Frame bundle

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

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.

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

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

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

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

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

The frame bundle of ${\displaystyle E}$, denoted by ${\displaystyle F(E)}$ or ${\displaystyle F_{\mathrm {GL} }(E)}$, is the disjoint 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 ${\displaystyle \mathrm {GL} (k,\mathbb {R} )}$ acts on ${\displaystyle F(E)}$ on the right as above. This action is clearly free and the orbits 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 ${\displaystyle (U_{i},\phi _{i})}$ be a local trivialization of ${\displaystyle E}$. Then for each x ∈ U_i one has a linear isomorphism ${\displaystyle \phi _{i,x}:E_{x}\to \mathbb {R} ^{k}}$. This data determines a bijection

${\displaystyle \psi _{i}:\pi ^{-1}(U_{i})\to U_{i}\times \mathrm {GL} (k,\mathbb {R} )}$

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 ${\displaystyle U_{i}\times \mathrm {GL} (k,\mathbb {R} )}$. The topology on ${\displaystyle F(E)}$ is the final 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 a principal fiber bundle over ${\displaystyle X}$ with structure group ${\displaystyle \mathrm {GL} (k,\mathbb {R} )}$ and local trivializations ${\displaystyle (\{U_{i}\},\{\psi _{i}\})}$. One can check that the transition 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 a smooth 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 ${\displaystyle F(E)}$ are associated bundles. Each one determines the other. The frame bundle ${\displaystyle F(E)}$ can be constructed from ${\displaystyle E}$ as above, or more abstractly using the fiber 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 ${\displaystyle \mathrm {GL} (k,\mathbb {R} )}$, where the action of structure group ${\displaystyle \mathrm {GL} (k,\mathbb {R} )}$ on the fiber ${\displaystyle \mathrm {GL} (k,\mathbb {R} )}$ is that of left multiplication.

Given any linear representation ${\displaystyle \rho :\mathrm {GL} (k,\mathbb {R} )\to \mathrm {GL} (V,\mathbb {F} )}$ 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 the equivalence relation ${\displaystyle (pg,v)\sim (p,\rho (g)v)}$ for all ${\displaystyle g}$ in ${\displaystyle \mathrm {GL} (k,\mathbb {R} )}$. Denote the equivalence classes by ${\displaystyle [p,v]}$.

The vector bundle ${\displaystyle E}$ is naturally isomorphic to the bundle ${\displaystyle F(E)\times _{\rho }\mathbb {R} ^{k}}$ where ${\displaystyle \rho }$ is the fundamental representation of ${\displaystyle \mathrm {GL} (k,\mathbb {R} )}$ 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 is well-defined.

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

The tangent frame bundle (or simply the frame bundle) of a smooth manifold ${\displaystyle M}$ is the frame bundle associated with the tangent bundle of ${\displaystyle M}$. The frame bundle of ${\displaystyle M}$ is often denoted ${\displaystyle FM}$ or ${\displaystyle \mathrm {GL} (M)}$ 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 ${\displaystyle \mathrm {GL} (n,\mathbb {R} )}$ bundle over ${\displaystyle M}$.

Local sections of the frame bundle of ${\displaystyle M}$ are called smooth 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 ${\displaystyle \psi :FU\to U\times \mathrm {GL} (n,\mathbb {R} )}$ 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 is parallelizable 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 ${\displaystyle (x^{1},\ldots ,x^{n})}$ the coordinate vector fields

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

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 the method 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 a vector-valued 1-form on ${\displaystyle FM}$ called the solder form (also known as the fundamental or tautological 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 the differential 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 ${\displaystyle \pi }$ and right equivariant in the sense that

${\displaystyle R_{g}^{*}\theta =g^{-1}\theta }$

where ${\displaystyle R_{g}}$ is right translation by ${\displaystyle g\in \mathrm {GL} (n,\mathbb {R} )}$. A form with these properties is called a basic or tensorial 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 smooth bundle maps ${\displaystyle TM\to TM}$ over ${\displaystyle M}$. Viewed in this light ${\displaystyle \theta }$ is just the identity 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 a Riemannian bundle metric then each fiber ${\displaystyle E_{x}}$ 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 ${\displaystyle E_{x}}$. An orthonormal frame for ${\displaystyle E_{x}}$ is an ordered orthonormal basis for ${\displaystyle E_{x}}$, or, equivalently, a linear isometry

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

where ${\displaystyle \mathbb {R} ^{k}}$ is equipped with the standard Euclidean metric. The orthogonal 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.

The orthonormal 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 ${\displaystyle E}$ is orientable then one can define the oriented orthonormal frame bundle of ${\displaystyle E}$, denoted ${\displaystyle F_{\mathrm {SO} }(E)}$, as the principal ${\displaystyle \mathrm {SO} (k)}$-bundle of all positively oriented orthonormal frames.

If ${\displaystyle M}$ is an ${\displaystyle n}$-dimensional Riemannian 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 ${\displaystyle F_{\mathrm {SO} }M}$.

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

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

is principal bundle map. One says that ${\displaystyle F_{\mathrm {O} }(E)}$ is a reduction of the structure group of ${\displaystyle F_{\mathrm {GL} }(E)}$ from ${\displaystyle \mathrm {GL} (n,\mathbb {R} )}$ 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 ${\displaystyle F_{\mathrm {GL} }(M)}$ to the orthogonal group ${\displaystyle \mathrm {O} (n)}$.

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

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

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.

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

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

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