Principal bundle

A principal $G$-bundle is a fiber bundle $\pi :P\to B$ with a free right action of a Lie group $G$ that is locally equivariantly isomorphic to $U\times G\to U$. Each fiber is a single $G$-orbit.

Write a point of Real projective space as a nonzero vector $(w_1,w_2)\in {\mathbb{C}}^2$ modulo real nonzero scaling. The map to Complex projective space sends its class to the complex line through that vector. A free right action of the circle group is $[w]\cdot u=[\lambda w]$ where ${\lambda }^2=u$; the two possible roots differ by a real sign and therefore give the same projective point. Local smooth square-root branches prove smoothness. Over $[z:1]$, the fibre coordinate is $(w_2/|w_2|{)}^2$; over $[1:\zeta ]$, it is $(w_1/|w_1|{)}^2$. On the overlap $\zeta =1/z$, the transition function of a principal bundle is $u\mapsto (z/\overline{z})u$. Its phase winds twice. Ordinary scalar multiplication by $u$ instead has a kernel of order two and is not a free action of a group.

For equivariant local trivializations of a right principal bundle, the change of coordinates is left multiplication in the group coordinate. Indeed an equivariant change $H$ satisfies $H(x,gh)=H(x,g)h$, so evaluating at the identity gives $H(x,g)=(x,\psi (x)g)$. Its inverse shows that $\psi$ takes values in the Lie group, and restriction to $(x,e)$ makes it smooth. Composition on triple overlaps gives ${\psi }_{ki}={\psi }_{kj}{\psi }_{ji}$.

The frame bundle of an $n$-dimensional smooth manifold has fiber over $x$ consisting of linear isomorphisms $u:{\mathbb{R}}^n\to T_{x}M$. Right composition by $GL(n,\mathbb{R})$ makes it a principal bundle. A global frame of a vector bundle is a global section of a fiber bundle and trivializes the tangent bundle. A metric restricts the allowed frames to the orthonormal frame bundle.

For a metric of signature $(r,s)$, the orthonormal frame bundle consists of all linear isometries $u:{\mathbb{R}}^{r,s}\to T_{x}M$. Right composition by $O(r,s)$ makes it a principal bundle. Its standard associated vector bundle is the tangent bundle, via $[u,v]\mapsto u(v)$. A global orthonormal frame is a global section and therefore trivializes this principal bundle. A Lie group with a left-invariant metric has such a frame by left translating an orthonormal basis at the identity.

Represent the Anti-de Sitter spacetime quadric by $x\cdot x=-{\ell }^2$ in ${\mathbb{R}}^{3,2}$. Appending its timelike unit normal $x/\ell$ to an oriented tangent pseudo-orthonormal frame identifies the frame bundle with $SO(3,2)$. The stabilizer of that normal is $SO(3,1)$, so projection realizes $SO(3,2)/SO(3,1)$ as the base of a principal bundle. For the commonly used universal cover of the anti-de Sitter quadric, pull this bundle back along the covering map; the quadric itself has a periodic timelike coordinate.

An oriented pseudo-orthonormal frame in Minkowski spacetime consists of its base point $a$ and a Lorentz matrix $L$ acting on a fixed reference frame. The pairs $(a,L)$ form the Poincaré group, and right multiplication by $(0,h)$ sends the frame to $(a,Lh)$ while fixing its base point. Thus the displayed projection is a principal bundle with group $SO(3,1)$. The full frame bundle uses $O(3,1)$; a specified time orientation restricts to the corresponding connected component.

Represent unit-radius de Sitter spacetime by the spacelike unit hyperboloid in five-dimensional Minkowski spacetime. Appending its position vector as the last column of an oriented pseudo-orthonormal frame produces a matrix in $SO(4,1)$. Projection to that last column identifies $SO(4,1)$ with the oriented orthonormal frame bundle, a principal bundle with group $SO(3,1)$. The full bundle allowing both orientations is $O(4,1)$ with group $O(3,1)$. Requiring a time orientation as well selects the corresponding identity components.

In global coordinates the de Sitter spacetime metric is $-d{\tau }^2+{\cosh }^2\tau g_{S^3}$. A global frame of a vector bundle orthonormal for the round metric $Y_1,Y_2,Y_3$ on $S^3$ gives the displayed global pseudo-orthonormal frame. For example, identify $S^3$ with the unit quaternions and use $qi,qj,qk$. Choosing the spatial orientation and the future direction consistently gives a section of a fiber bundle for the oriented, time-oriented frame bundle. The principal bundle trivialization by a global section then proves that bundle is trivial.

For a principal bundle $P\to B$ and a left action of its structure group on a manifold $F$, the associated bundle is $(P\times F)/\sim$, where $(pg,f)\sim (p,gf)$. Local sections identify it with $U\times F$; principal transition functions act on $F$. Linear actions give associated vector bundles.

A principal bundle is trivial exactly when it has a global section of a fiber bundle. Given such a section, the displayed map is a smooth equivariant isomorphism: the free transitive action on each fiber gives the unique inverse group coordinate, and local trivializations make it smooth. Conversely, a product trivialization has the section $x\mapsto (x,e)$.

A linear representation $\rho :G\to GL(V)$ of a principal bundle's structure group defines the quotient of $P\times V$ by $(p,v)\sim (pg,\rho (g{)}^{-1}v)$. It is a vector bundle over the same base. A principal connection induces parallel transport and a local covariant derivative $d+{\rho }_{\ast }(A)$ on its sections.

The adjoint bundle is the associated vector bundle for the Adjoint representation of a Lie group on its Lie algebra. The gauge curvature of a principal connection is globally a two-form with values in this bundle. Differences of principal connections and their infinitesimal variations are one-forms with values in the same bundle.

For a topological group $G$, a universal principal bundle $EG\to BG$ has contractible total space and classifies numerable principal $G$-bundles by pullback. Over a paracompact base, isomorphism classes correspond to homotopy classes of maps to $BG$. For a discrete group, its classifying space of a discrete group is a $K(G,1)$; for the circle, $B{S}^1\simeq {\mathbb{CP}}^{\infty }$.

The stabilizer of a line in $O(n)$ is $O(1)\times O(n-1)$, giving the homogeneous space ${\mathbb{RP}}^{n-1}$. Its classifying-space Serre fibration has rationally acyclic fiber for odd $n$, so its cohomological Serre spectral sequence has only its degree-zero row and gives the displayed restriction isomorphism. The factor $BO(1)={\mathbb{RP}}^{\infty }$ is itself rationally acyclic. For even $n$, the fiber has a top rational class, with determinant-sign monodromy, so the one-row argument fails. The top universal Pontryagin class $p_{n/2}$ is nonzero on $BO(n)$ but vanishes on restriction to $BO(n-1)$, witnessing genuine failure of stabilization.

Give a discrete group $G$ a contractible free CW complex $EG$. The quotient $BG=EG/G$ has universal cover $EG$, so it has fundamental group $G$ and no higher homotopy groups. It is thus an Eilenberg–MacLane space $K(G,1)$, even when $G$ is nonabelian. The cohomology of $BG$ with the appropriate local coefficients computes group cohomology.

Let $H(n)$ be the real vector space of Hermitian matrices. The polar decomposition of an invertible complex matrix gives a diffeomorphismIts inverse sends $g$ toRight multiplication by $U(n)$ changes only $u$, so the orbit space is smoothly $H(n)$ and the quotient map is a globally trivial principal bundle.

If a Lie group acts smoothly, freely, and properly on a smooth manifold $P$, then the quotient $P/G$ has a unique smooth structure for which $P\to P/G$ is a principal bundle. An action by a compact Lie group is automatically proper.

A principal connection can be specified by a $G$-equivariant horizontal distribution $H\subset TP$ complementary to the vertical bundle, or equivalently by a Lie-algebra-valued connection form $\mathcal{A}$ that reproduces infinitesimal generators and has $H=\ker \mathcal{A}$.

For a smooth strictly convex rigid body with a smooth unique contact point, a configuration consists of its orientation $R\in SO(3)$ and horizontal position $c_{\parallel }\in {\mathbb{R}}^2$; its height is fixed by contact. Translation makes this a principal bundle over $SO(3)$ with group ${\mathbb{R}}^2$. If $r(R)$ points from its reference point to contact and $\Omega$ is its spatial angular velocity, rolling without slipping requires $\dot{c}+\Omega \times r=0$. The displayed translation-valued principal connection vanishes on precisely these velocities. It is translation-invariant and equals the identity on vertical translation vectors. A prescribed orientation curve consequently has a unique horizontal lift from a given initial position. No-slip alone leaves spin free and does not specify a connection over the plane with orientation as its fiber; an additional no-twist constraint is needed for that familiar spherical model.

A connection on a principal circle bundle has local real one-form representatives $A$ transforming by $A\mapsto A+d\chi$. Its curvature is the globally defined two-form $F=dA$. In unit-charge normalization its holonomy around a closed loop is $e^{i\oint A}$, and curvature periods are integer multiples of $2\pi$ on closed surfaces. A circulation representative is defined modulo $2\pi$ under large gauge transformations.

A gauge transformation is an automorphism of a principal bundle covering the identity on the base and commuting with its right group action. Pullback relates gauge-equivalent principal connections. Locally it acts by the displayed formula; the gauge curvature transforms by conjugation. Gauge functions on overlapping charts satisfy the bundle's transition compatibility. Infinitesimally $g=e^{\epsilon }$ gives $\delta A=D_A\epsilon$.

Pulling a principal connection back along a local section $s$ gives a Lie-algebra-valued one-form on the base. If the section changes to $sg$, the form has the displayed inhomogeneous transformation. Its gauge curvature $F=dA+A\wedge A$ transforms homogeneously as $F^{\prime }=g^{-1}Fg$. The collection of such local forms encodes a global principal connection even when no single section exists globally.

A local principal connection form $A$ and a fiber coordinate $\gamma$ reconstruct the connection on a principal bundle in a local section. Changing that section by $s^{\prime }=su$ changes $\gamma$ to $u^{-1}\gamma$ and $A$ to $u^{-1}Au+u^{-1}du$. Substitution cancels the two terms involving $du$, so the reconstructed forms agree on overlaps. Here $A$ is Lie-algebra-valued and matrix conjugation denotes the adjoint action. Agreement across trivializations differs from the right-action equivariance $R_h^{\ast }\omega ={\mathrm{Ad}}_{h^{-1}}\omega$.

The horizontal distribution of a principal connection is the smooth complement $H$ to the tangent spaces of the group orbits. A tangent vector is horizontal exactly when the connection form $\mathcal{A}$ annihilates it.

On a coordinate trivialization of a principal bundle, the right-invariant vector fields $R_a(\gamma )=T_a\gamma$ satisfy $[R_a,R_b]=-c^d{}_{ab}{R}_d$. With $A=A_i^a{T}_{a}d{x}^i$, the displayed lifts project to the commuting coordinate fields and contract to zero with the principal connection. Taking their Lie brackets gives the local curvature of a principal connection coefficients $F_{ij}={\partial }_i{A}_j-{\partial }_j{A}_i+[A_i,A_j]$. Thus these lifts commute exactly when the connection is flat. Arbitrary base fields need not commute, and flatness alone does not supply a global coordinate frame or remove holonomy of a connection.

A local section $s:U\to P$ is horizontal when $ds(TU)\subseteq H$, equivalently when $s^{\ast }\mathcal{A}=0$. A flat principal connection has horizontal sections locally, while its holonomy can obstruct a global horizontal section.

The holonomy of a connection along a closed curve is the group element relating the endpoints of its horizontal lift. A flat connection can have nontrivial holonomy around a noncontractible loop.

An $SU(2)$ connection whose holonomy lies in a maximal circle, equivalently one preserving a splitting $E=L\oplus L^{-1}$. If its holonomy is the entire circle, parallel adjoint sections are exactly the one-real-dimensional diagonal Lie algebra. The off-diagonal real plane is the underlying real bundle of $L^2$ and has circle weight two.

// Destination: geometry-and-topology.bigb

In the round real projective plane, project the great-circle arc $(\cos t,\sin t,0)$ for $0\le t\le \pi$. Its endpoint and velocity match after antipodal identification, so it is a closed geodesic. On the sphere its tangent and the constant vector $(0,0,1)$ are parallel. Endpoint identification fixes the returned tangent and reverses the returned transverse vector. Its parallel transport is therefore a reflection. Nonorientability permits determinant-minus-one holonomy.

For the Riemannian metric $d{r}^2+f(r{)}^{2}d{\varphi }^2$, where $f>0$ and $\varphi$ has period $2\pi$, parallel transport around $r=r_0$ acts on the orthonormal basis components of a tangent vector by a rotation through $-2\pi f^{\prime }(r_0)$. Indeed, those components obey $u^{\prime }=f^{\prime }(r_0)v$ and $v^{\prime }=-f^{\prime }(r_0)u$ as functions of $\varphi$. Thus all tangent vectors return to themselves exactly when $f^{\prime }(r_0)\in \mathbb{Z}$. The integer criterion concerns holonomy and is stronger than local flatness.

On a connected regular radial interval, trivial holonomy around circular fibres forces the continuous function $f^{\prime }$ to be integer-valued, hence constant. The resulting Riemannian metrics have $f(r)=nr+b$, with $n\in \mathbb{Z}$ and $f>0$ on the interval. Their Gaussian curvature is $-f^{\prime \prime }/f=0$. For $n\ne 0$, $\rho =f/|n|$ and $\theta =|n|\varphi$ give a local isometry to the Euclidean plane; for $n=0$, $r,b\varphi$ give a flat cylindrical chart. The polar developing map is injective on the regular domain when $|n|=1$, and is a multiple covering when $|n|>1$. Trivial holonomy therefore does not imply a global Euclidean chart. The slope is a discrete parameter, not an unrestricted real one.

For a connected Riemannian manifold, the full holonomy group consists of parallel transport maps around all piecewise smooth loops based at $p$. Metric compatibility makes them orthogonal. Basepoint changes conjugate this group by parallel transport. This is full holonomy, not just the subgroup using contractible loops.

The holonomy representation is the natural action of the Riemannian holonomy group on $T_{p}M$. It induces representations on cotangent spaces and tensor powers. Irreducibility means that no nonzero proper real linear subspace of $T_{p}M$ is invariant.

A nonzero parallel one-form determines a nonzero parallel vector by metric duality. Its value is fixed by the full Riemannian holonomy group, so its span is an invariant line. This contradicts irreducibility when $\dim M\ge 2$. The dimension condition matters: the standard circle has trivial but irreducible one-dimensional holonomy representation and a nonzero parallel one-form.

Evaluation at a point identifies parallel tensor fields with tensors fixed by the full Riemannian holonomy group. A parallel field is fixed after transport around every loop. Conversely transport a fixed tensor along paths from the basepoint; loop invariance makes the result independent of the path, and local parallel transport shows smoothness and parallelness. Connectedness gives uniqueness.

For the principal $O(k)$-bundle of orthonormal $k$-frames over the real Grassmannian, the canonical horizontal vectors move every frame vector orthogonally to the spanned plane. If a frame is represented by a matrix $Q$ with $Q^{\mathsf{T}}Q=I$, the connection form is