# Frame bundle

In mathematics, a frame bundle is a principal fiber bundle $F(E)$ associated with any vector bundle $E$. The fiber of $F(E)$ over a point $x$ is the set of all ordered bases, or frames, for $E_{x}$. The general linear group acts naturally on $F(E)$ via a change of basis, giving the frame bundle the structure of a principal $\mathrm {GL} (k,\mathbb {R} )$-bundle (where k is the rank of $E$).
The frame bundle of a smooth manifold is the one associated with its tangent bundle. For this reason it is sometimes called the tangent frame bundle.

## Definition and construction

Let $E\to X$ be a real vector bundle of rank $k$ over a topological space $X$. A frame at a point $x\in X$ is an ordered basis for the vector space $E_{x}$. Equivalently, a frame can be viewed as a linear isomorphism
    $p:\mathbf {R} ^{k}\to E_{x}.$
The set of all frames at $x$, denoted $F_{x}$, has a natural right action by the general linear group $\mathrm {GL} (k,\mathbb {R} )$ of invertible $k\times k$ matrices: a group element $g\in \mathrm {GL} (k,\mathbb {R} )$ acts on the frame $p$ via composition to give a new frame
    $p\circ g:\mathbf {R} ^{k}\to E_{x}.$
This action of $\mathrm {GL} (k,\mathbb {R} )$ on $F_{x}$ is both free and transitive (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, $F_{x}$ is homeomorphic to $\mathrm {GL} (k,\mathbb {R} )$ although it lacks a group structure, since there is no "preferred frame". The space $F_{x}$ is said to be a $\mathrm {GL} (k,\mathbb {R} )$-torsor.
The frame bundle of $E$, denoted by $F(E)$ or $F_{\mathrm {GL} }(E)$, is the disjoint union of all the $F_{x}$:
    $\mathrm {F} (E)=\coprod _{x\in X}F_{x}.$
Each point in $F(E)$ is a pair (x, p) where $x$ is a point in $X$ and $p$ is a frame at $x$. There is a natural projection $\pi :F(E)\to X$ which sends $(x,p)$ to $x$. The group $\mathrm {GL} (k,\mathbb {R} )$ acts on $F(E)$ on the right as above. This action is clearly free and the orbits are just the fibers of $\pi$.

### Principal bundle structure

The frame bundle $F(E)$ can be given a natural topology and bundle structure determined by that of $E$. Let $(U_{i},\phi _{i})$ be a local trivialization of $E$. Then for each x ∈ U$_i$ one has a linear isomorphism $\phi _{i,x}:E_{x}\to \mathbb {R} ^{k}$. This data determines a bijection
    $\psi _{i}:\pi ^{-1}(U_{i})\to U_{i}\times \mathrm {GL} (k,\mathbb {R} )$
given by
    $\psi _{i}(x,p)=(x,\phi _{i,x}\circ p).$
With these bijections, each $\pi ^{-1}(U_{i})$ can be given the topology of $U_{i}\times \mathrm {GL} (k,\mathbb {R} )$. The topology on $F(E)$ is the final topology coinduced by the inclusion maps $\pi ^{-1}(U_{i})\to F(E)$.
With all of the above data the frame bundle $F(E)$ becomes a principal fiber bundle over $X$ with structure group $\mathrm {GL} (k,\mathbb {R} )$ and local trivializations $(\{U_{i}\},\{\psi _{i}\})$. One can check that the transition functions of $F(E)$ are the same as those of $E$.
The above all works in the smooth category as well: if $E$ is a smooth vector bundle over a smooth manifold $M$ then the frame bundle of $E$ can be given the structure of a smooth principal bundle over $M$.

## Associated vector bundles

A vector bundle $E$ and its frame bundle $F(E)$ are associated bundles. Each one determines the other. The frame bundle $F(E)$ can be constructed from $E$ as above, or more abstractly using the fiber bundle construction theorem. With the latter method, $F(E)$ is the fiber bundle with same base, structure group, trivializing neighborhoods, and transition functions as $E$ but with abstract fiber $\mathrm {GL} (k,\mathbb {R} )$, where the action of structure group $\mathrm {GL} (k,\mathbb {R} )$ on the fiber $\mathrm {GL} (k,\mathbb {R} )$ is that of left multiplication.
Given any linear representation $\rho :\mathrm {GL} (k,\mathbb {R} )\to \mathrm {GL} (V,\mathbb {F} )$ there is a vector bundle
    $\mathrm {F} (E)\times _{\rho }V$
associated with $F(E)$ which is given by product $F(E)\times V$ modulo the equivalence relation $(pg,v)\sim (p,\rho (g)v)$ for all $g$ in $\mathrm {GL} (k,\mathbb {R} )$. Denote the equivalence classes by $[p,v]$.
The vector bundle $E$ is naturally isomorphic to the bundle $F(E)\times _{\rho }\mathbb {R} ^{k}$ where $\rho$ is the fundamental representation of $\mathrm {GL} (k,\mathbb {R} )$ on $\mathbb {R} ^{k}$. The isomorphism is given by
    $[p,v]\mapsto p(v)$
where $v$ is a vector in $\mathbb {R} ^{k}$ and $p:\mathbb {R} ^{k}\to E_{x}$ is a frame at $x$. One can easily check that this map is well-defined.
Any vector bundle associated with $E$ can be given by the above construction. For example, the dual bundle of $E$ is given by $F(E)\times _{\rho ^{*}}(\mathbb {R} ^{k})^{*}$ where $\rho ^{*}$ is the dual of the fundamental representation. Tensor bundles of $E$ can be constructed in a similar manner.

## Tangent frame bundle

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

### Smooth frames

Local sections of the frame bundle of $M$ are called smooth frames on $M$. The cross-section theorem for principal bundles states that the frame bundle is trivial over any open set in $U$ in $M$ which admits a smooth frame. Given a smooth frame $s:U\to FU$, the trivialization $\psi :FU\to U\times \mathrm {GL} (n,\mathbb {R} )$ is given by
    $\psi (p)=(x,s(x)^{-1}\circ p)$
where $p$ is a frame at $x$. It follows that a manifold is parallelizable if and only if the frame bundle of $M$ admits a global section.
Since the tangent bundle of $M$ is trivializable over coordinate neighborhoods of $M$ so is the frame bundle. In fact, given any coordinate neighborhood $U$ with coordinates $(x^{1},\ldots ,x^{n})$ the coordinate vector fields
    $\left({\frac {\partial }{\partial x^{1}}},\ldots ,{\frac {\partial }{\partial x^{n}}}\right)$
define a smooth frame on $U$. 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 the method of moving frames.

### Solder form

The frame bundle of a manifold $M$ is a special type of principal bundle in the sense that its geometry is fundamentally tied to the geometry of $M$. This relationship can be expressed by means of a vector-valued 1-form on $FM$ called the solder form (also known as the fundamental or tautological 1-form). Let $x$ be a point of the manifold $M$ and $p$ a frame at $x$, so that
    $p:\mathbf {R} ^{n}\to T_{x}M$
is a linear isomorphism of $\mathbb {R} ^{n}$ with the tangent space of $M$ at $x$. The solder form of $FM$ is the $\mathbb {R} ^{n}$-valued 1-form $\theta$ defined by
    $\theta _{p}(\xi )=p^{-1}\mathrm {d} \pi (\xi )$
where ξ is a tangent vector to $FM$ at the point $(x,p)$, and $p^{-1}:T_{x}M\to \mathbb {R} ^{n}$ is the inverse of the frame map, and $d\pi$ is the differential of the projection map $\pi :FM\to M$. The solder form is horizontal in the sense that it vanishes on vectors tangent to the fibers of $\pi$ and right equivariant in the sense that
    $R_{g}^{*}\theta =g^{-1}\theta$
where $R_{g}$ is right translation by $g\in \mathrm {GL} (n,\mathbb {R} )$. A form with these properties is called a basic or tensorial form on $FM$. Such forms are in 1-1 correspondence with $TM$-valued 1-forms on $M$ which are, in turn, in 1-1 correspondence with smooth bundle maps $TM\to TM$ over $M$. Viewed in this light $\theta$ is just the identity map on $TM$.
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.

## Orthonormal frame bundle

If a vector bundle $E$ is equipped with a Riemannian bundle metric then each fiber $E_{x}$ is not only a vector space but an inner product space. It is then possible to talk about the set of all orthonormal frames for $E_{x}$. An orthonormal frame for $E_{x}$ is an ordered orthonormal basis for $E_{x}$, or, equivalently, a linear isometry
    $p:\mathbb {R} ^{k}\to E_{x}$
where $\mathbb {R} ^{k}$ is equipped with the standard Euclidean metric. The orthogonal group $\mathrm {O} (k)$ 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 $\mathrm {O} (k)$-torsor.
The orthonormal frame bundle of $E$, denoted $F_{\mathrm {O} }(E)$, is the set of all orthonormal frames at each point $x$ in the base space $X$. It can be constructed by a method entirely analogous to that of the ordinary frame bundle. The orthonormal frame bundle of a rank $k$ Riemannian vector bundle $E\to X$ is a principal $\mathrm {O} (k)$-bundle over $X$. Again, the construction works just as well in the smooth category.
If the vector bundle $E$ is orientable then one can define the oriented orthonormal frame bundle of $E$, denoted $F_{\mathrm {SO} }(E)$, as the principal $\mathrm {SO} (k)$-bundle of all positively oriented orthonormal frames.
If $M$ is an $n$-dimensional Riemannian manifold, then the orthonormal frame bundle of $M$, denoted $F_{\mathrm {O} }(M)$ or $\mathrm {O} (M)$, is the orthonormal frame bundle associated with the tangent bundle of $M$ (which is equipped with a Riemannian metric by definition). If $M$ is orientable, then one also has the oriented orthonormal frame bundle $F_{\mathrm {SO} }M$.
Given a Riemannian vector bundle $E$, the orthonormal frame bundle is a principal $\mathrm {O} (k)$-subbundle of the general linear frame bundle. In other words, the inclusion map
    $i:{\mathrm {F} }_{\mathrm {O} }(E)\to {\mathrm {F} }_{\mathrm {GL} }(E)$
is principal bundle map. One says that $F_{\mathrm {O} }(E)$ is a reduction of the structure group of $F_{\mathrm {GL} }(E)$ from $\mathrm {GL} (n,\mathbb {R} )$ to $\mathrm {O} (k)$.

## G-structures

If a smooth manifold $M$ comes with additional structure it is often natural to consider a subbundle of the full frame bundle of $M$ which is adapted to the given structure. For example, if $M$ is a Riemannian manifold we saw above that it is natural to consider the orthonormal frame bundle of $M$. The orthonormal frame bundle is just a reduction of the structure group of $F_{\mathrm {GL} }(M)$ to the orthogonal group $\mathrm {O} (n)$.
In general, if $M$ is a smooth $n$-manifold and $G$ is a Lie subgroup of $\mathrm {GL} (n,\mathbb {R} )$ we define a G-structure on $M$ to be a reduction of the structure group of $F_{\mathrm {GL} }(M)$ to $G$. Explicitly, this is a principal $G$-bundle $F_{G}(M)$ over $M$ together with a $G$-equivariant bundle map
    ${\mathrm {F} }_{G}(M)\to {\mathrm {F} }_{\mathrm {GL} }(M)$
over $M$.
In this language, a Riemannian metric on $M$ gives rise to an $\mathrm {O} (n)$-structure on $M$. The following are some other examples.
* Every oriented manifold has an oriented frame bundle which is just a $\mathrm {GL} ^{+}(n,\mathbb {R} )$-structure on $M$.
* A volume form on $M$ determines a $\mathrm {SL} (n,\mathbb {R} )$-structure on $M$.
* A $2n$-dimensional symplectic manifold has a natural $\mathrm {Sp} (2n,\mathbb {R} )$-structure.
* A $2n$-dimensional complex or almost complex manifold has a natural $\mathrm {GL} (n,\mathbb {C} )$-structure.
In many of these instances, a $G$-structure on $M$ uniquely determines the corresponding structure on $M$. For example, a $\mathrm {SL} (n,\mathbb {R} )$-structure on $M$ determines a volume form on $M$. However, in some cases, such as for symplectic and complex manifolds, an added integrability condition is needed. A $\mathrm {Sp} (2n,\mathbb {R} )$-structure on $M$ uniquely determines a nondegenerate 2-form on $M$, but for $M$ to be symplectic, this 2-form must also be closed.