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In `F33f`_`[mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mathematics]`_`f, a `!frame bundle`! is a `F33f`_`[principal fiber bundle`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Principal_fiber_bundle]`_`f F ( E ) {\\displaystyle F(E)} associated with any `F33f`_`[vector bundle`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Vector_bundle]`_`f `* E {\\displaystyle E} `*. The fiber of F ( E ) {\\displaystyle F(E)} over a point `* x {\\displaystyle x} `* is the set of all `F33f`_`[ordered bases`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Ordered_basis]`_`f, or `*frames`*, for `* E x {\\displaystyle E_{x}} `*. The `F33f`_`[general linear group`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=General_linear_group]`_`f acts naturally on F ( E ) {\\displaystyle F(E)} via a `F33f`_`[change of basis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Change_of_basis]`_`f, giving the frame bundle the structure of a principal `* G L ( k , R ) {\\displaystyle \\mathrm {GL} (k,\\mathbb {R} )} `*-bundle (where `*k`* is the rank of `* E {\\displaystyle E} `*).

The frame bundle of a `F33f`_`[smooth manifold`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Smooth_manifold]`_`f is the one associated with its `F33f`_`[tangent bundle`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Tangent_bundle]`_`f. For this reason it is sometimes called the `!tangent frame bundle`!.

>>Contents

• `F0af`_`[Definition and construction`#definition-and-construction]`_`f

• `F0af`_`[Principal bundle structure`#principal-bundle-structure]`_`f

• `F0af`_`[Associated vector bundles`#associated-vector-bundles]`_`f

• `F0af`_`[Tangent frame bundle`#tangent-frame-bundle]`_`f

• `F0af`_`[Smooth frames`#smooth-frames]`_`f

• `F0af`_`[Solder form`#solder-form]`_`f

• `F0af`_`[Orthonormal frame bundle`#orthonormal-frame-bundle]`_`f

• `F0af`_`[G -structures`#g-structures]`_`f

• `F0af`_`[References`#references]`_`f

-─

>>Definition and construction

Let `* E → → X {\\displaystyle E\\to X} `* be a real `F33f`_`[vector bundle`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Vector_bundle]`_`f of rank `* k {\\displaystyle k} `* over a `F33f`_`[topological space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Topological_space]`_`f `* X {\\displaystyle X} `*. A `!frame`! at a point `* x ∈ ∈ X {\\displaystyle x\\in X} `* is an `F33f`_`[ordered basis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Ordered_basis]`_`f for the vector space `* E x {\\displaystyle E_{x}} `*. Equivalently, a frame can be viewed as a `F33f`_`[linear isomorphism`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Linear_isomorphism]`_`f

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

The set of all frames at `* x {\\displaystyle x} `*, denoted `* F x {\\displaystyle F_{x}} `*, has a natural `F33f`_`[right action`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Group_action_(mathematics)]`_`f by the `F33f`_`[general linear group`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=General_linear_group]`_`f `* G L ( k , R ) {\\displaystyle \\mathrm {GL} (k,\\mathbb {R} )} `* of invertible `* k × × k {\\displaystyle k\\times k} `* matrices: a group element `* g ∈ ∈ G L ( k , R ) {\\displaystyle g\\in \\mathrm {GL} (k,\\mathbb {R} )} `* acts on the frame `* p {\\displaystyle p} `* via `F33f`_`[composition`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Function_composition]`_`f to give a new frame

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

This action of `* G L ( k , R ) {\\displaystyle \\mathrm {GL} (k,\\mathbb {R} )} `* on `* F x {\\displaystyle F_{x}} `* is both `F33f`_`[free`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Free_action]`_`f and `F33f`_`[transitive`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Transitive_action]`_`f (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 {\\displaystyle F_{x}} `* is `F33f`_`[homeomorphic`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Homeomorphic]`_`f to `* G L ( k , R ) {\\displaystyle \\mathrm {GL} (k,\\mathbb {R} )} `* although it lacks a group structure, since there is no "preferred frame". The space `* F x {\\displaystyle F_{x}} `* is said to be a `* G L ( k , R ) {\\displaystyle \\mathrm {GL} (k,\\mathbb {R} )} `*-`F33f`_`[torsor`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Torsor]`_`f.

The `!frame bundle`! of `* E {\\displaystyle E} `*, denoted by F ( E ) {\\displaystyle F(E)} or F G L ( E ) {\\displaystyle F_{\\mathrm {GL} }(E)} , is the `F33f`_`[disjoint union`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Disjoint_union]`_`f of all the `* F x {\\displaystyle F_{x}} `*:

F ( E ) = ∐ ∐ x ∈ ∈ X F x . {\\displaystyle \\mathrm {F} (E)=\\coprod _{x\\in X}F_{x}.}

Each point in F ( E ) {\\displaystyle F(E)} is a pair (`*x`*, `*p`*) where `* x {\\displaystyle x} `* is a point in `* X {\\displaystyle X} `* and `* p {\\displaystyle p} `* is a frame at `* x {\\displaystyle x} `*. There is a natural projection π π : F ( E ) → → X {\\displaystyle \\pi :F(E)\\to X} which sends `*`! ( x , p ) {\\displaystyle (x,p)} `!`* to `* x {\\displaystyle x} `*. The group `* G L ( k , R ) {\\displaystyle \\mathrm {GL} (k,\\mathbb {R} )} `* acts on F ( E ) {\\displaystyle F(E)} on the right as above. This action is clearly free and the `F33f`_`[orbits`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Orbit_(group_theory)]`_`f are just the fibers of `*`! π π {\\displaystyle \\pi } `!`*.

>>>Principal bundle structure

The frame bundle F ( E ) {\\displaystyle F(E)} can be given a natural topology and bundle structure determined by that of `* E {\\displaystyle E} `*. Let `*`! ( U i , ϕ ϕ i ) {\\displaystyle (U_{i},\\phi _{i})} `!`* be a `F33f`_`[local trivialization`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Local_trivialization]`_`f of `* E {\\displaystyle E} `*. Then for each `*x`* ∈ `*U`*`*i`* one has a linear isomorphism `*`! ϕ ϕ i , x : E x → → R k {\\displaystyle \\phi _{i,x}:E_{x}\\to \\mathbb {R} ^{k}} `!`*. This data determines a bijection

ψ ψ i : π π − − 1 ( U i ) → → U i × × G L ( k , R ) {\\displaystyle \\psi _{i}:\\pi ^{-1}(U_{i})\\to U_{i}\\times \\mathrm {GL} (k,\\mathbb {R} )}

given by

ψ ψ i ( x , p ) = ( x , ϕ ϕ i , x ∘ ∘ p ) . {\\displaystyle \\psi _{i}(x,p)=(x,\\phi _{i,x}\\circ p).}

With these bijections, each `*`! π π − − 1 ( U i ) {\\displaystyle \\pi ^{-1}(U_{i})} `!`* can be given the topology of `* U i × × G L ( k , R ) {\\displaystyle U_{i}\\times \\mathrm {GL} (k,\\mathbb {R} )} `*. The topology on F ( E ) {\\displaystyle F(E)} is the `F33f`_`[final topology`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Final_topology]`_`f coinduced by the inclusion maps `*`! π π − − 1 ( U i ) → → F ( E ) {\\displaystyle \\pi ^{-1}(U_{i})\\to F(E)} `!`*.

With all of the above data the frame bundle F ( E ) {\\displaystyle F(E)} becomes a `F33f`_`[principal fiber bundle`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Principal_fiber_bundle]`_`f over `* X {\\displaystyle X} `* with `F33f`_`[structure group`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Structure_group]`_`f `* G L ( k , R ) {\\displaystyle \\mathrm {GL} (k,\\mathbb {R} )} `* and local trivializations `*`! ( { U i } , { ψ ψ i } ) {\\displaystyle (\\{U_{i}\\},\\{\\psi _{i}\\})} `!`*. One can check that the `F33f`_`[transition functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Transition_map]`_`f of F ( E ) {\\displaystyle F(E)} are the same as those of `* E {\\displaystyle E} `*.

The above all works in the smooth category as well: if `* E {\\displaystyle E} `* is a smooth vector bundle over a `F33f`_`[smooth manifold`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Smooth_manifold]`_`f `* M {\\displaystyle M} `* then the frame bundle of `* E {\\displaystyle E} `* can be given the structure of a smooth principal bundle over `* M {\\displaystyle M} `*.

>>Associated vector bundles

A vector bundle `* E {\\displaystyle E} `* and its frame bundle F ( E ) {\\displaystyle F(E)} are `F33f`_`[associated bundles`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Associated_bundle]`_`f. Each one determines the other. The frame bundle F ( E ) {\\displaystyle F(E)} can be constructed from `* E {\\displaystyle E} `* as above, or more abstractly using the `F33f`_`[fiber bundle construction theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Fiber_bundle_construction_theorem]`_`f. With the latter method, F ( E ) {\\displaystyle F(E)} is the fiber bundle with same base, structure group, trivializing neighborhoods, and transition functions as `* E {\\displaystyle E} `* but with abstract fiber `* G L ( k , R ) {\\displaystyle \\mathrm {GL} (k,\\mathbb {R} )} `*, where the action of structure group `* G L ( k , R ) {\\displaystyle \\mathrm {GL} (k,\\mathbb {R} )} `* on the fiber `* G L ( k , R ) {\\displaystyle \\mathrm {GL} (k,\\mathbb {R} )} `* is that of left multiplication.

Given any `F33f`_`[linear representation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Linear_representation]`_`f `* ρ ρ : G L ( k , R ) → → G L ( V , F ) {\\displaystyle \\rho :\\mathrm {GL} (k,\\mathbb {R} )\\to \\mathrm {GL} (V,\\mathbb {F} )} `* there is a vector bundle

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

associated with F ( E ) {\\displaystyle F(E)} which is given by product F ( E ) × × V {\\displaystyle F(E)\\times V} modulo the `F33f`_`[equivalence relation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Equivalence_relation]`_`f `*`! ( p g , v ) ∼ ∼ ( p , ρ ρ ( g ) v ) {\\displaystyle (pg,v)\\sim (p,\\rho (g)v)} `!`* for all `* g {\\displaystyle g} `* in `* G L ( k , R ) {\\displaystyle \\mathrm {GL} (k,\\mathbb {R} )} `*. Denote the equivalence classes by `*`! [ p , v ] {\\displaystyle [p,v]} `!`*.

The vector bundle `* E {\\displaystyle E} `* is `F33f`_`[naturally isomorphic`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Naturally_isomorphic]`_`f to the bundle F ( E ) × × ρ ρ R k {\\displaystyle F(E)\\times _{\\rho }\\mathbb {R} ^{k}} where `*`! ρ ρ {\\displaystyle \\rho } `!`* is the fundamental representation of `* G L ( k , R ) {\\displaystyle \\mathrm {GL} (k,\\mathbb {R} )} `* on `*`! R k {\\displaystyle \\mathbb {R} ^{k}} `!`*. The isomorphism is given by

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

where `*`! v {\\displaystyle v} `!`* is a vector in `*`! R k {\\displaystyle \\mathbb {R} ^{k}} `!`* and `*`! p : R k → → E x {\\displaystyle p:\\mathbb {R} ^{k}\\to E_{x}} `!`* is a frame at `* x {\\displaystyle x} `*. One can easily check that this map is `F33f`_`[well-defined`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Well-defined]`_`f.

Any vector bundle associated with `* E {\\displaystyle E} `* can be given by the above construction. For example, the `F33f`_`[dual bundle`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dual_bundle]`_`f of `* E {\\displaystyle E} `* is given by F ( E ) × × ρ ρ ∗ ∗ ( R k ) ∗ ∗ {\\displaystyle F(E)\\times _{\\rho ^{*}}(\\mathbb {R} ^{k})^{*}} where ρ ρ ∗ ∗ {\\displaystyle \\rho ^{*}} is the `F33f`_`[dual`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dual_representation]`_`f of the fundamental representation. `F33f`_`[Tensor bundles`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Tensor_bundle]`_`f of `* E {\\displaystyle E} `* can be constructed in a similar manner.

>>Tangent frame bundle

The `!tangent frame bundle`! (or simply the `!frame bundle`!) of a `F33f`_`[smooth manifold`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Smooth_manifold]`_`f `* M {\\displaystyle M} `* is the frame bundle associated with the `F33f`_`[tangent bundle`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Tangent_bundle]`_`f of `* M {\\displaystyle M} `*. The frame bundle of `* M {\\displaystyle M} `* is often denoted `* F M {\\displaystyle FM} `* or `* G L ( M ) {\\displaystyle \\mathrm {GL} (M)} `* rather than `* F ( T M ) {\\displaystyle F(TM)} `*. In physics, it is sometimes denoted `* L M {\\displaystyle LM} `*. If `* M {\\displaystyle M} `* is `* n {\\displaystyle n} `*-dimensional then the tangent bundle has rank `* n {\\displaystyle n} `*, so the frame bundle of `* M {\\displaystyle M} `* is a principal `* G L ( n , R ) {\\displaystyle \\mathrm {GL} (n,\\mathbb {R} )} `* bundle over `* M {\\displaystyle M} `*.

>>>Smooth frames

`F33f`_`[Local sections`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Section_(fiber_bundle)]`_`f of the frame bundle of `* M {\\displaystyle M} `* are called `F33f`_`[smooth frames`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Smooth_frame]`_`f on `* M {\\displaystyle M} `*. The cross-section theorem for principal bundles states that the frame bundle is trivial over any open set in `* U {\\displaystyle U} `* in `* M {\\displaystyle M} `* which admits a smooth frame. Given a smooth frame `* s : U → → F U {\\displaystyle s:U\\to FU} `*, the trivialization `* ψ ψ : F U → → U × × G L ( n , R ) {\\displaystyle \\psi :FU\\to U\\times \\mathrm {GL} (n,\\mathbb {R} )} `* is given by

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

where `* p {\\displaystyle p} `* is a frame at `* x {\\displaystyle x} `*. It follows that a manifold is `F33f`_`[parallelizable`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Parallelizable_manifold]`_`f if and only if the frame bundle of `* M {\\displaystyle M} `* admits a global section.

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

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

define a smooth frame on `* U {\\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 `F33f`_`[method of moving frames`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Method_of_moving_frames]`_`f.

>>>Solder form

The frame bundle of a manifold `* M {\\displaystyle M} `* is a special type of principal bundle in the sense that its geometry is fundamentally tied to the geometry of `* M {\\displaystyle M} `*. This relationship can be expressed by means of a `F33f`_`[vector-valued 1-form`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Vector-valued_differential_form]`_`f on `* F M {\\displaystyle FM} `* called the `!`F33f`_`[solder form`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Solder_form]`_`f`! (also known as the `!fundamental`! or `F33f`_`[tautological 1-form`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Tautological_one-form]`_`f). Let `* x {\\displaystyle x} `* be a point of the manifold `* M {\\displaystyle M} `* and `* p {\\displaystyle p} `* a frame at `* x {\\displaystyle x} `*, so that

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

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

θ θ p ( ξ ξ ) = p − − 1 d π π ( ξ ξ ) {\\displaystyle \\theta _{p}(\\xi )=p^{-1}\\mathrm {d} \\pi (\\xi )}

where ξ is a tangent vector to `* F M {\\displaystyle FM} `* at the point `* ( x , p ) {\\displaystyle (x,p)} `*, and `* p − − 1 : T x M → → R n {\\displaystyle p^{-1}:T_{x}M\\to \\mathbb {R} ^{n}} `* is the inverse of the frame map, and `* d π π {\\displaystyle d\\pi } `* is the `F33f`_`[differential`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Pushforward_(differential)]`_`f of the projection map `* π π : F M → → M {\\displaystyle \\pi :FM\\to M} `*. The solder form is horizontal in the sense that it vanishes on vectors tangent to the fibers of `* π π {\\displaystyle \\pi } `* and `F33f`_`[right equivariant`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Equivariant]`_`f in the sense that

R g ∗ ∗ θ θ = g − − 1 θ θ {\\displaystyle R_{g}^{*}\\theta =g^{-1}\\theta }

where `* R g {\\displaystyle R_{g}} `* is right translation by `* g ∈ ∈ G L ( n , R ) {\\displaystyle g\\in \\mathrm {GL} (n,\\mathbb {R} )} `*. A form with these properties is called a basic or `F33f`_`[tensorial form`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Tensorial_form]`_`f on `* F M {\\displaystyle FM} `*. Such forms are in 1-1 correspondence with `* T M {\\displaystyle TM} `*-valued 1-forms on `* M {\\displaystyle M} `* which are, in turn, in 1-1 correspondence with smooth `F33f`_`[bundle maps`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bundle_map]`_`f `* T M → → T M {\\displaystyle TM\\to TM} `* over `* M {\\displaystyle M} `*. Viewed in this light `* θ θ {\\displaystyle \\theta } `* is just the `F33f`_`[identity map`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Identity_function]`_`f on `* T M {\\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.

>>Orthonormal frame bundle

If a vector bundle `* E {\\displaystyle E} `* is equipped with a `F33f`_`[Riemannian bundle metric`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Riemannian_bundle_metric]`_`f then each fiber `* E x {\\displaystyle E_{x}} `* is not only a vector space but an `F33f`_`[inner product space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Inner_product_space]`_`f. It is then possible to talk about the set of all `F33f`_`[orthonormal frames`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Orthonormal_frame]`_`f for `* E x {\\displaystyle E_{x}} `*. An orthonormal frame for `* E x {\\displaystyle E_{x}} `* is an ordered `F33f`_`[orthonormal basis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Orthonormal_basis]`_`f for `* E x {\\displaystyle E_{x}} `*, or, equivalently, a `F33f`_`[linear isometry`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Linear_isometry]`_`f

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

where `* R k {\\displaystyle \\mathbb {R} ^{k}} `* is equipped with the standard `F33f`_`[Euclidean metric`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Euclidean_metric]`_`f. The `F33f`_`[orthogonal group`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Orthogonal_group]`_`f `* O ( k ) {\\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 `* O ( k ) {\\displaystyle \\mathrm {O} (k)} `*-`F33f`_`[torsor`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Torsor]`_`f.

The `!orthonormal frame bundle`! of `* E {\\displaystyle E} `*, denoted `* F O ( E ) {\\displaystyle F_{\\mathrm {O} }(E)} `*, is the set of all orthonormal frames at each point `* x {\\displaystyle x} `* in the base space `* X {\\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 `* k {\\displaystyle k} `* Riemannian vector bundle `* E → → X {\\displaystyle E\\to X} `* is a principal `* O ( k ) {\\displaystyle \\mathrm {O} (k)} `*-bundle over `* X {\\displaystyle X} `*. Again, the construction works just as well in the smooth category.

If the vector bundle `* E {\\displaystyle E} `* is `F33f`_`[orientable`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Orientability]`_`f then one can define the `!oriented orthonormal frame bundle`! of `* E {\\displaystyle E} `*, denoted `* F S O ( E ) {\\displaystyle F_{\\mathrm {SO} }(E)} `*, as the principal `* S O ( k ) {\\displaystyle \\mathrm {SO} (k)} `*-bundle of all positively oriented orthonormal frames.

If `* M {\\displaystyle M} `* is an `* n {\\displaystyle n} `*-dimensional `F33f`_`[Riemannian manifold`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Riemannian_manifold]`_`f, then the orthonormal frame bundle of `* M {\\displaystyle M} `*, denoted `* F O ( M ) {\\displaystyle F_{\\mathrm {O} }(M)} `* or `* O ( M ) {\\displaystyle \\mathrm {O} (M)} `*, is the orthonormal frame bundle associated with the tangent bundle of `* M {\\displaystyle M} `* (which is equipped with a Riemannian metric by definition). If `* M {\\displaystyle M} `* is orientable, then one also has the oriented orthonormal frame bundle `* F S O M {\\displaystyle F_{\\mathrm {SO} }M} `*.

Given a Riemannian vector bundle `* E {\\displaystyle E} `*, the orthonormal frame bundle is a principal `* O ( k ) {\\displaystyle \\mathrm {O} (k)} `*-`F33f`_`[subbundle`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Subbundle]`_`f of the general linear frame bundle. In other words, the inclusion map

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

is principal `F33f`_`[bundle map`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bundle_map]`_`f. One says that `* F O ( E ) {\\displaystyle F_{\\mathrm {O} }(E)} `* is a `F33f`_`[reduction of the structure group`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Reduction_of_the_structure_group]`_`f of `* F G L ( E ) {\\displaystyle F_{\\mathrm {GL} }(E)} `* from `* G L ( n , R ) {\\displaystyle \\mathrm {GL} (n,\\mathbb {R} )} `* to `* O ( k ) {\\displaystyle \\mathrm {O} (k)} `*.

>>G -structures

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

In general, if `* M {\\displaystyle M} `* is a smooth `* n {\\displaystyle n} `*-manifold and `* G {\\displaystyle G} `* is a `F33f`_`[Lie subgroup`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Lie_subgroup]`_`f of `* G L ( n , R ) {\\displaystyle \\mathrm {GL} (n,\\mathbb {R} )} `* we define a `!`F33f`_`[G-structure`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=G-structure]`_`f`! on `* M {\\displaystyle M} `* to be a `F33f`_`[reduction of the structure group`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Reduction_of_the_structure_group]`_`f of `* F G L ( M ) {\\displaystyle F_{\\mathrm {GL} }(M)} `* to `* G {\\displaystyle G} `*. Explicitly, this is a principal `* G {\\displaystyle G} `*-bundle `* F G ( M ) {\\displaystyle F_{G}(M)} `* over `* M {\\displaystyle M} `* together with a `* G {\\displaystyle G} `*-equivariant `F33f`_`[bundle map`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bundle_map]`_`f

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

over `* M {\\displaystyle M} `*.

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

• Every `F33f`_`[oriented manifold`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Orientability]`_`f has an oriented frame bundle which is just a `* G L + ( n , R ) {\\displaystyle \\mathrm {GL} ^{+}(n,\\mathbb {R} )} `*-structure on `* M {\\displaystyle M} `*.

• A `F33f`_`[volume form`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Volume_form]`_`f on `* M {\\displaystyle M} `* determines a `* S L ( n , R ) {\\displaystyle \\mathrm {SL} (n,\\mathbb {R} )} `*-structure on `* M {\\displaystyle M} `*.

• A `* 2 n {\\displaystyle 2n} `*-dimensional `F33f`_`[symplectic manifold`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Symplectic_manifold]`_`f has a natural `* S p ( 2 n , R ) {\\displaystyle \\mathrm {Sp} (2n,\\mathbb {R} )} `*-structure.

• A `* 2 n {\\displaystyle 2n} `*-dimensional `F33f`_`[complex`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Complex_manifold]`_`f or `F33f`_`[almost complex manifold`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Almost_complex_manifold]`_`f has a natural `* G L ( n , C ) {\\displaystyle \\mathrm {GL} (n,\\mathbb {C} )} `*-structure.

In many of these instances, a `* G {\\displaystyle G} `*-structure on `* M {\\displaystyle M} `* uniquely determines the corresponding structure on `* M {\\displaystyle M} `*. For example, a `* S L ( n , R ) {\\displaystyle \\mathrm {SL} (n,\\mathbb {R} )} `*-structure on `* M {\\displaystyle M} `* determines a volume form on `* M {\\displaystyle M} `*. However, in some cases, such as for symplectic and complex manifolds, an added `F33f`_`[integrability condition`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Integrability_condition]`_`f is needed. A `* S p ( 2 n , R ) {\\displaystyle \\mathrm {Sp} (2n,\\mathbb {R} )} `*-structure on `* M {\\displaystyle M} `* uniquely determines a `F33f`_`[nondegenerate`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Nondegenerate_form]`_`f `F33f`_`[2-form`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=2-form]`_`f on `* M {\\displaystyle M} `*, but for `* M {\\displaystyle M} `* to be symplectic, this 2-form must also be `F33f`_`[closed`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Closed_differential_form]`_`f.

>>References

• `:citerefkobayashinomizu1996`aKobayashi, Shoshichi; Nomizu, Katsumi (1996), `*`F33f`_`[Foundations of Differential Geometry`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Foundations_of_Differential_Geometry]`_`f`*, vol. 1 (New ed.), `F33f`_`[Wiley Interscience`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Wiley_Interscience]`_`f, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-471-15733-3

• `:citerefkol-michorslov-k1993`aKolář, 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

• `:citerefsternberg1983`a`F33f`_`[Sternberg, S.`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Shlomo_Sternberg]`_`f (1983), `*Lectures on Differential Geometry`* ((2nd ed.) ed.), New York: Chelsea Publishing Co., `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-8218-1385-4

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