# Longitudinal Structure of Quark-Gluon Plasma Unveiled Through Nuclear
Deformations

Chunjian Zhang  Shengli Huang  Jiangyong Jia<EMAIL_ADDRESS>
###### Abstract

The study of quark-gluon plasma (QGP) is hindered by our limited understanding
of its initial conditions, particularly its longitudinal structure. We propose
a novel approach that entails analyzing collisions involving nuclei of similar
masses but different deformations. This strategy allows us to vary the initial
conditions and collective expansion of the QGP, while minimizing the influence
of non-flow correlations. Using a dynamical transport model, we have for the
first time extracted the complete longitudinal structure of elliptic flow
($v_{2}$). Our findings reveal that although deformation significantly
enhances the overall magnitude of $v_{2}$, it does not alter its longitudinal
profile. This approach not only enables the separation of the rapidity
dependence of flow from its rapidity decorrelations but also prompts further
investigation into other nuclear structural features, such as nuclear skin
thickness, to advance our understanding of the QGP’s initial conditions.

###### pacs:

25.75.Gz, 25.75.Ld, 25.75.-1

Introduction. High-energy collisions of atomic nuclei at RHIC and the LHC
create a hot and dense QGP, whose space-time expansion is well described by
the relativistic viscous hydrodynamic equations. Owing to quantum fluctuations
in nuclear wavefunctions and the stochastic nature of energy deposition, the
initial conditions and subsequent transverse expansion of the QGP vary
significantly from one event to another. These variations lead to a range of
phenomena accessible in experiments, offering insights into the dynamics and
properties of the QGP Romatschke and Romatschke (2019); Busza  _et al._
(2018). Yet, our understanding of the QGP is currently hindered by our limited
knowledge of its initial condition. Bridging this gap stands as a pressing
challenge in high-energy nuclear physics Achenbach  _et al._ (2024); Bally
_et al._ (2022).

The fluctuations in the initial state extend beyond the transverse ($xy$)
plane and are also evident along the longitudinal direction. In a typical
picture for nuclear collisions, the energy carried by colliding nucleons are
deposited in QCD strings, with significant fluctuations in their starting
positions and lengths along pseudorapidity $\eta$. Consequently, the shape and
size of the initial QGP and its collective flow fluctuate and decorrelate with
$\eta$ even within a single event Bozek  _et al._ (2011); Jia and Huo (2014);
Pang  _et al._ (2016); Jia  _et al._ (2016); Wu  _et al._ (2018); Shen and
Schenke (2018). This picture is supported by measurements of forward-backward
fluctuations of multiplicity and anisotropic flow Khachatryan  _et al._
(2015); Aaboud  _et al._ (2018); Aad  _et al._ (2021); Acharya  _et al._
(2018); Aaboud  _et al._ (2017). Quantitative assessments of the longitudinal
fluctuations (or longitudinal decorrelations), however, are impeded by the
limited detector coverage in $\eta$ and contaminations from non-flow
correlations such as jet fragmentation and resonance decays Xu  _et al._
(2022). Moreover, the method to disentangle the average variation of
observables as a function of $\eta$ from decorrelation effects remains
experimentally elusive.

In this paper, we propose a new approach to measure longitudinal
decorrelations by comparing collisions of atomic nuclei with similar mass
numbers but vastly different shapes. This data-driven approach completely
eliminates the influence of non-flow and other short-range correlations. We
unveil the complete longitudinal structure of elliptic flow and discuss its
implications for the initial condition and expansion dynamics of the QGP.

Method. We demonstrate the idea using elliptic flow $V_{2}=v_{2}e^{2i\Psi}$,
representing the anisotropic distribution of particles in the $xy$-plane:
$dN/d\phi\propto 1+2v_{2}\cos[2(\phi-\Psi)]$. $V_{2}$ captures the
hydrodynamic response to the elliptically-shaped overlap region, defined by
the eccentricity vector $\mathcal{E}_{2}=\varepsilon_{2}e^{2i\Phi}$. Both
$\mathcal{E}_{2}$ and $V_{2}$ fluctuate along $\eta$ within a single collision
event Jia and Huo (2014); Pang  _et al._ (2016). Experimentally, elliptic flow
is accessed using the two-particle correlation (2PC) method, which quantifies
the second moment of its event-by-event distribution,
$v_{2}\\{2\\}^{2}=V_{2\Delta}(\eta_{1},\eta_{2})=\left\langle
V_{2}(\eta_{1})V_{2}^{*}(\eta_{2})\right\rangle$. The decorrelations manifests
as deviations of a factorization ratio from unity:

$\displaystyle R(\eta_{1},\eta_{2})=\frac{\left\langle
V_{2}(\eta_{1})V_{2}^{*}(\eta_{2})\right\rangle}{\sqrt{\left\langle
V_{2}(\eta_{1})V_{2}^{*}(\eta_{1})\right\rangle\left\langle
V_{2}(\eta_{2})V_{2}^{*}(\eta_{2})\right\rangle}}<1$ (1)

The presence of non-flow correlations concentrated at small relative rapidity
$|\Delta\eta|\lesssim 1-2$ ($\Delta\eta=\eta_{1}-\eta_{2}$), however, impedes
a direct measurement of $R$. Historically, flow decorrelations were
investigated indirectly by correlating reference particles at forward rapidity
e.g. $\eta_{\mathrm{ref}}>4$, with particles of interest at mid-rapidity e.g.
$0<\eta<2$, using the correlator Khachatryan  _et al._ (2015):

$\displaystyle r_{2}(\eta)_{\eta_{\mathrm{ref}}}=\frac{\left\langle
V_{2}(-\eta)V_{2}^{*}(\eta_{\mathrm{ref}}\right\rangle}{\left\langle
V_{2}(\eta)V_{2}^{*}(\eta_{\mathrm{ref}})\right\rangle}\equiv\frac{R(-\eta,\eta_{\mathrm{ref}})}{R(\eta,\eta_{\mathrm{ref}})}\;.$
(2)

The second part of the equation is valid for symmetric systems. It is evident
that $r_{2}$ only captures a limited information of $R$. Moreover, an
assumption of $r_{2}$’s independence on $\eta_{\mathrm{ref}}$ is often made to
interpret $r_{2}$ as a measure of decorrelation between $\eta$ and $-\eta$.

The shape of the colliding nuclei also impacts the elliptic flow Filip  _et
al._ (2009); Shou  _et al._ (2015); Adamczyk  _et al._ (2015); STAR
Collaboration (2024); Giacalone (2023) and its longitudinal structure Nie  _et
al._ (2023). Most nuclei are non-spherical, exhibiting either prolate or
oblate shapes characterized by the quadrupole deformation parameter $\beta$.
The event-wise initial state eccentricity vector comprises a component
corresponding to the spherical nuclei and another arising from deformation:
$\mathcal{E}_{2}=\mathcal{E}_{2,\mathrm{sp}}+\mathcal{E}_{2,\beta}$. The
$\mathcal{E}_{2,\beta}\approx p(\Omega_{1},\Omega_{2})\beta$ depends on the
Euler angle $\Omega$ of the two colliding nuclei and is uncorrelated with
$\mathcal{E}_{2,\mathrm{sp}}$, e.g.
$\left\langle\mathcal{E}_{2,\mathrm{sp}}\mathcal{E}_{2,\beta}^{*}\right\rangle=0$
Jia (2022). Consequently, the elliptic flow in each collision event is the sum
of two uncorrelated components: $V_{2}=V_{2,\mathrm{sp}}+V_{2,\beta}$. The
flow measured via the 2PC method contains independent contributions from these
sources along with non-flow $\delta_{\mathrm{nf}}$ that primarily depends on
$\Delta\eta$:

$\displaystyle V_{2\Delta}$
$\displaystyle=V_{2\Delta,\mathrm{sp}}+V_{2\Delta,\beta}+\delta_{\mathrm{nf}}$
$\displaystyle=a(\eta_{1},\eta_{2})+b(\eta_{1},\eta_{2})\beta^{2}+\delta_{\mathrm{nf}}(|\eta_{1}-\eta_{2}|)$
(3)

Both spherical baseline $a$ and deformation component $b\beta^{2}$ may exhibit
rapidity dependencies. In ultra-central collisions, the
$\mathcal{E}_{2,\mathrm{sp}}$, and thus $V_{2\Delta,\mathrm{sp}}$, are driven
by nucleon random fluctuations, and are inversely proportional to the mass
number $A$, $V_{2\Delta,\mathrm{sp}}\propto 1/A$ Blaizot  _et al._ (2014).

Ideally, the rapidity structure of the deformation-induced component can be
isolated by comparing isobar pairs $X+X$ and $Y+Y$ with different $\beta$
values:

$\displaystyle
b(\eta_{1},\eta_{2})=(V_{2\Delta,\mathrm{X+X}}-V_{2\Delta,\mathrm{Y+Y}})/(\beta_{\mathrm{X+X}}^{2}-\beta_{\mathrm{Y+Y}}^{2})$
(4)

The subtraction of $V_{2\Delta}$ eliminates the spherical baseline and non-
flow contributions, facilitating direct explorations of the QGP’s longitudinal
structure. An avaliable pair for this purpose is 238U+238U and 197Au+197Au
collisions, acquired by the STAR experiment at RHIC. U has large prolate
deformation, while Au is mildly oblate. We focus on the most central
collisions, where the impact of the deformation is most significant.

Setup. Our analysis is performed using the transport model, AMPT Lin  _et al._
(2005), which simulates the full 3D evolution of heavy-ion collisions event by
event. The distribution of nucleons in nuclei is sampled from a Woods-Saxon
parameterization in polar coordinate:
$\rho(r,\theta)\propto[1+\exp((r-R_{0}(1+\beta Y_{2,0}(\theta))/a_{0})]^{-1}$.
Collisions for three configurations are simulated, each characterized by
distinct geometrical parameters as listed in Table 1. The collision events are
generated at $\mbox{$\sqrt{s_{\mathrm{NN}}}$}=193$ GeV for U+U and
$\mbox{$\sqrt{s_{\mathrm{NN}}}$}=200$ GeV for Au+Au, empolying the string-
melting mode with a partonic cross-section of 3 $m$b. We calculate
eccentricity for forward-going, backward-going and all participating nucleons,
denoted as $\mathcal{E}_{2,\mathrm{F}}$, $\mathcal{E}_{2,\mathrm{B}}$ and
$\mathcal{E}_{2}$, respectively. The particles for the 2PC analysis are chosen
from the transverse momentum range $0.2<p_{\mathrm{T}}<3$ GeV/$c$ and
pseudorapidity range $|\eta|<5$. Event centrality is defined by charged
particle multiplicity in $|\eta|<0.5$ ($N_{\mathrm{ch}}$).

System |  238U+238U |  238U+238U |  197Au+197Au
---|---|---|---
$\beta$,$R_{0}$(fm),$a_{0}$(fm) | 0.28, 6.81, 0.55 | 0, 6.81, 0.55 | -0.14, 6.62, 0.52
Table 1:  Nuclei species and their quadrupole deformation $\beta$, radius
$R_{0}$ and skin thickness $a_{0}$ used in AMPT simulation. Figure 1:  The
elliptic flow from 2PC method, $V_{2\Delta}(\eta_{1},\eta_{2})$, in 0–5% most
central U+U collisions with deformation $\beta=0.28$ (a), no deformation
$\beta=0$ (b), and their difference
$V_{2\Delta,\beta}=V_{2\Delta}\\{\beta=0.28\\}-V_{2\Delta}\\{\beta=0\\}$ (c).
The $V_{2\Delta,\beta}$ is separated into the $\eta$ dependence of elliptic
flow, $[v_{2}\\{2\\}(\eta)]^{2}\equiv V_{2\Delta,\beta}(\eta,\eta)$, in the
inset of panel (c) and the flow decorrelations in panel (d), captured by the
factorization ratio $R_{\beta}(\eta_{1},\eta_{2})$.

Result. Figure 1(a) shows the two-dimensional distribution of
$V_{2\Delta}(\eta_{1},\eta_{2})$ in 0–5% central U+U collisions without
deformation. A notable diagonal ridge along $\eta_{+}=(\eta_{1}+\eta_{2})/2$
within $|\Delta\eta|<2$ is observed, indicating contributions from non-flow
and/or other short-range correlations driven by localized hot spots in the
initial state Pang  _et al._ (2014). Beneath this diagonal ridge lies a
broader distribution whose amplitude decreases with $\eta_{+}$ and
$\Delta\eta$, reflecting a decrease of flow with $\eta$ and the effects of
flow decorrelations, respectively. Deformation enhances $V_{2\Delta}$ by about
a factor of two (Fig. 1(b)), and this enhancement (Fig. 1(c)) represents the
deformation-induced component $V_{2\Delta,\beta}=b\beta^{2}$. We found that
the shape of $V_{2\Delta,\beta}$ is indeed independent of the $\beta$ value
used.

Figure 2:  Projections of $R_{\beta}$ in bottom-right panel of Fig. 1 along
the vertical direction as a function of $\eta_{2}$ (a) and diagonal direction
as a function of $\Delta\eta$ (b), as well as the
$r_{2}=R_{\beta}(-\eta,\eta_{\mathrm{ref}})/R_{\beta}(\eta,\eta_{\mathrm{ref}})$
calculated using different $\eta_{\mathrm{ref}}$ (c).

The $V_{2\Delta,\beta}$ exhibits a broad shape in pseudorapidity. Its values
along the diagonal direction, $V_{2\Delta,\beta}(\eta,\eta)=\left\langle
V_{2,\beta}(\eta)V_{2,\beta}^{*}(\eta)\right\rangle$ (inset), capture the
$\eta$ dependence of flow without any decorrelation effects. This flow signal
remains nearly constant for $|\eta|<2$ but then decreases at larger $\eta$. To
isolate decorrelations of flow from its $\eta$ dependence, we compute the 2D
matrix in Eq. (1) and display the resulting $R(\eta_{1},\eta_{2})$ in Fig.
1(d).

To dissect the trends in Fig. 1(d), we depict the values of $R$ in narrow
slices along the vertical or diagonal directions in Fig. 2(a) and Fig. 2(b),
respectively. Vertical slices capture the decorrelation of particles at
$\eta_{2}$ relative to the reference particles at $\eta_{1}$. Notably, for
reference particles at far-backward rapidity $\eta_{1}\sim-5$, the
decorrelations are stronger towards positive $\eta_{2}$. Conversely, when
reference particles are situated at midrapidity $\eta_{1}\sim 0$, the
decorrelations are negligible up to $\eta_{2}\sim 2$ before turning on
quickly. These trends manifest differently in the diagonal slices in Fig.
2(b), which displays decorrelation as a function of $\eta_{+}$ for pairs at a
fixed $\Delta\eta$. Evidently, pairs at a given $\Delta\eta$ generally exhibit
less decorrelation around mid-rapidity than those at larger rapidity.

Figure 2(c) displays the $r_{2}$ calculated via Eq. (2). Since non-flow is
absent in $R(\eta_{1},\eta_{2})$, $r_{2}$ can be calculated for any
$\eta_{\mathrm{ref}}$. Remarkably, the values of $r_{2}$ depend on the choice
of $\eta_{\mathrm{ref}}$, in particular for $\eta_{\mathrm{ref}}\lesssim 3$.
This result indicates that the common notion Khachatryan  _et al._ (2015); Aad
_et al._ (2021); Behera  _et al._ (2020) that $r_{2}$ measures the
decorrelation between $\eta$ and $-\eta$ is incorrect.

Having investigated decorrelations of the deformation-induced component, we
next explore the decorrelations of the normal component
$V_{2\Delta,\mathrm{sp}}(\eta_{1},\eta_{2})$. As demonstrated below, there are
strong evidences that decorrelations of long-range part of
$V_{2\Delta,\mathrm{sp}}$ is quite similar to that of $V_{2\Delta,\beta}$.

Let’s first emphasize the well-established fact that the decorrelation of
$V_{2}$ is driven by the difference between $\mathcal{E}_{2,\mathrm{F}}$ and
$\mathcal{E}_{2,\mathrm{B}}$. This intepretation arises because the initial-
state geometry at forward (backward) rapidity aligns more with
$\mathcal{E}_{2,\mathrm{F}}$ ($\mathcal{E}_{2,\mathrm{B}}$), while at mid-
rapidity, it is driven by
$\mathcal{E}_{2}\sim(\mathcal{E}_{2,\mathrm{F}}+\mathcal{E}_{2,\mathrm{B}})/2$
Jia and Huo (2014); Behera  _et al._ (2020). Hence, $V_{2}$ decorrelation can
be approximated by its projection along $\mathcal{E}_{2,\mathrm{B}}$,

$\displaystyle v_{2,\varepsilon_{\mathrm{B}}}(\eta)\equiv\frac{\left\langle
V_{2}(\eta)\mathcal{E}_{2,\mathrm{B}}^{*}\right\rangle}{\sqrt{\left\langle\mathcal{E}_{2,\mathrm{B}}\mathcal{E}_{2,\mathrm{B}}^{*}\right\rangle}}\;$
(5)

Figure 3(a) shows the results in 0–5% centrality with (without) deformation,
$v_{2,\varepsilon_{\mathrm{B}}}(\eta)$
($v_{2,\varepsilon_{\mathrm{B,sp}}}(\eta)$), along with the deformation-
induced component calculated as

$\displaystyle
v_{2,\varepsilon_{\mathrm{B},\beta}}(\eta)\\!\equiv\\!\frac{\left\langle
V_{2}(\eta)\mathcal{E}_{2,{\mathrm{B}},\beta}^{*}\right\rangle}{\sqrt{\left\langle\mathcal{E}_{2,{\mathrm{B}},\beta}\mathcal{E}_{2,{\mathrm{B}},\beta}^{*}\right\rangle}}\\!=\\!\frac{\left\langle
V_{2}(\eta)\mathcal{E}_{2,{\mathrm{B}}}^{*}\right\rangle_{\beta\\!=0\\!.28}\\!-\\!\left\langle
V_{2}(\eta)\mathcal{E}_{2,{\mathrm{B}}}^{*}\right\rangle_{\beta\\!=\\!0}\\!}{\sqrt{\left\langle\mathcal{E}_{2,{\mathrm{B}}}\mathcal{E}_{2,{\mathrm{B}}}^{*}\right\rangle_{\beta\\!=\\!0.28}\\!-\\!\left\langle\mathcal{E}_{2,{\mathrm{B}}}\mathcal{E}_{2,{\mathrm{B}}}^{*}\right\rangle_{\beta\\!=\\!0}}\\!}\;,$
(6)

satisfying the relation
$[\\!v_{2,\varepsilon_{\mathrm{B}}}\\!\\{\\!\beta\\!=\\!0.28\\!\\}\\!]^{2}\\!=\\!v_{2,\varepsilon_{\mathrm{B}},\mathrm{sp}}^{2}\\!+\\!v_{2,\varepsilon_{\mathrm{B},\beta}}^{2}$.
Large asymmetries in $\eta$ are observed for all cases. We further overlay the
2PC flow from Fig. 1(c) for $\eta_{\mathrm{ref}}=4-5$:

$\displaystyle
v_{2,\eta_{\mathrm{ref},\beta}}(\eta)=\frac{V_{2\Delta,\beta}(\eta,\eta_{\mathrm{ref}})}{\sqrt{V_{2\Delta,\beta}(\eta_{\mathrm{ref}},\eta_{\mathrm{ref}})}}$
(7)

Since reference particles are chosen from very forward rapidity,
$v_{2,\eta_{\mathrm{ref},\beta}}$ should resemble
$v_{2,\varepsilon_{\mathrm{B}},\beta}$. In reality,
$v_{2,\varepsilon_{\mathrm{B}},\beta}\approx
0.75v_{2,\eta_{\mathrm{ref},\beta}}$, implying that part of the flow signal is
not driven by participant eccentricity (instead by local hot spots In
preparation ).

We quantify eccentricity-induced decorrelations via
$r_{2.\varepsilon_{\mathrm{B}}}=v_{2,\varepsilon_{\mathrm{B}}}(\eta)/v_{2,\varepsilon_{\mathrm{B}}}(-\eta)$,
defined similar to Eq. (2). The results in 0–5% centrality are displayed in
Fig. 3 (b). Significant decreases with $\eta$ are observed in all four cases:
eccentricity-driven flow with and without deformation
$v_{2,\varepsilon_{\mathrm{B},\beta}}$ and
$v_{2,\varepsilon_{\mathrm{B}},\mathrm{sp}}$, and the deformation-induced
components $v_{2,\varepsilon_{\mathrm{B},\beta}}$ and
$v_{2,\eta_{\mathrm{ref},\beta}}$. Remarkably, all cases show nearly identical
trends, suggesting flow component corresponding to spherical nuclei and the
component driven by deformation have similar longitudinal structure. However,
the decorrelation of spherical baseline is slightly larger than deformation-
induced component in 0–2% central collisions (see Appendix).

Figure 3:  Left: Comparison of the flow calculated using the initial state
eccentricity of backward-going nucleon participants
$v_{2,\varepsilon_{\mathrm{B}}}$ in Eq. (5) in U+U collisions with
$\beta=0.28$ (open-black circles) and $\beta=0$ (open-red circles), the
difference between the two (solid-blue circles). They reflect the magnitudes
of $V_{2}$ projected along $\mathcal{E}_{2,\mathrm{B}}$. The solid-pink
circles show the deformation-driven elliptic flow obtained directly from two-
particle methods via distributions like those in Fig. 1(c). Right: The ratios
of distribution $\mathcal{O}(\eta)$, calculated as
$r_{2}=\mathcal{O}(\eta)/\mathcal{O}(-\eta)$, for the four distributions in
the top panels, reflecting the flow decorrelations measured in the traditional
methods. Results are obtained in the 0–5% most central U+U collisions.

Our analysis hinges on the complete elimination of non-flow and other short-
range correlations to unveil the full longitudinal structure of $V_{2}$.
Ideally, this can be achieved only through a comparison between systems with
the same $A$, such as U+U with and without deformation. For the comparison
between 197Au+197Au and 238U+238U, the non-flow contributions at the same
centrality do not cancel, given a 20% difference in their masses. This can be
fixed by introduce a scale factor in the subtraction,

$\displaystyle
V_{2\Delta,\beta}^{\mathrm{U-Au}}=V_{2\Delta,\mathrm{U}}\\{\beta=0.28\\}-\frac{\left\langle
N_{\mathrm{ch}}\right\rangle_{\mathrm{Au}}}{\left\langle
N_{\mathrm{ch}}\right\rangle_{\mathrm{U}}}V_{2\Delta,\mathrm{Au}}\\{\beta=-0.14\\}\;.$
(8)

Here, $\left\langle N_{\mathrm{ch}}\right\rangle$ denotes the average charge
particle multiplicity in the chosen centrality range. This works because the
2PC non-flow is expected to scale inversely with $N_{\mathrm{ch}}$ in the
absence of medium effects: $\delta_{\mathrm{nf}}\propto
1/N_{\mathrm{ch}}\propto 1/A$ Chatrchyan  _et al._ (2013); Abdulhamid  _et
al._ (2023). This scaling factor also ensures the cancellation of the
spherical baseline since $V_{2\Delta,\mathrm{sp}}\propto 1/A$.

Figure 4:  Top: The factorization ratio $R_{\beta}$ for
$V_{2\Delta,\beta}^{\mathrm{U-Au}}$ obtained from the difference between U+U
and Au+Au collisions via Eq. (8) (similar to Fig. 1(d). Bottom: Slices of
$R_{\beta}$ along the vertical direction (b) as well as the $r_{2}$ calculated
using different $\eta_{\mathrm{ref}}$ (c). They are compared with results
obtained by subtracting U+U collisions with deformation from U+U collisions
without deformation in Fig. 2.

Figure 4(a) presents the $R(\eta_{1},\eta_{2})$ derived from
$V_{2\Delta,\beta}^{\mathrm{U-Au}}$ via Eq. (8). The shape of the distribution
resembles Fig. 1(d), indicating that non-flow correlations are effectively
eliminated 111Not using the scale factor in Eq. (8) would result in a shallow
minimum along the diagonal direction, indicative of over-subtraction of non-
flow.. Consequently, the decorrelations inferred from $R$ (Fig. 4(b)) and
$r_{2}$ (Fig. 4(c)) resembles the ideal cases with non-flow completely
elliminated.

Summary. We introduce a novel approach to probe the longitudinal structure of
QGP in central high-energy nuclear collisions by comparing two systems
involving nuclei with similar mass but differing shapes. By measuring the
difference of two-particle azimuthal correlation in 238U+238U and 197Au+197Au
collisions, we extract the component of the elliptic flow induced by the large
quadrupole deformation of 238U: $V_{2\Delta,\beta}(\eta_{1},\eta_{2})$. This
extraction via Eq. (8) effectively subtracts out non-flow correlations,
unveiling the decorrelation of $V_{2\Delta,\beta}$ over the full $\eta_{1}$
and $\eta_{2}$ space. Using a transport model simulation, we found that the
decorrelations for pairs with a given $\Delta\eta$ is stronger at large
rapidity than at mid-rapidity. This non-unform decorrelations results from
$\eta$-dependent eccentricity that interpolate between
$\mathcal{E}_{2,\mathrm{F}}$ and $\mathcal{E}_{2,\mathrm{B}}$, driven by the
forward-going and backward-going nucleons, respectively. Moreover, the
decorrelations of deformation-induced flow is found to be similar to
decorrelations of its spherical baseline. Our method extends the concept of
event-shape engineering Schukraft  _et al._ (2013) but circumvents its
inherent selection bias. Therefore, this “event-shape design” data-driven
approach could be used to map out the QGP’s longitudinal structure by
comparing 238U+238U and 197Au+197Au at RHIC, 129Xe+129Xe and 208Pb+208Pb at
the LHC, or the recently proposed small collision systems 16O+16O and
20Ne+20Ne Giacalone  _et al._ (2024).

The idea of probing initial state via nuclear structure knowledge goes beyond
nuclear shapes. In fact, other global properties of atomic nuclei, such as the
nuclear radius $R_{0}$ and nuclear skin $a_{0}$, can be similarly employed to
probe the initial conditions. The STAR experiment has collected data on
96Ru+96Ru and 96Zr+96Zr collisions, which, in addition to their differing
shapes, also possess distinct $a_{0}$ and $R_{0}$ values. The $R_{0}$ and
$a_{0}$ have been shown to impact several bulk observables in mid-central
collisions, including the $N_{\mathrm{ch}}$ distribution, $v_{2}$ and mean
$p_{\mathrm{T}}$ Jia and Zhang (2023); Jia  _et al._ (2023); Xu  _et al._
(2023). Investigating the rapidity dependence of bulk observables and their
differences between Ru+Ru and Zr+Zr collisions could further constrain the
longitudinal structure of QGP’s initial conditions in a broad centrality
range. We leave this intriguing prospect for future study.

We thank Somadutta Bhatta and Giuliano Giacalone for useful discussions. This
work is supported by DOE Research Grant Number DE-SC0024602.

## Appendix

Figure 5 displays the longtidudinal decorrelations in ultra-central 0–2%
collisions, similar to Fig. 3. While the general trends resemble those
observed in the 0–5% centrality, the overall strength of decorrelations are
stronger, consistent with previous findings regarding the centrality
dependence of flow decorrelations Khachatryan  _et al._ (2015). Meanwhile, the
$r_{2}\\{v_{2,\varepsilon_{\mathrm{B}}}\\}$ without deformation (open-red)
shows a somewhat stronger decrease compared to that with deformation (open-
black), implying that decorrelations of $V_{2\Delta,\mathrm{sp}}$ is also
stronger than that of $V_{2\Delta,\beta}$. This subtle difference between the
deformation-induced component and its spherical baseline is apparent only in
ultra-central collisions, attributable to the tendency of the symmetry axes of
the two nuclei to align in these collisions, which we discuss below.

Figure 5:  Left: Comparison of the flow calculated using the initial state
eccentricity of backward-going nucleon participants
$v_{2,\varepsilon_{\mathrm{B}}}$ in Eq. (5) in U+U collisions with
$\beta=0.28$ (open-black circles) and $\beta=0$ (open-red circles), the
difference between the two (solid-blue circles). They reflect the magnitudes
of $V_{2}$ projected along $\mathcal{E}_{2,\mathrm{B}}$. The solid-pink
circles show the deformation-driven elliptic flow obtained directly from two-
particle methods via distributions like those in Fig. 1(c). Right: The ratios
of distribution $\mathcal{O}(\eta)$, calculated as
$r_{2}=\mathcal{O}(\eta)/\mathcal{O}(-\eta)$, for the four distributions in
the top panels, reflecting the flow decorrelations measured in the traditional
methods. Results are obtained in the 0–2% most central U+U collisions.

The longitudinal structure of the QGP’s initial ellipticity interpolates
between $\mathcal{E}_{2,\mathrm{F}}$ at the forward rapidity and
$\mathcal{E}_{2,\mathrm{B}}$ at the backward rapidity. Therefore, the
decorrelations in Figs. 3 and 5 are expected to reflect the decorrelations
between $\mathcal{E}_{2,\mathrm{F}}$ and $\mathcal{E}_{2,\mathrm{B}}$
characterized by:

$\displaystyle
R_{\varepsilon}^{\mathrm{F-B}}=\frac{\left\langle\mathcal{E}_{2,{\mathrm{F}}}\mathcal{E}_{2,{\mathrm{B}}}^{*}\right\rangle}{\sqrt{\left\langle\mathcal{E}_{2,{\mathrm{F}}}\mathcal{E}_{2,{\mathrm{F}}}^{*}\right\rangle\left\langle\mathcal{E}_{2,{\mathrm{B}}}\mathcal{E}_{2,{\mathrm{B}}}^{*}\right\rangle}}\;.$
(9)

This quantity captures the origin of the flow decorrelations defined in Eq.
(1). The centrality dependence of $R_{\varepsilon}^{\mathrm{F-B}}$ is
displayed in Fig. 6 for U+U collisions with and without deformation. The
presence of deformation increases the values of
$R_{\varepsilon}^{\mathrm{F-B}}$ in 0–2% centrality range, implying less
decorrelations. Beyond the 2% centrality, the deformation has little impact on
$R_{\varepsilon}^{\mathrm{F-B}}$. These behaviors naturally explain the
ordering of the decorrelation seen in Fig. 5 (b).

Figure 6:  Centrality dependence of $R_{\varepsilon}^{\mathrm{F-B}}$ defined
in Eq. (9) for U+U collisions with and without deformation. The inset shows
the rms values of the $\varepsilon_{2}$ for the two cases.

On the other hand, the magnitudes of the eccentricity, in terms of its mean-
square values, are consistently enhanced by deformation over a much larger
centrality range, as shown in the inset. Evidently, deformation always
enhances the value of $\left\langle\varepsilon_{2}^{2}\right\rangle$, but may
not necessarily lead to a better alignment of the two eccentricity vectors
$\mathcal{E}_{2,\mathrm{F}}$ and $\mathcal{E}_{2,\mathrm{B}}$ 222Both
$\mathcal{E}_{2,\mathrm{F}}$ and $\mathcal{E}_{2,\mathrm{B}}$ have a
deformation component $\mathcal{E}_{2,\beta}$, statistically uncorrelated with
each other. In a Glauber model, it is easy to show that the mean square value
of total eccentrity
$\mathcal{E}_{2}=(\mathcal{E}_{2,\mathrm{F}}+\mathcal{E}_{2,\mathrm{B}})/2$ is
enhanced by
$\left\langle\mathcal{E}_{2,\beta}\mathcal{E}_{2,\beta}*\right\rangle/2$. For
full alignment, the enhancement would be
$\left\langle\mathcal{E}_{2,\beta}\mathcal{E}_{2,\beta}*\right\rangle$.. The
alignment of eccentricity vectors in 0–2% centrality is the result of
requiring large $N_{\mathrm{ch}}$ which prefers an alignment in the
orientations of their symmetry axes. Selection of central but not ultra-
central collisions does not fully enforce such alignment, and the orientations
of the two nuclei hence are partially random with respect to each other.

## References

  * Romatschke and Romatschke (2019) P. Romatschke and U. Romatschke,  _Relativistic Fluid Dynamics In and Out of Equilibrium_, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2019) arXiv:1712.05815 [nucl-th] .
  * Busza  _et al._ (2018) W. Busza, K. Rajagopal, and W. van der Schee, Ann. Rev. Nucl. Part. Sci. 68, 339 (2018), arXiv:1802.04801 [hep-ph] .
  * Achenbach  _et al._ (2024) P. Achenbach _et al._ , Nucl. Phys. A 1047, 122874 (2024), arXiv:2303.02579 [hep-ph] .
  * Bally  _et al._ (2022) B. Bally _et al._ , (2022), arXiv:2209.11042 [nucl-ex] .
  * Bozek  _et al._ (2011) P. Bozek, W. Broniowski, and J. Moreira, Phys. Rev. C 83, 034911 (2011), arXiv:1011.3354 [nucl-th] .
  * Jia and Huo (2014) J. Jia and P. Huo, Phys. Rev. C 90, 034915 (2014), arXiv:1403.6077 [nucl-th] .
  * Pang  _et al._ (2016) L.-G. Pang, H. Petersen, G.-Y. Qin, V. Roy, and X.-N. Wang, Eur. Phys. J. A 52, 97 (2016), arXiv:1511.04131 [nucl-th] .
  * Jia  _et al._ (2016) J. Jia, S. Radhakrishnan, and M. Zhou, Phys. Rev. C 93, 044905 (2016), arXiv:1506.03496 [nucl-th] .
  * Wu  _et al._ (2018) X.-Y. Wu, L.-G. Pang, G.-Y. Qin, and X.-N. Wang, Phys. Rev. C 98, 024913 (2018), arXiv:1805.03762 [nucl-th] .
  * Shen and Schenke (2018) C. Shen and B. Schenke, Phys. Rev. C 97, 024907 (2018), arXiv:1710.00881 [nucl-th] .
  * Khachatryan  _et al._ (2015) V. Khachatryan _et al._ (CMS), Phys. Rev. C 92, 034911 (2015), arXiv:1503.01692 [nucl-ex] .
  * Aaboud  _et al._ (2018) M. Aaboud _et al._ (ATLAS), Eur. Phys. J. C 78, 142 (2018), arXiv:1709.02301 [nucl-ex] .
  * Aad  _et al._ (2021) G. Aad _et al._ (ATLAS), Phys. Rev. Lett. 126, 122301 (2021), arXiv:2001.04201 [nucl-ex] .
  * Acharya  _et al._ (2018) S. Acharya _et al._ (ALICE), Phys. Lett. B 781, 20 (2018), arXiv:1710.07975 [nucl-ex] .
  * Aaboud  _et al._ (2017) M. Aaboud _et al._ (ATLAS), Phys. Rev. C 95, 064914 (2017), arXiv:1606.08170 [hep-ex] .
  * Xu  _et al._ (2022) Z. Xu, X. Wu, C. Sword, G. Wang, S. A. Voloshin, and H. Z. Huang, Phys. Rev. C 105, 024902 (2022), arXiv:2012.06689 [nucl-ex] .
  * Filip  _et al._ (2009) P. Filip, R. Lednicky, H. Masui, and N. Xu, Phys. Rev. C 80, 054903 (2009).
  * Shou  _et al._ (2015) Q. Y. Shou, Y. G. Ma, P. Sorensen, A. H. Tang, F. Videbæk, and H. Wang, Phys. Lett. B 749, 215 (2015), arXiv:1409.8375 [nucl-th] .
  * Adamczyk  _et al._ (2015) L. Adamczyk _et al._ (STAR), Phys. Rev. Lett. 115, 222301 (2015), arXiv:1505.07812 [nucl-ex] .
  * STAR Collaboration (2024) STAR Collaboration, (2024), arXiv:2401.06625 [nucl-ex] .
  * Giacalone (2023) G. Giacalone, Eur. Phys. J. A 59, 297 (2023), arXiv:2305.19843 [nucl-th] .
  * Nie  _et al._ (2023) M. Nie, C. Zhang, Z. Chen, L. Yi, and J. Jia, Phys. Lett. B 845, 138177 (2023), arXiv:2208.05416 [nucl-th] .
  * Jia (2022) J. Jia, Phys. Rev. C 105, 044905 (2022), arXiv:2109.00604 [nucl-th] .
  * Blaizot  _et al._ (2014) J.-P. Blaizot, W. Broniowski, and J.-Y. Ollitrault, Phys. Rev. C 90, 034906 (2014), arXiv:1405.3274 [nucl-th] .
  * Lin  _et al._ (2005) Z.-W. Lin, C. M. Ko, B.-A. Li, B. Zhang, and S. Pal, Phys. Rev. C 72, 064901 (2005), arXiv:nucl-th/0411110 .
  * Pang  _et al._ (2014) L. Pang, Q. Wang, and X.-N. Wang, Phys. Rev. C 89, 064910 (2014), arXiv:1309.6735 [nucl-th] .
  * Behera  _et al._ (2020) A. Behera, M. Nie, and J. Jia, Phys. Rev. Res. 2, 023362 (2020), arXiv:2003.04340 [nucl-th] .
  * (28) In preparation, .
  * Chatrchyan  _et al._ (2013) S. Chatrchyan _et al._ (CMS), Phys. Lett. B 724, 213 (2013), arXiv:1305.0609 [nucl-ex] .
  * Abdulhamid  _et al._ (2023) M. I. Abdulhamid _et al._ (STAR), Phys. Rev. Lett. 130, 242301 (2023), arXiv:2210.11352 [nucl-ex] .
  * Schukraft  _et al._ (2013) J. Schukraft, A. Timmins, and S. A. Voloshin, Phys. Lett. B 719, 394 (2013), arXiv:1208.4563 [nucl-ex] .
  * Giacalone  _et al._ (2024) G. Giacalone _et al._ , (2024), arXiv:2402.05995 [nucl-th] .
  * Jia and Zhang (2023) J. Jia and C. Zhang, Phys. Rev. C 107, L021901 (2023), arXiv:2111.15559 [nucl-th] .
  * Jia  _et al._ (2023) J. Jia, G. Giacalone, and C. Zhang, Phys. Rev. Lett. 131, 022301 (2023), arXiv:2206.10449 [nucl-th] .
  * Xu  _et al._ (2023) H.-j. Xu, W. Zhao, H. Li, Y. Zhou, L.-W. Chen, and F. Wang, Phys. Rev. C 108, L011902 (2023), arXiv:2111.14812 [nucl-th] .


# The Euler Class from a General Connection, Relative to a Metric

Brian Klatt  Rutgers University
Department of Mathematics
New Brunswick, NJ
United States<EMAIL_ADDRESS>
###### Abstract.

We extend the well-known formula for the Euler class of a real oriented even-
dimensional vector bundle in terms of the curvature of a metric connection to
the case of a general linear connection provided a metric is present. We
rewrite the classical Gauss-Bonnet theorem in dimension two in light of this
formula. We also discuss a potential application to a conjecture of Chern, and
make a brief digression to discuss $m$-quasi-Einstein manifolds.

###### Key words and phrases:

Euler class, Gauss-Bonnet

###### 2010 Mathematics Subject Classification:

Primary: 53C05

##  1\. Introduction

The Gauss-Bonnet theorem is one of the most remarkable and beautiful theorems
in geometry, and in all of mathematics. It states (see e.g. [9]) that if
$(M^{2},g)$ is a closed oriented surface with metric $g$ and Gauss curvature
$K_{g}$ then

$\int_{M^{2}}\,K_{g}\,\mathrm{d}V=2\pi\chi(M^{2})$

where $\chi$ denotes the Euler characteristic. Thus, knowing a single
geometric quantity, the total Gauss curvature, completely determines the
topology of the surface; or alternatively, knowing the topology restricts the
range of possibilities for a metric on the surface.

This theorem was generalized by Allendoerfer & Weil to the case of a closed
oriented $2m$-dimensional Riemannian manifold [1], and given a profoundly
simple intrinsic proof shortly after by Chern [6]. Their result is that

$\int_{M^{2m}}\,\mathrm{Pf}(\Omega)=(2\pi)^{m}\chi(M^{2m})$

where $\Omega=\Omega^{i}{}_{j}$ is the endomorphism-valued curvature $2$-form
and

$\mathrm{Pf}(\Omega)=\frac{1}{2^{m}m!}\displaystyle\sum_{\sigma\in
S_{2m}}(-1)^{\sigma}\Omega^{\alpha_{\sigma(1)}}{}_{\alpha_{\sigma(2)}}\wedge...\wedge\Omega^{\alpha_{\sigma(2m-1)}}{}_{\alpha_{\sigma(2m)}}$

is a well-defined $2m$-form when computed in an oriented orthonormal basis,
called the Pfaffian form.

In fact, this form makes perfect sense if $\Omega$ is the curvature $2$-form
of a connection compatible with a fibre metric $g$ on an oriented rank-$2k$
vector bundle $E$, where the sum defining the Pfaffian is now taken over
$S_{2k}$. We then have the well-known theorem (which we will refer to simply
as the Chern-Gauss-Bonnet theorem) that

$(2\pi)^{k}e(E)=\mathrm{Pf}(\Omega)=\frac{1}{2^{k}k!}\displaystyle\sum_{\sigma\in
S_{2k}}(-1)^{\sigma}\Omega^{\alpha_{\sigma(1)}}{}_{\alpha_{\sigma(2)}}\wedge...\wedge\Omega^{\alpha_{\sigma(2k-1)}}{}_{\alpha_{\sigma(2k)}}$

in the cohomology ring of $M$, where $e(E)$ is the Euler class, defined as the
pullback of the Thom class of $E$ by the zero section (see [3] for a statement
and proof).

It is natural to wonder whether the Euler class can be computed from a general
(i.e. not necessarily metric) connection, but a famed inequality of Milnor
([12], but see also [8]) implies that there are flat oriented plane bundles
with nonzero Euler class over a surface. Thus the Euler class of such a bundle
cannot be computed purely from its curvature. This does not, however, rule out
the possibility of a natural formula for the Euler class from a general
connection; there may exist such a formula, provided it includes non-curvature
terms. It is the primary purpose of this paper to state and prove just such a
formula. This we accomplish in sections 2 and 4; section 3 contains some
preliminaries to the proof on the Pfaffian of matrices.

We then, in section 5, record the global and local Gauss-Bonnet formulas in
light of our general formula.

Section 6 contains miscellanea related to the main theorem, in two
subsections. The first part calls attention to a potential application to a
conjecture of Chern on the Euler characteristic of affine manifolds. The
second is an invitation to further research on ``generalized $m$-quasi-
Einstein manifolds.'' This is related to the main theorem by happenstance; it
is through this topic that the author was led to discover the main theorem (in
particular, by thinking about open problem (vi) of [4]).

Finally section 7 is an appendix in two parts: The first serves to fix
notations and contains some standard formulas of differential geometry so as
to avoid cluttering the flow of argumentation in the main part of the paper,
and the second is an extensive compendium of geometric quantities associated
to a particular class of tangent bundle connections.

##  2\. Statement of the Formula

Our setup will be a general linear connection $\nabla$ and fibre metric $g$ on
an oriented rank-$2k$ real vector bundle $E$. The simplest way to realize a
formula of the kind we desire is to find the correct ``error term" to insert
into $(2\pi)^{k}e(E)=\mathrm{Pf}(\Omega)$, such that the error term vanishes
when the connection $\nabla$ is metric. Thus the obvious building block for
our error term is $\nabla g$. This is a symmetric matrix of $1$-forms with
lower bundle indices: $\nabla g=\nabla_{i}g_{\alpha\beta}$. We can naturally
construct from this an object of the same tensorial type as
$\Omega=\Omega^{\alpha}{}_{\beta}=\Omega_{ij}{}^{\alpha}{}_{\beta}$ by using
the inverse metric to first construct $g^{-1}\nabla g=(g^{-1}\nabla
g)_{i}{}^{\alpha}{}_{\beta}=g^{\alpha\gamma}\nabla_{i}g_{\gamma\beta}$ and
then squaring this matrix, using the wedge product to multiply the one-form
entries:

######  Definition 2.1.

The endomorphism-valued two-form $(g^{-1}\nabla g)^{2}$ is defined by

$\displaystyle(g^{-1}\nabla g)^{2}$ $\displaystyle=(g^{-1}\nabla
g)^{2})_{ij}{}^{\alpha}{}_{\beta}$ $\displaystyle=(g^{-1}\nabla
g)_{i}{}^{\alpha}{}_{\gamma}\wedge(g^{-1}\nabla g)_{j}{}^{\gamma}{}_{\beta}$
$\displaystyle=g^{\alpha\gamma}\nabla_{i}g_{\gamma\delta}\,g^{\delta\epsilon}\nabla_{j}g_{\epsilon\beta}-g^{\alpha\gamma}\nabla_{j}g_{\gamma\delta}\,g^{\delta\epsilon}\nabla_{i}g_{\epsilon\beta}$

With these considerations in mind, one can do explicit computations with a
simple non-metric tangent bundle connection like, say,
$\Gamma_{i}{}^{k}{}_{j}=\overline{\Gamma}_{i}{}^{k}{}_{j}+a\,\varphi_{i}\delta^{k}_{j}+b\,\varphi_{j}\delta^{k}_{i}+c\,\varphi^{k}g_{ij}$
($\overline{\Gamma}_{i}{}^{k}{}_{j}$ are the Christoffel symbols for the Levi-
Civita connection $\nabla_{g}$ of $g$ and $\varphi$ is a $1$-form) to guess
the error term, which leads to our main theorem.

######  Theorem 2.2.

Let $E$ be an oriented rank-$2k$ real vector bundle with connection $\nabla$,
curvature $\Omega=\Omega_{\nabla}$, and fibre metric $g$. Then

$(2\pi)^{k}e(E)=\mathrm{Pf}(\Omega-\tfrac{1}{4}(g^{-1}\nabla g)^{2})$

where $e(E)$ is the Euler class of the bundle, and the Pfaffian is computed
using any oriented orthonormal basis for the metric $g$.

######  Remark 2.3.

Theorem 2.2 brings attention to an interesting class of connections $\nabla$
with respect to a given metric $g$: those satisfying $(g^{-1}\nabla g)^{2}=0$.
This condition is strictly weaker than metric compatibility because it is
satisfied if (but not only if), for example,
$\nabla_{i}g_{\alpha\beta}=\varphi_{i}g_{\alpha\beta}$; on the tangent bundle,
this includes Weyl connections
$\nabla=\nabla_{g}-\tfrac{\varphi_{i}}{2}\delta^{k}_{j}-\tfrac{\varphi_{j}}{2}\delta^{k}_{i}+\tfrac{\varphi^{k}}{2}g_{ij}$
[14]. Therefore for Weyl connections, and any connection such that
$(g^{-1}\nabla g)^{2}=0$, one can compute the Euler class directly from the
curvature without any error term via $(2\pi)^{k}e(E)=\mathrm{Pf}(\Omega)$.

##  3\. Preliminary for the Proof

We only need a brief interlude on properties of the Pfaffian before we can
proceed to the proof. (For further reference see Volume V of [17].)

######  Definition 3.1.

Let $M_{1}$, …, $M_{k}$ be $2k$-by-$2k$ matrices with entries in a commutative
algebra $R$ over $\mathbb{Q}$. We define the Pfaffian of these matrices by

$\mathrm{Pf}(M_{1},...,M_{k})=\frac{1}{2^{k}k!}\displaystyle\sum_{\sigma\in
S_{2k}}(-1)^{\sigma}(M_{1})^{i_{\sigma(1)}}_{i_{\sigma(2)}}...(M_{k})^{i_{\sigma(2k-1)}}_{i_{\sigma(2k)}}$

By setting all $M_{i}=M$ we recover $\mathrm{Pf}(M)$, the usual Pfaffian of
$M$.

The Pfaffian is obviously multilinear with respect to matrix addition and
multiplication of matrices by elements of $R$. We also have the following

######  Proposition 3.2.

The Pfaffian satisfies the following properties:

  1. (1)

$\mathrm{Pf}(M_{1},...,(M_{i})^{T},...,M_{k})=-\mathrm{Pf}(M_{1},...,M_{i},...,M_{k})$

  2. (2)

$\mathrm{Pf}(M_{1},...,M_{i},...,M_{k})=\mathrm{Pf}(M_{1},...,(M_{i})_{A},...,M_{k})$
where $(M_{i})_{A}=\frac{1}{2}(M_{i}-(M_{i})^{T})$

  3. (3)

$\mathrm{Pf}(M_{1},...,M_{k})=\mathrm{Pf}((M_{1})_{A},...,(M_{k})_{A})$ and
$\mathrm{Pf}(M)=\mathrm{Pf}(M_{A})$

###### Proof.

  1. (1)

This is obvious from the definition, since a transposition has sign equal to
$-1$.

  2. (2)

With $(M_{i})_{A}=\frac{1}{2}(M_{i}-(M_{i})^{T})$ and
$(M_{i})_{S}=\frac{1}{2}(M_{i}+(M_{i})^{T})$, compute

$\displaystyle\mathrm{Pf}(M_{1},...,M_{i},...,M_{k})$
$\displaystyle=\mathrm{Pf}(M_{1},...,(M_{i})_{A}+(M_{i})_{S},...,M_{k})$
$\displaystyle=\mathrm{Pf}(M_{1},...,(M_{i})_{A},...,M_{k})+\mathrm{Pf}(M_{1},...,(M_{i})_{S},...,M_{k})$

and note by (1) that

$\displaystyle\mathrm{Pf}(M_{1},...,(M_{i})_{S},...,M_{k})$
$\displaystyle=\mathrm{Pf}(M_{1},...,((M_{i})_{S})^{T},...,M_{k})$
$\displaystyle=-\mathrm{Pf}(M_{1},...,(M_{i})_{S},...,M_{k})$

so

$\mathrm{Pf}(M_{1},...,(M_{i})_{S},...,M_{k})=0,$

and (2) follows.

  3. (3)

Use induction on (2).

∎

##  4\. Proof of the Main Theorem and Developments

We are now ready for the proof of Thm. 2.2.

###  4.1. Proof of Theorem 2.2

###### Proof.

Let $\\{E_{\alpha}\\}$ be a local oriented orthonormal basis of $E$, and
$\omega=\omega^{\alpha}{}_{\beta}=\omega_{i}{}^{\alpha}{}_{\beta}$ and
$\Omega=\Omega^{\alpha}{}_{\beta}=\Omega_{ij}{}^{\alpha}{}_{\beta}$ be the
connection and curvature matrices, respectively, in this frame. We split the
matrices of forms

$\omega=\omega_{A}+\omega_{S}$

and

$\Omega=\Omega_{A}+\Omega_{S}$

into antisymmetric (subscript ``A'') and symmetric (subscript ``S'') parts.

One can readily check that

$\Omega_{A}=\mathrm{d}\omega_{A}+\omega_{A}\wedge\omega_{A}+\omega_{S}\wedge\omega_{S}$

and

$\Omega_{S}=\mathrm{d}\omega_{S}+\omega_{A}\wedge\omega_{S}+\omega_{S}\wedge\omega_{A}.$

We also compute in our local orthonormal frame that

$g^{-1}\nabla
g=\delta^{\alpha\gamma}(\mathrm{d}\delta_{\gamma\beta}-\omega^{\epsilon}{}_{\gamma}\delta_{\epsilon\beta}-\omega^{\epsilon}{}_{\beta}\delta_{\gamma\epsilon})=-(\omega^{\alpha}{}_{\beta}+\omega^{\beta}{}_{\alpha})=-2\,\omega_{S}$

which immediately implies

$\tfrac{1}{4}(g^{-1}\nabla g)^{2}=\omega_{S}\wedge\omega_{S}.$

Therefore

$\mathrm{Pf}(\Omega-\tfrac{1}{4}(g^{-1}\nabla
g)^{2})=\mathrm{Pf}(\mathrm{d}\omega_{A}+\omega_{A}\wedge\omega_{A}+\Omega_{S})=\mathrm{Pf}(\mathrm{d}\omega_{A}+\omega_{A}\wedge\omega_{A})$

where we neglected the symmetric part in the last equality by (3) of Prop.
3.2.

All that remains of the proof is to show that $\omega_{A}$ defines a metric
connection, for then

$\mathrm{Pf}(\Omega-\tfrac{1}{4}(g^{-1}\nabla
g)^{2})=\mathrm{Pf}(\mathrm{d}\omega_{A}+\omega_{A}\wedge\omega_{A})=\mathrm{Pf}(\Omega_{\omega_{A}})=(2\pi)^{k}e(E)$

by the Chern-Gauss-Bonnet theorem. This follows easily from

$\omega_{A}=\omega-\omega_{S}=\omega+\tfrac{1}{2}g^{-1}\nabla g.$

In fact, define $\nabla^{g}=\nabla+\frac{1}{2}g^{-1}\nabla g$. This is clearly
a connection with connection matrix $\omega_{A}$ in our local orthonormal
frame, and since this matrix is antisymmetric, it's a metric connection. This
completes the proof. ∎

###  4.2. The Canonical Associated Metric Connection

We record an observation from the proof.

######  Definition 4.1.

Suppose $E$ is a real vector bundle with linear connection $\nabla$ and fibre
metric $g$. Then we call

$\nabla^{g}=\nabla+\tfrac{1}{2}g^{-1}\nabla g$

the _canonical g-metric connection associated to_ $\nabla$.

######  Proposition 4.2.

$\nabla^{g}$ is indeed a metric connection with respect to $g$.

We give a proof independent of our work in the proof of Thm 2.2.

###### Proof.

Choose an arbitrary local frame $\\{E_{\alpha}\\}$ and let
$\omega^{\alpha}{}_{\beta}$ be the matrix of connection $1$-forms of $\nabla$.
Then the connection $1$-forms of $\nabla^{g}$ are
$\eta^{\alpha}{}_{\beta}=\omega^{\alpha}{}_{\beta}+\frac{1}{2}g^{\alpha\gamma}\nabla
g_{\gamma\beta}$, and we can simply compute

$\displaystyle\nabla^{g}g$
$\displaystyle=\mathrm{d}g_{\alpha\beta}-\eta^{\gamma}{}_{\alpha}g_{\gamma\beta}-\eta^{\gamma}{}_{\beta}g_{\alpha\gamma}$
$\displaystyle=\mathrm{d}g_{\alpha\beta}-(\omega^{\gamma}{}_{\alpha}+\tfrac{1}{2}g^{\gamma\delta}\nabla
g_{\delta\alpha})g_{\gamma\beta}-(\omega^{\gamma}{}_{\beta}+\tfrac{1}{2}g^{\gamma\delta}\nabla
g_{\delta\beta})g_{\alpha\gamma}$
$\displaystyle=\mathrm{d}g_{\alpha\beta}-\omega^{\gamma}{}_{\alpha}g_{\gamma\beta}-\omega^{\gamma}{}_{\beta}g_{\alpha\gamma}-\nabla
g_{\alpha\beta}$ $\displaystyle=0$

∎

In brief, then, Thm. 2.2 tells us that to compute the Euler class from
$\nabla$ in the presence of a metric $g$, one should compute the Pfaffian form
of the canonical metric connection induced by $\nabla$ and $g$.

Our next proposition gives an alternative characterization of the connection
$\nabla^{g}$ as the connection nearest to $\nabla$, at each point, in the
affine subspace of $g$-metric connections. Thus $\nabla^{g}$ is a sort of
orthogonal projection of $\nabla$ onto the space of $g$-metric connections.

######  Proposition 4.3.

Let $E$ be a real vector bundle, $\nabla$ a linear connection, and $g$ a fibre
metric. Also, arbitrarily fix a Riemannian metric $h$ on the underlying
manifold, and define length in $T^{*}M\otimes E\otimes E^{*}$ with the metric
$\mu=h^{-1}\otimes g\otimes g^{-1}$. If $\nabla^{\prime}$ is a $g$-metric
connection, then $|\nabla-\nabla^{g}|\leq|\nabla-\nabla^{\prime}|$ with
equality if and only if $\nabla^{\prime}=\nabla^{g}$.

###### Proof.

The basic point is that $\nabla^{g}-\nabla=\frac{1}{2}g^{-1}\nabla g$ is
$g$-symmetric, which is orthogonal to the $g$-antisymmetric directions that
define the metric connections.

To be precise, let $\nabla^{\prime}$ be a $g$-metric connection. Then

$|\nabla-\nabla^{\prime}|^{2}=|(\nabla-\nabla^{g})-(\nabla^{\prime}-\nabla^{g})|^{2}$

We'll compute in a local $g$-orthonormal frame $\\{E_{\alpha}\\}$ of $E$ and a
local $h$-orthonormal frame $\\{e_{i}\\}$ of $TM$, so
$g_{\alpha\beta}=\delta_{\alpha\beta}$ and $h_{ij}=\delta_{ij}$. Denote by
$\omega$ the matrix of connection $1$-forms. As discussed in the proof of Thm
2.2, $\nabla-\nabla^{g}=\omega_{S}$, a symmetric matrix of $1$-forms, while
$\nabla^{\prime}-\nabla^{g}=T_{i}{}^{\alpha}{}_{\beta}$ is an antisymmetric
matrix of $1$-forms since each connection in the difference is metric. Now
compute

$\mu(\omega_{S},T)=\displaystyle\sum_{i,\,\alpha,\,\beta}(\omega_{S})_{i}{}^{\alpha}{}_{\beta}\,T_{i}{}^{\alpha}{}_{\beta}=-\displaystyle\sum_{i,\,\alpha,\,\beta}(\omega_{S})_{i}{}^{\beta}{}_{\alpha}\,T_{i}{}^{\beta}{}_{\alpha}=-\mu(\omega_{S},T)$

Thus $\mu(\omega_{S},T)=0$ and we find that

$\displaystyle|\nabla-\nabla^{\prime}|^{2}$
$\displaystyle=|(\nabla-\nabla^{g})-(\nabla^{\prime}-\nabla^{g})|^{2}$
$\displaystyle=|(\nabla-\nabla^{g})|^{2}+|(\nabla^{\prime}-\nabla^{g})|^{2}$
$\displaystyle=|\omega_{S}|^{2}+|(\nabla^{\prime}-\nabla^{g})|^{2}$

and so the conclusion clearly follows. ∎

##  5\. The Formula on the Tangent Bundle of a Surface

We will now give Thm 2.2 explicitly on the tangent bundle of a compact
oriented surface, so as to compare it to the classical Gauss-Bonnet formula.
Of course the integrands will differ by a divergence term; we will determine
it exactly. We also give the corresponding local Gauss-Bonnet theorem.

###  5.1. Global Gauss-Bonnet Theorem

We begin by specializing our setup to the case where $E=TM$, and so $g$ is a
Riemannian metric on $M$, with Levi-Civita connection $\nabla_{g}$. Given a
connection $\nabla$ (not assumed to be metric or torsion-free) we have a
``right triangle'' (by Prop. 4.3) of connections with vertices $\nabla$,
$\nabla^{g}$, and $\nabla_{g}$. If we write $\nabla=\nabla_{g}-D$ (so $D$ is
the hypotenuse of the triangle), then straightforward computation from the
definition of $\nabla^{g}$ shows $\nabla^{g}=\nabla+D_{S}$ and therefore
$\nabla^{g}=\nabla_{g}-D_{A}$ where

$D_{S}=\tfrac{1}{2}(D+g^{-1}D^{T}g)=\tfrac{1}{2}(D_{i}{}^{k}{}_{j}+g^{kl}D_{i}{}^{m}{}_{l}g_{mj})$

and

$D_{A}=\tfrac{1}{2}(D-g^{-1}D^{T}g)=\tfrac{1}{2}(D_{i}{}^{k}{}_{j}-g^{kl}D_{i}{}^{m}{}_{l}g_{mj})$

Now we set $B=D_{A}$ and $H=\Omega(\nabla^{g})$ for convenience, so we have in
particular $\nabla^{g}=\nabla_{g}-B$. The difference tensor $B$ has one
independent trace
$B_{i}=B_{k}{}^{k}{}_{i}=B_{kki}=\tfrac{1}{2}(D_{jji}-D_{jij})$, and standard
differential geometric calculations (see the appendix) reveal that

$\displaystyle\mathrm{Pf}(H)$
$\displaystyle=(K_{g}-\mathrm{d}^{*}_{g}(B_{i}))\mathrm{d}V_{g}$
$\displaystyle=(K_{g}-\tfrac{1}{2}\mathrm{d}^{*}_{g}(D_{jji}-D_{jij}))\mathrm{d}V_{g}$

which by Thm. 2.2 yields the following

######  Proposition 5.1 (Global Gauss-Bonnet Theorem).

Suppose $\nabla$ is a connection with curvature $\Omega$ on the tangent bundle
of a closed oriented surface $(M^{2},g)$ with Levi-Civita connection
$\nabla_{g}$, Gauss curvature $K_{g}$, and volume form $\mathrm{d}V_{g}$. If
we write $\nabla=\nabla_{g}-D$ and $\nabla^{g}=\nabla_{g}-B$, then

$\displaystyle 2\pi\,e(TM^{2})$
$\displaystyle=\mathrm{Pf}(\Omega-\tfrac{1}{4}(g^{-1}\nabla g)^{2})$
$\displaystyle=(K_{g}-\mathrm{d}^{*}_{g}(B_{i}))\mathrm{d}V_{g}$
$\displaystyle=(K_{g}-\tfrac{1}{2}\mathrm{d}^{*}_{g}(D_{jji}-D_{jij}))\mathrm{d}V_{g}$

and so

$\displaystyle\int_{M^{2}}(K_{g}-\tfrac{1}{2}\mathrm{d}^{*}_{g}(D_{jji}-D_{jij}))\mathrm{d}V_{g}=2\pi\,\chi(M^{2})$

###  5.2. Local Gauss-Bonnet Theorem

Considering the divergence theorem with inward-pointing normal $N$,

$\int_{D}-\mathrm{d}^{*}_{g}(B_{i})\mathrm{d}V_{g}=\int_{D}\mathrm{div}_{g}(B^{i})\mathrm{d}V_{g}=-\int_{\partial
D}g_{ij}B^{i}N^{j}\mathrm{d}s=-\int_{\partial D}B_{i}(N)\mathrm{d}s$

the following proposition is the natural local version of the above Global
Gauss-Bonnet theorem (we borrow some phrasing and notation from Theorem 9.3 of
[9]).

######  Proposition 5.2 (Local Gauss-Bonnet Theorem).

Suppose $\gamma$ is a curved polygon on an oriented surface $(M^{2},g)$,
positively oriented as the boundary of an open set $D$ with compact closure,
with inward pointing normal $N$, and exterior angles $\epsilon_{i}$. Let
$\overline{\nabla}$ be the Levi-Civita connection,
$\kappa_{N}=g(\overline{\nabla}_{\dot{\gamma}}\dot{\gamma},N)$ be the signed
curvature of $\gamma$, and $B_{i}$ be a $1$-form. Then

(5.1)
$\int_{D}(K_{g}-\mathrm{d}^{*}_{g}(B_{i}))\mathrm{d}A_{g}+\int_{\gamma}(\kappa_{N}+B_{i}(N))\mathrm{d}s+\displaystyle\sum_{i}\epsilon_{i}=2\pi$

If $\nabla^{\prime}=\overline{\nabla}-B_{i}{}^{k}{}_{j}$ is a metric
connection, i.e. $B_{ikj}=g_{km}B_{i}{}^{m}{}_{j}$ satisfies
$B_{ikj}=-B_{ijk}$, and $B_{i}=B_{k}{}^{k}{}_{i}=B_{kki}$, then
$K_{g}-\mathrm{d}^{*}_{g}(B_{i})=\mathrm{Pf}(\Omega_{\nabla^{\prime}})$ is the
``Gauss curvature'' of $\nabla^{\prime}$ and
$\kappa_{N}+B_{i}(N)=g(\nabla^{\prime}_{\dot{\gamma}}\dot{\gamma},N)$ is the
``signed curvature of $\gamma$ with respect to $\nabla^{\prime}$.'' If
$\nabla=\overline{\nabla}-D_{i}{}^{k}{}_{j}$ is a general connection, then
according to Thm. 2.2 we apply this to the metric connection
$\nabla^{g}=\overline{\nabla}-\tfrac{1}{2}(D_{i}{}^{k}{}_{j}-g^{kl}D_{i}{}^{m}{}_{l}g_{mj})$.

###### Proof.

For any $1$-form $B_{i}$, Eqn 5.1 follows from the standard local Gauss-Bonnet
theorem, the divergence theorem (i.e. Stokes' theorem), and an argument to
approximate $D$ by smooth domains like the one at the end of the proof of the
Gauss-Bonnet formula (Theorem 9.3) in [9].

If $\nabla^{\prime}=\overline{\nabla}-B_{i}{}^{k}{}_{j}$ is a metric
connection, then
$K_{g}-\mathrm{d}^{*}_{g}(B_{i})=\mathrm{Pf}(\Omega_{\nabla^{\prime}})$ is
just Eqn. 7.3 of the appendix. We can also compute

$g(\nabla^{\prime}_{\dot{\gamma}}\dot{\gamma},N)=g(\overline{\nabla}_{\dot{\gamma}}\dot{\gamma}-B_{\dot{\gamma}}\dot{\gamma},N)=\kappa_{N}-g(B_{\dot{\gamma}}\dot{\gamma},N)$

where in any orthonormal basis, say $\\{\dot{\gamma},N\\}$,
$B=[\begin{smallmatrix}0&b\\\ -b&0\end{smallmatrix}]$ as an endomorphism-
valued $1$-form, with $b=-\star_{1}B_{i}$ (see Eqn. 7.4 in the Appendix). Thus

$B_{\dot{\gamma}}\dot{\gamma}=-b(\dot{\gamma})N=-(\star_{1}b)(N)N=-B_{i}(N)N,$

which yields

$g(\nabla^{\prime}_{\dot{\gamma}}\dot{\gamma},N)=\kappa_{N}+B_{i}(N)$

∎

######  Remark 5.3.

We will not go into the details, but the interested reader may wish to check
the following: if one tracks the proof of Gauss-Bonnet given in [9] but
substitutes a general connection for the Levi-Civita connection and writes
this connection as a metric connection plus difference tensor, any
contribution due to the difference tensor cancels out of the calculations. One
is thereby left with Prop. 5.1. It is precisely when the metric connection is
the canonical associated metric connection that we are being ``least
wasteful'' in what gets cancelled out.

######  Remark 5.4.

We presume that Prop. 5.1 is not really ``new.'' For instance, Corwin and
Morgan in [7] prove a Gauss-Bonnet theorem on a smooth disk in a ``surface
with densities'': $\mathrm{d}s=\delta_{1}\mathrm{d}s_{0}$ and
$\mathrm{d}A=\delta_{2}\mathrm{d}A_{0}$ where the $\delta_{i}$ are positive
density functions. They define $K^{\prime}=K_{g}-\Delta\log\delta_{1}$ and
$\kappa=\tfrac{\delta_{1}}{\delta_{2}}\kappa_{N}-\tfrac{1}{\delta_{2}}g(\nabla\delta_{1},N)$
and show that

$\int_{\gamma}\tfrac{\delta_{2}}{\delta_{1}}\kappa\,\mathrm{d}s_{0}+\int_{D}K^{\prime}\,\mathrm{d}A_{0}=2\pi$

However, this is clearly equivalent to

$\int_{\gamma}(\kappa_{N}-\mathrm{d}(\log\delta_{1})(N))\,\mathrm{d}s_{0}+\int_{D}(K_{g}+\mathrm{d}^{*}_{g}(\mathrm{d}\log\delta_{1}))\,\mathrm{d}A_{0}=2\pi$

since $\Delta=-\mathrm{d}^{*}_{g}\mathrm{d}$. This is Prop. 5.1 when
$B_{i}=-\mathrm{d}(\log\delta_{1})$. If we write $-\log\delta_{1}=f$, then one
nice choice of connection $\nabla$ whose canonical metric connection yields
this $B_{i}$ is

$\nabla=\overline{\nabla}+(m-1)\mathrm{d}f_{i}\,\delta^{k}_{j}+(m-1)\mathrm{d}f_{j}\,\delta^{k}_{i}+(m+1)\overline{\nabla}f^{k}g_{ij}$

It's ``nice'' because it's torsion-free and the vanishing of its traceless
Ricci tensor (with respect to $g$) is related to the surface being a
generalized $m$-quasi-Einstein manifold, a notion which seems to be naturally
related to that of ``manifold with density.'' See e.g. [13] and [5], and the
next section.

##  6\. Applications & Explorations

Given that $\mathrm{Pf}(\Omega-\frac{1}{4}(g^{-1}\nabla
g)^{2})=\mathrm{Pf}(\Omega_{\nabla^{g}})$ and $\nabla^{g}$ is a metric
connection, one may legitimately ask why we have bothered to frame Thm. 2.2 in
terms of general connections. The real reason is that this is the form in
which the theorem was discovered, and we prefer it aesthetically. However we
also believe there are substantial reasons to frame the theorem this way, as
follows.

For one, we may be given a connection with special properties a priori
unrelated to a metric and want to explore the topology of our manifold; we can
then introduce a metric and use Thm. 2.2 to do so. One such instance is
discussed in the next subsection.

Another possibility is that we could have a metric and some other differential
geometric data which is not obviously subordinate to the metric. It is not
clear in such a scenario that one should default to the Levi-Civita
connection, or any particular metric connection for that matter; perhaps some
other, non-metric, connection is better adapted to the situation, or can
suggest natural geometric quantities to study. If so, Thm. 2.2 can be used to
explore the relation between the geometric data and the Euler characteristic.
A potential example of this is discussed in the second subsection below.

###  6.1. Chern's Conjecture on the Euler Characteristic of Affine Manifolds

An advantage of phrasing Thm. 2.2 in terms of general connections is that we
hope it may find an application in the study of a conjecture of Chern. This
conjecture posits that the Euler characteristic of a closed manifold with a
torsion-free flat connection on its tangent bundle should have vanishing Euler
characteristic (see [11] for an overview). If the connection is metric then
the conjecture is true, even without the torsion-free assumption, by the
Chern-Gauss-Bonnet theorem. The conjecture is also true in dimension two by
Milnor's inequality [12], again without the torsion-free assumption. However,
examples of Smillie in all even dimensions greater than two ([16], [8]) show
that in general the torsion-free assumption is necessary for the truth of the
conjecture.

By Thm. 2.2, it suffices to show that $\mathrm{Pf}((g^{-1}\nabla g)^{2})$
integrates to $0$ for any Riemannian metric $g$ when $\nabla$ is a torsion-
free flat connection; or equivalently to show that $\mathrm{Pf}((g^{-1}\nabla
g)^{2})$ is exact. The author has made attempts at this in dimension two
without noteworthy success. Perhaps instead of working with an arbitrary
metric $g$, one should try to be more discerning and choose a $g$ adapted in
some way to $\nabla$. Imposing $\nabla g=0$ is too strong, but what other
equations are natural? One obvious idea based on the concepts discussed here
is to demand that the metric $g$ minimize the distance between $\nabla$ and
$\nabla^{g}$ globally, or what amounts to the same, minimize the magnitude of
$\nabla g$. This suggests minimizing a scale-invariant functional like

$\mathcal{F}_{\nabla}(g)=\int_{M^{n}}|\nabla g|^{n}\,\mathrm{d}V_{g}$

or minimizing, say, the functional

$\mathcal{K}_{\nabla}(g)=\int_{M^{n}}|\nabla g|^{2}\,\mathrm{d}V_{g}$

subject to the constraint $\int_{M^{n}}\mathrm{d}V_{g}=1$. (Note that
$\mathcal{F}_{\nabla}=\mathcal{K}_{\nabla}$ when $n=2$, so we don't need the
volume constraint when using $\mathcal{K}_{\nabla}$ on a surface; is this
possibly related to not needing the torsion-free assumption when $n=2$?)
Perhaps the minima of these functionals, if they exist, will be well-adapted
to the problem at hand. We will not, however, discuss these ideas any further
here, and merely hope that they may prove fruitful in future efforts on this
conjecture.

###  6.2. Natural Connections Incompatible with a Given Metric: Generalized
$m$-Quasi-Einstein Manifolds

In this subsection, we will discuss a situation in which a metric $g$ occurs
alongside other differential geometric data, making the introduction of a
connection $\nabla$ incompatible with $g$ plausibly natural.

####  6.2.1. Generalized $m$-quasi-Einstein Manifolds

A _generalized $m$-quasi-Einstein manifold_ [2] is a Riemannian manifold such
that

(6.1)
$r_{g}+\tfrac{1}{2}\pounds_{\varphi^{\\#}}g-\tfrac{1}{m}\varphi\otimes\varphi=\lambda
g$

where $r_{g}$ is the Ricci tensor of $g$, $\varphi\in\Omega^{1}(M^{n})$,
$\pounds$ denotes the Lie derivative, $m$ is an extended real number, and
$\lambda\in C^{\infty}(M^{n})$. To understand the significance of this
equation, it helps to look at some special cases. When $m=0$ we can
understand, by convention, that the equation simplifies to $r_{g}=\lambda g$,
which also occurs when $\varphi=0$ for any $m$; this is an Einstein manifold
when $n\geq 3$. When $m=\pm\infty$ we have $\frac{1}{m}=0$, so if $\lambda$ is
constant, Eqn. 6.1 becomes the equation of a Ricci soliton, a notion which
occurs in the study of singularities of the Ricci flow [4]. In the case
$m=-(n-2)$, the equation describes an Einstein-Weyl manifold (compare to the
equation on page 100 of [14]), a notable concept in conformal geometry.
Finally, when $\varphi=\mathrm{d}f$, $m$ is a non-zero integer, and $\lambda$
is constant, Eqn. 6.1 is related to constructing Einstein warped products and
conformal Einstein metrics, depending on $m$ (see [5] for more information and
references).

As a Riemannian geometer working with Eqn. 6.1, it is tempting to default to
using the Levi-Civita connection $\nabla_{g}$, which perhaps gives undue
primacy to the metric $g$ and places $\varphi$ in an undeservedly subordinate
role. We propose that one way to unify these two objects which places them on
more equal footing is to consider a connection $\nabla$ which depends on both
quantities and generates Eqn. 6.1 in an appropriate sense. Namely, we demand
that Eqn. 6.1 be equivalent to $\mathring{r}_{(ij)}=0$, where
$\mathring{r}_{()}$ is the traceless (with respect to $g$) symmetric part of
the Ricci tensor of $R=\Omega(\nabla)$. (Our perspective here is strongly
influenced by the philosophy, inherited from E. Cartan and espoused in [15],
that differential geometry is the study of connections on principal bundles.)

####  6.2.2. The Connection Ansatz

What would this connection $\nabla$ look like? Answering this question comes
down to determining the difference tensor $\nabla-\nabla_{g}$. It's easy to
see based on Eqn. 6.1 that this three-index difference tensor should be a
tensor algebraic in $g$ and $\varphi$. Denoting the connection coefficients of
$\nabla$ by $\Gamma_{i}{}^{k}{}_{j}$, and the Christoffel symbols of the Levi-
Civita connection $\nabla_{g}$ by $\overline{\Gamma}_{i}{}^{k}{}_{j}$, our
ansatz is

(6.2)
$\Gamma_{i}{}^{k}{}_{j}=\overline{\Gamma}_{i}{}^{k}{}_{j}+a\varphi_{i}\delta^{k}_{j}+b\varphi_{j}\delta^{k}_{i}+c\varphi^{k}g_{ij}$

The torsion of $\nabla$ vanishes if and only if $a=b$ but, without a full
geometric understanding of the issue, we prefer to leave open the possibility
that torsion may play some role.

The formula for the traceless Ricci tensor of this connection is (see the
Appendix)

$\displaystyle\mathring{r}_{()}=$
$\displaystyle\,\overline{r}-\tfrac{1}{2}((n-1)b+c)\pounds_{\varphi^{\\#}}\,g+((n-1)b^{2}-c^{2})\varphi\otimes\varphi+$
$\displaystyle-\tfrac{1}{n}(\overline{s}+((n-1)b+c)\mathrm{d}^{*}_{g}\varphi+((n-1)b^{2}-c^{2})|\varphi|^{2})g$

From this we see clearly that Eqn 6.1 is identical to $\mathring{r}_{()}=0$
provided

$(n-1)b+c=-1$ $(n-1)b^{2}-c^{2}=-\tfrac{1}{m}$

or equivalently

$c=-1-(n-1)b$ $(n-1)(n-2)b^{2}+2(n-1)b+(1-\tfrac{1}{m})=0$

When $n=2$, these equations have the unique solution

$b=-\tfrac{1}{2}(1-\tfrac{1}{m})$ $c=-\tfrac{1}{2}(1+\tfrac{1}{m})$

When $n>2$, the discriminant of the quadratic satisfied by $b$ is
$\Delta=4(n-1)(1+\tfrac{n-2}{m})$ so only when $m=-(n-2)$ is there a unique
solution, given by

$b=-\tfrac{1}{n-2}$ $c=\tfrac{1}{n-2}$

When $a=b$, so that torsion vanishes, this is precisely the case in which the
manifold is an Einstein-Weyl manifold, and the connection $\nabla$ is torsion-
free and preserves the conformal class of the metric $g$.

Otherwise, there is no unique solution and given Eqn. 6.1 we can construct
_two_ torsion-free connections with our ansatz such that Eqn 6.1 is equivalent
to $\mathring{r}_{()}=0$. The explicit solutions are

$b=\tfrac{1}{n-2}\big{(}-1\pm\sqrt{\tfrac{m+n-2}{m(n-1)}}\,\big{)}$
$c=-\tfrac{1}{n-2}\big{(}-1\pm\sqrt{\tfrac{(m+n-2)(n-1)}{m}}\,\big{)}$

Intriguingly, the radical in the expression for $b$ (which is equivalent to
$b+c$, see below) seems to occur in the integration of certain differential
equations when constructing a class of quasi-Einstein manifolds; see the proof
of Theorem 5.7 in [5].

Despite the non-uniqueness, there is one particularly noteworthy case which
has been singled-out previously. This occurs when $a=b$ and $c=0$, which
identifies the torsion-free connections among the family 6.2 which are
projectively equivalent to (i.e. have the same unparametrized geodesics as)
the Levi-Civita connection. From our system of equations for $b$, $c$, and $m$
we find the values $b=-\tfrac{1}{n-1}$ and $m=-(n-1)$. This was studied by
Wylie and Yeroshkin in [18], primarily in the case where $\varphi$ is an exact
form. Their interest seems to have been driven by the fact that when
$a=b=-\tfrac{1}{n-1}$, $c=0$, and $\varphi$ is closed, the Ricci tensor of the
torsion-free connection $\nabla$ is symmetric and particularly simple,
satisfying (see the appendix)

$r=r_{g}+\tfrac{1}{2}\pounds_{\varphi^{\\#}}g+\tfrac{1}{n-1}\varphi\otimes\varphi$

This is one of the so-called Bakry-Émery Ricci tensors.

We should also note that the family 6.2 with $a=b$ and $\varphi$ an exact form
was noted in [10] along with a computation of the Ricci tensor, but no
connection was made to $m$-quasi-Einstein manifolds. (The author considered
the connections 6.2 for the reasons outlined above before becoming aware of
[18] or [10].)

We hope that this brief introduction might interest others in investigating
the connections defined by Eqn. 6.2 and what role they might play, if any, in
the study of quasi-Einstein manifolds. It's possible that they are only
noteworthy in the cases already studied (that is, the torsion-free cases in
which the connection preserves at least the conformal class of the metric, or
is projectively equivalent to the Levi-Civita connection), but the general
case is likely worthy of further scrutiny. One possible point of entry to
evaluating the significance of the connection $\nabla$ when it is torsion-
free, given the attention already afforded to the conformal and projective
special cases, is to write,

$\displaystyle\Gamma_{i}{}^{k}{}_{j}=\,\,$
$\displaystyle\overline{\Gamma}_{i}{}^{k}{}_{j}-c\varphi_{i}\delta^{k}_{j}-c\varphi_{j}\delta^{k}_{i}+c\varphi^{k}g_{ij}$
$\displaystyle+(b+c)\varphi_{i}\delta^{k}_{j}+(b+c)\varphi_{j}\delta^{k}_{i}$

where we can easily compute

$b+c=\mp\sqrt{\tfrac{m+n-2}{m(n-1)}}$

This realizes the connection $\nabla$ as being projectively equivalent to a
connection

$\nabla^{\prime}=\nabla_{g}-c\varphi_{i}\delta^{k}_{j}-c\varphi_{j}\delta^{k}_{i}+c\varphi^{k}g_{ij}$

which preserves the conformal class of $g$, and such that the $1$-form which
determines the rescaling of the metric $g$ under $\nabla^{\prime}$-parallel
transport is a constant multiple of the $1$-form which determines the
reparametrization of the $\nabla^{\prime}$-geodesics. When
$\varphi=\mathrm{d}f$, we have

$\nabla=\nabla_{e^{-2cf}g}\mp\sqrt{\tfrac{m+n-2}{m(n-1)}}(\partial_{i}f\delta^{k}_{j}+\partial_{j}f\delta^{k}_{i})$

which seems to single out the conformal metric $e^{-2cf}g$ for consideration.
(Though, recall that there are really _two_ values of $c$ which are roots of a
quadratic, so we have conformal _metrics_ to consider; this suggests that
additionally the midpoint $\tfrac{1}{n-2}$, and the corresponding conformal
metric $e^{\frac{-2f}{n-2}}g$, may be worthy of consideration. This is perhaps
further suggested by the fact that if one writes $\mathring{r}_{(ij)}=0$ in
terms of $\tilde{g}=e^{\frac{-2f}{n-2}}g$ quantities, then second derivatives
of $f$ don't appear explicitly.)

The author has not fully pursued these ideas in favor of applying Thm. 2.2
with the connections in Eqn. 6.2 to investigate the Euler characteristic of
generalized $m$-quasi-Einstein $4$-manifolds; this will appear in a future
work. We encourage others to take up the general line of thought in this
section, and have provided in the appendix an extensive compendium of formulae
as an aid to those interested.

##  7\. Appendix

###  7.1. Relations Between Curvatures

On a general vector bundle with connections $\nabla$ and $\nabla^{\prime}$
related by $\nabla=\nabla^{\prime}-B$, we can easily calculate that the
relation

(7.1) $\Omega_{\nabla}=\Omega_{\nabla^{\prime}}-\nabla^{\prime}\circ B+B\wedge
B$

exists between the curvatures. Note that $\nabla^{\prime}\circ
B=\mathrm{d}B+\omega^{\prime}\wedge B+B\wedge\omega^{\prime}$ denotes the
$\nabla^{\prime}$-covariant derivative of $B$ as an $\mathrm{End}(E)$-valued
$1$-form; thus it is an $\mathrm{End}(E)$-valued $2$-form. Specializing to
$E=TM$ one computes that

$(\nabla^{\prime}\circ
B)_{ij}{}^{k}{}_{l}=\nabla^{\prime}_{i}B_{j}{}^{k}{}_{l}-\nabla^{\prime}_{j}B_{i}{}^{k}{}_{l}+T(\nabla^{\prime})_{i}{}^{m}{}_{j}\,B_{m}{}^{k}{}_{l}$

where
$T(\nabla^{\prime})=\omega^{\prime}_{i}{}^{k}{}_{j}-\omega^{\prime}_{j}{}^{k}{}_{i}$
is the torsion of $\nabla^{\prime}$.

Suppose now that $(M,g)$ is a Riemannian manifold, $\tilde{\nabla}$ is a
metric connection on the tangent bundle, and $\overline{\nabla}=\nabla_{g}$ is
the Levi-Civita connection. Write $\tilde{\nabla}=\overline{\nabla}-B$, and
denote the curvatures by $H=\Omega(\tilde{\nabla})$,
$\overline{R}=\Omega(\overline{\nabla})$. Then the formula above reads

$H_{ij}{}^{k}{}_{l}=\overline{R}_{ij}{}^{k}{}_{l}-\overline{\nabla}_{i}B_{j}{}^{k}{}_{l}+\overline{\nabla}_{j}B_{i}{}^{k}{}_{l}+B_{i}{}^{k}{}_{m}B_{j}{}^{m}{}_{l}-B_{j}{}^{k}{}_{m}B_{i}{}^{m}{}_{l}$

We can convert this into a formula with all indices down by defining
$H_{ijkl}=g_{km}H_{ij}{}^{m}{}_{l}$,
$\overline{R}_{ijkl}=g_{km}\overline{R}_{ij}{}^{m}{}_{l}$, and
$B_{ikj}=g_{km}B_{i}{}^{m}{}_{j}$ yielding

(7.2)
$H_{ijkl}=\overline{R}_{ijkl}-\overline{\nabla}_{i}B_{jkl}+\overline{\nabla}_{j}B_{ikl}+B_{ikm}B_{jml}-B_{jkm}B_{iml}$

where we're using an extended Einstein summation convention such that
summation over repeated lower indices is understood and denotes contraction
with the inverse metric.

####  7.1.1. On the Tangent Bundle of a Surface

Consider now Eqn 7.2 on a surface, where we work in a local orthonormal basis
$\\{e_{1},e_{2}\\}$. We have only the single scalar equation

$H_{1212}=\overline{R}_{1212}-\overline{\nabla}_{1}B_{212}+\overline{\nabla}_{2}B_{112}+B_{11m}B_{2m2}-B_{21m}B_{1m2}$

Considering the skew-symmetry of $B$ in the second and third indices (since
it's the difference between metric connections), each of the last two terms
vanishes, so this is in fact just

$H_{1212}=K_{g}-\overline{\nabla}_{1}B_{212}+\overline{\nabla}_{2}B_{112}$

by also substituting in the Gauss curvature $K_{g}$ of $g$. Now we trace
$-\overline{\nabla}_{i}B_{jkl}+\overline{\nabla}_{j}B_{ikl}$ over $i,k$ and
$j,l$ to find

$-\overline{\nabla}_{i}B_{jij}+\overline{\nabla}_{j}B_{iij}=-\overline{\nabla}_{1}B_{212}+\overline{\nabla}_{2}B_{112}-\overline{\nabla}_{2}B_{121}+\overline{\nabla}_{1}B_{221}=2(-\overline{\nabla}_{1}B_{212}+\overline{\nabla}_{2}B_{112})$

by using the skew-symmetry of $B$, while also

$-\overline{\nabla}_{i}B_{jij}+\overline{\nabla}_{j}B_{iij}=2\overline{\nabla}_{i}B_{jji}=-2\,\mathrm{d}^{*}_{g}(B_{jji})=-2\,\mathrm{d}^{*}_{g}(B_{i})$

by again using skew-symmetry and renaming dummy indices in the first equality,
defining $B_{jji}=B_{i}$ in the last, and where we are denoting by
$\mathrm{d}^{*}_{g}$ the formal $L^{2}$-adjoint of $\mathrm{d}$ which is
induced by $g$;
$\mathrm{d}^{*}_{g}\psi=-\star_{2}\mathrm{d}\star_{1}\psi=-\nabla_{i}\psi_{i}$
on $1$-forms, where $\star_{i}$ is the Hodge star operator on $i$-forms. Thus

(7.3) $H_{1212}=K_{g}-\mathrm{d}^{*}_{g}(B_{i})$

Alternatively we can proceed from Eqn. 7.1

$H=\bar{R}-\nabla^{\prime}\circ B+B\wedge
B=\bar{R}-(\mathrm{d}B+\bar{\Gamma}\wedge B+B\wedge\bar{\Gamma})+B\wedge B$

in our orthonormal frame $\\{e_{1},e_{2}\\}$, in which case

$\bar{\Gamma}=\begin{bmatrix}0&\bar{\gamma}\\\
-\bar{\gamma}&0\end{bmatrix},\hskip 7.22743ptB=\begin{bmatrix}0&b\\\
-b&0\end{bmatrix}$

where $\bar{\gamma}$ is a locally-defined $1$-form and $b=\mathrm{Pf}(B)$ is a
globally-defined $1$-form if $M^{2}$ is oriented and our frame is consistent
with the orientation, but $b$ generally has an ambiguous sign. It's trivial to
compute that $\bar{\Gamma}\wedge\bar{\Gamma}$, $B\wedge B$, and the
(super)commutator $\bar{\Gamma}\wedge B+B\wedge\bar{\Gamma}$ vanish, so

$\displaystyle H$ $\displaystyle=\bar{R}-\mathrm{d}B$
$\displaystyle=\mathrm{d}\bar{\Gamma}-\mathrm{d}B$
$\displaystyle=\begin{bmatrix}0&\mathrm{d}\bar{\gamma}-\mathrm{d}b\\\
-(\mathrm{d}\bar{\gamma}-\mathrm{d}b)&0\end{bmatrix}$
$\displaystyle=\begin{bmatrix}0&K_{g}\mathrm{d}V_{g}-\mathrm{d}b\\\
-(K_{g}\mathrm{d}V_{g}-\mathrm{d}b)&0\end{bmatrix}$

Comparing to Eqn 7.3 reveals

$\mathrm{d}b=\mathrm{d}^{*}_{g}(B_{i})\mathrm{d}V_{g};$

this reflects, and is a consequence of, the easily computed fact that

(7.4) $b=-\star_{1}B_{i}\hskip 7.22743pt\Leftrightarrow\hskip
7.22743pt\star_{1}b=B_{i}$

which sheds more light on our earlier observation that $b$ has an ambiguous
local sign.

In any case, putting this all together,

$\displaystyle\mathrm{Pf}(H)$ $\displaystyle=H_{1212}\,e^{1}\wedge e^{2}$
$\displaystyle=(K_{g}-\mathrm{d}^{*}_{g}(B_{i}))\mathrm{d}V_{g}$

###  7.2. The Curvature of the Levi-Civita Connection Plus a Pure-Trace Tensor

Let $\overline{\Gamma}_{i}{}^{k}{}_{j}$ denotes the Christoffel symbols of the
Levi-Civita connection $\nabla_{g}$ of a metric $g$. We consider a connection
$\nabla$ whose coefficients are given by

$\Gamma_{i}{}^{k}{}_{j}=\overline{\Gamma}_{i}{}^{k}{}_{j}+\alpha_{i}\delta^{k}_{j}+\beta_{j}\delta^{k}_{i}+\gamma^{k}g_{ij}$

where $\alpha$, $\beta$, $\gamma$ are $1$-forms. We will record various
quantities associated to this connection.

####  7.2.1. Torsion, Non-Metricity, Canonical Metric Connection

The torsion is

$T(\nabla)=(\alpha-\beta)_{i}\delta^{k}_{j}-(\alpha-\beta)_{j}\delta^{k}_{i}$

which clearly vanishes exactly when $\alpha=\beta$, while the non-metricity is

$\nabla_{i}g_{jk}=-2\alpha_{i}g_{jk}-(\beta+\gamma)_{j}g_{ik}-(\beta+\gamma)_{k}g_{ij}$

which is easily shown to vanish if and only if $\alpha=0$ and $\beta=-\gamma$,
and to be proportional to the metric $g_{jk}$ exactly when $\beta=-\gamma$;
this latter situation is equivalent to $\nabla$ preserving the conformal class
of $g$.

It's additionally immediate from the non-metricity tensor that the canonical
associated metric tensor is

$\nabla^{g}=\nabla_{g}-B$

where

$B_{i}{}^{k}{}_{j}=-\tfrac{(\beta-\gamma)_{j}}{2}\delta^{k}_{i}+\tfrac{(\beta-\gamma)^{k}}{2}g_{ij}$

or (with indices lowered)

$B_{ikj}=-\tfrac{(\beta-\gamma)_{j}}{2}g_{ik}+\tfrac{(\beta-\gamma)_{k}}{2}g_{ij}$

This tensor has one independent trace,

$B_{j}=B_{iij}=-\tfrac{n-1}{2}(\beta-\gamma)_{j}$

####  7.2.2. Curvature of $\nabla$

The curvature of $\nabla$ is

$\displaystyle R_{ij}{}^{k}{}_{l}=\,$
$\displaystyle\overline{R}_{ij}{}^{k}{}_{l}+(\partial_{i}\alpha_{j}-\partial_{j}\alpha_{i})\delta^{k}_{l}+\overline{\nabla}_{i}\beta_{l}\delta^{k}_{j}-\overline{\nabla}_{j}\beta_{l}\delta^{k}_{i}+\overline{\nabla}_{i}\gamma^{k}g_{jl}-\overline{\nabla}_{j}\gamma^{k}g_{il}$
$\displaystyle+(\delta^{k}_{i}\beta_{j}-\delta^{k}_{j}\beta_{i})\beta_{l}+(\gamma_{i}g_{jl}-\gamma_{j}g_{il})\gamma^{k}+g(\beta,\gamma)(\delta^{k}_{i}g_{jl}-\delta^{k}_{j}g_{il})$

with index-lowered variant $R_{ijkl}=g_{km}R_{ij}{}^{m}{}_{l}$ given by

$\displaystyle R_{ijkl}=$
$\displaystyle\overline{R}_{ijkl}+(\partial_{i}\alpha_{j}-\partial_{j}\alpha_{i})g_{kl}+\overline{\nabla}_{i}\beta_{l}g_{kj}-\overline{\nabla}_{j}\beta_{l}g_{ki}+\overline{\nabla}_{i}\gamma_{k}g_{jl}-\overline{\nabla}_{j}\gamma_{k}g_{il}$
$\displaystyle+(g_{ik}\beta_{j}-g_{jk}\beta_{i})\beta_{l}+(\gamma_{i}g_{jl}-\gamma_{j}g_{il})\gamma_{k}+g(\beta,\gamma)(g_{ik}g_{jl}-g_{jk}g_{il})$

Since $\nabla$ is not necessarily torsion-free or metric, the curvature tensor
lacks the familiar symmetries of a Riemannian curvature tensor, and there are
three independent traces of $R$, not the usual one: the Ricci tensor
$r_{jl}=R_{ij}{}^{i}{}_{l}$, $\rho_{jk}=R_{ijki}$, and
$\zeta_{ij}=R_{ij}{}^{k}{}_{k}$

We first examine the Ricci tensor:

$\displaystyle r=$
$\displaystyle\,\,\overline{r}+\mathrm{d}\alpha-\tfrac{1}{2}\mathrm{d}((n-1)\beta+\gamma)-\tfrac{1}{2}\pounds_{((n-1)\beta+\gamma)^{\\#}}\,g$
$\displaystyle+(n-1)\beta\otimes\beta-\gamma\otimes\gamma+(-\mathrm{d}^{*}_{g}(\gamma)+g((n-1)\beta+\gamma,\gamma))g$

Clearly the symmetric part of $r$ is

$r_{()}=\overline{r}-\tfrac{1}{2}\pounds_{((n-1)\beta+\gamma)^{\\#}}\,g+(n-1)\beta\otimes\beta-\gamma\otimes\gamma+(-\mathrm{d}^{*}_{g}(\gamma)+g((n-1)\beta+\gamma,\gamma))g$

and the antisymmetric part is

$r_{[\,]}=\mathrm{d}\alpha-\tfrac{1}{2}\mathrm{d}((n-1)\beta+\gamma)$

with trace $s=g^{ij}r_{ij}=g^{ij}r_{(ij)}$ satisfying

$s=\overline{s}+(n-1)\mathrm{d}^{*}_{g}(\beta-\gamma)+(n-1)(|\beta|^{2}+n\,g(\beta,\gamma)+|\gamma|^{2})$

Therefore the following equation holds for the traceless symmetric Ricci
tensor $\mathring{r}_{()}=r_{()}-\tfrac{s}{n}g$:

$\displaystyle\mathring{r}_{()}=$
$\displaystyle\,\,\overline{r}-\tfrac{1}{2}\pounds_{((n-1)\beta+\gamma)^{\\#}}\,g+(n-1)\beta\otimes\beta-\gamma\otimes\gamma+$
$\displaystyle-\tfrac{1}{n}(\overline{s}+\mathrm{d}^{*}_{g}((n-1)\beta+\gamma)+(n-1)|\beta|^{2}-|\gamma|^{2})g$

Now consider the tensor $\rho$:

$\displaystyle\rho=$
$\displaystyle-\overline{r}-\mathrm{d}\alpha-\tfrac{1}{2}\mathrm{d}(\beta+(n-1)\gamma)-\tfrac{1}{2}\pounds_{(\beta+(n-1)\gamma)^{\\#}}\,g$
$\displaystyle+\beta\otimes\beta-(n-1)\gamma\otimes\gamma+(-\mathrm{d}^{*}_{g}\beta-g(\beta,\beta+(n-1)\gamma))g$

The symmetric part is

$\rho_{()}=-\overline{r}-\tfrac{1}{2}\pounds_{(\beta+(n-1)\gamma)^{\\#}}\,g+\beta\otimes\beta-(n-1)\gamma\otimes\gamma+(-\mathrm{d}^{*}_{g}\beta-g(\beta,\beta+(n-1)\gamma))g$

while the antisymmetric part is

$\rho_{[\,]}=-\mathrm{d}\alpha-\tfrac{1}{2}\mathrm{d}(\beta+(n-1)\gamma)$

and the trace $\sigma=g^{ij}\rho_{ij}=g^{ij}\rho_{(ij)}$ is fully determined
by the trace of the Ricci tensor:

$\sigma=-s$

Thus for the traceless symmetric part we have

$\displaystyle\mathring{\rho}_{()}=$
$\displaystyle\,-\overline{r}-\tfrac{1}{2}\pounds_{(\beta+(n-1)\gamma)^{\\#}}\,g+\beta\otimes\beta-(n-1)\gamma\otimes\gamma$
$\displaystyle-\tfrac{1}{n}(-\overline{s}+\mathrm{d}^{*}_{g}(\beta+(n-1)\gamma)+|\beta|^{2}-(n-1)|\gamma|^{2})g$

Some interesting simplifications occur if we consider $\tfrac{1}{2}(r-\rho)$
and $\tfrac{1}{2}(r+\rho)$ (these are natural as traces of $R$ after breaking
the $kl$ index pair into antisymmetric and symmetric parts). The traces are
$s$ and $0$ respectively, while the other components are as follows:

$\displaystyle\tfrac{1}{2}(\mathring{r}-\mathring{\rho})_{()}=$
$\displaystyle\,\,\overline{r}+\tfrac{n-2}{2}(-\tfrac{1}{2}\pounds_{(\beta-\gamma)^{\\#}}g+\beta\otimes\beta+\gamma\otimes\gamma)$
$\displaystyle-\tfrac{1}{n}(\overline{s}+\tfrac{n-2}{2}(\mathrm{d}^{*}_{g}(\beta-\gamma)+|\beta|^{2}+|\gamma|^{2}))g$
$\displaystyle\tfrac{1}{2}(r-\rho)_{[\,]}=$
$\displaystyle\,\,\mathrm{d}\alpha-\tfrac{n-2}{4}\mathrm{d}(\beta-\gamma)$
$\displaystyle\tfrac{1}{2}(r+\rho)_{()}=$
$\displaystyle\,\tfrac{1}{2}(\mathring{r}+\mathring{\rho})_{()}$
$\displaystyle=$
$\displaystyle-\tfrac{n}{4}\pounds_{(\beta+\gamma)^{\\#}}g+\tfrac{n}{2}(\beta+\gamma)\cdot(\beta-\gamma)-\tfrac{1}{2}(\mathrm{d}^{*}_{g}(\beta+\gamma)+g(\beta+\gamma,\beta-\gamma))g$
$\displaystyle\tfrac{1}{2}(r+\rho)_{[\,]}=$
$\displaystyle-\tfrac{n}{4}\mathrm{d}(\beta+\gamma)$

Finally we have the antisymmetric tensor $\zeta$:

$\zeta=n\mathrm{d}\alpha+2\mathrm{d}(\beta+\gamma)$

####  7.2.3. Curvature of $\nabla^{g}$

The canonical metric connection of $\nabla$ is $\nabla^{g}$, whose Christoffel
symbols $\eta_{i}{}^{k}{}_{j}$ satisfy (see Section 7.1)

$\eta_{i}{}^{k}{}_{j}=\overline{\Gamma}_{i}{}^{k}{}_{j}+\tfrac{(\beta-\gamma)_{j}}{2}\delta^{k}_{i}-\tfrac{(\beta-\gamma)^{k}}{2}g_{ij}$

This is the same type of connection that we considered above, with $\alpha=0$
and $\beta$ and $-\gamma$ replaced by $\tfrac{1}{2}(\beta-\gamma)$. Denote the
curvature of $\eta$ by $H=H_{ijkl}$. This tensor is antisymmetric in the $ij$
and $kl$ pairs since $\nabla^{g}$ is compatible with $g$, but we can't expect
further symmetries. This is, however, enough to imply that there is only one
independent trace; we'll use the Ricci trace over the $ik$ indices and call it
$h$. We then have the formulas that follow.

(1,3)-Curvature Tensor:

$\displaystyle H_{ij}{}^{k}{}_{l}=\,$
$\displaystyle\overline{R}_{ij}{}^{k}{}_{l}+\tfrac{1}{2}\overline{\nabla}_{i}(\beta-\gamma)_{l}\delta^{k}_{j}-\tfrac{1}{2}\overline{\nabla}_{j}(\beta-\gamma)_{l}\delta^{k}_{i}-\tfrac{1}{2}\overline{\nabla}_{i}(\beta-\gamma)^{k}g_{jl}+\tfrac{1}{2}\overline{\nabla}_{j}(\beta-\gamma)^{k}g_{il}$
$\displaystyle+\tfrac{1}{4}(\delta^{k}_{i}(\beta-\gamma)_{j}-\delta^{k}_{j}(\beta-\gamma)_{i})(\beta-\gamma)_{l}+\tfrac{1}{4}((\beta-\gamma)_{i}g_{jl}-(\beta-\gamma)_{j}g_{il})(\beta-\gamma)^{k}$
$\displaystyle-\tfrac{1}{4}|\beta-\gamma|^{2}(\delta^{k}_{i}g_{jl}-\delta^{k}_{j}g_{il})$

(0,4)-Curvature Tensor:

$\displaystyle H_{ijkl}=$
$\displaystyle\overline{R}_{ijkl}+\tfrac{1}{2}\overline{\nabla}_{i}(\beta-\gamma)_{l}g_{kj}-\tfrac{1}{2}\overline{\nabla}_{j}(\beta-\gamma)_{l}g_{ki}-\tfrac{1}{2}\overline{\nabla}_{i}(\beta-\gamma)_{k}g_{jl}+\tfrac{1}{2}\overline{\nabla}_{j}(\beta-\gamma)_{k}g_{il}$
$\displaystyle+\tfrac{1}{4}g\owedge[(\beta-\gamma)\otimes(\beta-\gamma)]_{ijkl}-\tfrac{1}{2}|\beta-\gamma|^{2}g\owedge
g_{ijkl}$

Ricci tensor:

$\displaystyle h=\,\,$
$\displaystyle\overline{r}-\tfrac{n-2}{4}\mathrm{d}(\beta-\gamma)-\tfrac{n-2}{4}\pounds_{(\beta-\gamma)^{\\#}}g$
$\displaystyle+\tfrac{n-2}{4}(\beta-\gamma)\otimes(\beta-\gamma)+(\tfrac{1}{2}\mathrm{d}^{*}_{g}(\beta-\gamma)-\tfrac{n-2}{4}|\beta-\gamma|^{2})g$

Symmetric Ricci tensor:

$\displaystyle h_{()}=\,\,$
$\displaystyle\overline{r}-\tfrac{n-2}{4}\pounds_{(\beta-\gamma)^{\\#}}g+\tfrac{n-2}{4}(\beta-\gamma)\otimes(\beta-\gamma)$
$\displaystyle+(\tfrac{1}{2}\mathrm{d}^{*}_{g}(\beta-\gamma)-\tfrac{n-2}{4}|\beta-\gamma|^{2})g$

Antisymmetric Ricci tensor:

$h_{[\,]}=-\tfrac{n-2}{4}\mathrm{d}(\beta-\gamma)$

Trace:

$\tau=\overline{s}+(n-1)\mathrm{d}^{*}_{g}(\beta-\gamma)-\tfrac{(n-1)(n-2)}{4}|\beta-\gamma|^{2}$

Traceless symmetric Ricci tensor:

$\displaystyle\mathring{h}_{()}=\,\,$
$\displaystyle\overline{r}-\tfrac{n-2}{4}\pounds_{(\beta-\gamma)^{\\#}}g+\tfrac{n-2}{4}(\beta-\gamma)\otimes(\beta-\gamma)$
$\displaystyle-\tfrac{1}{n}(\overline{s}+\tfrac{n-2}{2}\mathrm{d}^{*}_{g}(\beta-\gamma)+\tfrac{n-2}{4}|\beta-\gamma|^{2})g$

####  7.2.4. Specializing the equations

If we substitute $\alpha=a\varphi$, $\beta=b\varphi$, $\gamma=c\varphi$ we
obtain further specializations:

$\nabla$ tensors:

(1,3)-Curvature Tensor:

$\displaystyle R_{ij}{}^{k}{}_{l}=\,$
$\displaystyle\overline{R}_{ij}{}^{k}{}_{l}+a(\partial_{i}\varphi_{j}-\partial_{j}\varphi_{i})\delta^{k}_{l}+b\overline{\nabla}_{i}\varphi_{l}\delta^{k}_{j}-b\overline{\nabla}_{j}\varphi_{l}\delta^{k}_{i}+c\overline{\nabla}_{i}\varphi^{k}g_{jl}-c\overline{\nabla}_{j}\varphi^{k}g_{il}$
$\displaystyle+b^{2}(\delta^{k}_{i}\varphi_{j}-\delta^{k}_{j}\varphi_{i})\varphi_{l}+c^{2}(\varphi_{i}g_{jl}-\varphi_{j}g_{il})\varphi^{k}+bc|\varphi|^{2}(\delta^{k}_{i}g_{jl}-\delta^{k}_{j}g_{il})$

(0,4)-Curvature Tensor:

$\displaystyle R_{ijkl}=$
$\displaystyle\overline{R}_{ijkl}+a(\partial_{i}\varphi_{j}-\partial_{j}\varphi_{i})g_{kl}+b\overline{\nabla}_{i}\varphi_{l}g_{kj}-b\overline{\nabla}_{j}\varphi_{l}g_{ki}+c\overline{\nabla}_{i}\varphi_{k}g_{jl}-c\overline{\nabla}_{j}\varphi_{k}g_{il}$
$\displaystyle+b^{2}(g_{ik}\varphi_{j}-g_{jk}\varphi_{i})\varphi_{l}+c^{2}(\varphi_{i}g_{jl}-\varphi_{j}g_{il})\varphi_{k}+bc|\varphi|^{2}(g_{ik}g_{jl}-g_{jk}g_{il})$

Ricci tensor:

$\displaystyle r=$
$\displaystyle\,\,\overline{r}+a\mathrm{d}\varphi-\tfrac{1}{2}((n-1)b+c)\mathrm{d}\varphi-((n-1)b+c)\tfrac{1}{2}\pounds_{\varphi^{\\#}}\,g$
$\displaystyle+((n-1)b^{2}-c^{2})\varphi\otimes\varphi+c(-\mathrm{d}^{*}_{g}\varphi+((n-1)b+c)|\varphi|^{2})g$

Symmetric Ricci tensor:

$r_{()}=\overline{r}-((n-1)b+c)\tfrac{1}{2}\pounds_{\varphi^{\\#}}\,g+((n-1)b^{2}-c^{2})\varphi\otimes\varphi+c(-\mathrm{d}^{*}_{g}\varphi+((n-1)b+c)|\varphi|^{2})g$

Antisymmetric Ricci tensor:

$r_{[\,]}=a\mathrm{d}\varphi-\tfrac{1}{2}((n-1)b+c)\mathrm{d}\varphi$

Trace of Ricci tensor:

$\displaystyle s$
$\displaystyle=(\overline{s}+((n-1)b+c)\mathrm{d}^{*}_{g}\varphi+((n-1)b^{2}-c^{2})|\varphi|^{2})+n\cdot
c(-\mathrm{d}^{*}_{g}\varphi+((n-1)b+c)|\varphi|^{2})$
$\displaystyle=\overline{s}+(n-1)(b-c)\mathrm{d}^{*}_{g}\varphi+(n-1)(b^{2}+nbc+c^{2})|\varphi|^{2}$

Traceless symmetric Ricci tensor:

$\displaystyle\mathring{r}_{()}=$
$\displaystyle\,\overline{r}-\tfrac{1}{2}((n-1)b+c)\pounds_{\varphi^{\\#}}\,g+((n-1)b^{2}-c^{2})\varphi\otimes\varphi+$
$\displaystyle-\tfrac{1}{n}(\overline{s}+((n-1)b+c)\mathrm{d}^{*}_{g}\varphi+((n-1)b^{2}-c^{2})|\varphi|^{2})g$

$\rho$ tensor:

$\displaystyle\rho=$
$\displaystyle-\overline{r}-a\mathrm{d}\varphi-\tfrac{1}{2}(b+(n-1)c)\mathrm{d}\varphi-\tfrac{1}{2}(b+(n-1)c)\pounds_{\varphi^{\\#}}\,g$
$\displaystyle+(b^{2}-(n-1)c^{2})\varphi\otimes\varphi+b(-\mathrm{d}^{*}_{g}\varphi-(b+(n-1)c)|\varphi|^{2})g$

Symmetric $\rho$:

$\rho_{()}=-\overline{r}-\tfrac{1}{2}(b+(n-1)c)\pounds_{\varphi^{\\#}}\,g+(b^{2}-(n-1)c^{2})\varphi\otimes\varphi+b(-\mathrm{d}^{*}_{g}\varphi-(b+(n-1)c)|\varphi|^{2})g$

Antisymmetric $\rho$:

$\rho_{[\,]}=-a\mathrm{d}\varphi-\tfrac{1}{2}(b+(n-1)c)\mathrm{d}\varphi$

Trace of $\rho$:

$\sigma=-s$

Traceless symmetric $\rho$:

$\displaystyle\mathring{\rho}_{()}=$
$\displaystyle\,-\overline{r}-\tfrac{1}{2}(b+(n-1)c)\pounds_{\varphi^{\\#}}\,g+(b^{2}-(n-1)c^{2})\varphi\otimes\varphi$
$\displaystyle-\tfrac{1}{n}(-\overline{s}+(b+(n-1)c)\mathrm{d}^{*}_{g}\varphi+(b^{2}-(n-1)c^{2})|\varphi|^{2})g$

Linear combinations of $r$ and $\rho$:

$\displaystyle\tfrac{1}{2}(\mathring{r}-\mathring{\rho})_{()}=$
$\displaystyle\,\,\overline{r}+\tfrac{n-2}{2}(-(b-c)\tfrac{1}{2}\pounds_{\varphi^{\\#}}g+(b^{2}+c^{2})\varphi\otimes\varphi)$
$\displaystyle-\tfrac{1}{n}(\overline{s}+\tfrac{n-2}{2}((b-c)\mathrm{d}^{*}_{g}\varphi+(b^{2}+c^{2})|\varphi|^{2})g$
$\displaystyle\tfrac{1}{2}(r-\rho)_{[\,]}=$
$\displaystyle\,\,a\mathrm{d}\varphi-\tfrac{n-2}{4}(b-c)\mathrm{d}\varphi$
$\displaystyle\tfrac{1}{2}(r+\rho)_{()}=$
$\displaystyle\,\tfrac{1}{2}(\mathring{r}+\mathring{\rho})_{()}$
$\displaystyle=$
$\displaystyle(b+c)(-\tfrac{n}{4}\pounds_{\varphi^{\\#}}g+\tfrac{n}{2}(b-c)\varphi\otimes\varphi-\tfrac{1}{2}(\mathrm{d}^{*}_{g}\varphi+(b-c)|\varphi|^{2})g)$
$\displaystyle\tfrac{1}{2}(r+\rho)_{[\,]}=$
$\displaystyle-\tfrac{n}{4}(b+c)\mathrm{d}\varphi$

Antisymmetric $\zeta$ tensor:

$\zeta=n\cdot a\,\mathrm{d}\varphi+2(b+c)\mathrm{d}\varphi$

$\nabla^{g}$ tensors:

(1,3)-Curvature Tensor:

$\displaystyle H_{ij}{}^{k}{}_{l}=\,$
$\displaystyle\overline{R}_{ij}{}^{k}{}_{l}+\tfrac{b-c}{2}(\overline{\nabla}_{i}\varphi_{l}\delta^{k}_{j}-\overline{\nabla}_{j}\varphi_{l}\delta^{k}_{i}-\overline{\nabla}_{i}\varphi^{k}g_{jl}+\overline{\nabla}_{j}\varphi^{k}g_{il})$
$\displaystyle+\tfrac{(b-c)^{2}}{4}(\delta^{k}_{i}\varphi_{j}-\delta^{k}_{j}\varphi_{i})\varphi_{l}+\tfrac{(b-c)^{2}}{4}(\varphi_{i}g_{jl}-\varphi_{j}g_{il})\varphi^{k}$
$\displaystyle-\tfrac{(b-c)^{2}}{4}|\varphi|^{2}(\delta^{k}_{i}g_{jl}-\delta^{k}_{j}g_{il})$

(0,4)-Curvature Tensor:

$\displaystyle H_{ijkl}=$
$\displaystyle\overline{R}_{ijkl}+\tfrac{b-c}{2}(\overline{\nabla}_{i}\varphi_{l}g_{kj}-\overline{\nabla}_{j}\varphi_{l}g_{ki}-\overline{\nabla}_{i}\varphi_{k}g_{jl}+\overline{\nabla}_{j}\varphi_{k}g_{il})$
$\displaystyle+\tfrac{(b-c)^{2}}{4}g\owedge[\varphi\otimes\varphi]_{ijkl}-\tfrac{(b-c)^{2}}{2}|\varphi|^{2}g\owedge
g_{ijkl}$

Ricci tensor:

$\displaystyle h=\,\,$
$\displaystyle\overline{r}-\tfrac{n-2}{4}(b-c)\mathrm{d}\varphi-\tfrac{n-2}{4}(b-c)\pounds_{\varphi^{\\#}}g$
$\displaystyle+\tfrac{n-2}{4}(b-c)^{2}\varphi\otimes\varphi+(\tfrac{b-c}{2}\mathrm{d}^{*}_{g}\varphi-\tfrac{n-2}{4}(b-c)^{2}|\varphi|^{2})g$

Symmetric Ricci tensor:

$\displaystyle h_{()}=\,\,$
$\displaystyle\overline{r}-\tfrac{n-2}{4}(b-c)\pounds_{\varphi^{\\#}}g+\tfrac{n-2}{4}(b-c)^{2}\varphi\otimes\varphi$
$\displaystyle+(\tfrac{b-c}{2}\mathrm{d}^{*}_{g}\varphi-\tfrac{n-2}{4}(b-c)^{2}|\varphi|^{2})g$

Antisymmetric Ricci tensor:

$h_{[\,]}=-\tfrac{n-2}{4}(b-c)\mathrm{d}\varphi$

Trace:

$\tau=\overline{s}+(n-1)(b-c)\mathrm{d}^{*}_{g}\varphi-\tfrac{(n-1)(n-2)}{4}(b-c)^{2}|\varphi|^{2}$

Traceless symmetric Ricci tensor:

$\displaystyle\mathring{h}_{()}=\,\,$
$\displaystyle\overline{r}-\tfrac{n-2}{4}(b-c)\pounds_{\varphi^{\\#}}g+\tfrac{n-2}{4}(b-c)^{2}\varphi\otimes\varphi$
$\displaystyle-\tfrac{1}{n}(\overline{s}+\tfrac{n-2}{2}(b-c)\mathrm{d}^{*}_{g}\varphi+\tfrac{n-2}{4}(b-c)^{2}|\varphi|^{2})g$

####  7.2.5. Assuming $a=b$, $(n-1)b+c=-1$, $(n-1)b^{2}-c^{2}=-\tfrac{1}{m}$

If we assume $a=b$, $(n-1)b+c=-1$, and $(n-1)b^{2}-c^{2}=-\tfrac{1}{m}$, then
when $n>2$ we have two solutions $(b_{+},c_{+})$ and $(b_{-},c_{-})$, given
explicitly by

$b_{\pm}=\tfrac{1}{n-2}\big{(}-1\pm\sqrt{\tfrac{m+n-2}{m(n-1)}}\,\big{)}$
$c_{\pm}=-\tfrac{1}{n-2}\big{(}-1\pm\sqrt{\tfrac{(m+n-2)(n-1)}{m}}\,\big{)}$

This produces two torsion-free connections

$\nabla_{+}=\nabla_{g}+b_{+}\varphi_{i}\delta^{k}_{j}+b_{+}\varphi_{j}\delta^{k}_{i}+c_{+}\varphi^{k}g_{ij}$
$\nabla_{-}=\nabla_{g}+b_{-}\varphi_{i}\delta^{k}_{j}+b_{-}\varphi_{j}\delta^{k}_{i}+c_{-}\varphi^{k}g_{ij}$

with identical and particularly simple traceless symmetric Ricci tensors

$\mathring{r}_{\pm}{}_{()}=\overline{r}+\tfrac{1}{2}\pounds_{\varphi^{\\#}}\,g-\tfrac{1}{m}\varphi\otimes\varphi-\tfrac{1}{n}(\overline{s}-\mathrm{d}^{*}_{g}\varphi-\tfrac{1}{m}|\varphi|^{2})g$

despite the fact that their symmetric Ricci tensors differ due to an explicit
dependence on $c$:

$r_{\pm}{}_{()}=\overline{r}+\tfrac{1}{2}\pounds_{\varphi^{\\#}}\,g-\tfrac{1}{m}\varphi\otimes\varphi+c_{\pm}(-\mathrm{d}^{*}_{g}\varphi-|\varphi|^{2})g$

However, if one considers the average of the symmetric Ricci tensors,

$\tfrac{1}{2}(r_{+}{}_{()}+r_{-}{}_{()})=\overline{r}+\tfrac{1}{2}\pounds_{\varphi^{\\#}}\,g-\tfrac{1}{m}\varphi\otimes\varphi+\tfrac{1}{n-2}(-\mathrm{d}^{*}_{g}\varphi-|\varphi|^{2})g$

(using $\tfrac{1}{2}(c_{+}+c_{-})=\tfrac{1}{n-2}$) then this does not depend
on choosing between $(b_{+},c_{+})$ and $(b_{-},c_{-})$. From this we find
also

$\tfrac{1}{2}(s_{+}+s_{-})=\overline{s}-2\mathrm{d}^{*}_{g}\varphi-\tfrac{m+1}{m}|\varphi|^{2}-\tfrac{2}{n-2}(\mathrm{d}^{*}_{g}\varphi+|\varphi|^{2})$

The expression
$\overline{s}-2\mathrm{d}^{*}_{g}\varphi-\tfrac{m+1}{m}|\varphi|^{2}$ is an
analogue of the notion of weighted scalar curvature in [5].

Perhaps studying the averages of these and other curvature tensors will prove
worthwhile; it's not hard to see that the quantities $b_{+}+b_{-}$,
$c_{+}+c_{-}$, $b_{+}^{2}+b_{-}^{2}$, $c_{+}^{2}+c_{-}^{2}$, and
$b_{+}c_{+}+b_{-}c_{-}$ don't involve any root extraction, so the averages of
curvature tensors won't involve any root expressions. However, it's not
entirely clear what underlying principle would lead to the consideration of
such averages. We leave such explorations for the interested reader to embark
upon.

##  8\. Acknowledgements

The author would like to thank the Rutgers University, New Brunswick
Mathematics Department for their financial support while this work was
completed.

## References

  * [1] Allendoerfer, C. B.,& Weil, A. (1943). The Gauss-Bonnet theorem for Riemannian polyhedra. Transactions of the American Mathematical Society, 53(1), 101-129.
  * [2] Barros, A. A., & Gomes, J. N. (2017). Triviality of compact m-quasi-Einstein manifolds. Results in Mathematics, 71(1-2), 241-250.
  * [3] Bell, D. (2006). The Gauss-Bonnet theorem for vector bundles. Journal of Geometry, 85(1-2), 15-21.
  * [4] Cao, H. D. (2009). Recent progress on Ricci solitons. arXiv preprint arXiv:0908.2006.
  * [5] Case, J. S. (2012). Smooth metric measure spaces and quasi-Einstein metrics. International Journal of Mathematics, 23(10), 1250110.
  * [6] Chern, S. S. (1944). A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds. Annals of mathematics, 747-752.
  * [7] Corwin, I., & Morgan, F. (2012). The Gauss-Bonnet formula on surfaces with densities. Involve, a Journal of Mathematics, 4(2), 199-202.
  * [8] Goldman, W. M. (2011). Two papers which changed my life: Milnor's seminal work on flat manifolds and flat bundles. arXiv preprint arXiv:1108.0216.
  * [9] Lee, J. M. (2006). Riemannian manifolds: an introduction to curvature (Vol. 176). Springer Science & Business Media.
  * [10] Li, J., & Xia, C. (2017). An integral formula for affine connections. The Journal of Geometric Analysis, 27(3), 2539-2556.
  * [11] Martínez Madrid, D. (2017). On Chern's conjecture about the Euler characteristic of affine manifolds. Instituto de Matemática Pura y Aplicada.
  * [12] Milnor, J. (1977). On fundamental groups of complete affinely flat manifolds. Advances in Mathematics, 25(2), 178-187.
  * [13] Morgan, F. (2009). Manifolds with density and Perelman's proof of the Poincaré conjecture. The American Mathematical Monthly, 116(2), 134-142.
  * [14] Pedersen, H., & Swann, A. (1993). Einstein-Weyl geometry, the Bach tensor and conformal scalar curvature. J. reine angew. Math, 441, 99-113.
  * [15] Sharpe, R. W. (2000). Differential geometry: Cartan's generalization of Klein's Erlangen program (Vol. 166). Springer Science & Business Media.
  * [16] Smillie, J. The Euler characteristic of flat bundles. preprint.
  * [17] Spivak, M. D. (1970). A comprehensive introduction to differential geometry. Publish or perish.
  * [18] Wylie, W., & Yeroshkin, D. (2016). On the geometry of Riemannian manifolds with density. arXiv preprint arXiv:1602.08000.