Inmathematics, aframe bundle is aprincipal fiber bundle${\displaystyle F(E)}$ associated with anyvector bundle${\displaystyle E}$. The fiber of${\displaystyle F(E)}$ over a point${\displaystyle x}$ is the set of allordered bases, orframes, for${\displaystyle E_x}$. Thegeneral linear group acts naturally on${\displaystyle F(E)}$ via achange of basis, giving the frame bundle the structure of a principal${\displaystyle \mathrm{G}\mathrm{L}(k,\mathbb{R})}$-bundle (wherek is the rank of${\displaystyle E}$).

The frame bundle of asmooth manifold is the one associated with itstangent bundle. For this reason it is sometimes called thetangent frame bundle.

Let${\displaystyle E\to X}$ be a realvector bundle of rank${\displaystyle k}$ over atopological space${\displaystyle X}$. Aframe at a point${\displaystyle x\in X}$ is anordered basis for the vector space${\displaystyle E_x}$. Equivalently, a frame can be viewed as alinear isomorphism

${\displaystyle p:{\mathbf{R}}^k\to E_x.}$

The set of all frames at${\displaystyle x}$, denoted${\displaystyle F_x}$, has a naturalright action by thegeneral linear group${\displaystyle \mathrm{G}\mathrm{L}(k,\mathbb{R})}$ of invertible${\displaystyle k\times k}$ matrices: a group element${\displaystyle g\in \mathrm{G}\mathrm{L}(k,\mathbb{R})}$ acts on the frame${\displaystyle p}$ viacomposition to give a new frame

${\displaystyle p\circ g:{\mathbf{R}}^k\to E_x.}$

This action of${\displaystyle \mathrm{G}\mathrm{L}(k,\mathbb{R})}$ on${\displaystyle F_x}$ is bothfree andtransitive (this follows from the standard linear algebra result that there is a unique invertible linear transformation sending one basis onto another). As a topological space,${\displaystyle F_x}$ ishomeomorphic to${\displaystyle \mathrm{G}\mathrm{L}(k,\mathbb{R})}$ although it lacks a group structure, since there is no "preferred frame". The space${\displaystyle F_x}$ is said to be a${\displaystyle \mathrm{G}\mathrm{L}(k,\mathbb{R})}$-torsor.

Theframe bundle of${\displaystyle E}$, denoted by${\displaystyle F(E)}$ or${\displaystyle F_{\mathrm{G}\mathrm{L}}(E)}$, is thedisjoint union of all the${\displaystyle F_x}$:

${\displaystyle \mathrm{F}(E)=\coprod _{x\in X}{F}_x.}$

Each point in${\displaystyle F(E)}$ is a pair (x,p) where${\displaystyle x}$ is a point in${\displaystyle X}$ and${\displaystyle p}$ is a frame at${\displaystyle x}$. There is a natural projection${\displaystyle \pi :F(E)\to X}$ which sends${\displaystyle (x,p)}$ to${\displaystyle x}$. The group${\displaystyle \mathrm{G}\mathrm{L}(k,\mathbb{R})}$ acts on${\displaystyle F(E)}$ on the right as above. This action is clearly free and theorbits are just the fibers of${\displaystyle \pi }$.

The frame bundle${\displaystyle F(E)}$ can be given a natural topology and bundle structure determined by that of${\displaystyle E}$. Let${\displaystyle (U_i,{\varphi }_i)}$ be alocal trivialization of${\displaystyle E}$. Then for eachx ∈U_i one has a linear isomorphism${\displaystyle {\varphi }_{i,x}:E_x\to {\mathbb{R}}^k}$. This data determines a bijection

${\displaystyle {\psi }_i:{\pi }^{-1}(U_i)\to U_i\times \mathrm{G}\mathrm{L}(k,\mathbb{R})}$

given by

${\displaystyle {\psi }_i(x,p)=(x,{\varphi }_{i,x}\circ p).}$

With these bijections, each${\displaystyle {\pi }^{-1}(U_i)}$ can be given the topology of${\displaystyle U_i\times \mathrm{G}\mathrm{L}(k,\mathbb{R})}$. The topology on${\displaystyle F(E)}$ is thefinal topology coinduced by the inclusion maps${\displaystyle {\pi }^{-1}(U_i)\to F(E)}$.

With all of the above data the frame bundle${\displaystyle F(E)}$ becomes aprincipal fiber bundle over${\displaystyle X}$ withstructure group${\displaystyle \mathrm{G}\mathrm{L}(k,\mathbb{R})}$ and local trivializations${\displaystyle (\{U_i\},\{{\psi }_i\})}$. One can check that thetransition functions of${\displaystyle F(E)}$ are the same as those of${\displaystyle E}$.

The above all works in the smooth category as well: if${\displaystyle E}$ is a smooth vector bundle over asmooth manifold${\displaystyle M}$ then the frame bundle of${\displaystyle E}$ can be given the structure of a smooth principal bundle over${\displaystyle M}$.

A vector bundle${\displaystyle E}$ and its frame bundle${\displaystyle F(E)}$ areassociated bundles. Each one determines the other. The frame bundle${\displaystyle F(E)}$ can be constructed from${\displaystyle E}$ as above, or more abstractly using thefiber bundle construction theorem. With the latter method,${\displaystyle F(E)}$ is the fiber bundle with same base, structure group, trivializing neighborhoods, and transition functions as${\displaystyle E}$ but with abstract fiber${\displaystyle \mathrm{G}\mathrm{L}(k,\mathbb{R})}$, where the action of structure group${\displaystyle \mathrm{G}\mathrm{L}(k,\mathbb{R})}$ on the fiber${\displaystyle \mathrm{G}\mathrm{L}(k,\mathbb{R})}$ is that of left multiplication.

Given anylinear representation${\displaystyle \rho :\mathrm{G}\mathrm{L}(k,\mathbb{R})\to \mathrm{G}\mathrm{L}(V,\mathbb{F})}$ there is a vector bundle

${\displaystyle \mathrm{F}(E){\times }_{\rho }V}$

associated with${\displaystyle F(E)}$ which is given by product${\displaystyle F(E)\times V}$ modulo theequivalence relation${\displaystyle (pg,v)\sim (p,\rho (g)v)}$ for all${\displaystyle g}$ in${\displaystyle \mathrm{G}\mathrm{L}(k,\mathbb{R})}$. Denote the equivalence classes by${\displaystyle [p,v]}$.

The vector bundle${\displaystyle E}$ isnaturally isomorphic to the bundle${\displaystyle F(E){\times }_{\rho }{\mathbb{R}}^k}$ where${\displaystyle \rho }$ is the fundamental representation of${\displaystyle \mathrm{G}\mathrm{L}(k,\mathbb{R})}$ on${\displaystyle {\mathbb{R}}^k}$. The isomorphism is given by

${\displaystyle [p,v]\mapsto p(v)}$

where${\displaystyle v}$ is a vector in${\displaystyle {\mathbb{R}}^k}$ and${\displaystyle p:{\mathbb{R}}^k\to E_x}$ is a frame at${\displaystyle x}$. One can easily check that this map iswell-defined.

Any vector bundle associated with${\displaystyle E}$ can be given by the above construction. For example, thedual bundle of${\displaystyle E}$ is given by${\displaystyle F(E){\times }_{{\rho }^{\ast }}({\mathbb{R}}^k{)}^{\ast }}$ where${\displaystyle {\rho }^{\ast }}$ is thedual of the fundamental representation.Tensor bundles of${\displaystyle E}$ can be constructed in a similar manner.

Thetangent frame bundle (or simply theframe bundle) of asmooth manifold${\displaystyle M}$ is the frame bundle associated with thetangent bundle of${\displaystyle M}$. The frame bundle of${\displaystyle M}$ is often denoted${\displaystyle FM}$ or${\displaystyle \mathrm{G}\mathrm{L}(M)}$ rather than${\displaystyle F(TM)}$. In physics, it is sometimes denoted${\displaystyle LM}$. If${\displaystyle M}$ is${\displaystyle n}$-dimensional then the tangent bundle has rank${\displaystyle n}$, so the frame bundle of${\displaystyle M}$ is a principal${\displaystyle \mathrm{G}\mathrm{L}(n,\mathbb{R})}$ bundle over${\displaystyle M}$.

Local sections of the frame bundle of${\displaystyle M}$ are calledsmooth frames on${\displaystyle M}$. The cross-section theorem for principal bundles states that the frame bundle is trivial over any open set in${\displaystyle U}$ in${\displaystyle M}$ which admits a smooth frame. Given a smooth frame${\displaystyle s:U\to FU}$, the trivialization${\displaystyle \psi :FU\to U\times \mathrm{G}\mathrm{L}(n,\mathbb{R})}$ is given by

${\displaystyle \psi (p)=(x,s(x{)}^{-1}\circ p)}$

where${\displaystyle p}$ is a frame at${\displaystyle x}$. It follows that a manifold isparallelizable if and only if the frame bundle of${\displaystyle M}$ admits a global section.

Since the tangent bundle of${\displaystyle M}$ is trivializable over coordinate neighborhoods of${\displaystyle M}$ so is the frame bundle. In fact, given any coordinate neighborhood${\displaystyle U}$ with coordinates${\displaystyle (x^1,\dots ,x^n)}$ the coordinate vector fields

${\displaystyle \left(\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^1},\dots ,\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^n}\right)}$

define a smooth frame on${\displaystyle U}$. One of the advantages of working with frame bundles is that they allow one to work with frames other than coordinates frames; one can choose a frame adapted to the problem at hand. This is sometimes called themethod of moving frames.

The frame bundle of a manifold${\displaystyle M}$ is a special type of principal bundle in the sense that its geometry is fundamentally tied to the geometry of${\displaystyle M}$. This relationship can be expressed by means of avector-valued 1-form on${\displaystyle FM}$ called thesolder form (also known as thefundamental ortautological 1-form). Let${\displaystyle x}$ be a point of the manifold${\displaystyle M}$ and${\displaystyle p}$ a frame at${\displaystyle x}$, so that

${\displaystyle p:{\mathbf{R}}^n\to T_{x}M}$

is a linear isomorphism of${\displaystyle {\mathbb{R}}^n}$ with the tangent space of${\displaystyle M}$ at${\displaystyle x}$. The solder form of${\displaystyle FM}$ is the${\displaystyle {\mathbb{R}}^n}$-valued 1-form${\displaystyle \theta }$ defined by

${\displaystyle {\theta }_p(\xi )=p^{-1}\mathrm{d}\pi (\xi )}$

where ξ is a tangent vector to${\displaystyle FM}$ at the point${\displaystyle (x,p)}$, and${\displaystyle p^{-1}:T_{x}M\to {\mathbb{R}}^n}$ is the inverse of the frame map, and${\displaystyle d\pi }$ is thedifferential of the projection map${\displaystyle \pi :FM\to M}$. The solder form is horizontal in the sense that it vanishes on vectors tangent to the fibers of${\displaystyle \pi }$ andright equivariant in the sense that

${\displaystyle R_g^{\ast }\theta =g^{-1}\theta }$

where${\displaystyle R_g}$ is right translation by${\displaystyle g\in \mathrm{G}\mathrm{L}(n,\mathbb{R})}$. A form with these properties is called a basic ortensorial form on${\displaystyle FM}$. Such forms are in 1-1 correspondence with${\displaystyle TM}$-valued 1-forms on${\displaystyle M}$ which are, in turn, in 1-1 correspondence with smoothbundle maps${\displaystyle TM\to TM}$ over${\displaystyle M}$. Viewed in this light${\displaystyle \theta }$ is just theidentity map on${\displaystyle TM}$.

As a naming convention, the term "tautological one-form" is usually reserved for the case where the form has a canonical definition, as it does here, while "solder form" is more appropriate for those cases where the form is not canonically defined. This convention is not being observed here.

If a vector bundle${\displaystyle E}$ is equipped with aRiemannian bundle metric then each fiber${\displaystyle E_x}$ is not only a vector space but aninner product space. It is then possible to talk about the set of allorthonormal frames for${\displaystyle E_x}$. An orthonormal frame for${\displaystyle E_x}$ is an orderedorthonormal basis for${\displaystyle E_x}$, or, equivalently, alinear isometry

${\displaystyle p:{\mathbb{R}}^k\to E_x}$

where${\displaystyle {\mathbb{R}}^k}$ is equipped with the standardEuclidean metric. Theorthogonal group${\displaystyle \mathrm{O}(k)}$ acts freely and transitively on the set of all orthonormal frames via right composition. In other words, the set of all orthonormal frames is a right${\displaystyle \mathrm{O}(k)}$-torsor.

Theorthonormal frame bundle of${\displaystyle E}$, denoted${\displaystyle F_{\mathrm{O}}(E)}$, is the set of all orthonormal frames at each point${\displaystyle x}$ in the base space${\displaystyle X}$. It can be constructed by a method entirely analogous to that of the ordinary frame bundle. The orthonormal frame bundle of a rank${\displaystyle k}$ Riemannian vector bundle${\displaystyle E\to X}$ is a principal${\displaystyle \mathrm{O}(k)}$-bundle over${\displaystyle X}$. Again, the construction works just as well in the smooth category.

If the vector bundle${\displaystyle E}$ isorientable then one can define theoriented orthonormal frame bundle of${\displaystyle E}$, denoted${\displaystyle F_{\mathrm{S}\mathrm{O}}(E)}$, as the principal${\displaystyle \mathrm{S}\mathrm{O}(k)}$-bundle of all positively oriented orthonormal frames.

If${\displaystyle M}$ is an${\displaystyle n}$-dimensionalRiemannian manifold, then the orthonormal frame bundle of${\displaystyle M}$, denoted${\displaystyle F_{\mathrm{O}}(M)}$ or${\displaystyle \mathrm{O}(M)}$, is the orthonormal frame bundle associated with the tangent bundle of${\displaystyle M}$ (which is equipped with a Riemannian metric by definition). If${\displaystyle M}$ is orientable, then one also has the oriented orthonormal frame bundle${\displaystyle F_{\mathrm{S}\mathrm{O}}M}$.

Given a Riemannian vector bundle${\displaystyle E}$, the orthonormal frame bundle is a principal${\displaystyle \mathrm{O}(k)}$-subbundle of the general linear frame bundle. In other words, the inclusion map

${\displaystyle i:{\mathrm{F}}_{\mathrm{O}}(E)\to {\mathrm{F}}_{\mathrm{G}\mathrm{L}}(E)}$

is principalbundle map. One says that${\displaystyle F_{\mathrm{O}}(E)}$ is areduction of the structure group of${\displaystyle F_{\mathrm{G}\mathrm{L}}(E)}$ from${\displaystyle \mathrm{G}\mathrm{L}(n,\mathbb{R})}$ to${\displaystyle \mathrm{O}(k)}$.

If a smooth manifold${\displaystyle M}$ comes with additional structure it is often natural to consider a subbundle of the full frame bundle of${\displaystyle M}$ which is adapted to the given structure. For example, if${\displaystyle M}$ is a Riemannian manifold we saw above that it is natural to consider the orthonormal frame bundle of${\displaystyle M}$. The orthonormal frame bundle is just a reduction of the structure group of${\displaystyle F_{\mathrm{G}\mathrm{L}}(M)}$ to the orthogonal group${\displaystyle \mathrm{O}(n)}$.

In general, if${\displaystyle M}$ is a smooth${\displaystyle n}$-manifold and${\displaystyle G}$ is aLie subgroup of${\displaystyle \mathrm{G}\mathrm{L}(n,\mathbb{R})}$ we define aG-structure on${\displaystyle M}$ to be areduction of the structure group of${\displaystyle F_{\mathrm{G}\mathrm{L}}(M)}$ to${\displaystyle G}$. Explicitly, this is a principal${\displaystyle G}$-bundle${\displaystyle F_G(M)}$ over${\displaystyle M}$ together with a${\displaystyle G}$-equivariantbundle map

${\displaystyle {\mathrm{F}}_G(M)\to {\mathrm{F}}_{\mathrm{G}\mathrm{L}}(M)}$

over${\displaystyle M}$.

In this language, a Riemannian metric on${\displaystyle M}$ gives rise to an${\displaystyle \mathrm{O}(n)}$-structure on${\displaystyle M}$. The following are some other examples.

• Everyoriented manifold has an oriented frame bundle which is just a${\displaystyle {\mathrm{G}\mathrm{L}}^{+}(n,\mathbb{R})}$-structure on${\displaystyle M}$.
• Avolume form on${\displaystyle M}$ determines a${\displaystyle \mathrm{S}\mathrm{L}(n,\mathbb{R})}$-structure on${\displaystyle M}$.
• A${\displaystyle 2n}$-dimensionalsymplectic manifold has a natural${\displaystyle \mathrm{S}\mathrm{p}(2n,\mathbb{R})}$-structure.
• A${\displaystyle 2n}$-dimensionalcomplex oralmost complex manifold has a natural${\displaystyle \mathrm{G}\mathrm{L}(n,\mathbb{C})}$-structure.

In many of these instances, a${\displaystyle G}$-structure on${\displaystyle M}$ uniquely determines the corresponding structure on${\displaystyle M}$. For example, a${\displaystyle \mathrm{S}\mathrm{L}(n,\mathbb{R})}$-structure on${\displaystyle M}$ determines a volume form on${\displaystyle M}$. However, in some cases, such as for symplectic and complex manifolds, an addedintegrability condition is needed. A${\displaystyle \mathrm{S}\mathrm{p}(2n,\mathbb{R})}$-structure on${\displaystyle M}$ uniquely determines anondegenerate2-form on${\displaystyle M}$, but for${\displaystyle M}$ to be symplectic, this 2-form must also beclosed.

• Kobayashi, Shoshichi; Nomizu, Katsumi (1996),Foundations of Differential Geometry, vol. 1 (New ed.),Wiley Interscience,ISBN 0-471-15733-3
• Kolář, Ivan; Michor, Peter; Slovák, Jan (1993),Natural operators in differential geometry(PDF), Springer-Verlag, archived fromthe original(PDF) on 2017-03-30, retrieved2008-08-02
• Sternberg, S. (1983),Lectures on Differential Geometry (2nd ed.), New York: Chelsea Publishing Co.,ISBN 0-8218-1385-4

• Fiber bundles
• Vector bundles

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