{hor}}\in H_p$ and a unique $\xi \in \mathfrak{g}$ (uniqueness because $\xi \mapsto {\zeta }_{\xi }(p)$ is a linear isomorphism $\mathfrak{g}\to V_p$, the action being free). Define ${\omega }_P(v):=\xi$. Then ${\omega }_P({\zeta }_{\eta })=\eta$ for all $\eta$, and $\ker {\omega }_P=H$. Equivariance of $H$ translates into $R_h^{\ast }{\omega }_P={\mathrm{Ad}}_{h^{-1}}{\omega }_P$: for $v\in T_{p}P$, applying $(d{R}_h)$ sends the vertical part ${\zeta }_{\xi }(p)$ to ${\zeta }_{{\mathrm{Ad}}_{h^{-1}}\xi }(p\cdot h)$ (the equivariance of fundamental fields), so ${\omega }_P((d{R}_h)v)={\mathrm{Ad}}_{h^{-1}}\xi ={\mathrm{Ad}}_{h^{-1}}{\omega }_P(v)$. Conversely a form ${\omega }_P$ with the two stated properties has $\ker {\omega }_P$ a $G$-invariant complement to $V$, recovering $H$. These constructions are mutually inverse. Composing the three correspondences gives the bijection. $\square$

Bridge. The theorem identifies the splitting $\sigma$ of 03.05.22 with the horizontal distribution of 03.05.07 and with the equivariant connection one-form, so the single algebraic notion "right-splitting of an exact sequence of vector bundles" is exactly the geometric notion of a principal connection; this builds toward the curvature theorem below, where the failure of $\sigma$ to respect the bracket on $\mathrm{At}(P)$ is computed against the structure equation $\Omega =d\omega +\frac{1}{2}[\omega ,\omega ]$, and it appears again in the flatness criterion, where $\sigma$ being a Lie-algebroid morphism, the horizontal distribution being integrable, and the curvature vanishing are shown to be one condition viewed three ways. The central insight is that the existence of a connection is the splittability of the Atiyah sequence, which always holds smoothly because a vector-bundle epimorphism over a manifold admits a section; this is exactly why connections always exist, while the harder existence question for flat connections is the obstruction to the splitting being compatible with the brackets. Putting these together, the Atiyah-sequence picture identifies the affine space of connections with the affine space of splittings, whose difference space is ${\Omega }^1(M;\mathrm{ad}P)$, the model for gauge potentials.

Exercises Intermediate+

Lean formalization Intermediate+

lean_status: none — Mathlib has no LieAlgebroid structure and no Atiyah-algebroid construction, so the central bijection of this unit (connections $\leftrightarrow$ splittings of the Atiyah sequence) is not expressible against current Mathlib. The intended statement is sketched in pseudo-Lean to indicate the missing structure; it does not compile, which is why no Lean module is declared.

-- Pseudo-Lean: target structure, not in Mathlib.
variable {G M P : Type*} [LieGroup G] [Manifold M] [Manifold P]
variable (π : P → M) [PrincipalBundle G π]

-- the Atiyah sequence 0 → ad P → At P → TM → 0 (from 03.05.22)
def AtiyahSeq : ShortExact (adBundle π) (AtiyahBundle π) (TangentBundle M) := ...

-- a connection IS a right-splitting of that sequence
structure Connection (π) where
  σ        : VectorBundleHom (TangentBundle M) (AtiyahBundle π)
  splits   : anchor π ∘ σ = id                       -- π_* ∘ σ = id

-- curvature = bracket defect, valued in ad P
def curvature (c : Connection π) (v w : VectorField M) : Section (adBundle π) :=
  c.σ ⁅v, w⁆ - ⁅c.σ v, c.σ w⁆                        -- lands in ker(anchor) = ad P

-- flat ↔ σ is an algebroid morphism ↔ horizontal distribution integrable
theorem flat_iff_morphism (c : Connection π) :
    (∀ v w, curvature c v w = 0) ↔ IsLieAlgebroidHom c.σ := ...

The first genuine obstacle is upstream of this unit: the quotient vector bundle $TP/G$, the Lie-algebroid structure, and the notion of a splitting of a short exact sequence of vector bundles are all absent, so even the type Connection cannot be stated until the Atiyah algebroid of 03.05.22 is formalised.

Advanced results Master

The bracket defect equals the structure-equation curvature. Write ${\omega }_P\in {\Omega }^1(P,\mathfrak{g}{)}^G$ for the connection one-form of the splitting $\sigma$, and define the $\mathfrak{g}$-valued $2$-form $\Omega :=d{\omega }_P+\frac{1}{2}[{\omega }_P,{\omega }_P]$ on $P$. The horizontal projection makes $\Omega$ tensorial and $\mathrm{Ad}$-equivariant, hence descends to $\bar{\Omega }\in {\Omega }^2(M;\mathrm{ad}P)$. The algebroid curvature $F(v,w)=\sigma [v,w]-[\sigma v,\sigma w]$ and the descended structure-equation curvature $\bar{\Omega }$ coincide: $F=\bar{\Omega }$ ^{[Kobayashi-Nomizu Ch. II §5]}. The computation is the Maurer-Cartan identity read on horizontal lifts. For $v,w\in \mathfrak{X}(M)$ with horizontal lifts $\sigma v,\sigma w$, the field $[\sigma v,\sigma w]$ has horizontal part the lift of $[v,w]$, namely $\sigma [v,w]$, and vertical part the fundamental field ${\zeta }_{-{\omega }_P([\sigma v,\sigma w])}$. The intrinsic formula $d{\omega }_P(X,Y)=X({\omega }_{P}Y)-Y({\omega }_{P}X)-{\omega }_P[X,Y]$ on horizontal $X=\sigma v$, $Y=\sigma w$ collapses, since ${\omega }_P$ kills horizontal fields, to $d{\omega }_P(\sigma v,\sigma w)=-{\omega }_P[\sigma v,\sigma w]$. As $[{\omega }_P,{\omega }_P]$ vanishes on horizontal arguments, $\Omega (\sigma v,\sigma w)=-{\omega }_P[\sigma v,\sigma w]$, and $\iota$-transporting ${\omega }_P([\sigma v,\sigma w])$ into $\mathrm{ad}(P)$ returns exactly $\sigma [v,w]-[\sigma v,\sigma w]$. So the two curvatures agree.

Bianchi identity as algebroid $d_{\nabla }$-closedness. A connection $\sigma$ induces a covariant exterior derivative $d_{\nabla }$ on $\mathrm{ad}(P)$-valued forms, the $\mathrm{ad}(P)$-coefficient version of the de Rham differential twisted by the connection on $\mathrm{ad}(P)=(P\times \mathfrak{g})/G$ that $\sigma$ supplies. The second Bianchi identity is the single equation $$ d_\nabla F = 0 . $$ In structure-equation form this is $d\Omega +[{\omega }_P,\Omega ]=0$, obtained by applying $d$ to $\Omega =d{\omega }_P+\frac{1}{2}[{\omega }_P,{\omega }_P]$ and using $d^2=0$ with the graded Jacobi identity for $[\cdot ,\cdot ]$. In algebroid form it is the Jacobi identity for the bracket on $\mathrm{At}(P)$ evaluated on $\sigma v,\sigma w,\sigma u$: the cyclic sum of $[\sigma v,[\sigma w,\sigma u]]$ vanishes, and projecting the horizontal-lift discrepancies onto $\mathrm{ad}(P)$ packages this cyclic sum as $d_{\nabla }F=0$.

Existence and the obstruction to flatness. The smooth Atiyah sequence always splits, because a surjective bundle morphism ${\pi }_{\ast }:\mathrm{At}(P)\to TM$ over a manifold admits a section: take local sections over a cover and average them against a subordinate partition of unity, the average of right-splittings being a right-splitting. So connections always exist, and the space of them is the affine space modelled on ${\Omega }^1(M;\mathrm{ad}P)$. A flat connection is a splitting that is a Lie-algebroid morphism, equivalently one whose horizontal distribution is Frobenius-integrable, equivalently $F=0$. Such a splitting need not exist: its existence forces $P$ to be associated to a representation of ${\pi }_1(M)$, since the integrable horizontal distribution foliates $P$ and the holonomy of the foliation is a flat-bundle monodromy. The obstruction is cohomological — a flat connection on a $G$-bundle over $M$ corresponds to a conjugacy class of representations ${\pi }_1(M)\to G$ — and this is the gateway to the character-variety description of moduli of flat connections.

A curved splitting: the Hopf bundle $S^3\to S^2$. The Hopf fibration is a principal $U(1)$-bundle $\pi :S^3\to S^2$ with $\dim M=2$, $\dim G=1$, so its Atiyah sequence reads $0\to \mathrm{ad}(S^3)\to \mathrm{At}(S^3)\to T{S}^2\to 0$ with $\mathrm{ad}(S^3)=S^3{\times }_{U(1)}\mathfrak{u}(1)$ a real line bundle, the product line bundle since $U(1)$ is abelian, so its adjoint action on $\mathfrak{u}(1)=i\mathbb{R}$ is the identity. Realise $S^3\subset {\mathbb{C}}^2$ as unit vectors $(z_1,z_2)$ with action $(z_1,z_2)\cdot e^{i\theta }=(z_1{e}^{i\theta },z_2{e}^{i\theta })$. The standard connection one-form is ${\omega }_P=\frac{1}{2}\mathrm{Im}({\bar{z}}_{1}d{z}_1+{\bar{z}}_{2}d{z}_2)\cdot i\in {\Omega }^1(S^3,\mathfrak{u}(1))$, the round-metric normal to the fibres; it satisfies ${\omega }_P({\zeta }_{\xi })=\xi$ and $U(1)$-invariance, so it is a left-splitting and determines a connection $\sigma$. Its curvature is $\bar{\Omega }=-i{\pi }^{\ast }{\omega }_{\mathrm{FS}}$, a nonzero multiple of the Fubini-Study area form on $S^2$, so this splitting is not flat. The bracket defect is forced to be nonzero: were $\sigma$ a morphism, $S^3$ would carry an integrable horizontal $2$-plane field foliating it by flat $U(1)$-bundle leaves, but ${\int }_{S^2}\bar{\Omega }=-2\pi$ (the Euler number of the Hopf bundle is $\pm 1$) is a topological obstruction to any flat $U(1)$-connection. So the Hopf bundle is the smallest example where the splitting exists but no flat splitting can.

The Atiyah-Bott affine-space picture. Atiyah and Bott organised gauge theory over a Riemann surface around exactly this affine structure: the space $\mathcal{A}$ of connections on a fixed bundle is an affine space modelled on ${\Omega }^1(\Sigma ;\mathrm{ad}P)$, the gauge group acts on it, and the curvature $F:\mathcal{A}\to {\Omega }^2(\Sigma ;\mathrm{ad}P)$ is an equivariant moment map for the symplectic structure on $\mathcal{A}$ given by integrating the wedge-and-trace pairing ^{[Atiyah-Bott 1982]}. The Yang-Mills functional $\Vert F{\Vert }^2$ becomes an equivariant Morse function whose critical points are the Yang-Mills connections; flat connections are the absolute minima. That the moment map is the curvature is the symplectic shadow of the present unit: curvature is the bracket-defect of a splitting, and the failure of a splitting to be flat is what the moment map measures.

Atiyah's holomorphic obstruction. For a holomorphic principal bundle over a complex manifold the Atiyah sequence is a sequence of holomorphic vector bundles, and a holomorphic connection is a holomorphic right-splitting ^{[Atiyah 1957]}. The smooth sequence always splits; the holomorphic one need not, and the obstruction is the extension class $a(P)\in H^1(M;\mathrm{ad}(P)\otimes {\Omega }_M^1)$, the Atiyah class. For a line bundle $a(P)$ is the first Chern class in Hodge guise, and Atiyah's theorem is that a compact Kähler $M$ admits a holomorphic connection iff $a(P)=0$. The same exact sequence governs both the always-soluble smooth splitting problem and the obstructed holomorphic one; the splitting language is what makes the two parallel.

Synthesis. A principal connection is one datum wearing three faces, and the central insight of this unit is that all three are the single algebraic act of right-splitting the Atiyah sequence $0\to \mathrm{ad}(P)\to \mathrm{At}(P)\to TM\to 0$: the horizontal distribution of 03.05.07 is the image of the splitting $\sigma$, the connection one-form ${\omega }_P\in {\Omega }^1(P,\mathfrak{g}{)}^G$ is the complementary left-splitting, and the splitting datum lives in the Atiyah algebroid of 03.05.22. Curvature is the failure of $\sigma$ to respect the bracket, $F=\sigma [\cdot ,\cdot ]-[\sigma \cdot ,\sigma \cdot ]\in {\Omega }^2(M;\mathrm{ad}P)$, and this is exactly the structure-equation curvature $\Omega =d{\omega }_P+\frac{1}{2}[{\omega }_P,{\omega }_P]$ once descended, so the algebroid and Cartan formalisms compute one tensor. The second Bianchi identity is the algebroid $d_{\nabla }F=0$, the Jacobi identity of $\mathrm{At}(P)$ in disguise; flatness is the condition that $\sigma$ be a Lie-algebroid morphism, which is dual to the horizontal distribution being Frobenius-integrable, the central insight being that integrability of the splitting and vanishing of the bracket defect are the same statement. Putting these together, the space of connections is the affine torsor under ${\Omega }^1(M;\mathrm{ad}P)$, and curvature is the moment map of Atiyah-Bott; this identifies the gauge-theoretic affine space with the splittings of a single exact sequence, generalises the Maurer-Cartan structure equation to arbitrary principal bundles, and in the holomorphic category identifies the obstruction to a holomorphic connection with the extension class of the very sequence whose smooth splittings are the connections.

Full proof set Master

Proposition (right-splittings and left-splittings of the Atiyah sequence correspond bijectively). Let $0\to \mathrm{ad}(P)\stackrel{\iota }{\to }\mathrm{At}(P)\stackrel{{\pi }_{\ast }}{\to }TM\to 0$ be the Atiyah sequence. The assignment $\sigma \mapsto {\omega }_{\sigma }$, where ${\omega }_{\sigma }$ is the projection onto $\mathrm{ad}(P)$ along $\mathrm{im}\sigma$, is a bijection from right-splittings $\sigma$ (with ${\pi }_{\ast }\sigma =\mathrm{id}$) to left-splittings $\omega$ (with $\omega \iota =\mathrm{id}$).

Proof. Given a right-splitting $\sigma$, the condition ${\pi }_{\ast }\sigma =\mathrm{id}$ forces $\mathrm{im}\sigma \cap \ker {\pi }_{\ast }=0$ and, by a rank count $\mathrm{rk}\mathrm{At}(P)=\mathrm{rk}TM+\mathrm{rk}\mathrm{ad}(P)$, $\mathrm{At}(P)=\mathrm{im}\sigma \oplus \iota (\mathrm{ad}(P))$. Define ${\omega }_{\sigma }:\mathrm{At}(P)\to \mathrm{ad}(P)$ as ${\iota }^{-1}$ composed with the projection onto the second summand. Then ${\omega }_{\sigma }\iota =\mathrm{id}$, so ${\omega }_{\sigma }$ is a left-splitting. Conversely a left-splitting $\omega$ gives a complement $\ker \omega$ to $\iota (\mathrm{ad}(P))$, and ${\pi }_{\ast }$ restricts to an isomorphism $\ker \omega \stackrel{\sim }{\to }TM$ (injective since $\ker \omega \cap \ker {\pi }_{\ast }=\ker \omega \cap \iota (\mathrm{ad}(P))=0$, surjective by rank); its inverse is a right-splitting ${\sigma }_{\omega }$. The maps $\sigma \mapsto {\omega }_{\sigma }$ and $\omega \mapsto {\sigma }_{\omega }$ are mutually inverse because each recovers the same direct-sum decomposition $\mathrm{At}(P)=\mathrm{im}\sigma \oplus \iota (\mathrm{ad}(P))$ with $\ker \omega =\mathrm{im}\sigma$. $\square$

Proposition (the bracket defect is tensorial and $\mathrm{ad}(P)$-valued). For a connection $\sigma$, the map $F(v,w)=\sigma [v,w]-[\sigma v,\sigma w]$ is a well-defined $C^{\infty }(M)$-bilinear antisymmetric section of $\mathrm{Hom}({\Lambda }^{2}TM,\mathrm{ad}(P))$, i.e. $F\in {\Omega }^2(M;\mathrm{ad}P)$.

Proof. First $F(v,w)\in \Gamma (\mathrm{ad}P)$: apply the anchor, ${\pi }_{\ast }F(v,w)={\pi }_{\ast }\sigma [v,w]-{\pi }_{\ast }[\sigma v,\sigma w]=[v,w]-[{\pi }_{\ast }\sigma v,{\pi }_{\ast }\sigma w]=[v,w]-[v,w]=0$, using that ${\pi }_{\ast }$ is a bracket homomorphism 03.04.16 and ${\pi }_{\ast }\sigma =\mathrm{id}$; so $F(v,w)\in \ker {\pi }_{\ast }=\Gamma (\mathrm{ad}P)$. Antisymmetry is inherited from both brackets. Tensoriality: with $f\in C^{\infty }(M)$, $\sigma [v,fw]=f\sigma [v,w]+(vf)\sigma w$ and $[\sigma v,f\sigma w]=f[\sigma v,\sigma w]+(({\pi }_{\ast }\sigma v)f)\sigma w=f[\sigma v,\sigma w]+(vf)\sigma w$, by the algebroid Leibniz rule and ${\pi }_{\ast }\sigma v=v$. Subtraction cancels the $(vf)\sigma w$ terms, leaving $F(v,fw)=fF(v,w)$; antisymmetry gives the first slot. So $F$ is $C^{\infty }(M)$-bilinear and antisymmetric, hence a bundle map ${\Lambda }^{2}TM\to \mathrm{ad}(P)$. $\square$

Proposition (flatness $\iff$ morphism $\iff$ integrable horizontal distribution). For a connection $\sigma$ with horizontal distribution $H=\mathrm{im}\sigma$ (pulled to $P$): $F\equiv 0$ iff $\sigma$ is a Lie-algebroid morphism iff $H$ is involutive.

Proof. $F\equiv 0$ unwinds to $\sigma [v,w]=[\sigma v,\sigma w]$ for all $v,w\in \mathfrak{X}(M)$. Since $\sigma$ already intertwines anchors (${\pi }_{\ast }\sigma =\mathrm{id}=$ identity anchor on $TM$ followed by ${\pi }_{\ast }$), preservation of brackets is precisely the condition that $\sigma$ be a morphism of Lie algebroids; so $F\equiv 0\iff \sigma$ a morphism. For involutivity: $H$ is spanned, near each point, by horizontal lifts $\sigma v$ of base fields. For $X=\sigma v,Y=\sigma w$, the bracket $[X,Y]$ is horizontal iff its vertical part vanishes, and its vertical part is $\iota$-image of $-F(v,w)$ because ${\pi }_{\ast }[X,Y]=[v,w]$ while the horizontal lift of $[v,w]$ is $\sigma [v,w]$, so $[X,Y]-\sigma [v,w]=-F(v,w)$ is exactly the vertical defect. Hence $H$ is closed under bracket iff $F\equiv 0$. By the Frobenius theorem, involutivity of the constant-rank distribution $H$ is integrability. So all three are equivalent. $\square$

Proposition (curvature change under a gauge of the splitting). If ${\sigma }^{\prime }=\sigma +\iota \circ a$ with $a\in {\Omega }^1(M;\mathrm{ad}P)$, then $F^{\prime }=F+d_{\nabla }a+\frac{1}{2}[a,a]$, where $d_{\nabla }$ is the covariant derivative of the connection $\sigma$ on $\mathrm{ad}(P)$ and $[a,a]$ uses the bracket on $\mathrm{ad}(P)$.

Proof. Compute $F^{\prime }(v,w)={\sigma }^{\prime }[v,w]-[{\sigma }^{\prime }v,{\sigma }^{\prime }w]$ with ${\sigma }^{\prime }=\sigma +\iota a$. Expanding, $$ \sigma'[v,w] = \sigma[v,w] + \iota a([v,w]), $$ $$ [\sigma'v,\sigma'w] = [\sigma v,\sigma w] + [\sigma v,\iota a(w)] + [\iota a(v),\sigma w] + [\iota a(v),\iota a(w)]. $$ The cross terms $[\sigma v,\iota a(w)]-[\sigma w,\iota a(v)]$, plus the $a([v,w])$ term, assemble into $-d_{\nabla }a(v,w)$ by the intrinsic formula for the covariant exterior derivative, where $d_{\nabla }$ uses the connection-induced bracket action of $\sigma v$ on sections of $\mathrm{ad}(P)$. The last term $[\iota a(v),\iota a(w)]$ is the bracket of two $\mathrm{ad}(P)$-valued sections, contributing $-\frac{1}{2}[a,a]$ after antisymmetrisation. Collecting signs against $F(v,w)=\sigma [v,w]-[\sigma v,\sigma w]$ gives $F^{\prime }=F+d_{\nabla }a+\frac{1}{2}[a,a]$, the gauge-transformation law for curvature, here read as the change of the bracket defect under a shift of the splitting. $\square$

Connections Master

The Atiyah algebroid 03.05.22 is the ambient object this unit splits: its sections are the $G$-invariant vector fields, its anchor ${\pi }_{\ast }$ is the descent of $d\pi$, and its kernel $\mathrm{ad}(P)$ is the home of curvature. A connection is a right-splitting of the Atiyah sequence built there, so the present unit is the connection-theoretic completion of the algebroid construction, and the bijection it proves is the reason that unit's exact sequence carries the whole curvature theory.

The principal connection 03.05.07 in horizontal-distribution and connection-form language is matched here term by term with the splitting: the horizontal distribution is $\mathrm{im}\sigma$, the connection one-form is the complementary projection $\omega$, and the structure-equation curvature $\Omega =d\omega +\frac{1}{2}[\omega ,\omega ]$ is the descended bracket defect. The two units describe the same object; this one supplies the algebroid translation and the existence-by-splittability argument.

The Lie algebroid 03.04.16 supplies the bracket and Leibniz machinery that make the curvature tensorial and that phrase flatness as the morphism condition. The general fact that the anchor of a Lie algebroid is a bracket homomorphism is exactly what forces ${\pi }_{\ast }F=0$, landing curvature in $\mathrm{ad}(P)$; flatness is $\sigma$ being a morphism in the category of Lie algebroids.

Historical & philosophical context Master

The reading of a connection as a splitting of a short exact sequence originates with Michael Atiyah's 1957 study of complex analytic connections, where, for a holomorphic principal bundle over a complex manifold, a holomorphic connection is a holomorphic splitting of the sequence now bearing his name, and the obstruction to splitting is an extension class in a first cohomology group, the Atiyah class ^{[Atiyah 1957]}. Atiyah's concern was the existence of holomorphic connections; the splitting formulation made existence a cohomological vanishing condition. The smooth differential-geometric theory of connections as horizontal distributions and equivariant one-forms, with curvature read from the structure equation, was codified in Kobayashi and Nomizu's 1963 foundations, the standard reference for the connection-form and horizontal-lift pictures matched here against the splitting ^{[Kobayashi-Nomizu Ch. II §5]}.

The algebroid synthesis — the Atiyah sequence as the anchor sequence of the Lie algebroid $TP/G$, connections as splittings, curvature as the bracket defect, and flatness as the morphism condition — is due to Kirill Mackenzie, whose lecture notes and monograph made the splitting of the Atiyah sequence the organising device for connection theory in the groupoid-algebroid framework ^{[Mackenzie Ch. V §1]}. The affine-space and moment-map reading of the same data, with the curvature as the moment map for the gauge-group action on the space of connections, is Atiyah and Bott's 1982 work on the Yang-Mills equations over Riemann surfaces, which turned this connection theory into an infinite-dimensional symplectic geometry ^{[Atiyah-Bott 1982]}. The splitting picture thus links Atiyah's holomorphic obstruction theory, the classical Cartan-Ehresmann connection calculus, and the symplectic gauge theory of the 1980s through one exact sequence.

Bibliography Master

@article{atiyah1957,
  author    = {Atiyah, Michael F.},
  title     = {Complex analytic connections in fibre bundles},
  journal   = {Transactions of the American Mathematical Society},
  volume    = {85},
  pages     = {181--207},
  year      = {1957}
}

@article{atiyahbott1982,
  author    = {Atiyah, Michael F. and Bott, Raoul},
  title     = {The Yang-Mills equations over Riemann surfaces},
  journal   = {Philosophical Transactions of the Royal Society of London A},
  volume    = {308},
  number    = {1505},
  pages     = {523--615},
  year      = {1982}
}

@book{mackenzie2005,
  author    = {Mackenzie, Kirill C. H.},
  title     = {General Theory of Lie Groupoids and Lie Algebroids},
  series    = {London Mathematical Society Lecture Note Series},
  volume    = {213},
  publisher = {Cambridge University Press},
  year      = {2005}
}

@book{mackenzie1987,
  author    = {Mackenzie, Kirill},
  title     = {Lie Groupoids and Lie Algebroids in Differential Geometry},
  series    = {London Mathematical Society Lecture Note Series},
  volume    = {124},
  publisher = {Cambridge University Press},
  year      = {1987}
}

@book{kobayashinomizu1963,
  author    = {Kobayashi, Shoshichi and Nomizu, Katsumi},
  title     = {Foundations of Differential Geometry, Volume I},
  publisher = {Interscience Publishers, John Wiley \& Sons},
  year      = {1963}
}

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Connection as splitting of the Atiyah algebroid — a principal connection on $P\to M$ is a right-splitting $\sigma :TM\to \mathrm{At}(P)$ of the Atiyah sequence $0\to \mathrm{ad}(P)\to \mathrm{At}(P)\to TM\to 0$, equivalently a connection one-form $\omega \in {\Omega }^1(P,\mathfrak{g}{)}^G$; curvature is the bracket defect $F=\sigma [\cdot ,\cdot ]-[\sigma \cdot ,\sigma \cdot ]\in {\Omega }^2(M;\mathrm{ad}P)$, agreeing with $\Omega =d\omega +\frac{1}{2}[\omega ,\omega ]$; flatness is $\sigma$ a Lie-algebroid morphism, equivalently Frobenius integrability of the horizontal distribution.