Generalised Exterior Curvature

• Generalised exterior curvature is a framework that extends classical curvature to nontraditional geometric settings, including pullback Courant algebroids and exterior differential forms.
• It derives generalized Gauß–Codazzi equations, addresses constraint conditions in generalized Einstein equations, and employs canonical connections via generalized metrics and divergence operators.
• Its applications span discrete and combinatorial geometry, generalized Cartan structures, and analytic settings like optimal transport, broadening curvature’s classical role.

Generalised exterior curvature denotes a family of curvature constructions in which classical curvature is reformulated on extended geometric objects, most prominently exact Courant algebroids, tensor-valued differential forms, hybrid simplicial–dual cells, extended tangent bundles, and other nonclassical configuration spaces. In the most direct recent usage, an exact Courant algebroid $E\to M$ together with a hypersurface immersion $\iota:N\hookrightarrow M$ supports a notion of generalised exterior curvature built from the pullback Courant algebroid $\iota^!E$, the pullback of generalised metrics and divergence operators, and hypersurface invariants including the generalised second fundamental form and the generalised mean curvature; this framework yields generalised Gauß–Codazzi equations, constraint equations for the initial value formulation of the generalised Einstein equations, and restriction results for generalised Kähler and hyper-Kähler structures (Cortés et al., 16 Jul 2025). Across adjacent literatures, the same expression or closely related ones also refer to exterior-form formulations of curvature, generalized Cartan-curvature hierarchies, discrete curvature measures on piecewise-flat manifolds, and curvature-like obstructions in transport, network, PDE, and Radon-transform settings. This suggests that the term is best understood as a research-level umbrella for curvature notions that extend classical differential-geometric exterior constructions beyond the tangent-bundle and smooth-tensor setting.

1. Generalised exterior curvature in exact Courant algebroids

The most literal recent formulation arises in the setting of exact Courant algebroids. Given an exact Courant algebroid $E\to M$ and an immersion $\iota:N\hookrightarrow M$, there is a well-known construction of an exact Courant algebroid $\iota^!E\to N$, the pullback of $E$. In this setting, the pullback of generalised metrics and divergence operators can be described, and, assuming $N$ is a hypersurface, one obtains a notion of generalised exterior curvature together with the generalised second fundamental form and the generalised mean curvature (Cortés et al., 16 Jul 2025).

The same framework produces generalised versions of the Gauß–Codazzi equations. As stated in the abstract of "Exterior Generalised Geometry" (Cortés et al., 16 Jul 2025), these equations are then applied to the constraint equations for the initial value formulation of the generalised Einstein equations. The paper further states that there is a generalised geometry version of the fundamental theorem for hypersurfaces and that generalised Kähler and hyper-Kähler structures restrict to submanifolds compatible with the generalised almost complex structure. It also characterises exact semi-Riemannian Courant algebroids which are flat with respect to the canonical generalised connection, and these play the role of the ambient space in the fundamental theorem mentioned there (Cortés et al., 16 Jul 2025).

A plausible implication is that, in this usage, generalised exterior curvature functions as the hypersurface-level analogue of classical immersion theory, but now on exact Courant algebroids rather than on ordinary semi-Riemannian tangent bundles. The available source text does not provide the explicit definitions or formulas, but it identifies the principal ingredients: pullback Courant algebroids, generalised metrics, divergence operators, second fundamental form, mean curvature, and Gauß–Codazzi structure (Cortés et al., 16 Jul 2025).

2. Canonical generalised Levi-Civita curvature and curvature invariants

A closely related development concerns curvature of exact Courant algebroids equipped with a generalised metric $\mathcal G$ and a divergence operator $\mathrm{div}$. In this setting, the exact Courant algebroid has the form

$\iota:N\hookrightarrow M$0

with the standard Courant structure. A generalized metric is an orthogonal involution of $\iota:N\hookrightarrow M$1 such that

$\iota:N\hookrightarrow M$2

with $\iota:N\hookrightarrow M$3 positive definite and $\iota:N\hookrightarrow M$4 negative definite with respect to the pairing, while a divergence operator is a first-order operator on sections of $\iota:N\hookrightarrow M$5 satisfying the usual Leibniz rule with respect to functions and the anchor (Cortés et al., 23 Jul 2025).

The central issue is that generalized Levi-Civita connections are generally not unique. The construction of a canonical connection

$\iota:N\hookrightarrow M$6

resolves this by supplementing the generalized metric with the divergence operator. This connection is characterized by compatibility with the generalized metric, metric compatibility with the Courant pairing, vanishing generalized torsion in the Courant sense, and prescribed divergence

$\iota:N\hookrightarrow M$7

Its curvature can then be treated canonically, rather than as an artefact of an arbitrary choice inside an affine space of torsion-free metric-compatible generalized connections (Cortés et al., 23 Jul 2025).

Within this paper, the meaning of “generalised exterior curvature” is broadened. The curvature operator is defined by

$\iota:N\hookrightarrow M$8

with the Courant bracket interpreted appropriately. The details emphasize that curvature should not be thought of only as the classical $\iota:N\hookrightarrow M$9 of a connection: because the Courant bracket is not a Lie bracket, the curvature structure naturally includes extra components and contraction operations that lead to several Ricci-type objects. The generalised Riemann tensor associated to $\iota^!E$0 admits a master decomposition into classical Riemannian curvature, $\iota^!E$1-flux contributions, and divergence-dependent correction terms. The corresponding scalar curvature has the schematic form

$\iota^!E$2

so the curvature invariants of $\iota^!E$3 package ordinary curvature, flux, and generalized volume data in a single Courant-algebroid formalism (Cortés et al., 23 Jul 2025).

This relationship is significant for the exterior-curvature language of hypersurfaces. The hypersurface theory of (Cortés et al., 16 Jul 2025) explicitly invokes generalised metrics, divergence operators, and a canonical generalised connection, while (Cortés et al., 23 Jul 2025) supplies the canonical curvature tool-kit for such data. This suggests that generalised exterior curvature in Courant geometry is naturally embedded in a wider canonical curvature theory for the pair $\iota^!E$4.

3. Generalised Cartan geometry and curvature hierarchies

A second major line of work interprets generalized curvature through Cartan geometry. In "Current Algebra and Generalised Cartan Geometry" (Hassler et al., 2024), the usual Cartan-geometric package

$\iota^!E$5

is lifted to

$\iota^!E$6

where $\iota^!E$7 is a principal $\iota^!E$8-bundle and $\iota^!E$9 is a quasi-Lie bialgebra with an isotropic subalgebra $E\to M$0. Ordinary Cartan curvature is

$E\to M$1

whereas generalized Cartan curvature is defined by

$E\to M$2

Its components contain generalized torsion, generalized Riemann curvature, and additional higher tensors (Hassler et al., 2024).

In the generalized metric formalism, the generalized tangent space is split using

$E\to M$3

and the generalized Levi-Civita type connections satisfy $E\to M$4, vanishing generalized torsion, and compatibility with $E\to M$5 and the dilaton density. The generalized Riemann tensor is extracted from the generalized Cartan curvature by

$E\to M$6

while a five-index tensor $E\to M$7 organizes higher generalized tensors. The central result is that most of these additional generalized tensors are actually covariant derivatives of the generalized Riemann tensor (Hassler et al., 2024).

A closely related but higher-algebraic generalization appears in "Gauged Extended Field Theory and Generalised Cartan Geometry" (Hassler et al., 4 Sep 2025) and "Generalised Cartan Geometry" (Osten, 20 May 2026). There the extended tangent bundle is organized by a tensor hierarchy,

$E\to M$8

and the generalized curvature is extracted from the brane current algebra through

$E\to M$9

At linearized level, the curvature of the generalized spin connection is

$\iota:N\hookrightarrow M$0

and for $\iota:N\hookrightarrow M$1,

$\iota:N\hookrightarrow M$2

These are the higher curvatures required by the tensor hierarchy (Hassler et al., 4 Sep 2025).

Taken together, these works show that in generalized Cartan geometry the closest analogue of exterior curvature is not a single curvature 2-form but a hierarchy of torsion and curvature tensors defined on an extended tangent bundle, with Bianchi identities that are themselves hierarchical rather than closed on one curvature tensor alone (Osten, 20 May 2026).

4. Exterior-form curvature in gravity and pseudo-Riemannian geometry

Another established meaning of generalized exterior curvature is curvature expressed directly as a tensor-valued differential form. In "The Exterior Calculus of Quadratic Gravity" (Arık et al., 2024), the Levi-Civita connection 1-forms $\iota:N\hookrightarrow M$3 satisfy Cartan’s first structure equation

$\iota:N\hookrightarrow M$4

and the curvature 2-forms are defined by Cartan’s second structure equation

$\iota:N\hookrightarrow M$5

These obey

$\iota:N\hookrightarrow M$6

In this formalism, curvature is handled not as a single tensor-component object but as a tensor-valued differential form (Arık et al., 2024).

A central feature is the irreducible decomposition

$\iota:N\hookrightarrow M$7

where $\iota:N\hookrightarrow M$8 is the Weyl 2-form,

$\iota:N\hookrightarrow M$9

and

$\iota^!E\to N$0

The formalism then builds the Bach 1-forms

$\iota^!E\to N$1

and the scalar-sector analogue

$\iota^!E\to N$2

The field equations of general quadratic curvature gravity are written as

$\iota^!E\to N$3

Here the curvature is “generalized” in three senses stated explicitly in the source: tensor-valued forms instead of coordinate tensors, irreducible decomposition of curvature, and generalized Bach-type conserved forms (Arık et al., 2024).

This exterior-form approach remains within pseudo-Riemannian geometry, but it enlarges the operational meaning of curvature by elevating the exterior calculus itself to the primary carrier of geometric information. Relative to the Courant-algebroid and generalized Cartan settings, the underlying manifold is ordinary, yet curvature is generalized at the level of representation and variational structure rather than at the level of the tangent bundle.

5. Discrete and combinatorial exterior curvature

Piecewise-flat geometry supplies a further extension in which curvature is encoded by integrated measures over hybrid support domains rather than by pointwise smooth tensors. In "On exterior calculus and curvature in piecewise-flat manifolds" (McDonald et al., 2012), curvature is concentrated at codimension-2 hinges $\iota^!E\to N$4, with defect angle

$\iota^!E\to N$5

For a loop enclosing an area $\iota^!E\to N$6 in the plane orthogonal to $\iota^!E\to N$7, the sectional curvature is

$\iota^!E\to N$8

The Einstein–Hilbert integrand is exact after integration over a local hinge neighborhood,

$\iota^!E\to N$9

and globally

$E$0

The resulting discrete curvature is therefore an integrated hinge measure (McDonald et al., 2012).

Discrete exterior calculus is built from simplices, circumcentric duals, and hybrid cells. For a $E$1-simplex $E$2, the hybrid support volume is

$E$3

and the discrete Hodge dual is encoded by equality of average densities on primal and dual cells. The Riemann tensor is represented as a bivector-valued 2-form, and in the piecewise-flat setting it has effectively rank one in the space of 2-forms, with nontrivial curvature only in the plane orthogonal to the hinge $E$4. Ricci and scalar curvature then arise by contraction and volume-weighted averaging over hinge neighborhoods (McDonald et al., 2012).

In network geometry, a comparable shift occurs from smooth tensors to curvature derived from curves and 2-cells. "A Simple Differential Geometry for Networks and its Generalizations" (Saucan et al., 2019) starts from Menger curvature and Haantjes curvature. Haantjes curvature is defined by

$E$5

and for a graph path $E$6,

$E$7

Using a local Gauss–Bonnet idea, the Haantjes sectional curvature of a 2-cell $E$8 along an edge $E$9 is

$N$0

and the Haantjes-Ricci curvature of $N$1 is

$N$2

The paper explicitly frames Menger curvature as a classical exterior curvature analogue for triples of points and Haantjes curvature as a discrete analogue of geodesic curvature (Saucan et al., 2019).

These discrete theories differ in technical realization, but both replace local smooth curvature fields by integrated or cellwise geometric data derived from an exterior-calculus viewpoint. This suggests a broad discrete meaning of generalised exterior curvature: curvature encoded by measures on simplicial, dual, or network support domains rather than by pointwise differential invariants.

6. Curvature-like generalizations outside classical differential geometry

Several other literatures use “generalized curvature” in ways that retain the structural role of curvature while departing from submanifold or tensor-form geometry. In optimal transport induced by a Tonelli Lagrangian $N$3, the generalized curvature is a scalar quantity

$N$4

where $N$5 and $N$6 are defined from second derivatives of the Lagrangian and Jacobi-type linearization along action-minimizing curves. It enters the displacement Hessian formula

$N$7

and non-negativity of $N$8 implies convexity of the entropy functional along suitable $N$9 interpolants (Yang, 2023). The source explicitly states that this object is not a tensor in the classical sense, even though it is intrinsic.

In analysis of Radon-like operators of intermediate dimension, generalized curvature is encoded by a determinant weight on a triple-incidence manifold. The curvature weight is

$\mathcal G$0

where $\mathcal G$1. This replaces classical Phong–Stein rotational curvature in a higher-codimension setting and governs sublevel-set estimates that yield $\mathcal G$2-improving bounds (Gressman, 2016).

In exterior overdetermined problems, curvature enters through mean-curvature-dependent boundary data rather than through an internal curvature tensor. For a bounded domain $\mathcal G$3, the normalized problem

$\mathcal G$4

ties Neumann data to the normalized mean curvature $\mathcal G$5. The paper proves rigidity of the spherical solution in several parameter regimes, including all bounded domains when $\mathcal G$6 and all bounded domains when $\mathcal G$7, while for $\mathcal G$8 rigidity is proved under star-shapedness (Niebel, 8 Apr 2026). The source text explicitly places this within the theme of generalized exterior curvature problems.

These examples show that the curvature vocabulary can be generalized in at least three distinct ways: by replacing the underlying bundle, by replacing pointwise tensors with exterior or integrated geometric objects, or by retaining the functional role of curvature as the quantity controlling convexity, rigidity, or analytic improvement.

7. Conceptual synthesis and scope

The literature does not present a single universal definition of generalised exterior curvature. Instead, several recurring motifs appear.

Setting	Curvature carrier	Representative role
Exact Courant algebroids	Pullback Courant algebroid, generalised metric, divergence, canonical connection	Hypersurface invariants, Gauß–Codazzi, generalised Einstein constraints
Generalised Cartan geometry	Extended tangent bundle and tensor hierarchy	Hierarchy of generalized torsions and curvatures
Exterior calculus and discrete geometry	Curvature 2-forms, hybrid cells, deficit angles, cell curvatures	Variational gravity, piecewise-flat curvature, network Ricci/scalar analogues

First, curvature is repeatedly packaged in exterior-calculus form rather than solely as a component tensor. This is explicit in curvature 2-forms $\mathcal G$9 (Arık et al., 2024), in bivector-valued discrete Riemann forms on hybrid cells (McDonald et al., 2012), and in generalized Cartan curvature extracted from current algebras (Hassler et al., 2024).

Second, the ambient geometric object is often enlarged. In exact Courant algebroids, $\mathrm{div}$0 is replaced by

$\mathrm{div}$1

and hypersurface geometry is reformulated through $\mathrm{div}$2 (Cortés et al., 16 Jul 2025). In generalized Cartan geometry, the tangent bundle becomes $\mathrm{div}$3 or a full tensor-hierarchy bundle $\mathrm{div}$4 (Hassler et al., 2024). In discrete theories, the local support of curvature is a hybrid cell rather than a point (McDonald et al., 2012).

Third, generalized curvature typically resolves a structural inadequacy of the classical framework. In exact Courant geometry, a divergence operator removes the non-uniqueness of generalized Levi-Civita connections and makes Ricci and scalar curvature canonical invariants of $\mathrm{div}$5 (Cortés et al., 23 Jul 2025). In generalized Cartan geometry, higher curvatures are required because the curvature of the generalized spin connection is not covariant on $\mathrm{div}$6 alone (Hassler et al., 4 Sep 2025). In network geometry, Haantjes curvature is preferred because it is better suited as a discrete version of classical geodesic curvature and does not require a fixed background geometry (Saucan et al., 2019).

A common misconception would be to treat all these constructions as variants of a single generalized Riemann tensor. The sources do not support that identification. Some constructions are hypersurface-theoretic and explicitly involve generalised second fundamental form and generalised mean curvature (Cortés et al., 16 Jul 2025); some are curvature hierarchies on extended bundles (Osten, 20 May 2026); some are conserved fourth-order forms in quadratic gravity (Arık et al., 2024); some are integrated hinge measures (McDonald et al., 2012); and some are curvature-like scalars or weights governing convexity or analytic estimates (Yang, 2023, Gressman, 2016).

The most stable encyclopedia-level characterization is therefore structural rather than definitional: generalised exterior curvature is a class of curvature concepts that preserve the exterior, variational, or obstruction-theoretic role of classical curvature while extending its domain of definition to generalized bundles, higher algebraic structures, discrete hybrid cells, or other nonclassical geometric settings.