Frame bundle

Let $E\; \backslash to\; X$ be a real vector bundle of rank $k$ over a topological space $X$. A frame at a point $x\; \backslash in\; X$ is an ordered basis for the vector space $E\_x$. Equivalently, a frame can be viewed as a linear isomorphism

:$p\; :\; \backslash mathbf\{R\}^k\; \backslash to\; E\_x.$

The set of all frames at $x$, denoted $F\_x$, has a natural right action by the general linear group $\backslash mathrm\{GL\}(k,\backslash mathbb\{R\})$ of invertible $k\; \backslash times\; k$ matrices: a group element $g\; \backslash in\; \backslash mathrm\{GL\}(k,\backslash mathbb\{R\})$ acts on the frame $p$ via composition to give a new frame

:$p\backslash circ\; g:\backslash mathbf\{R\}^k\backslash to\; E\_x.$

This action of $\backslash mathrm\{GL\}(k,\backslash mathbb\{R\})$ 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 $\backslash mathrm\{GL\}(k,\backslash mathbb\{R\})$ although it lacks a group structure, since there is no "preferred frame". The space $F\_x$ is said to be a $\backslash mathrm\{GL\}(k,\backslash mathbb\{R\})$-torsor.

The frame bundle of $E$, denoted by $F(E)$ or $F\_\{\backslash mathrm\{GL\}\}(E)$, is the disjoint union of all the $F\_x$:

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

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 $\backslash pi:\; F(E)\backslash to\; X$ which sends $(x,p)$ to $x$. The group $\backslash mathrm\{GL\}(k,\backslash mathbb\{R\})$ acts on $F(E)$ on the right as above. This action is clearly free and the orbits are just the fibers of $\backslash pi$.

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

:$\backslash psi\_i\; :\; \backslash pi^\{-1\}(U\_i)\backslash to\; U\_i\backslash times\; \backslash mathrm\{GL\}(k,\; \backslash mathbb\{R\})$

given by

:$\backslash psi\_i(x,p)\; =\; (x,\backslash phi\_\{i,x\}\backslash circ\; p).$

With these bijections, each $\backslash pi^\{-1\}(U\_i)$ can be given the topology of $U\_i\; \backslash times\; \backslash mathrm\{GL\}(k,\backslash mathbb\{R\})$. The topology on $F(E)$ is the final topology coinduced by the inclusion maps $\backslash pi^\{-1\}(U\_i)\; \backslash to\; F(E)$.

With all of the above data the frame bundle $F(E)$ becomes a principal fiber bundle over $X$ with structure group $\backslash mathrm\{GL\}(k,\backslash mathbb\{R\})$ and local trivializations $(\backslash \{U\_i\backslash \},\backslash \{\backslash psi\_i\backslash \})$. 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 $M$.

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 $\backslash mathrm\{GL\}(k,\backslash mathbb\{R\})$, where the action of structure group $\backslash mathrm\{GL\}(k,\backslash mathbb\{R\})$ on the fiber $\backslash mathrm\{GL\}(k,\backslash mathbb\{R\})$ is that of left multiplication.

Given any linear representation $\backslash rho:\; \backslash mathrm\{GL\}(k,\backslash mathbb\{R\})\; \backslash to\; \backslash mathrm\{GL\}(V,\backslash mathbb\{F\})$ there is a vector bundle

:$\backslash mathrm\; F(E)\backslash times\_\{\backslash rho\}V$

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

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

:$[p,v]\backslash mapsto\; p(v)$

where $v$ is a vector in $\backslash mathbb\{R\}^k$ and $p:\; \backslash mathbb\{R\}^k\; \backslash to\; E\_x$ 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 $F(E)\; \backslash times\_\{\backslash rho^*\}\; (\backslash mathbb\{R\}^k)^*$ where $\backslash rho^*$ is the dual of the fundamental representation. Tensor bundles of $E$ can be constructed in a similar manner.

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 $\backslash mathrm\{GL\}(M)$ 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 $\backslash mathrm\{GL\}(n,\backslash mathbb\{R\})$ bundle over $M$.

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 $s:\; U\; \backslash to\; FU$, the trivialization $\backslash psi:\; FU\; \backslash to\; U\; \backslash times\; \backslash mathrm\{GL\}(n,\backslash mathbb\{R\})$ is given by

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

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 $(x^1,\backslash ldots,x^n)$ the coordinate vector fields

:$\backslash left(\backslash frac\{\backslash partial\}\{\backslash partial\; x^1\},\backslash ldots,\backslash frac\{\backslash partial\}\{\backslash partial\; x^n\}\backslash right)$

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.

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

:$p\; :\; \backslash mathbf\{R\}^n\backslash to\; T\_xM$

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

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

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

:$R\_g^*\backslash theta\; =\; g^\{-1\}\backslash theta$

where $R\_g$ is right translation by $g\; \backslash in\; \backslash mathrm\{GL\}(n,\backslash mathbb\{R\})$. 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 $TM\; \backslash to\; TM$ over $M$. Viewed in this light $\backslash theta$ 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.

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

:$p:\backslash mathbb\{R\}^k\; \backslash to\; E\_x$

where $\backslash mathbb\{R\}^k$ is equipped with the standard Euclidean metric. The orthogonal group $\backslash mathrm\{O\}(k)$ 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 $\backslash mathrm\{O\}(k)$-torsor.

The orthonormal frame bundle of $E$, denoted $F\_\{\backslash mathrm\{O\}\}(E)$, 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 $E\; \backslash to\; X$ is a principal $\backslash mathrm\{O\}(k)$-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 $F\_\{\backslash mathrm\{SO\}\}(E)$, as the principal $\backslash mathrm\{SO\}(k)$-bundle of all positively oriented orthonormal frames.

If $M$ is an $n$-dimensional Riemannian manifold, then the orthonormal frame bundle of $M$, denoted $F\_\{\backslash mathrm\{O\}\}(M)$ or $\backslash mathrm\{O\}\; (M)$, 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 $F\_\{\backslash mathrm\{SO\}\}M$.

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

:$i:\{\backslash mathrm\; F\}\_\{\backslash mathrm\; O\}(E)\; \backslash to\; \{\backslash mathrm\; F\}\_\{\backslash mathrm\{GL\}\}(E)$

is principal bundle map. One says that $F\_\{\backslash mathrm\{O\}\}(E)$ is a reduction of the structure group of $F\_\{\backslash mathrm\{GL\}\}(E)$ from $\backslash mathrm\{GL\}(n,\backslash mathbb\{R\})$ to $\backslash mathrm\{O\}(k)$.

{{see also|G-structure}}

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 $F\_\{\backslash mathrm\{GL\}\}(M)$ to the orthogonal group $\backslash mathrm\{O\}(n)$.

In general, if $M$ is a smooth $n$-manifold and $G$ is a Lie subgroup of $\backslash mathrm\{GL\}(n,\backslash mathbb\{R\})$ we define a G-structure on $M$ to be a reduction of the structure group of $F\_\{\backslash mathrm\{GL\}\}(M)$ to $G$. Explicitly, this is a principal $G$-bundle $F\_\{G\}(M)$ over $M$ together with a $G$-equivariant bundle map

:$\{\backslash mathrm\; F\}\_\{G\}(M)\; \backslash to\; \{\backslash mathrm\; F\}\_\{\backslash mathrm\{GL\}\}(M)$

over $M$.

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

• Every oriented manifold has an oriented frame bundle which is just a $\backslash mathrm\{GL\}^+(n,\backslash mathbb\{R\})$-structure on $M$.
• A volume form on $M$ determines a $\backslash mathrm\{SL\}(n,\backslash mathbb\{R\})$-structure on $M$.
• A $2n$-dimensional symplectic manifold has a natural $\backslash mathrm\{Sp\}(2n,\backslash mathbb\{R\})$-structure.
• A $2n$-dimensional complex or almost complex manifold has a natural $\backslash mathrm\{GL\}(n,\backslash mathbb\{C\})$-structure.

In many of these instances, a $G$-structure on $M$ uniquely determines the corresponding structure on $M$. For example, a $\backslash mathrm\{SL\}(n,\backslash mathbb\{R\})$-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 $\backslash mathrm\{Sp\}(2n,\backslash mathbb\{R\})$-structure on $M$ uniquely determines a nondegenerate 2-form on $M$, but for $M$ to be symplectic, this 2-form must also be closed.

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Category:Fiber bundles

Category:Vector bundles