Connection on a principal bundle

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1 Definition

Let $G$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}G$ be a Lie group with Lie algebra $\frak{g}$$\frak{g}$ and $F \to M$$F \to M$ a principal bundle for $G$$G$ over a smooth manifold $M$$M$. A connection on $F$$F$ is a distribution (a subbundle of the tangent bundle) $\mathcal{H} \subset TF$$\mathcal{H} \subset TF$ on $F$$F$, called the "horizontal distribution", which is $G$$G$-invariant and complementary to the vertical distribution $\mathcal{V}$$\mathcal{V}$ on $F$$F$.

The decomposition $TF = \mathcal{V} \oplus \mathcal{H}$$TF = \mathcal{V} \oplus \mathcal{H}$ can be given by the projection $\pi_\mathcal{V} : TF \to \mathcal{V}$$\pi_\mathcal{V} : TF \to \mathcal{V}$ onto the vertical distribution. Since each vertical space $\mathcal{V}_f$$\mathcal{V}_f$ can be identified with $\frak{g}$$\frak{g}$ (see Principal bundle), this map $\pi_V$$\pi_V$ can be viewed as a $\frak{g}$$\frak{g}$-valued 1-form on $F$$F$, a linear map $\omega : TF \to \frak{g}$$\omega : TF \to \frak{g}$; this is called the connection form.

The $\frak{g}$$\frak{g}$-valued 2-form $\Omega := d\omega + [\omega,\omega]$$\Omega := d\omega + [\omega,\omega]$ is called curvature form and measures the non-integrability of the distribution $\mathcal{H}$$\mathcal{H}$, see the theory page Connections for details.

A connection $\mathcal{H}$$\mathcal{H}$ on a $G$$G$-principal bundles $F$$F$ induces a distribution on any associated bundle $E = (F \times E_o)/G$$E = (F \times E_o)/G$ (see Principal bundle) since $\mathcal{H}$$\mathcal{H}$ passes trivially to $F \times E_o$$F \times E_o$ and by $G$$G$-invariance to $E$$E$. The induced distribution is called a connection on $E$$E$. If $E_o$$E_o$ is a vector bundle (the action of $G$$G$ on $E_o$$E_o$ is linear), the connection on $E$$E$ is closely related to a covariant derivative (see Connections).

2 Examples

A (semi-)Riemannian metric on $M$$M$ defines a connection on the orthonormal frame bundle $F$$F$, the Levi-Civita connection: The horizontal space $\mathcal{H}_f$$\mathcal{H}_f$ at some orthonormal basis $f$$f$ of $T_pM$$T_pM$ consists of the derivatives of all curves $f(t)$$f(t)$ in $F$$F$ which are parallel along their base point curve $p(t)$$p(t)$ in $M$$M$.

Another type of example is the canonical connection on the principal bundle $G \to G/H$$G \to G/H$ of a reductive homogeneous space $M = G/H$$M = G/H$.

For further information see [Kobayashi&Nomizu1963].

References

• [Kobayashi&Nomizu1963] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. MR1393940 (97c:53001a) Zbl 0508.53002