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\begin{document}

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\section*{frame bundle}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{frame_and_coframe_bundles}{}\section*{{Frame and coframe bundles}}\label{frame_and_coframe_bundles}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{traditional}{Traditional}\dotfill \pageref*{traditional} \linebreak
\noindent\hyperlink{InDifferentialCohesion}{In differential cohesion}\dotfill \pageref*{InDifferentialCohesion} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{TheCanonical1Form}{The canonical differential 1-form}\dotfill \pageref*{TheCanonical1Form} \linebreak
\noindent\hyperlink{relation_to_structures}{Relation to $G$-structures}\dotfill \pageref*{relation_to_structures} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{traditional}{}\subsubsection*{{Traditional}}\label{traditional}

Given a $k$-[[vector bundle]] $p\colon E \to M$ of finite [[rank]] $n$, its \textbf{frame bundle} (or bundle of [[frame of a vector space|frames]] in $E \to M$) is the bundle $F E \to M$ over the same base whose [[fiber]] over $x \in M$ is the set of all vector space [[basis|bases]] of $E_x = p^{-1}(x)$. The frame bundle has a natural action of $GL_n(k)$ given by an ordered change of basis which is free and transitive, i. e., the frame bundle is a [[principal bundle|principal]] $GL_n(k)$-bundle.

The \textbf{frame bundle of a [[manifold]]} $M$ is the [[principal bundle]] $F T M \to M$ (also denoted $F M \to M$) of frames in the [[tangent bundle]] $T M$.

In the finite-dimensional case, the dual $GL_n$-principal bundle $(F T)^* M$ is the \textbf{coframe bundle} of the manifold. This means that $F^* M = (F T)^* M$ is the associated bundle to $F T M \times_{GL_n(k)}GL_n(k)$ where the left action of $GL_n(k)$ on $GL_n(k)$ is given by right multiplication by inverses $g. h = h\cdot g^{-1}$. Also $F T M\cong (F T)^* M\times_{GL_n(k)} GL_n(k)$ using the same formula. Furthermore, the right action of $GL_n(k)$ on this associated bundle is given by left multiplication by inverses on $GL_n(k)$ factor.

Coframe bundle $F^* M$ has the following independent description. One looks at the set $\mathcal{U}(M)$ of tuples of the form $(p,(U,h))$ where $p\in U$ and $(U,h)$ is chart of the smooth structure on $M$, $U\subset M$, $h : U\to \mathbf{R}^n$ (an atlas where $U$-s make a basis of topology suffices). $GL_n(k)$ acts on the right on $\mathcal{U}(M)$ by

\begin{displaymath}
(p, (U, h)) A := (p, (U, A^{-1} h)).
\end{displaymath}
Then $((p,(U,h))A)A' = (p,(U,h)) (AA')$ holds. The total space $F^* M$ of the coframe bundle by the definition, as a set, consists of classes of equivalence of tuples in $\mathcal{U}(M)$ where $(p,(U,h)) \sim (p',(U',h'))$ iff $p = p'$ and the [[Jacobian matrix]] of the transition between charts at $h'(p)$ is the unit matrix: $J_{h'(p)}(h\circ (h')^{-1}) = I$. The left action of $GL_n(k)$ is induced on the quotient. There is an obvious projection $\pi: [(p,(U,h)]\mapsto p$. To define the differential and principal bundle structure one charts $F^* M\to M$ with local trivializations from the neighborhoods of the form $U\times GL_n(k)$, transfers the structure and checks that the transition functions are of the appropriate smoothness class and right $GL_n(k)$-equivariant. The basic prescription is that to every chart $(U,h)$ one defines a map

\begin{displaymath}
\phi_{h} = \pi^{-1}(U)\to U \times GL_n(k),\,\,\,\,\,\,z\mapsto (\pi(z), J_{h(\pi(z))}(h'\circ h^{-1})),
\end{displaymath}
where $z = [(\pi(z), (U',h'))]$ with $\pi(z)\in U'\cap U$. This does not depend on the choice of the chart $(U',h')$ around $\pi(z)$. There is an equivariance

\begin{displaymath}
J_{h(\pi(z A))}(h'\circ h^{-1})) = A^{-1} J_{h(\pi(z))}(h'\circ h^{-1}))
\end{displaymath}
and on intersection of $(U,h)$ and $(V,g)$

\begin{displaymath}
J_{h(\pi(z))}(h'\circ h^{-1})) = J_{g(\pi(z))}(h'\circ g^{-1})J_{h(\pi(z))}(g\circ h^{-1})
\end{displaymath}
Then $\phi_h$ is onto and

\begin{displaymath}
(\phi_h \circ (\phi_g)^{-1})(p,A) = (p, A J_{h(p)}(g\circ h^{-1})
\end{displaymath}
what shows that the transition functions are smooth (where $GL_n(k)$ has the standard differential structure).

\hypertarget{InDifferentialCohesion}{}\subsubsection*{{In differential cohesion}}\label{InDifferentialCohesion}

In a context of [[differential cohesion]], then the frame bundle (or [[higher order frame bundle]]) of a $V$-[[manifold]] is the [[principal bundle]] ([[principal infinity-bundle]]) to which the [[infinitesimal disk bundle]] is the canonically [[associated bundle]] ([[associated infinity-bundle]])

See at \emph{\href{differential+cohesion#GLnTangentBundles}{differential cohesion -- Frame bundles}}.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{TheCanonical1Form}{}\subsubsection*{{The canonical differential 1-form}}\label{TheCanonical1Form}

The frame bundle $Fr(X)$ carries a canonical [[differential 1-form]] with values in $\mathbb{R}^n$.

\begin{displaymath}
\alpha \in \Omega^1(Fr(X), \mathbb{R}^n)
\end{displaymath}
This is defined as follows. Let $p \in Fr(X)$ be a point in the frame bundle $\pi \colon Fr(X)\to X$ over some point $x \in X$, hence a linear isomorphism $p \colon T_x \simeq \mathbb{R}^n$. For $v \in T_p Fr(X)$ a [[tangent vector]] to the frame bundle, its projection $\pi_\ast v \in T_x X$ is a [[tangent vector]] to $X$. Then the value of $\alpha$ on $v$ is the image of this $\pi_\ast(v)$ under the isomorphism $p$

\begin{displaymath}
\alpha(v) \coloneqq p(\pi_\ast(v))
  \,.
\end{displaymath}
(\hyperlink{Sternberg64}{Sternberg 64, section VII, (2.2)})

\hypertarget{relation_to_structures}{}\subsubsection*{{Relation to $G$-structures}}\label{relation_to_structures}

A choice sub-bundle of a frame bundle which is a $G$-[[principal bundle]] for $G\hookrightarrow GL(n)$ defines a \emph{[[G-structure]]}. See there for more.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item a [[section]] of a frame bundle is also called a \emph{[[frame field]]}.


\item [[higher order frame bundle]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Wikipedia (English) \href{http://en.wikipedia.org/wiki/Frame_bundle}{frame bundle}


\item [[Shlomo Sternberg]], \emph{Lectures on differential geometry}, Prentice Hall 1964; Russian transl. Mir 1970



\end{itemize}
[[!redirects coframe bundle]] [[!redirects frame bundles]]



\end{document}