---
title: First Gray Curvature Condition Overview
url: https://www.emergentmind.com/topics/first-gray-curvature-condition
type: topic
---

# First Gray Curvature Condition Overview

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 [1612.02330].

## 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,
\[
V_X\rho(X,X) \;=\; \frac{2}{n+2}\,X(\mathrm{Scal})\,g(X,X).
\tag{0.1}
\]
This identity is presented as the Riemannian avatar of what is often called the first Gray curvature condition in the Hermitian context [1612.02330].

A key reformulation is that \((M,g)\) is an \(AC^\perp\) manifold if and only if there exists a tensor \(S\) of type \((1,1)\) such that
\[
\mathrm{Ric} = S + \frac{2}{n+2}\,\mathrm{Scal}\,\mathrm{Id}
\]
and \(S\) is a Killing tensor:
\[
g\big((\nabla S)(X,X),X\big)=0 \quad \text{for all } X\in TM.
\]
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 \(\rho\) to be parallel controlled by \(\nabla \mathrm{Scal}\). When the scalar curvature is constant, the condition reduces to the \(A\)-manifold condition, namely Killing Ricci tensor [1612.02330].

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 \(J\). The \(AC^\perp\) reformulation isolates a purely Riemannian condition that no longer refers to \(J\) 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 \(AC^\perp\) scalar–Ricci identity.

## 2. Structure under a two-point Ricci spectrum

A principal application of the \(AC^\perp\) condition concerns manifolds whose Ricci tensor has exactly two eigenvalues, one of multiplicity \(1\) and the other of multiplicity \(n-1\). On the open set
\[
V :=\{x\in M : \lambda(x) \neq \mu(x)\},
\]
the tangent bundle splits orthogonally as
\[
TM|_V \;=\; D_\lambda \oplus D_\mu,
\]
where \(D_\lambda\) has rank \(n-1\) and \(D_\mu\) has rank \(1\). If \(\xi_0\) is a unit vector field spanning \(D_\mu\), \(\pi:TM\to D_\mu\) is the orthogonal projection, and \(m=g(\pi\cdot,\pi\cdot)\), then
\[
\rho = \lambda\, g + (\mu-\lambda)\, m.
\]
This expresses the Ricci tensor as isotropic on the codimension-one distribution \(D_\lambda\) and distinguished along the line field \(D_\mu\) [1612.02330].

In the \(AC^\perp\) decomposition
\[
\mathrm{Ric} = S + \frac{2}{n+2}\,\mathrm{Scal}\,\mathrm{Id},
\]
if \(\lambda',\mu'\) denote the eigenvalues of the Killing tensor \(S\) on \(D_\lambda\) and \(D_\mu\), then
\[
\lambda' = \lambda - \frac{2}{n+2}\mathrm{Scal}, \qquad
\mu' = \mu - \frac{2}{n+2}\mathrm{Scal}.
\]
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
\[
(1-2/n)\,d\lambda'(X)=0,\quad d\lambda'(X)=0 \text{ along }D_{\lambda'},
\]
and, for \(X\in D_{\lambda'}\), \(Y\in D_{\mu'}\),
\[
g(\nabla_X X,Y)=\frac{1}{2}\frac{d\lambda'(Y)}{\lambda'-\mu'}\,|X|^2.
\]
These formulas constrain the variation of the eigenvalue functions and the geometry of the eigendistributions [1612.02330].

The same framework yields linear relations among the Ricci eigenvalues in the presence of the conformal vector field \(\xi\) constructed later. If \(\xi\) is conformal and non-Killing, then
\[
n\mu - 2(n-1)\lambda = C_0 = \text{const}.
\]
If \(\xi\) is Killing, then
\[
(n-4)\lambda + 2\mu = C_0 = \text{const}.
\]
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 \(AC^\perp\) manifolds with the above two-eigenvalue Ricci spectrum, the first main structural result is the existence of a globally defined vector field \(\xi\) such that
\[
\mathcal{L}_\xi g=\alpha g,
\qquad
(\mathrm{Ric}-\mu\,\mathrm{Id})\xi=0,
\qquad
g(\xi,\xi)=|\lambda'-\mu'|.
\]
Thus \(\xi\) is simultaneously conformal and an eigenfield of the Ricci operator. Moreover, either \(\xi\) is Killing, in which case \(\lambda'\) is constant, or \(\xi\) is conformal and non-Killing, in which case \(\mu'\) is constant and \(\xi\) is a closed gradient conformal field [1612.02330].

The proof proceeds through the tensor
\[
T(X,Y):=g(SX,Y),
\]
where
\[
T=\rho-\frac{2}{n+2}\,\mathrm{Scal}\,g=\lambda' g+(\mu'-\lambda')m.
\]
Writing \(m=\omega\otimes\omega\) on a component of \(V\), the Killing condition for \(T\) becomes equivalent to an identity
\[
\lambda'(X)g(X,X)-h(X,X)\,\omega(X)=0,
\]
where
\[
h(X,Y)=(\nabla_X\omega)(Y)+(\nabla_Y\omega)(X).
\]
From this one derives \(h=\mathcal{L}_\xi g\), so \(\xi\) 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 \((M,g)\) is real analytic, simply connected, \(AC^\perp\), with Ricci eigenvalues \(\lambda,\mu\) of multiplicities \(n\) and \(1\) respectively, and such that
\[
\mu - \frac{2}{n+3}\,\mathrm{Scal} = \text{constant},
\qquad
\lambda - \frac{2}{n+3}\,\mathrm{Scal} \text{ is non-constant},
\]
Theorem 2 gives a complete warped-product description [1612.02330].

If \(|\lambda-\mu|>0\) everywhere, then
\[
M \cong \mathbb{R}\times_f M_*,
\qquad
g=dt^2+f(t)^2\,g_{M_*},
\]
where \(M_*\) is Einstein. If the set \(N=\{x\in M:\lambda=\mu\}\) is nonempty, then \(N\) consists of one or two points and \(M\) is a warped product over an interval, with spherical fiber \(S^n\) and round metric in the cases recorded in the paper. In all cases, the warping function satisfies
\[
-(n-1)\left( \frac{f''}{f} + \frac{(n-2)f'^2}{f^2}\right)
         = \frac{\tau}{f^2} - C f^2,
\tag{2.4}
\]
where \(\tau\) is the Einstein constant of \(M_*\) and \(C\in\mathbb{R}\setminus\{0\}\). 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 \(AC^\perp\) geometry to Einstein–Weyl structures. If \(([g],D)\) is an Einstein–Weyl structure and \(g_0\in[g]\) is the Gauduchon metric, then \((M,g_0)\) is \(AC^\perp\). If one has a pair of Einstein–Weyl structures \((g,\omega)\) and \((g,-\omega)\) on the same Riemannian manifold, then \((M,g)\) is \(AC^\perp\) with two Ricci eigenvalues of multiplicities \(n-1\) and \(1\) [1612.02330].

The Einstein–Weyl condition is
\[
\rho + \frac{n-2}{2}\,D\omega = \Lambda g,
\tag{1.5}
\]
with
\[
D\omega(X,Y)=(\nabla_X\omega)(Y)+(\nabla_Y\omega)(X)+\omega(X)\omega(Y),
\]
and
\[
\Lambda=\frac{1}{2(n)}\Big(\mathrm{Scal}+\operatorname{div}\omega-\tfrac{n-2}{2}|\omega^\#|^2\Big).
\tag{1.6}
\]
For \(AC^\perp\) manifolds with two Ricci eigenvalues and conformal eigenfield \(\xi\), one has \(\omega(X)=g(\xi,X)\). In the gradient case, \(\omega\) is closed, and both Einstein–Weyl structures are conformally Einstein:
\[
g_1=e^{+\phi}g,\qquad g_2=e^{-\phi}g
\]
when \(\omega=d\phi\).

This leads to explicit families of \(AC^\perp\) metrics. On the sphere \(S^{n+1}\) there exists a one-parameter family
\[
g_A=dt^2+f(t)^2 g_{\mathrm{can}},
\]
where \(f\) solves either
\[
(f')^2=1+Af^2+f^4,\qquad A\in(-\infty,-2),
\tag{2.6}
\]
or
\[
(f')^2=1+Af^2-f^4,\qquad A\in\mathbb{R}.
\tag{2.7}
\]
In the case \((2.6)\), the corresponding \(AC^\perp\) manifold has \(\lambda-\mu=Cf^2>0\), admits a pair of Einstein–Weyl structures \((g,\omega)\), \((g,-\omega)\) with \(\omega\) closed, and the Einstein metrics \(e^{\pm\phi}g\) are standard metrics of constant sectional curvature on \(S^{n+1}\) [1612.02330].

For noncompact complete examples, if \((M_*,g_{M_*})\) is Einstein with \(\mathrm{Ric}_{M_*}=\tau g_{M_*}\) and \(\tau<0\), then on
\[
M=\mathbb{R}\times_f M_*
\]
there is a one-parameter family of complete \(AC^\perp\) metrics
\[
g_A=dt^2+f(t)^2g_{M_*},
\]
with
\[
(f')^2=-\frac{\tau}{n-1}+Af^2-f^4,\qquad A^2>-4\tau,\ \tau<0,
\tag{2.8}
\]
equivalently
\[
f''=Af-2f^3.
\tag{2.9}
\]
These satisfy the first Gray condition while having \(\lambda\le\mu\), so they do not admit a pair of Einstein–Weyl structures \((g,\omega)\), \((g,-\omega)\) 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 \(3\)-connection on a manifold \(X\) is a triple
\[
(A,B,C),\qquad
A\in\Omega^1(X,\mathfrak g),\quad
B\in\Omega^2(X,\mathfrak h),\quad
C\in\Omega^3(X,\mathfrak l),
\]
where
\[
\mathfrak l \xrightarrow{\delta} \mathfrak h \xrightarrow{\alpha} \mathfrak g
\]
is a differential \(2\)-crossed module equipped with the actions and Peiffer lifting required in the theory [1311.3796].

The associated curvature forms are
\[
\Omega_1=dA+A\wedge A \in \Omega^2(X,\mathfrak g),
\]
\[
\Omega_2=dB+A\triangleright B \in \Omega^3(X,\mathfrak h),
\]
\[
\Omega_3=dC+A\triangleright C+B\mathbin{\{\}B \in \Omega^4(X,\mathfrak l).
\]
The fake curvatures are then defined by
\[
F_1=\Omega_1-\alpha(B)=dA+A\wedge A-\alpha(B),
\]
\[
F_2=\Omega_2-\delta(C)=dB+A\triangleright B-\delta(C).
\]
When the \(3\)-connection arises from a smooth Gray-functor
\[
F:\mathcal P_3(X)\longrightarrow \mathcal G^{\mathscr L},
\]
the paper shows that one has identically
\[
F_1=0,\qquad F_2=0.
\]
Accordingly, it is natural in this setting to call
\[
dA+A\wedge A-\alpha(B)=0
\]
the **first Gray curvature condition**, and
\[
dB+A\triangleright B-\delta(C)=0
\]
the **second Gray curvature condition** [1311.3796].

These identities are not merely definitions. They arise from Gray-functoriality of holonomy. The local correspondence is
\[
\text{Gray-functor }F \quad \Longleftrightarrow \quad \text{\(3\)-connection }(A,B,C),
\]
and lax-natural transformations between Gray-functors correspond to \(3\)-gauge transformations of \(3\)-connections. The first condition expresses the compatibility between \(1\)-holonomy and \(2\)-holonomy: differentiating the naturality identity on small \(2\)-paths yields
\[
dA+A\wedge A-\alpha(B)=0.
\]
Similarly, differentiating the naturality identity on small \(3\)-paths yields
\[
dB+A\triangleright B-\delta(C)=0.
\]
Thus the first Gray curvature condition is the infinitesimal coherence law connecting the curvature of the \(1\)-connection \(A\) to the \(2\)-form \(B\).

A characteristic feature of this \(3\)-gauge theory is that the \(3\)-curvature contains the Peiffer term \(B\mathbin{\{\}B\). The paper explains this by the Gray-categorical interchange law: in a Gray-category there are two horizontal compositions of \(2\)-cells, connected by a nontrivial interchanging \(3\)-arrow. In the Gray \(3\)-groupoid built from a \(2\)-crossed module, the third component of the interchanging \(3\)-arrow is precisely the Peiffer lifting. Infinitesimally, this produces the term \(B\mathbin{\{\}B\) in \(\Omega_3\) [1311.3796].

## 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 \(3\)-gauge transformation determined by \(g:X\to G\), \(\phi\in\Omega^1(X,\mathfrak h)\), and \(\psi\in\Omega^2(X,\mathfrak l)\), the transformed \(3\)-curvature satisfies
\[
\Omega_3' = g^{-1}\triangleright\Omega_3 - (F_1')\triangleright'\psi.
\]
The paper states that the \(3\)-curvature \(4\)-form is covariant under the gauge transformations if the fake \(1\)- and fake \(2\)-curvatures vanish. Thus, when the first and second Gray curvature conditions hold,
\[
\Omega_3' = g^{-1}\triangleright\Omega_3,
\]
and likewise \(\Omega_1\) and \(\Omega_2\) transform covariantly by the \(G\)-action [1311.3796].

The same paper defines a \(3\)-dimensional holonomy as the image under a Gray-functor of the boundary of a small \(4\)-path. If
\[
H_F(\Omega_{x_1,x_2,x_3,x_4})=K_1(x)\cdots K_{14}(x)\in L,
\]
then differentiating at the origin gives
\[
\left.\frac{\partial^4}{\partial x_1\partial x_2\partial x_3\partial x_4}\right|_{0}
\big(K_1K_2\cdots K_{14}\big)
=\Omega_3(v_1,v_2,v_3,v_4).
\]
This shows that the derivative of the \(3\)-holonomy yields the \(3\)-curvature \(4\)-form. Because the holonomy is constructed through a Gray-functor and the gauge transformations arise from lax-natural transformations, the \(3\)-dimensional holonomy is \(3\)-gauge invariant. Infinitesimally, this invariance is compatible with the curvature covariance only when the Gray curvature conditions \(F_1=F_2=0\) are imposed.

A separate, only partially related usage appears in subelliptic analysis. For a Hörmander-type diffusion operator
\[
L:=\sum_{i=1}^n X_i^2 + X_0
\]
with carré du champ \(\Gamma\), a multi-field generalized curvature condition is introduced:
\[
\Gamma_2(f)+\sum_{i=1}^l r_i\,\Gamma_2^{(i)}(f)
\ge \sum_{i=0}^l K_i(r_1,\dots,r_l)\,\Gamma^{(i)}(f),
\qquad f\in C^3(M),\ r_i>0.
\tag{1.5}
\]
Coupled with Lyapunov-growth control,
\[
LW\le CW,\qquad \tilde\Gamma(W)\le CW^2,\qquad
\tilde\Gamma:=\sum_{i=0}^l \Gamma^{(i)},
\]
this becomes the standing generalized curvature condition used throughout the paper [1202.0778].

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
\[
\Gamma_2(f)+r\,\Gamma_2^Z(f)\ge (\rho_1-r)\Gamma(f)+\rho_2\Gamma^Z(f),
\qquad f\in C^2(M),\ r>0,
\tag{1.3}
\]
while the paper’s main innovation is the multi-field version \((1.5)\). 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 \(AC^\perp\) Riemannian geometry it denotes the cyclic scalar–Ricci identity characterizing Gray manifolds; in \(3\)-gauge theory it denotes the first fake-curvature constraint
\[
dA+A\wedge A=\alpha(B),
\]
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.

Source: https://www.emergentmind.com/topics/first-gray-curvature-condition