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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• Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry (PDF), Springer-Verlag, archived from the original (PDF) on 2017-03-30, retrieved 2008-08-02
• Sternberg, S. (1983), Lectures on Differential Geometry ((2nd ed.) ed.), New York: Chelsea Publishing Co., ISBN 0-8218-1385-4