Generalised Frame Bundles

• Generalised frame bundles are a family of constructions that extend classical frame bundles by modifying admissible directions, structure groups, or quotient structures.
• They enable the lifting of maps and submersions with strong geometric conditions, ensuring properties like horizontal conformality and harmonicity.
• These constructions find applications in differential geometry, supergravity truncations, Cartan geometry, and compactified or branched moduli spaces.

In the cited literature, generalised frame bundles appear in several distinct constructions extending the classical bundle of linear or orthonormal frames. These extensions include subbundles adapted to a distribution and used to lift submersions, Cartan-type coframed manifolds that are only locally equivalent to frame bundles, $O(d,d)$-generalised Leibniz parallelisable spaces, branched and Grassmannian replacements of ordinary framings, and frame-bundle analogues for semi-principal, double, and quasi-principal bundle theories (Niedzialomski et al., 2024, Maujouy, 2022, Anderson, 2017, Billon, 2023, Bhosle et al., 2012, Pap et al., 2020, Lang et al., 2016, Doubrov et al., 2012). This suggests that the common theme is not a single canonical definition, but a family of constructions that preserve the organising role of “frames” while modifying the admissible directions, the structure group, the global quotient structure, or the very meaning of a frame.

1. Classical frame bundles as the reference model

For an $n$-dimensional manifold $M$, the ordinary linear frame bundle is

$$L(M)=\{(u_1,\dots,u_n)\mid (u_1,\dots,u_n)\text{ is a basis of }T_xM\text{ for some }x\in M\},$$

with structure group $GL(n)$. In the Riemannian case, the orthonormal frame bundle $O(M)\subset L(M)$ is the subbundle of frames orthonormal with respect to $g$, with structure group $O(n)$. For an oriented Riemannian manifold, the oriented orthonormal frame bundle is written $F(M)=\SO(M)$ (Niedzialomski et al., 2024, Limbeek, 2014).

The classical lift of a local diffeomorphism $\varphi:M\to N$ is defined by

$n$0

This construction presupposes that $n$1 is invertible on all of $n$2, and it therefore serves as the baseline from which several generalisations depart (Niedzialomski et al., 2024).

On $n$3, the Levi-Civita connection determines the canonical splitting

$n$4

and the Sasaki–Mok–O’Neill metric is

$n$5

Its fibers are totally geodesic, and this metric is rigid enough that, for closed oriented connected Riemannian $n$6-manifolds with equal fiber-volume normalisation and $n$7, the isometry type of $n$8 determines the isometry type of $n$9 (Limbeek, 2014). In this sense, ordinary frame bundles already encode substantial base-manifold geometry.

2. Distribution-adapted frame subbundles and lifts of submersions

A direct differential-geometric generalisation replaces the full tangent bundle by a chosen distribution $M$0. The construction in "Lifts of maps to frame bundles" defines subbundles $M$1 and $M$2 adapted to the splitting

$M$3

When $M$4, one recovers the classical bundles $M$5 and $M$6. In the orthonormal case, the structure group of $M$7 is

$M$8

so $M$9 is a genuine $$L(M)=\{(u_1,\dots,u_n)\mid (u_1,\dots,u_n)\text{ is a basis of }T_xM\text{ for some }x\in M\},$$0-structure subbundle (Niedzialomski et al., 2024).

The motivating case is a submersion $$L(M)=\{(u_1,\dots,u_n)\mid (u_1,\dots,u_n)\text{ is a basis of }T_xM\text{ for some }x\in M\},$$1. Its horizontal distribution is

$$L(M)=\{(u_1,\dots,u_n)\mid (u_1,\dots,u_n)\text{ is a basis of }T_xM\text{ for some }x\in M\},$$2

and the restriction

$$L(M)=\{(u_1,\dots,u_n)\mid (u_1,\dots,u_n)\text{ is a basis of }T_xM\text{ for some }x\in M\},$$3

is an isomorphism. This makes it possible to define a lift even when $$L(M)=\{(u_1,\dots,u_n)\mid (u_1,\dots,u_n)\text{ is a basis of }T_xM\text{ for some }x\in M\},$$4 is not a local diffeomorphism: $$L(M)=\{(u_1,\dots,u_n)\mid (u_1,\dots,u_n)\text{ is a basis of }T_xM\text{ for some }x\in M\},$$5 The main case studied is

$$L(M)=\{(u_1,\dots,u_n)\mid (u_1,\dots,u_n)\text{ is a basis of }T_xM\text{ for some }x\in M\},$$6

and this lift is itself a submersion (Niedzialomski et al., 2024).

The geometry of the lift is developed using the Mok metric, also called the diagonal lift metric. On $$L(M)=\{(u_1,\dots,u_n)\mid (u_1,\dots,u_n)\text{ is a basis of }T_xM\text{ for some }x\in M\},$$7,

$$L(M)=\{(u_1,\dots,u_n)\mid (u_1,\dots,u_n)\text{ is a basis of }T_xM\text{ for some }x\in M\},$$8

$$L(M)=\{(u_1,\dots,u_n)\mid (u_1,\dots,u_n)\text{ is a basis of }T_xM\text{ for some }x\in M\},$$9

A key technical device is the endomorphism

$GL(n)$0

which satisfies

$GL(n)$1

hence is an isomorphism (Niedzialomski et al., 2024).

Within this framework, the conformality and harmonicity of the lifted map become rigid. Theorem 4.1 states

$GL(n)$2

Theorem 4.2 states

$GL(n)$3

The construction therefore extends frame-bundle lifting from local diffeomorphisms to submersions, but only at the cost of replacing the full frame bundle by a distribution-adapted one and imposing strong geometric conditions on the base map (Niedzialomski et al., 2024).

3. Cartan-type local generalised frame bundles

A second major use of the term concerns manifolds equipped with a Lie-algebra-valued coframe, without any principal-bundle structure assumed a priori. In the most general form, one starts with a smooth manifold $GL(n)$4 and a coframe

$GL(n)$5

valued in a semidirect product Lie algebra

$GL(n)$6

satisfying a weakened Maurer–Cartan equation

$GL(n)$7

The defect from the ordinary Maurer–Cartan equation is horizontal: curvature and torsion are allowed, but only in the directions spanned by the solder part $GL(n)$8 (Maujouy, 9 Sep 2025).

The coframe determines an infinitesimal Lie algebra action. For $GL(n)$9,

$O(M)\subset L(M)$0

and the equivariance condition yields

$O(M)\subset L(M)$1

Thus the coframe does not merely encode tensor fields; it produces the infinitesimal symmetry algebra of the would-be frame bundle. If the fundamental vector fields are complete, if univalence holds, and if the integrated action is free and proper, then $O(M)\subset L(M)$2 is a smooth manifold and $O(M)\subset L(M)$3 is a principal $O(M)\subset L(M)$4-bundle. If isotropy varies, the quotient may instead be an orbifold or more singular orbispace (Maujouy, 9 Sep 2025).

The Lorentzian $O(M)\subset L(M)$5-dimensional version is formulated on a structure-less differentiable $O(M)\subset L(M)$6-manifold $O(M)\subset L(M)$7 with a nondegenerate $O(M)\subset L(M)$8-valued coframe

$O(M)\subset L(M)$9

The local structure equations are written as

$g$0

Under global hypotheses—integrability/globalisability of the Lie algebra action, properness, and principal orbit type on a dense open set—the manifold becomes a genuine frame or spin frame bundle

$g$1

with the solder form identifying the associated $g$2-bundle with $g$3 (Maujouy, 2022).

This local-to-global viewpoint has two notable consequences. First, generalised frame bundles need not be globally principal bundles at all: twisted-torus and $g$4 examples show that the underlying orbit space can have conical singularities (Maujouy, 9 Sep 2025). Second, in the Einstein–Cartan–Dirac setting, a variational principle on the $g$5-manifold enforces the generalised frame-bundle equations and then descends to the usual spacetime equations on the reconstructed base. The resulting spacetime system is the Einstein–Cartan–Dirac theory, with torsion algebraically determined by the spin current and satisfying

$g$6

in the paper’s notation (Maujouy, 2022, Maujouy, 2023).

4. Generalised frames in $g$7 geometry and in compactified or branched settings

In $g$8 generalised geometry, the relevant replacement for the tangent bundle is the generalised tangent bundle

$g$9

with pairing

$O(n)$0

and generalised metric

$O(n)$1

For normal bundles of adjoint orbits $O(n)$2 in a semisimple Lie group $O(n)$3, one can construct global generalised frames

$O(n)$4

satisfying a generalised Leibniz algebra with constant structure coefficients. For regular orbits, $O(n)$5 is abelian, the normal bundle is flat, and after compactifying the normal directions one obtains compact spaces suitable for generalised Scherk–Schwarz reduction and consistent supergravity truncations (Anderson, 2017).

A different line of generalisation appears in complex geometry through higher-jet frame bundles. For a holomorphic projective structure on a Riemann surface $O(n)$6, the principal bundle is the bundle of projective $O(n)$7-frames

$O(n)$8

For a branched projective structure with branching divisor

$O(n)$9

the fibre over $F(M)=\SO(M)$0 is replaced by $F(M)=\SO(M)$1-jet data, leading to the bundle of branched projective $F(M)=\SO(M)$2-frames

$F(M)=\SO(M)$3

where

$F(M)=\SO(M)$4

is the space of branching classes. The associated $F(M)=\SO(M)$5-bundle

$F(M)=\SO(M)$6

depends on the branching class, and branched projective connections of fixed type $F(M)=\SO(M)$7 form an affine space directed by

$F(M)=\SO(M)$8

whenever nonempty (Billon, 2023).

Grassmannian framed bundles provide yet another compactified version of ordinary framings. A usual framing of a rank $F(M)=\SO(M)$9 bundle $\varphi:M\to N$0 at a marked point $\varphi:M\to N$1 is essentially an isomorphism $\varphi:M\to N$2. The compactified datum is an $\varphi:M\to N$3-plane

$\varphi:M\to N$4

and the pair $\varphi:M\to N$5 is called a Grassmannian framed vector bundle. The integers

$\varphi:M\to N$6

measure the failure of $\varphi:M\to N$7 to be the graph of an isomorphism; $\varphi:M\to N$8 is exactly the ordinary framing case. The resulting moduli spaces are constructed both algebraically, as GIT quotients, and symplectically, by compactifying Jeffrey’s extended moduli spaces, with a Hitchin–Kobayashi correspondence between the two descriptions (Bhosle et al., 2012).

5. Extensions beyond ordinary principal bundles

Some generalisations change not the local linear algebra of frames, but the class of bundles to which frame-bundle ideas apply. In semi-principal bundle theory, the fibre is not a torsor but a free $\varphi:M\to N$9-space with discrete orbit space. Such a fibre is a semi-torsor,

$n$00

and a basis of $n$01 is precisely a choice of one representative in each orbit. The symmetry group of the basis space is the wreath product

$n$02

and for a semi-principal $n$03-bundle $n$04 the frame bundle

$n$05

is a principal $n$06-bundle over $n$07. Parallel transport and connection theory carry over, and the frame-bundle assignment defines a functor from semi-principal bundles to ordinary principal bundles (Pap et al., 2020).

For double vector bundles, the appropriate analogue of a frame bundle is a double principal bundle. A DPB is a compatible square of principal bundles

$n$08

with a core principal bundle

$n$09

whose structure group is

$n$10

If $n$11 is a double vector bundle of fibre dimension $n$12, then its frame bundle

$n$13

is a DPB with structure group $n$14, and the associated bundle

$n$15

is naturally isomorphic to $n$16. The “frame” of a double vector space includes frames of the two side bundles, a frame of the core, and a decomposition $n$17, so ordinary frame bundles do not capture this data (Lang et al., 2016).

A further weakening is the quasi-principal frame bundle. Let $n$18, where $n$19 is the principal bundle of graded frames adapted to a fixed Tanaka symbol $n$20. For $n$21, the vertical tangent space is encoded by

$n$22

Unlike the principal case, $n$23 may vary with $n$24; the defining condition is that the graded spaces $n$25 are constant. Under this hypothesis, the Tanaka prolongation procedure still works: if the universal prolongation $n$26 is finite-dimensional, one obtains affine bundles

$n$27

with $n$28 carrying a canonical frame. This framework is designed for flag structures arising in ODEs, distributions, and Hamiltonian geometry, where the first-step moving-frame bundle is usually not principal (Doubrov et al., 2012).

6. Metrics, rigidity, and invariant theory

Generalised frame-bundle constructions are also tied to refined metric geometry on ordinary frame bundles. Using the maps

$n$29

the horizontal distribution of $n$30 and each vertical component $n$31 can be identified with the corresponding horizontal and vertical distributions on $n$32. This makes it possible to transport natural metrics from $n$33 to $n$34. A prominent class is

$n$35

$n$36

For these metrics one has explicit formulas for the Levi–Civita connection, curvature tensor, sectional curvature, and scalar curvature. In the special choice

$n$37

if $n$38 has constant curvature

$n$39

then the sectional curvature of $n$40 is nonnegative (Niedzialomski, 2012).

In a different but related direction, framed fibre bundles with closed manifold fibres admit configuration-space-integral characteristic classes. For a smooth submersion

$n$41

with a framing of the vertical tangent bundle

$n$42

the paper constructs a partition function

$n$43

and, after truncation, a natural cdga map

$n$44

These maps generalise Kontsevich’s characteristic classes from punctured homology-sphere fibres to closed framed fibres, showing that framing data continues to support a rich invariant theory far beyond the ordinary tangent-frame setting (Prigge, 7 May 2025).

Taken together, these developments show that generalised frame bundles function in several roles: as adapted domains for lifting maps, as local Cartan-geometric data from which principal bundles may or may not be reconstructed, as global generalised frames on $n$45, as compactified or branched replacements of ordinary framings, and as frame theories for bundle categories more general than principal bundles. A persistent structural lesson is that once the rigid requirement “frame bundle = principal bundle of full linear frames” is relaxed, one gains access to submersions, singular quotients, higher-jet or compactified moduli, double and wreath-product symmetries, and prolongation theories whose output is still a canonical frame, but no longer in the classical sense (Niedzialomski et al., 2024, Maujouy, 9 Sep 2025, Anderson, 2017, Lang et al., 2016, Doubrov et al., 2012).