First Gray Curvature Condition Overview

• The First Gray Curvature Condition is a cyclic scalar–Ricci identity in differential geometry that characterizes Gray manifolds through a linear relation between the Ricci tensor and scalar curvature.
• In higher gauge theory, it manifests as a fake-curvature constraint (dA+A∧A−α(B)=0), ensuring coherence between 1-holonomy and 2-holonomy via Gray-functoriality.
• It also appears in subelliptic analysis as a generalized curvature inequality, underscoring its cross-disciplinary significance in connecting geometric and gauge-theoretic frameworks.

Searching arXiv for the topic and the cited papers. The expression First Gray Curvature Condition has multiple technical meanings in contemporary mathematical literature. In differential geometry it most commonly refers to one of Alfred Gray’s curvature identities, or to a Riemannian reformulation of Gray’s $AC^\perp$ condition in terms of the Ricci tensor and scalar curvature. In higher gauge theory, the same phrase is used naturally for the first fake-curvature constraint forced by Gray-functoriality in $3$-gauge theory, namely $dA+A\wedge A-\alpha(B)=0$. In subelliptic analysis, the terminology “Gray” does not appear in the cited paper, but a standing generalized curvature condition is described as a broadened non-Riemannian curvature framework and is sometimes colloquially associated with Gray-type generalization. These usages are mathematically distinct and belong to different research programs (Jelonek, 2016).

1. Riemannian meaning: Gray’s $AC^\perp$ condition

In the purely Riemannian setting, the relevant Gray condition is formulated directly in terms of the Ricci tensor $\rho$ and scalar curvature $\mathrm{Scal}$. A Riemannian manifold $(M,g)$ of dimension $n$ is an $AC^\perp$ (Gray) manifold if its Ricci tensor satisfies

$\sum_{\text{cyclic in }X,Y,Z} \nabla_X \rho(Y,Z) \;=\; \frac{1}{n+2}\sum_{\text{cyclic} X(\mathrm{Scal})\,g(Y,Z). \tag{1.1}$

Equivalently,

$3$0

This identity is presented as the Riemannian avatar of what is often called the first Gray curvature condition in the Hermitian context (Jelonek, 2016).

A key reformulation is that $3$1 is an $3$2 manifold if and only if there exists a tensor $3$3 of type $3$4 such that

$3$5

and $3$6 is a Killing tensor: $3$7 Accordingly, in this usage the first Gray condition is a linear relation between covariant derivatives of the Ricci tensor and the gradient of scalar curvature, with the failure of $3$8 to be parallel controlled by $3$9. When the scalar curvature is constant, the condition reduces to the $dA+A\wedge A-\alpha(B)=0$0-manifold condition, namely Killing Ricci tensor (Jelonek, 2016).

Gray’s “Einstein-like” curvature conditions originated in almost Hermitian geometry, where one imposes algebraic symmetries on the Riemann curvature tensor involving an almost complex structure $dA+A\wedge A-\alpha(B)=0$1. The $dA+A\wedge A-\alpha(B)=0$2 reformulation isolates a purely Riemannian condition that no longer refers to $dA+A\wedge A-\alpha(B)=0$3 explicitly. This suggests that the phrase “first Gray curvature condition” is best understood contextually: in Hermitian geometry it refers to one of Gray’s classical identities, whereas in the Riemannian setting treated here it designates the $dA+A\wedge A-\alpha(B)=0$4 scalar–Ricci identity.

2. Structure under a two-point Ricci spectrum

A principal application of the $dA+A\wedge A-\alpha(B)=0$5 condition concerns manifolds whose Ricci tensor has exactly two eigenvalues, one of multiplicity $dA+A\wedge A-\alpha(B)=0$6 and the other of multiplicity $dA+A\wedge A-\alpha(B)=0$7. On the open set

$dA+A\wedge A-\alpha(B)=0$8

the tangent bundle splits orthogonally as

$dA+A\wedge A-\alpha(B)=0$9

where $AC^\perp$0 has rank $AC^\perp$1 and $AC^\perp$2 has rank $AC^\perp$3. If $AC^\perp$4 is a unit vector field spanning $AC^\perp$5, $AC^\perp$6 is the orthogonal projection, and $AC^\perp$7, then

$AC^\perp$8

This expresses the Ricci tensor as isotropic on the codimension-one distribution $AC^\perp$9 and distinguished along the line field $\rho$0 (Jelonek, 2016).

In the $\rho$1 decomposition

$\rho$2

if $\rho$3 denote the eigenvalues of the Killing tensor $\rho$4 on $\rho$5 and $\rho$6, then

$\rho$7

These eigenvalues satisfy the general Killing-tensor identities recalled in the paper. For example, on the open set where the number of distinct eigenvalues is locally constant, the eigendistributions of a Killing tensor satisfy relations such as

$\rho$8

and, for $\rho$9, $\mathrm{Scal}$0,

$\mathrm{Scal}$1

These formulas constrain the variation of the eigenvalue functions and the geometry of the eigendistributions (Jelonek, 2016).

The same framework yields linear relations among the Ricci eigenvalues in the presence of the conformal vector field $\mathrm{Scal}$2 constructed later. If $\mathrm{Scal}$3 is conformal and non-Killing, then

$\mathrm{Scal}$4

If $\mathrm{Scal}$5 is Killing, then

$\mathrm{Scal}$6

These relations reduce the degrees of freedom in the Ricci spectrum and are central to the classification.

3. Conformal eigenfields and warped-product classification

For complete, simply connected, real analytic $\mathrm{Scal}$7 manifolds with the above two-eigenvalue Ricci spectrum, the first main structural result is the existence of a globally defined vector field $\mathrm{Scal}$8 such that

$\mathrm{Scal}$9

Thus $(M,g)$0 is simultaneously conformal and an eigenfield of the Ricci operator. Moreover, either $(M,g)$1 is Killing, in which case $(M,g)$2 is constant, or $(M,g)$3 is conformal and non-Killing, in which case $(M,g)$4 is constant and $(M,g)$5 is a closed gradient conformal field (Jelonek, 2016).

The proof proceeds through the tensor

$(M,g)$6

where

$(M,g)$7

Writing $(M,g)$8 on a component of $(M,g)$9, the Killing condition for $n$0 becomes equivalent to an identity

$n$1

where

$n$2

From this one derives $n$3, so $n$4 is conformal, and then distinguishes the Killing and closed-gradient cases.

In the non-Killing conformal case, the geometry is described by a warped product. Under the hypotheses that $n$5 is real analytic, simply connected, $n$6, with Ricci eigenvalues $n$7 of multiplicities $n$8 and $n$9 respectively, and such that

$AC^\perp$0

Theorem 2 gives a complete warped-product description (Jelonek, 2016).

If $AC^\perp$1 everywhere, then

$AC^\perp$2

where $AC^\perp$3 is Einstein. If the set $AC^\perp$4 is nonempty, then $AC^\perp$5 consists of one or two points and $AC^\perp$6 is a warped product over an interval, with spherical fiber $AC^\perp$7 and round metric in the cases recorded in the paper. In all cases, the warping function satisfies

$AC^\perp$8

where $AC^\perp$9 is the Einstein constant of $\sum_{\text{cyclic in }X,Y,Z} \nabla_X \rho(Y,Z) \;=\; \frac{1}{n+2}\sum_{\text{cyclic} X(\mathrm{Scal})\,g(Y,Z). \tag{1.1}$0 and $\sum_{\text{cyclic in }X,Y,Z} \nabla_X \rho(Y,Z) \;=\; \frac{1}{n+2}\sum_{\text{cyclic} X(\mathrm{Scal})\,g(Y,Z). \tag{1.1}$1. The classification shows that under the first Gray condition, a two-eigenvalue Ricci tensor is rigid enough to force essentially cohomogeneity-one geometry.

4. Explicit families and Einstein–Weyl interpretation

The same Riemannian theory connects $\sum_{\text{cyclic in }X,Y,Z} \nabla_X \rho(Y,Z) \;=\; \frac{1}{n+2}\sum_{\text{cyclic} X(\mathrm{Scal})\,g(Y,Z). \tag{1.1}$2 geometry to Einstein–Weyl structures. If $\sum_{\text{cyclic in }X,Y,Z} \nabla_X \rho(Y,Z) \;=\; \frac{1}{n+2}\sum_{\text{cyclic} X(\mathrm{Scal})\,g(Y,Z). \tag{1.1}$3 is an Einstein–Weyl structure and $\sum_{\text{cyclic in }X,Y,Z} \nabla_X \rho(Y,Z) \;=\; \frac{1}{n+2}\sum_{\text{cyclic} X(\mathrm{Scal})\,g(Y,Z). \tag{1.1}$4 is the Gauduchon metric, then $\sum_{\text{cyclic in }X,Y,Z} \nabla_X \rho(Y,Z) \;=\; \frac{1}{n+2}\sum_{\text{cyclic} X(\mathrm{Scal})\,g(Y,Z). \tag{1.1}$5 is $\sum_{\text{cyclic in }X,Y,Z} \nabla_X \rho(Y,Z) \;=\; \frac{1}{n+2}\sum_{\text{cyclic} X(\mathrm{Scal})\,g(Y,Z). \tag{1.1}$6. If one has a pair of Einstein–Weyl structures $\sum_{\text{cyclic in }X,Y,Z} \nabla_X \rho(Y,Z) \;=\; \frac{1}{n+2}\sum_{\text{cyclic} X(\mathrm{Scal})\,g(Y,Z). \tag{1.1}$7 and $\sum_{\text{cyclic in }X,Y,Z} \nabla_X \rho(Y,Z) \;=\; \frac{1}{n+2}\sum_{\text{cyclic} X(\mathrm{Scal})\,g(Y,Z). \tag{1.1}$8 on the same Riemannian manifold, then $\sum_{\text{cyclic in }X,Y,Z} \nabla_X \rho(Y,Z) \;=\; \frac{1}{n+2}\sum_{\text{cyclic} X(\mathrm{Scal})\,g(Y,Z). \tag{1.1}$9 is $3$00 with two Ricci eigenvalues of multiplicities $3$01 and $3$02 (Jelonek, 2016).

The Einstein–Weyl condition is

$3$03

with

$3$04

and

$3$05

For $3$06 manifolds with two Ricci eigenvalues and conformal eigenfield $3$07, one has $3$08. In the gradient case, $3$09 is closed, and both Einstein–Weyl structures are conformally Einstein: $3$10 when $3$11.

This leads to explicit families of $3$12 metrics. On the sphere $3$13 there exists a one-parameter family

$3$14

where $3$15 solves either

$3$16

or

$3$17

In the case $3$18, the corresponding $3$19 manifold has $3$20, admits a pair of Einstein–Weyl structures $3$21, $3$22 with $3$23 closed, and the Einstein metrics $3$24 are standard metrics of constant sectional curvature on $3$25 (Jelonek, 2016).

For noncompact complete examples, if $3$26 is Einstein with $3$27 and $3$28, then on

$3$29

there is a one-parameter family of complete $3$30 metrics

$3$31

with

$3$32

equivalently

$3$33

These satisfy the first Gray condition while having $3$34, so they do not admit a pair of Einstein–Weyl structures $3$35, $3$36 in the sense used earlier.

5. Higher gauge theory meaning: fake curvature and Gray-functoriality

In higher gauge theory, the phrase first Gray curvature condition refers to a different object. A $3$37-connection on a manifold $3$38 is a triple

$3$39

where

$3$40

is a differential $3$41-crossed module equipped with the actions and Peiffer lifting required in the theory (Wang, 2013).

The associated curvature forms are

$3$42

$3$43

$3$44

The fake curvatures are then defined by

$3$45

$3$46

When the $3$47-connection arises from a smooth Gray-functor

$3$48

the paper shows that one has identically

$3$49

Accordingly, it is natural in this setting to call

$3$50

the first Gray curvature condition, and

$3$51

the second Gray curvature condition (Wang, 2013).

These identities are not merely definitions. They arise from Gray-functoriality of holonomy. The local correspondence is

$3$52

and lax-natural transformations between Gray-functors correspond to $3$53-gauge transformations of $3$54-connections. The first condition expresses the compatibility between $3$55-holonomy and $3$56-holonomy: differentiating the naturality identity on small $3$57-paths yields

$3$58

Similarly, differentiating the naturality identity on small $3$59-paths yields

$3$60

Thus the first Gray curvature condition is the infinitesimal coherence law connecting the curvature of the $3$61-connection $3$62 to the $3$63-form $3$64.

A characteristic feature of this $3$65-gauge theory is that the $3$66-curvature contains the Peiffer term $3$67. The paper explains this by the Gray-categorical interchange law: in a Gray-category there are two horizontal compositions of $3$68-cells, connected by a nontrivial interchanging $3$69-arrow. In the Gray $3$70-groupoid built from a $3$71-crossed module, the third component of the interchanging $3$72-arrow is precisely the Peiffer lifting. Infinitesimally, this produces the term $3$73 in $3$74 (Wang, 2013).

6. Gauge covariance, holonomy, and a subelliptic analogue

The higher-gauge interpretation of the first Gray curvature condition is tied to covariance. Under a general $3$75-gauge transformation determined by $3$76, $3$77, and $3$78, the transformed $3$79-curvature satisfies

$3$80

The paper states that the $3$81-curvature $3$82-form is covariant under the gauge transformations if the fake $3$83- and fake $3$84-curvatures vanish. Thus, when the first and second Gray curvature conditions hold,

$3$85

and likewise $3$86 and $3$87 transform covariantly by the $3$88-action (Wang, 2013).

The same paper defines a $3$89-dimensional holonomy as the image under a Gray-functor of the boundary of a small $3$90-path. If

$3$91

then differentiating at the origin gives

$3$92

This shows that the derivative of the $3$93-holonomy yields the $3$94-curvature $3$95-form. Because the holonomy is constructed through a Gray-functor and the gauge transformations arise from lax-natural transformations, the $3$96-dimensional holonomy is $3$97-gauge invariant. Infinitesimally, this invariance is compatible with the curvature covariance only when the Gray curvature conditions $3$98 are imposed.

A separate, only partially related usage appears in subelliptic analysis. For a Hörmander-type diffusion operator

$3$99

with carré du champ $dA+A\wedge A-\alpha(B)=0$00, a multi-field generalized curvature condition is introduced: $dA+A\wedge A-\alpha(B)=0$01 Coupled with Lyapunov-growth control,

$dA+A\wedge A-\alpha(B)=0$02

this becomes the standing generalized curvature condition used throughout the paper (Wang, 2012).

That work explicitly states that it does not use the word “Gray,” but notes that “generalized curvature condition” and “generalized curvature-dimension condition” are sometimes colloquially dubbed “Gray” conditions in the sense of a broadened, non-Riemannian curvature notion. In this context, the historically first generalized condition is the one-field inequality

$dA+A\wedge A-\alpha(B)=0$03

while the paper’s main innovation is the multi-field version $dA+A\wedge A-\alpha(B)=0$04. A plausible implication is that the term “First Gray Curvature Condition” should not be used in subelliptic diffusion theory without qualification, because the cited paper itself treats the relevant object as a generalized Bakry–Émery-type curvature condition rather than a Gray condition in a standard terminological sense.

In summary, the phrase First Gray Curvature Condition is not univocal. In $dA+A\wedge A-\alpha(B)=0$05 Riemannian geometry it denotes the cyclic scalar–Ricci identity characterizing Gray manifolds; in $dA+A\wedge A-\alpha(B)=0$06-gauge theory it denotes the first fake-curvature constraint

$dA+A\wedge A-\alpha(B)=0$07

forced by Gray-functoriality; and in subelliptic analysis it has at most a colloquial analogue in generalized curvature inequalities. The common theme is a compatibility law constraining curvature data beyond the ordinary Riemannian or gauge-theoretic setting, but the precise mathematical content depends entirely on the ambient category of structures under consideration.