# The quasar main sequence and its potential for cosmology

P. Marziani1 D. Dultzin2 A. Del Olmo3111AdO acknowledges financial support
from Spanish grants AYA2016-76682-C3-1-P and .SEV-2017-0709. M. D’Onofrio4
J. A. de Diego2 G. M. Stirpe5 E. Bon6 N. Bon6 B. Czerny7
J. Perea3 S. Panda7,8 M. L. Martinez-Aldama3,7 C. A. Negrete2 1INAF,
Osservatorio Astronomico di Padova, Italy, email<EMAIL_ADDRESS>2Instituto de Astronomía, UNAM, Mexico; 3IAA (CSIC), Granada, Spain;
4Università di Padova, Italy; 5INAF, OASS, Bologna, Italy; 6 Belgrade
Observatory, Serbia; 7Center For Theoretical Physics, Polish Academy of
Sciences, Warsaw, Poland; 8 Nicolaus Copernicus Astronomical Center, Polish
Academy of Sciences, Warsaw, Poland.

(2019)

###### Abstract

The main sequence offers a method for the systematization of quasar spectral
properties. Extreme FeII emitters (or extreme Population A, xA) are believed
to be sources accreting matter at very high rates. They are easily
identifiable along the quasar main sequence, in large spectroscopic surveys
over a broad redshift range. The very high accretion rate makes it possible
that massive black holes hosted in xA quasars radiate at a stable, extreme
luminosity-to-mass ratio. After reviewing the basic interpretation of the main
sequence, we report on the possibility of identifying virial broadening
estimators from low-ionization line widths, and provide evidence of the
conceptual validity of redshift-independent luminosities based on virial
broadening for a known luminosity-to-mass ratio.

###### keywords:

quasars: general, quasars: emission lines, line: profiles, cosmological
parameters, distance scale, dark matter

††journal: Nuclear Activity in Galaxies across Cosmic Time††volume:
356††editors: M. Povic et al.

##  1 Introduction

The concept of the quasar main sequence originated from a Principal Component
Analysis of parameters measured on the optical spectra of $\sim$80 PG quasars
([Boroson & Green 1992, Boroson & Green 1992]). The first eigenvector 1 (E1)
computed by the analysis was found to be mainly associated with two anti-
correlations, between strength of FeII$\lambda$4570 and prominence of
[OIII]$\lambda\lambda 4959,5007$, and strength of FeII$\lambda$4570 and FWHM
of the HI Balmer line H$\beta$. In the plane FWHM H$\beta$ vs. prominence of
the optical FeII emission (where the FeII prominence is measured by the
parameter $R_{\mathrm{FeII}}$ defined as the intensity ratio of the FeII
$\lambda$4570 blend and H$\beta$), the occupation of data points representing
low redshift quasars takes the form of an elbow-shaped sequence (Fig. 1).
Quasar spectra show a wide range of line widths, profile shapes,
$R_{\mathrm{FeII}}$, line shifts, line intensities which imply differences in
line emitting gas dynamics and ionization levels: the main sequence organizes
different properties [Sulentic et al.(2000a), (Sulentic et al. 2000a)].

Why are these two parameters – FWHM H$\beta$ and $R_{\mathrm{FeII}}$ – so
important? FeII emission extends from UV to the IR and can dominate the
thermal balance of the low-ionization part of the broad-line region (BLR,
[Marinello et al.(2016), Marinello et al. 2016]). FeII emission is self-
similar (at least to a first approximation) in quasars but FeII intensity with
respect to H$\beta$ changes from object to object. The FWHM(H$\beta$) is
explained mainly by Doppler broadening, and is related to projection of the
velocity field in the low-ionization BLR along the line of sight. There has
been a growing consensus that the low-ionization BLR is predominantly
virialized, since the early results of the first major reverberation mapping
campaigns [Peterson & Wandel 1999, (Peterson & Wandel 1999)]. Therefore
$R_{\mathrm{FeII}}$ and FWHM H$\beta$ can be considered tracers of the
physical and dynamical conditions in the low-ionization BLR. [Sulentic et
al.(2000a), Sulentic et al. (2000)] introduced the distinction between two
Populations. The FWHM H$\beta$ $\leq$ 4000 km s-1 condition selects narrower
sources that are preferentially moderate-to-strong FeII emitters (Population
A). Broader sources (FWHM H$\beta$ $>$ 4000 km s-1, Population B) tend to have
low FeII emission and are believed to be predominantly sources radiating at
lower Eddington ratio ($L/L_{\mathrm{Edd}}$) than the quasars of Population A
[Marziani et al.(2018a), (Marziani et al. 2018a)].

##  2 The main driver of the quasar main sequence

Several approaches consistently support a relation between Eddington ratio and
$R_{\mathrm{FeII}}$. For instance, according to the fundamental plane of
accreting massive black holes for reverberation-mapped active galactic nuclei
(AGNs) – a relation connecting $R_{\mathrm{FeII}}$, $L/L_{\mathrm{Edd}}$ and a
parameter $D$ dependent on line shape ([Du et al.(2016), Du et al. 2016]) –
$R_{\mathrm{FeII}}$ $>$1 implies $L/L_{\mathrm{Edd}}>$ 1\. An independent
confirmation is provided by the analysis of the stellar velocity dispersion of
the host spheroid, $\sigma_{\star}$. The $\sigma_{\star}$ has been used as a
proxy for the black hole mass in accordance with established scaling laws
connecting black hole and host spheroid mass [Magorrian et al.(1998), (e.g.,
Magorrian et al. 1998]). If the AGNs are subdivided in narrow luminosity bins,
the $\sigma_{\star}$ is found to decrease as a function of
$R_{\mathrm{FeII}}$, implying that $R_{\mathrm{FeII}}$ increases as a function
of $L/L_{\mathrm{Edd}}$ [Sun, & Shen(2015), (Sun & Shen 2015)]. The origin of
this connection remains unclear at the time of writing. The structure of the
emitting regions is probably influenced by the balance between gravitation and
radiation forces [Ferland et al.(2009), (Ferland et al. 2009)], and by a
change in accretion mode with increasing $L/L_{\mathrm{Edd}}$ [wangetal14,
(Wang et al. 2014)]. So the correlation may be a secondary effect of
structural changes induced on the BLR [Panda et al.(2019), (Panda et al.
2019)]. Since $R_{\mathrm{FeII}}$ is also dependent on metallicity [Panda et
al.(2018), (Panda et al. 2018)], high $L/L_{\mathrm{Edd}}$ might be associated
with enriched gas provided to the emitting region by nuclear and circumnuclear
star formation [D’Onofrio, & Marziani(2018), (D’Onofrio & Marziani 2018)].

Figure 1: The optical plane of the E1 MS, FWHM of the broad component of
H$\beta$ vs $R_{\mathrm{FeII}}$. Isodensity curves and shading represent the
source occupation from the sample of [zamfiretal10, Zamfir et al. (2010)],
with $\approx$ 300 quasars. The labels identify the loci of the main
populations: Population A with FWHM $\leq 4000$ km s-1; Population B with FWHM
$>4000$ km s-1; extreme Population A, $R_{\mathrm{FeII}}>1$. See text for more
details.

Outflows from mildly-ionized gases producing blue shifted lines from ionic
species of IP $\lesssim 50$ eV are ubiquitous in AGNs [richardsetal11,
(Richards et al. 2011)]. The extent and the energetics of the outflows are not
yet fully appreciated. For example, it has been possible only in recent year
to consider high velocity outflows from very hot gas (the so-called ultra-fast
outflows, UFOs, [Tombesi et al.(2011), Tombesi et al. 2011]). However, the
prominence of outflows traced by the mildly-ionized gas emitting
CIV$\lambda$1549 and [OIII]$\lambda\lambda$4959,5007 increases along the main
sequence (MS, see Figure 4 of [Sulentic et al.(2000), Sulentic et al. 2000 for
CIV]) and reaches a maximum in correspondence of the strongest FeII emitters,
at extreme values of $L/L_{\mathrm{Edd}}$.

Several interpretations of the MS (not necessarily in contradiction among
themselves) have been proposed. The occupation of data points can be accounted
for by a trend between Eddington ratio and $R_{\mathrm{FeII}}$, convolved with
the effect of orientation on the FWHM of H$\beta$ [marzianietal01, shenho14,
(Marziani et al. 2001, Shen & Ho 2014)]. This is especially true in the case
of a small range of black hole masses. There is a degeneracy between the
effect of black hole mass and orientation as both tend to increase the FWHM of
H$\beta$ [Marziani et al.(2018a), (Marziani et al. 2018a)]. [Panda et
al.(2019), Panda et al. (2019)] showed that, for a fixed black hole mass,
there is a limit in FWHM beyond which the orientation effects cannot go. Even
if limits to the FWHM range spanned by orientation broadening can be set, the
overlap of mass and orientation effects implies that in the 2D representation
of the MS make it impossible to retrieve independent values of these
parameters (viewing angle, $L/L_{\mathrm{Edd}}$, black hole mass) for
individual quasars. A 3D representation of the MS or additional constraints
are necessary. Another important side effect of the degeneracy between mass
and viewing angle is that a broad range of black hole mass leads to the MS
with a wedge-shaped occupation area [shenho14, (Shen & Ho 2014)].

An alternative view considers that Population B quasars are more massive and
radiate at lower values of the $L/L_{\mathrm{Edd}}$ than quasars belonging to
Population A. This difference between the two populations helps define the
shape of the main sequence as a sequence in the optical plane, and is most
likely a consequence of the down-sizing of nuclear activity at low redshift
[Fraix-Burnet et al.(2017), (Fraix-Burnet et al. 2017)]. So it is possible to
establish an evolutionary connection from the sources of extreme Population A
to the ones of Population B, where the cosmic arrow of time is provided by the
black hole mass which can only grow: the larger the mass the older the source.

##  3 Highly-accreting quasars

###  3.1 Extreme observational and physical properties

Extreme Population A (xA) quasars satisfy the condition $R_{\mathrm{FeII}}$
$>$ 1; they are those $\sim$10% of quasars in low-$z$ ($\lesssim 1$),
optically selected samples with extreme FeII emission. Their prevalence is
steeply decreasing toward higher $R_{\mathrm{FeII}}$ values: in the range 1.0
$\leq$ $R_{\mathrm{FeII}}$ $<1.5$ we find $\approx 7$ % of all quasars; in the
range 1.5 $\leq$ $R_{\mathrm{FeII}}$ $<2,\approx 3$ %. Quasars with
$R_{\mathrm{FeII}}$ $>$ 2 account for less than 1% of the optically-selected
quasar population ([Marziani et al.(2013), Marziani et al. 2013]). xA quasars
show distinctive features: their UV continuum is usually not reddened, they
are often with extremely weak UV emission lines (i.e., weak-lined quasars with
W(CIV)$\lambda$1549 $\leq$ 10Å following [Diamond-Stanic et al.(2009),
Diamond-Stanic et al. 2009]). The prominent AlIII$\lambda$1860, and very weak
CIII]$\lambda$1909 allow for easy UV selection criteria: if (1)
$R_{\mathrm{FeII}}>$ 1.0 is satisfied, then (2) UV AlIII
$\lambda$1860/SiIII]$\lambda$1892 $>$ 0.5 &
SiIII]$\lambda$1892/CIII]$\lambda$1909 $>$ 1.

Physical parameters are correspondingly extreme. First of all, their
$L/L_{\mathrm{Edd}}$ is at the high end of the distribution along the main
sequence, with small dispersion [Marziani, & Sulentic(2014), (Marziani &
Sulentic 2014)]. In other words xA quasars selected according to criteria 1
and 2 are extreme radiators, with maximum radiative output per unit mass close
to their Eddington limit. This condition is predicted by accretion disk theory
at high (possibly super-Eddington) accretion rates: radiative efficiency
should be low, and $L/L_{\mathrm{Edd}}$ saturate toward a limiting value
[sadowskietal14, (Sadowski et al. 2014, and references therein)]. The star
formation rate (SFR) in the host galaxy as estimated from radio observations
can be up to $\sim$ a few $10^{3}$M⊙ yr-1 ([Ganci et al.(2019), Ganci et al.
2019], $z\lesssim 1$). The broad emission line intensity ratios in the UV
suggest extremely high values for density ($n\gtrsim 10^{12}-10^{13}$ cm-3),
very low ionization (ionization parameter $\sim 10^{-3}-10^{-2.5}$, [Negrete
et al.(2012), Negrete et al. 2012]), and high metallicity ($Z\gtrsim$ 20 Z⊙,
[martinez-aldamaetal18, Martinez-Aldama et al. 2018]).

Broad emission lines in xA sources are produced by gas apparently enriched by
a circumnuclear Starburst. It is tempting to consider that xAs could be the
first unobscured stage emerging from obscured stages of AGN evolution. This
hypothesis fits an evolutionary sequence ([sandersetal88, Dultzin-Hacyan et
al.(2003), Sanders et al. 1988, Dultzin-Hacyan et al. 2003]): merging and
strong interaction lead to accumulation of gas in the galaxy central parsecs,
and to coeval Starburst and nuclear activity, whereas winds from massive stars
and supernovæ provide enriched material. Feedback effects on the host galaxies
induced by the extreme outflows of xA quasars can be significant if the AGN
luminosity is high, as the outflow thrust and kinetic powers are dependent on
the emission line luminosity. This means that xA quasars could likely be a
major factor in galactic evolution at moderate-to-high redshift ($z\gtrsim
1$). It is however important to stress that xA sources are not necessarily
luminous sources – xAs are the low-mass (relatively rare) high accretors in
the local Universe as well as the most distant quasars at $z\gtrsim 6$ ([Wang
et al.(2019), Wang et al. 2019]).

Almost symmetric H$\beta$ and AlIII$\lambda$1860 line profiles coexist with a
CIV$\lambda$1549 profile shifted by several thousand km/s, even at the highest
luminosity and when radiation forces predominate over gravity, i.e., when
$L/L_{\mathrm{Edd}}$ is high [Sulentic et al.(2017), vietrietal17, Bischetti
et al.(2017), (Sulentic et al. 2017, Vietri et al. 2017, Bischetti et al.
2017)]. A virialized subsystem emitting low ionization lines and a subsystem
due to winds or outflows constitute two regions that are, at least in part,
appearing as kinematically disjoint in the line profiles.

In addition, we can count on xA quasars’ spectral invariance: intensity ratios
remain the same; only the line width increases with luminosity. This implies
that the radius of the emitting region should rigorously scale as
$L^{\frac{1}{2}}$: if not, the ionization parameter should change with
luminosity. No significant spectral change from very high luminosity to low
luminosity sources has been detected yet. Putting together (1)
$L/L_{\mathrm{Edd}}=const.$, (2) $r\propto L^{\frac{1}{2}}$ and the virial
condition (3) $M_{\mathrm{BH}}\propto r$FWHM2, we obtain a relation linking
luminosity and line width as $L\propto$ FWHM4 [Marziani, & Sulentic(2014),
(Marziani & Sulentic 2014)].

Building the Hubble diagram distance modulus versus redshift (see e.g.,
[Risaliti, & Lusso(2015), Risaliti & Lusso 2015]) from the redshift-
independent virial luminosity estimates with xA quasars over the redshift
range $0.1\lesssim z\lesssim 3$, we obtain a distribution consistent with
concordance $\Lambda$CDM (Fig. 2). Constraints on the energy density of matter
$\Omega_{\mathrm{M}}$ (0.30 $\pm$ 0.06) are better than the ones from
supernovæ, because of the $z\sim 2$ coverage of the quasar sample. The
significant scatter of individual measurements ($\sim$ 1.1 - 1.3 mag) may be
associated with (a) uncertainty of FWHM, which enters with the 4th power in
the luminosity relation; (b) orientation that is likely to be the main source
of scatter in the classical scaling relations [Marziani et al.(2019),
(Marziani et al. 2019)]; (c) differences in intrinsic properties of the xA
quasars i.e., spectral energy distribution, ionizing photon flux, etc.

Figure 2: The Hubble diagram distance modulus $\mu$ vs $z$ for a sample of
quasars ([Dultzin et al.(2020), Dultzin et al. 2020]). Individual measurements
have been averaged over redshift bins. Error bars are sample standard
deviations for $z$ and $\mu$. Yellow data point refer to the use of the
H$\beta$ FWHM as a virial broadening estimator, magenta ones are computed from
the FWHM of the AlIII$\lambda$1860 doublet individual components. Error bars
show sample standard deviations in $\mu$ and $z$ for each bin.

##  4 Conclusion

The MS offer contextualization of quasar observational and physical properties
([Marziani et al.(2018b), Marziani et al. 2018b]). Several MS trends are
ultimately associated with Eddington ratio which apparently reaches an extreme
value in correspondence of the extreme Population A (xA). xA quasars include
the strongest FeII emitters, satisfying the condition $R_{\mathrm{FeII}}>1$.
They show a relatively high prevalence (10%) and are easily recognizable
thanks to their peculiar and luminosity-invariant spectral properties. In
addition, they are sources that tend to show low intrinsic variability ([Du et
al.(2018), Du et al. 2018]), and their low ionization lines are apparently
emitted in a virialized BLR (see also Swayamtrupta Panda’s contributions in
this volume). These properties make xA quasars suitable as possible Eddington
standard candles, where the invariant properties is not intrinsic luminosity,
but Eddington ratio.

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# Some inequalities and applications of Simons’ type formulas in Riemannian,
affine and statistical geometry

Barbara Opozda  Faculty of Mathematics and Computer Science UJ, ul.
Łojasiewicza 6, 30-348 Cracow, Poland<EMAIL_ADDRESS>
###### Abstract.

A few formulas and theorems for statistical structures are proved. They deal
with various curvatures as well as with metric properties of the cubic form or
its covariant derivative. Some of them generalize formulas and theorems known
in the case of Lagrangian submanifolds or affine hypersurfaces.

###### Key words and phrases:

statistical structure, affine hypersurface, Lagrangian submanifold, Simons’
formula, Yau’s maximum principle

###### 1991 Mathematics Subject Classification:

Primary: 53C05, 53A15, 53D12. 53C20

##  1\. Introduction

The name ”statistical geometry” is relatively new, although this geometry has
existed for long and in various editions. From a geometric viewpoint, a
statistical structure is nothing but a Codazzi pair. Starting from locally
strongly convex hypersurfaces in the Euclidean space, through locally strongly
convex equiaffine hypersurfces in the affine space $\mathbf{R}^{n+1}$ and
Lagrangian submanifolds in complex space forms to Hessian manifolds – all
these examples are statistical manifolds. Note that the structures of the
subclasses have very different properties and, therefore, the intersections of
the subclasses are small. For instance, if a statistical structure can be
realized on a Lagrangian submanifold and on an affine hypersurface, then its
Riemannian sectional curvature, its $\nabla$-sectional curvature and its
$K$-sectional curvature are all constant. Note also that the category of
statistical structures is much larger than the union of all the specific
subclasses mentioned above. In particular, results proved for affine
hypersurfaces or for Lagrangian submanifolds usually are rarely generalizable
to the general case of statistical structures.

The aim of this paper is to prove some global and local theorems for
statistical manifolds in the general setting, with extensive references to
affine hypersurfaces or Lagrangian submanifolds.

By a statistical structure we mean a pair $(g,\nabla)$, where $g$ is a
Riemannian metric tensor field and $\nabla$ is a torsion-free affine
connection such that the cubic form $\nabla g$ is symmetric. Let
$A=-\frac{1}{2}\nabla g$ and $\tau$ be the Czebyshev form defined as
$\tau(X)=\hbox{\rm tr}\,_{g}A(X,\cdot,\cdot)$. Set $K=A^{\sharp}$. The Levi-
Civita connection for $g$ will be denoted by $\hat{\nabla}$. $\hat{R}$ and
$\widehat{\hbox{\rm Ric}}\,$ will stand for the corresponding curvature and
Ricci tensors. A statistical structure is called trace-free if $\tau=0$. If
$\hat{\nabla}A$ is symmetric, then the statistical structure is called
conjugate symmetric. Basic information on statistical structures, affine
hypersurfaces, and Lagrangian submanifolds, needed in this paper, is contained
in Section 2.

In Section 3, we prove some algebraic formulas for $\|A\|$, $\|\tau\|$,
$\tau$, and $K$. These formulas yield inequalities for the Ricci tensors and
scalar curvatures of statistical structures.

The main part of this paper deals with the following Simons’ formula

(1) $\frac{1}{2}\Delta(\|s\|^{2})=g(\Delta s,s)+\|\hat{\nabla}s\|^{2},$

where $s$ is a tensor field and $\Delta s$ is a specially defined Laplacian on
$s$. The first assumption which should be made for computing the term
$g(\Delta s,s)$ is the symmetry of $\hat{\nabla}s$. Therefore, when we apply
this formula to the cubic form $A$, we shall assume that the statistical
structure is conjugate symmetric. Using various properties of statistical
structures, this term can be computed in various ways. For example, one can
get the equality, see (72),

(2)
$\frac{1}{2}\Delta(\|A\|^{2})=\|\hat{\nabla}A\|^{2}+g(\hat{\nabla}^{2}\tau,A)+g(\hat{R}-R,\hat{R})+g(\widehat{\hbox{\rm
Ric}\,},g(K_{\cdot},K_{\cdot})),$

where $R$ is the curvature tensor of $\nabla$. Computing this term in another
way one gets the following inequalities valid for conjugate symmetric trace-
free statistical structures, see (90),

(3)
$\begin{array}[]{rcl}&&\frac{1}{2}\Delta\|A\|^{2}\geq(n+1)H\|A\|^{2}+\frac{n+1}{n(n-1)}\|A\|^{4}+\|\hat{\nabla}A\|^{2},\\\
&&\frac{1}{2}\Delta\|A\|^{2}\leq(n+1)H\|A\|^{2}+\frac{3}{2}\|A\|^{4}+\|\hat{\nabla}A\|^{2}\end{array}$

in the case where $R=HR_{0}$, $R_{0}(X,Y)Z=g(Y,Z)X-g(X,Z)Y$, and $\dim M=n$.
In Section 5, we employ (3) for proving some estimations for the functions
$\|A\|^{2}$, $\|\hat{\nabla}A\|^{2}$ in the case where $g$ is complete. A
model theorem here is the following result due to Calabi

######  Theorem 1.1.

For a complete hyperbolic affine Blaschke sphere, the Ricci tensor of $g$ is
negative semi-definite and, consequently,

(4) $\|A\|^{2}\leq-\rho,$

where $\rho$ is the affine scalar curvature.

In the applications of Simons’ formula (1) the term $\|\hat{\nabla}s\|^{2}$ is
usually just nonnegative. We propose new estimations involving also
$\inf\|\hat{\nabla}A\|^{2}$ and $\hbox{\rm sup}\|\hat{\nabla}A\|^{2}$, see
Theorems 5.5, 5.7, and 5.8. For proving these theorems, we use Yau’s maximum
principle. For example, we prove (see Theorem 5.7) that if for a trace-free
statistical structure with complete metric on an $n$-dimensional manifold we
have $R=HR_{0}$, where $H$ is a negative number, then

(5) $\inf\,\|\hat{\nabla}A\|^{2}\leq\frac{n(n^{2}-1)H^{2}}{4}.$

In the last section, we prove a theorem on conjugate symmetric statistical
structures with nonnegative sectional curvature of the metric $g$. The theorem
generalizes a theorem known for minimal Lagrangian submanifolds, see [11].

##  2\. Preliminaries

In this section, we introduce the notation and collect basic information on
the geometry of statistical structures, affine hypersurfaces, and Lagrangian
submanifolds. All details for this section can be found in [13], [12], and
[4].

In this paper we consider only torsion-free connections. If needed, manifolds
are connected and orientable. If $g$ is a metric tensor field (positive-
definite) on a manifold $M$, then div will stand for a divergence relative to
the Levi-Civita connection $\hat{\nabla}$ of $g$. In particular, if $s$ is a
tensor field of type $(1,k)$, then $\hbox{\rm div}\,s$ is a tensor field of
type $(0,k)$ defined as

(6) $(\hbox{\rm div}\,s)(X_{1},...,X_{k})=\hbox{\rm
tr}\,\\{Y\to(\hat{\nabla}_{Y}s)(X_{1},...,X_{k})\\}.$

We shall use the following Laplacian relative to $g$ acting on tensor fields
(usually symmetric) of type $(0,k)$ for various values of $k$. Namely, we set

(7) $(\Delta s)(X_{1},...,X_{k})=\hbox{\rm
tr}\,_{g}(\hat{\nabla}^{2}s)(\cdot,\cdot,X_{1},...,X_{k})=\sum_{i}(\hat{\nabla}^{2}s)(e_{i},e_{i},X_{1},...,X_{k}),$

where $e_{1},...,e_{n}$ is an orthonormal basis of $T_{x}M$ and
$X_{1},...,X_{n}\in T_{x}M$, $x\in M$. Moreover,
$\hat{\nabla}^{2}s=\hat{\nabla}(\hat{\nabla}s)$, where $\hat{\nabla}s$ is a
tensor field of type $(0,k+1)$ given by the formula

$\hat{\nabla}s(X,...)=(\hat{\nabla}_{X}s)(...).$

In the case of differential forms, the above Laplacian is not the Hodge
Laplacian (defined as $d\delta+\delta d$). The two Laplacians are related by
the Weitzenb${\rm\ddot{o}}$ck formulas, see e.g., (63). In this paper, the
codifferential $\delta$ of a differential $k$-form $\omega$ is defined as

(8) $\delta\omega(X_{1},...,X_{k-1})=+\hbox{\rm
tr}\,_{g}(\hat{\nabla}w)(\cdot,\cdot,X_{1},...,X_{k-1}).$

Because of these agreements, Simons’ formula has the form as in (1). The
Laplacian defined in (7) is usually denoted by $\hat{\nabla}^{*}\hat{\nabla}$,
but we shall use the symbol $\Delta$ for a simpler notation.

###  2.1. Statistical structures

By a statistical structure on a manifold $M$ we mean a pair $(g,\nabla)$,
where $g$ is a metric tensor and $\nabla$ is a torsion-free connection on $M$
such that the cubic form $\nabla g$ is symmetric. The difference tensor
defined by

(9) $K(X,Y)=K_{X}Y=\nabla_{X}Y-\hat{\nabla}_{X}Y$

is symmetric and symmetric relative to $g$. A statistical structure is called
trivial if $\nabla=\hat{\nabla}$, that is, $K\equiv 0$. It is clear that a
statistical structure can be also defined as a pair $(g,K)$, where $K$ is a
symmetric and symmetric relative to $g$ $(1,2)$-tensor field. Namely, the
connection $\nabla$ is defined by (9). Alternatively, instead of $K$ one can
use the symmetric cubic form $A(X,Y,Z)=g(K(X,Y)Z)$. The cubic forms $\nabla g$
and $A$ are related as follows

(10) $\nabla g=-2A.$

A statistical structure is called trace-free if $E:=\hbox{\rm tr}\,_{g}K=0$.
Having a metric tensor $g$ and a connection $\nabla$ on a manifold $M$, one
defines a conjugate connection $\overline{\nabla}$ by the formula

(11) $g(\nabla_{X}Y,Z)+g(Y,\overline{\nabla}_{X}Z)=Xg(Y,Z).$

A pair $(g,\nabla)$ is a statistical structure if and only if
$(g,\overline{\nabla})$ is. The pairs are also simultaneously trace-free
because if $K$ is the difference tensor for $(g,\nabla)$, then $-K$ is the
difference tensor for $\overline{\nabla}$.

The curvature tensors for $\nabla$, $\overline{\nabla}$, $\hat{\nabla}$ will
be denoted by $R$, $\overline{R}$ and $\hat{R}$ respectively. The
corresponding Ricci tensors will be denoted by Ric , $\overline{\hbox{\rm
Ric}}\,$ and $\widehat{\hbox{\rm Ric}}\,$. In general, the curvature tensor
$R$ does not satisfy the equality $g(R(X,Y)Z,W)=-g(R(X,Y)W,Z)$. If it does, we
say that the statistical structure is conjugate symmetric. For a statistical
structure, we always have

(12) $g(R(X,Y)Z,W)=-g(\overline{R}(X,Y)W,Z).$

The following conditions are equivalent:
1) $R=\overline{R}$,
2) $\hat{\nabla}K$ is symmetric (equiv. $\hat{\nabla}A$ is symmetric),
3) $g(R(X,Y)Z,W)$ is skew-symmetric relative to $Z,W$,

Hence, each of the above three conditions characterizes a conjugate symmetric
statistical structure. The above equivalences follow from the following well-
known formula

(13)
$R(X,Y)=\hat{R}(X,Y)+(\hat{\nabla}_{X}K)_{Y}-(\hat{\nabla}_{Y}K)_{X}+[K_{X},K_{Y}].$

Writing the same equality for $\overline{\nabla}$ and adding both equalities,
we get

(14) $R+\overline{R}=2\hat{R}+2[K,K],$

where $[K,K]$ is a $(1,3)$-tensor field defined as
$[K,K](X,Y)Z=[K_{X},K_{Y}]Z$. In particular, if $R=\overline{R}$, then

(15) $R=\hat{R}+[K,K].$

The $(1,3)$-tensor field $[K,K]$ is a curvature tensor, that is, it satisfies
the first Bianchi identity and has the same symmetries as the Riemannian
curvature tensor. Define its Ricci tensor $\hbox{\rm Ric}\,^{K}$ as follows
$\hbox{\rm Ric}\,^{K}(Y,Z)=\hbox{\rm tr}\,\\{X\to[K,K](X,Y)Z\\}$. In [13], the
following formula was proved

(16) $\hbox{\rm Ric}\,^{K}(Y,Z)=\tau(K(Y,Z))-g(K_{Y},K_{Z}),$

where $\tau$ is the Czebyshev $1$-form defined as $\tau(X)=\hbox{\rm
tr}\,K_{X}=g(E,X)$. From (13) we receive, see [13],

(17) $\hbox{\rm Ric}\,=\widehat{\hbox{\rm Ric}}\,+\hbox{\rm
div}\,K-\hat{\nabla}\tau+\hbox{\rm Ric}\,^{K}.$

The $1$-form $\tau$ is closed (i.e., $\hat{\nabla}\tau$ is symmetric) if and
only if the Ricci tensor Ric is symmetric. For a conjugate symmetric
statistical structure, the Ricci tensor is symmetric. More generally, if for a
statistical structure $\hbox{\rm Ric}\,=\overline{\hbox{\rm Ric}}\,$, then
$d\tau=0$. Indeed, by writing (17) for $\overline{\nabla}$ and comparing with
(17), we see that $\hbox{\rm Ric}\,=\overline{\hbox{\rm Ric}}\,$ if and only
if $\hbox{\rm div}\,K=\hat{\nabla}\tau$. Therefore, if $\hbox{\rm
Ric}\,=\overline{\hbox{\rm Ric}}\,$, then $d\tau=0$ (because $\hbox{\rm
div}\,K$ is symmetric). Using (17) and the analogous formula for the
connection $\overline{\nabla}$ one also gets

(18) $\hbox{\rm Ric}\,+\overline{\hbox{\rm Ric}\,}=2\widehat{\hbox{\rm
Ric}\,}+2\tau\circ K-2g(K_{\cdot},K_{\cdot}).$

In particular, if $(g,\nabla)$ is trace-free, then

(19) $2\widehat{\hbox{\rm Ric}\,}\geq\hbox{\rm Ric}\,+\overline{\hbox{\rm
Ric}\,}.$

Recall now the scalar curvatures for statistical structures. We have the
Riemannian scalar curvature $\hat{\rho}$ for $g$ and the scalar curvature
$\rho$ for $\nabla$: $\rho=\hbox{\rm tr}\,_{g}\hbox{\rm Ric}\,$. It turns out
(e.g., by (12)) that if we define the analogous scalar curvature for
$\overline{\nabla}$, then it is equal to $\rho$. We also have the scalar
curvature $\rho^{K}=\hbox{\rm tr}\,_{g}\hbox{\rm Ric}\,^{K}$. From the above
formulas, one easily gets

(20) $\rho^{K}=\|E\|^{2}-\|K\|^{2},$ (21)
$\hat{\rho}=\rho+\|K\|^{2}-\|E\|^{2}.$

Recall now the sectional curvatures in statistical geometry. Of course, we
have the usual Riemannian sectional curvature of the metric tensor $g$. We
shall denote it by $\hat{k}(\pi)$ if $\pi$ is a vector plane in a tangent
space. In [13] (see also [6]) another sectional curvature was introduced. It
was called the sectional $\nabla$-curvature. Namely, it was observed that the
the tensor field $R+\overline{R}$ has all algebraic properties needed to
produce the sectional curvature. However, in general, this sectional curvature
does not have the same properties as the Riemannian sectional curvature. For
instance, Schur’s lemma does not hold, in general. However, it holds for
conjugate symmetric statistical structures. The sectional $\nabla$-curvature
is defined as follows

(22) $k(\pi)=\frac{1}{2}g((R+\overline{R})(e_{1},e_{2})e_{2},e_{1}),$

where $e_{1},e_{2}$ is an orthonormal basis of $\pi$. Another good curvature
tensor, which exists in statistical geometry is $[K,K]$. It again has all
algebraic properties to define a sectional curvature. This sectional curvature
was introduced and studied in [14]. It was called the sectional $K$-curvature.
If $\pi$ is a vector plane in a tangent space $T_{x}M$, then the sectional
$K$-curvature is equal to $g([K,K](e_{1},e_{2})e_{2},e_{1})$ for any
orthonormal basis $e_{1},e_{2}$ of $\pi$. Similarly to the case of the
sectional $\nabla$-curvature Schur’s lemma holds for conjugate symmetric
statistical structures.

A statistical structure is called Hessian if $\nabla$ is flat, that is, if
$R=0$. As it will be noticed below, all Hessian structures are locally
realizable on parabolic equiaffine spheres. Since $\hbox{\rm Ric}\,=0$ for a
Hessian structure and $\nabla$ is torsion-free, we know that the Koszul form
$\beta=\nabla\tau$ is symmetric. A Hessian structure is called Einstein-
Hessian if $\nabla\tau=\lambda g$, see [17]. For a Hessian structure we have

(23) $\beta=\hat{\nabla}\tau-\tau\circ K,\ \ \ \ \hbox{\rm
tr}\,_{g}\beta=\delta\tau-\|\tau\|^{2}.$

By (18), we also have

(24) $\widehat{\hbox{\rm Ric}}\,=g(K_{\cdot},K_{\cdot})-\tau\circ K.$

In geometry, the richest sources of statistical manifolds seem to be the
theory of affine hypersurfaces in $\mathbf{R}^{n+1}$ and that of Lagrangian
submanifolds in complex space forms. We shall shortly recall the basic facts
from these theories.

###  2.2. Equiaffine locally strongly convex hipersurfaces in
$\mathbf{R}^{n+1}$

For the theory of affine hypersurfaces, we refer to [12]. Let
$f:M\to\mathbf{R}^{n+1}$ be an immersed locally strongly convex hypersurface
of the affine space $\mathbf{R}^{n+1}$. Denote the standard flat affine
connection (the operator of ordinary differentiation) on $\mathbf{R}^{n+1}$ by
$D$. Assume that the hypersurface is equipped with a transversal vector field
$\xi$ such that $D_{X}\xi$ is tangent to the hypersurface for every $X\in
T_{x}M$, $x\in M$. Such a tranversal vector field is called equiaffine and a
hypersurface endowed with such a tranversal vector field is called an
equiaffine hypersurface. All hypersurfaces considered in this paper will be
equiaffine. By the following formulas of Gauss and Weingarten, one gets the
induced connection $\nabla$, the second fundamental form $g$, and the shape
operator $S$

(25) $\begin{array}[]{rcl}&&D_{X}f_{*}Y=f_{*}(\nabla_{X}Y)+g(X,Y)\xi,\\\
&&D_{X}\xi=-f_{*}(SX).\end{array}$

A hypersurface is locally strongly convex if and only if $g$ is definite. If
an equiaffine hypersurface is locally strongly convex, the sign of $\xi$ is
chosen in such a way that $g$ is positive definite. It turns out that for an
equiaffine hypersurface the cubic form $\nabla g$ is symmetric. The symmetry
of $\nabla g$ is the so-called first Codazzi equation for an equiaffine
hypersurface. It means that the induced structure $(g,\nabla)$ is a
statistical structure. We also have the following Gauss equation

(26) $R(X,Y)Z=g(Y,Z)\mathcal{S}X-g(X,Z)\mathcal{S}Y,$

The so-called Ricci equation for equiaffine immersions states that the Ricci
tensor Ric is symmetric. The symmetry of the Ricci tensor is the first
obstruction for a statistical structure to be realizable as the induced
structure on a hypersurface in $\mathbf{R}^{n+1}$. The second obstruction is
that the dual connection $\overline{\nabla}$ must be projectively flat.
Namely, an important theorem on the realizability of statistical structures on
equiaffine hypersurfaces is the following fundamental theorem proved in [5]

######  Theorem 2.1.

Let $(g,\nabla)$ be a statistical structure on a simply connected manifold $M$
and satisfy the following conditions:
1) the Ricci tensor of $\nabla$ is symmetric,
2) the dual connection $\overline{\nabla}$ is projectively flat.
Then there is a locally strongly convex immersion $f:M\to\mathbf{R}^{n+1}$ and
its equiaffine transversal vector field $\xi$ such that $\nabla$ is the
induced connection and $g$ is the second fundamental form for the equiaffine
hypersurface $(f,\xi)$.

From now on, we shall always assume that a hypersurface is locally strongly
convex without mentioning this each time.

An equiaffine hypersurface is called an equiaffine sphere if the shape
operator is a multiple of the identity, i.e., $S=H\hbox{\rm id}\,$, where id
is the identity $(1,1)$-tensor field on $M$. In such a case, $H$ must be
constant on connected $M$. A hypersphere is called elliptic if $H>0$,
hyperbolic if $H<0$, and parabolic if $H=0$. For an equiaffine sphere we have
the equality $R=HR_{0}$, where $R_{0}$ is the curvature tensor defined by
$R_{0}(X,Y)Z=g(Y,Z)X-g(X,Z)Y.$ It is now clear that for an equaiaffine sphere
its statistical structure is conjugate symmetric. The converse is also true,
that is, if the statistical structure on an equiaffine hypersurface is
conjugate symmetric, then the hypersurface must be a sphere (Lemma 12.5 in
[13]).

It is also known that if we have a statistical structure (on an
$n$-dimensional connected manifold $M$) whose curvature tensor $R$ satisfies
the equality $R=HR_{0}$, where $H$ is possibly a function, and $n>2$, then $H$
must be a constant (Proposition 12.7 in [13]). If $R=HR_{0}$ then
$g(R(X,Y)Z,W)$ is skew-symmetric for $Z,W$, hence $\overline{R}=R=HR_{0}$. If
moreover, $H$ is constant $\overline{\nabla}$ is clearly projectively flat. A
statistical structure for which $R=HR_{0}$, where $H$ is a constant, was
called in [7] a statistical structure of constant curvature. By Theorem 2.1,
we now have that statistical structures of constant curvature are (from a
local viewpoint) exactly the induced structures on equiaffine spheres. Let us
rewrite (15) in the case where $R=HR_{0}$:

(27) $HR_{0}=\hat{R}+[K,K].$

Among the equiaffine hypersuraces, the best known and historically first
discovered are those which additionally satisfy the so-called apolarity
condition. This condition is equivalent to the trace-freeness of the induced
statistical structure (in the terminology we are using in this paper). The
importance of this class of equiaffine hypersurfaces follows from the
following classical fact: For a locally strongly convex hypersurface, there is
a unique (up to a constant) equiaffine transversal vector field, for which the
induced statistical structure is trace-free. An equiaffine hypersurface whose
induced statistical structure is trace-free is called a Blaschke hypersurface.
An equiaffine sphere whose induced structure is trace-free is called a
Blaschke affine sphere. In contrast to Riemannian geometry, the category of
affine spheres is very rich and not well recognized by now. Trivial Blaschke
hypersurfaces are quadrics. In particular, a parabolic affine sphere with
vanishing cubic form, that is, whose induced statistical structure is trivial,
must be a part of an elliptic paraboloid.

###  2.3. Lagrangian submanifolds

We shall now briefly recall some facts and introduce some notations for
Lagrangian submanifolds. For this part we refer to [4]. Let $\tilde{M}(4c)$ be
a complex space form of holomorphic sectional curvature $4c$ and let $M$ be
its Lagrangian submanifold, that is, $M$ is a real submanifold of $\tilde{M}$,
$\dim M=\dim_{\mathbf{C}}\tilde{M}$ and the bundle $JTM$ is orthogonal to
$TM$, where $J$ is the complex structure on $\tilde{M}$. The induced metric
tensor field on $M$ will be denoted by $g$. If $\sigma$ is the second
fundamental tensor, then we set $K=J\sigma$. The tensor field $K$ is symmetric
and symmetric relative to $g$. Hence, the pair $(g,K)$ is a statistical
structure. We shall call it the induced statistical structure on a Lagrangian
submanifold. For this structure we shall use all the technique we use for
statistical structure. The Codazzi equation for a Lagrangian submanifold says
that the $(1,3)$-tensor field $\hat{\nabla}K$ is symmetric. Hence, the
statistical structure on a Lagrangian submanifold in a complex space form is
conjugate symmetric. We have the following Gauss equation for a Lagrangian
submanifold in $\tilde{M}(4c)$

(28) $cR_{0}=\hat{R}-[K,K].$

This equality is different than the equality (27) holding for equiaffine
spheres. By comparing the two equalities, we see that a statistical structure
on a Lagrangian submanifold can be also realized (locally) on an equiaffine
hypersurface only when the Riemannian sectional curvature for $g$ and the
sectional $K$-curvature are constant.

Instead of the mean curvature vector of a Lagrangian submanifold, one can
consider the tangent vector field $E=\hbox{\rm tr}\,_{g}K$. In particular, a
Lagrangian submanifold of $\tilde{M}(4c)$ is minimal if the induced
statistical structure is trace-free. The mean curvature tensor $\mu$ is equal
to $-\frac{JE}{n}$. Hence it is parallel if and only if $\hat{\nabla}E=0$. By
(28), we have

(29) $\widehat{\hbox{\rm Ric}}\,=c(n-1)g-g(K_{\cdot},K_{\cdot})+\tau\circ K$

and

(30)
$\hat{\rho}=cn(n-1)+\|E\|^{2}-\|A\|^{2}=cn(n-1)+n^{2}\|\mu\|^{2}-\|A\|^{2}.$

##  3\. Algebraic and curvature inequalities

The notation $\sum_{i\neq j}$ stands for the sum over indices $i,j$ such that
$i\neq j$, whereas $\sum_{j;j\neq i}$ means that the sum is for the index $j$
such that $j\neq i$ for a fixed $i$.

######  Theorem 3.1.

For a statistical structure we have

(31) $(\tau\circ K)(U,U)-g(K_{U},K_{U})\leq\frac{1}{4}\|\tau\|^{2}g(U,U)$

for every vector $U$. If $U\neq 0$, the equality holds if and only if $\tau=0$
and $K_{U}=0$.

If $g(K(U,U),U)=0$, then

(32) $(\tau\circ K)(U,U)-g(K_{U},K_{U})\leq\frac{1}{8}\|\tau\|^{2}g(U,U).$

If $U$ is unit, the equality holds if and only if $g(K(U,V),W)=0$ for $V,W$
perpendicular to $U$, $E$ is perpendicular to $U$ and $E=4K(U,U)$.

Proof. We shall prove (31) for an arbitrary unit vector $U\in T_{x}M$. We
extend a given vector $U$ to an orthonormal basis $e_{1}=U$, $e_{2},...,e_{n}$
of $T_{x}M$. We set $A_{ijk}=A(e_{i},e_{j},e_{k})$ and $a_{ir}=A_{iir}$. We
want to show

(33)
$0\leq\frac{1}{4}\sum_{ijr}A_{iir}A_{jjr}+\sum_{ir}A_{1ir}^{2}-\sum_{ir}A_{iir}A_{11r}.$

The last formula can be rewritten as follows

(34)
$0\leq\frac{1}{4}\sum_{ijr}a_{ir}a_{jr}+a_{11}^{2}+2\sum_{r=2}^{n}a_{1r}^{2}+\sum_{i\neq
1r\neq 1}A_{1ir}^{2}-\sum_{ir}a_{ir}a_{1r}.$

Since $\sum_{i\neq 1r\neq 1}A_{1ir}^{2}\geq 0$, it is sufficient to prove

(35) $0\leq
a_{11}^{2}+2\sum_{r=2}^{n}a_{1r}^{2}+\frac{1}{4}\sum_{ijr}a_{ir}a_{jr}-\sum_{ir}a_{ir}a_{1r}.$

The last inequality is equivalent to

(36) $0\leq a_{11}^{2}+2\sum_{r=2}^{n}a_{1r}^{2}+\frac{1}{4}\sum_{rj;j\neq
1}a_{1r}a_{jr}+\frac{1}{4}\sum_{r=1}^{n}\left(\sum_{j=2}^{n}a_{jr}\right)^{2}-\frac{3}{4}\sum_{ir}a_{ir}a_{1r}.$

For a fixed $r$, set $c_{r}=\sum_{j=2}^{n}a_{jr}$. We have

(37)
$0\leq\frac{1}{4}(a_{1r}-c_{r})^{2}=a_{1r}^{2}+\frac{1}{4}a_{1r}c_{r}+\frac{1}{4}c_{r}^{2}-\frac{3}{4}a_{1r}(c_{r}+a_{1r}).$

By comparing the last inequality with (36), we obtain (35). Indeed, for $r=1$
we get in (37)

$0\leq
a_{11}^{2}+\frac{1}{4}a_{11}c_{1}+\frac{1}{4}c_{1}^{2}-\frac{3}{4}a_{11}(c_{1}+a_{11}),$

hence this term appearing also in (36) is nonnegative and equal to $0$ if and
only if $a_{11}=c_{1}$. Consider now $r>1$. For such $r$ we have in (36) the
term

(38)
$2a_{1r}^{2}+\frac{1}{4}a_{1r}c_{r}+\frac{1}{4}c_{r}^{2}-\frac{3}{4}a_{1r}(c_{r}+a_{1r}).$

By comparing this with (37), we see that this term is nonnegative and equal to
$0$ if and only if $a_{1r}=0$ and $a_{1r}=c_{r}$. The inequality (31) has been
proved and moreover, one sees see that the equality holds if and only if
$A_{11r}=0$, $A_{22r}+...+A_{nnr}=0$ for every $r>1$,
$A_{111}=A_{221}+...+A_{nn1}$ and $A_{1ir}=0$ $\forall i\neq 1,r\neq 1$. In
particular, $A_{1ii}=0$ for $i>1$. These conditions imply that $K_{e_{1}}=0$
and $\tau=0$.

Assume now that $g(K(U,U),U)=0$, that is, $a_{11}=0$. In (33), we replace
$\frac{1}{4}$ by $\frac{1}{8}$ and we obtain the following formula analogous
to (36)

(39) $0\leq 2\sum_{r=2}^{n}a_{1r}^{2}+\frac{1}{8}\sum_{jr;j\neq
1}a_{1r}a_{jr}+\frac{1}{8}\sum_{r=1}^{n}\left(\sum_{j=2}^{n}a_{jr}\right)^{2}-\frac{7}{8}\sum_{ir}a_{ir}a_{1r}.$

Again we fix $r$. The only possibly non-vanishing term of the right hand side
in (39) with $r=1$ is $\frac{1}{8}c_{1}^{2}$. For $r>1$ we write

(40)
$0\leq\frac{1}{8}(3a_{1r}-c_{r})^{2}=2a_{1r}^{2}+\frac{1}{8}a_{1r}c_{r}+\frac{1}{8}c_{r}^{2}-\frac{7}{8}a_{1r}(c_{r}+a_{1r}).$

Comparing (40) with (39), we get the desired inequality.

If we have the equality in (32), then $c_{1}=0$. Since $a_{11}=0$, we have
$g(E,e_{1})=0$. By (40), we have $3A_{11r}=A_{22r}+...+A_{nnr}$. It follows
that $E=4K(e_{1},e_{1})$. As in the first part of the proof, we have that
$A_{1ij}=0$ for $i,j>1$. If $E=0$, then $K_{e_{1}}=0$. Assume that $E\neq 0$
and $e_{2}=\frac{E}{\|E\|}$. If $K(e_{1},e_{1})=\lambda e_{2}$, then
$g(K_{e_{1}},K_{e_{1}})=2\lambda^{2}$, $\tau(K(e_{1},e_{1}))=4\lambda^{2}$ and
$\|\tau\|^{2}=16\lambda^{2}$. 2

As an immediate consequence of the inequality (31) and the formula (18), we
get a generalization of (19)

######  Theorem 3.2.

For any statistical structure we have

(41) $2\widehat{\hbox{\rm Ric}}\,\geq\hbox{\rm Ric}\,+\overline{\hbox{\rm
Ric}}\,-\frac{1}{2}\|\tau\|^{2}g.$

For a statistical structure on an equiaffine sphere with $R=HR_{0}$, we have

(42) $\widehat{\hbox{\rm
Ric}}\,\geq\left(H(n-1)-\frac{1}{4}\|\tau\|^{2}\right)g.$

For a Lagrangian submanifold of a complex space form $\tilde{M}(4c)$, we have

(43) $\widehat{\hbox{\rm
Ric}}\,\leq\left(c(n-1)+\frac{n^{2}}{4}\|\mu\|^{2}\right)g,$

where $\mu$ is the mean curvature vector of the submanifold.

The last inequality was first proved in [3].

######  Theorem 3.3.

For any statistical structure, we have

(44) $\frac{n+2}{3}\|A\|^{2}-\|E\|^{2}\geq 0.$

Proof. We use the same notation as in the proof of the above theorem. To prove
(44), we first compute $\|A\|^{2}-\|E\|^{2}$. Let $e_{1},...,e_{n}$ be any
orthonormal basis of $T_{x}M$, $x\in M$. We have

(45)
$\begin{array}[]{rcl}&&\|E\|^{2}=\sum_{i}\left(\sum_{j}A_{jji}\right)^{2}\\\
&&\ \ \ \ =\sum_{i}A_{iii}^{2}+2\sum_{i\neq j}A_{iii}A_{jji}+\sum_{j\neq
i}A_{jji}^{2}+2\sum_{i\neq j,l\neq i,l<j}A_{jji}A_{lli}\\\ &&\ \ \ \ \ \ \
=\sum_{i}a_{ii}^{2}+2\sum_{i\neq j}a_{ii}a_{ji}+\sum_{j\neq
i}a_{ji}^{2}+2\sum_{i\neq j,l\neq i,l<j}a_{ji}a_{li}.\end{array}$ (46)
$\|A\|^{2}=\sum_{ijl}A_{ijl}^{2}=\sum_{i}A_{iii}^{2}+3\sum_{i\neq
j}A_{jji}^{2}+\epsilon=\sum_{i}a_{ii}^{2}+3\sum_{i\neq j}a_{ji}^{2}+\epsilon,$

where $\epsilon=\sum_{i\neq j,i\neq r\j\neq i}A_{ijr}^{2}$. We now have

(47) $\|A\|^{2}-\|E\|^{2}=2\sum_{i\neq j}a_{ji}^{2}-2\sum_{i\neq j,i\neq
l,l<j}a_{li}a_{ji}-2\sum_{i\neq j}a_{ii}a_{ji}+\epsilon.$

For any real numbers $b_{1},...,b_{k}$, the following formula holds

(48) $\sum_{l<j\leq k}(b_{l}-b_{j})^{2}=(k-1)\sum_{j}b_{j}^{2}-2\sum_{l<j\leq
k}b_{l}b_{j}.$

Hence, for a fixed $i$ we have

(49) $(n-2)\sum_{j;j\neq i}a_{ji}^{2}-2\sum_{jl;j\neq i,l\neq
i,l<j}a_{li}a_{ji}\geq 0.$

Denote the left hand side of the last inequality by $c_{i}$ and set
$c=\sum_{i}c_{i}$. By (46), (47), and (49), we now obtain

(50)
$\frac{n-4}{3}\|A\|^{2}+(\|A\|^{2}-\|E\|^{2})=\frac{n-4}{3}\sum_{i}a_{ii}^{2}-2\sum_{j\neq
i}a_{ii}a_{ji}+c+\frac{n-1}{3}\epsilon.$

For a fixed $i$ and a positive number $\mu$, we have

(51) $\begin{array}[]{rcl}&&0\leq\sum_{j;j\neq i}\left(\mu
a_{ii}-\frac{1}{\mu}a_{ji}\right)^{2}=\sum_{j;j\neq i}\left[(\mu
a_{ii})^{2}-2(\mu
a_{ii})\left(\frac{1}{\mu}a_{ji}\right)+\left(\frac{1}{\mu}a_{ji}\right)^{2}\right]\\\
&&\ \ \ \ \ \ \ \ =(n-1)\mu^{2}a_{ii}^{2}-2\sum_{j,j\neq
i}a_{ii}a_{ji}+\frac{1}{\mu^{2}}\sum_{j;j\neq i}a_{ji}^{2}.\end{array}$

We now add to the left hand side of (50) $\|A\|^{2}$ and we set
$\mu^{2}=\frac{1}{3}$. By (50), (46), and (51), we obtain

(52)
$\begin{array}[]{rcl}&&\frac{n+2}{3}\|A\|^{2}-\|E\|^{2}=\|A\|^{2}+\frac{n-4}{3}\|A\|^{2}+(\|A\|^{2}-\|E\|^{2})\\\
&&\ \ \ \ \ \ =\frac{n-1}{3}\sum_{i}a_{ii}^{2}-2\sum_{i\neq
j}a_{ii}a_{ji}+3\sum_{i\neq j}a_{ji}^{2}+\frac{n+2}{3}\epsilon+c\geq
0.\end{array}$

Observe additionally that in the last line the equality holds if and only if
for every $i$ and $j$ such that $i\neq j$ we have $A_{iii}=3A_{jji}$ and
$A_{ijr}=0$ for $i\neq j,i\neq r,j\neq r$.

2

######  Example 3.4.

Consider a statistical structure $(g,K)$ on $\mathbf{R}^{2}$, where $g$ is the
standard metric and the difference tensor $K$ is defined as follows

(53) $K(e_{1},e_{1})=e_{2},\ \ \ K(e_{1},e_{2})=e_{1},\ \ \
K(e_{2},e_{2})=3e_{2},$

where $e_{1},e_{2}$ is the canonical basis of $\mathbf{R}^{2}$. The vector
$U=e_{1}$ satisfies the equality in (32). We also see that for this structure
the equality holds in (44).

######  Theorem 3.5.

For any statistical structure, we have

(54) $\hat{\rho}-\rho=\|A\|^{2}-\|E\|^{2}\geq-\frac{n-1}{3}\|A\|^{2},$ (55)
$\hat{\rho}-\rho=\|A\|^{2}-\|E\|^{2}\geq-\frac{n-1}{n+2}\|E\|^{2}.$

For a statistical structure on a Lagrangian submanifold of a complex space
form $\tilde{M}(4c)$, we have

(56) $\hat{\rho}\leq\frac{n(n-1)}{n+2}\left(c(n+2)+n\|\mu\|^{2}\right),$ (57)
$\hat{\rho}\leq\frac{n-1}{3}\left(3cn+\|A\|^{2}\right).$

For a statistical structure on an equiaffine sphere with $R=HR_{0}$, we have

(58) $\hat{\rho}\geq\frac{n-1}{n+2}\left(Hn(n+2)-\|E\|^{2}\right),$

(59) $\hat{\rho}\geq\frac{n-1}{3}\left(3Hn-\|A\|^{2}\right).$

##  4\. Using Simons’ formulas

We shall use the following version of Simons’ formula. Let $g$ be a positive-
definite metric tensor field on $M$. For any tensor field $s$ on a manifold
$M$ we have

(60) $\frac{1}{2}\Delta(\|s\|^{2})=g(\Delta s,s)+\|\hat{\nabla}s\|^{2},$

where the meaning of the Laplacians is explained in Preliminaries. We shall
consider tensor fields of type $(0,k)$. Recall the Ricci identity:

(61)
$(\hat{\nabla}^{2}s)(X,Y,...)-(\hat{\nabla}^{2}s)(Y,X,...)=(\hat{R}(X,Y)\cdot
s)(...).$

In particular, if $\hat{\nabla}^{2}s$ is symmetric, then $\hat{R}(X,Y)\cdot
s=0$ for every $X,Y$. We shall employ (60) for studying the statistical cubic
form $A$, but first, we shall make some comments on the applications of (60)
to $1$-forms and bilinear symmetric forms.

###  4.1. For $1$-forms

If $\tau$ is a $1$-form on $M$, then we have

(62)
$(\hat{\nabla}^{2}\tau)(X,Y,Z)-(\hat{\nabla}^{2}\tau)(Y,X,Z)=-\tau(\hat{R}(X,Y)Z)=g(\hat{R}(X,Y)E,Z),$

where $E=\tau^{\sharp}$. From the last formula we immediately get

######  Proposition 4.1.

If $\tau$ is a closed $1$-form and $\hat{R}=0$, then the cubic form
$\hat{\nabla}^{2}\tau$ is symmetric.

######  Proposition 4.2.

Assume that $\hat{\nabla}^{2}\tau$ is a symmetric cubic form. Then $\hbox{\rm
im}\,\hat{R}\subset\hbox{\rm ker}\,\tau$. In particular, $\widehat{\hbox{\rm
Ric}}\,(X,E)=0$ for every $X$, hence $\tau=0$ if $\widehat{\hbox{\rm Ric}}\,$
is non-degenerate.

By (62), we also get

(63) $\hbox{\rm
tr}\,_{g}(\hat{\nabla}^{2}\tau)(\cdot,\cdot,X)=(d\delta\tau+\delta
d\tau)(X)+\widehat{\hbox{\rm Ric}}\,(X,E),$

Simons’ formula (60) now yields

(64) $\frac{1}{2}\Delta(\|\tau\|^{2})=g((d\delta+\delta
d)\tau,\tau)+\widehat{\hbox{\rm Ric}\,}(E,E)+\|\hat{\nabla}\tau\|^{2}.$

By Proposition 4.2, (63), and (64), we obtain

######  Theorem 4.3.

Let $\tau$ be a $1$-form such that $\hat{\nabla}^{2}\tau=0$. Then
$\widehat{\hbox{\rm Ric}}\,(X,E)=0$ for every $X$ and $\tau$ is harmonic.
If, additionally, $M$ is compact, or $\|\tau\|$ is constant, then
$\hat{\nabla}\tau=0$. If $\widehat{\hbox{\rm Ric}}\,$ is non-degenerate at a
point of $M$, then $\tau=0$.

Of course, (64) immediately implies the following classical Bochner’s theorem

######  Theorem 4.4.

Let $(M,g)$ be a compact Riemannian manifold. Assume that $\widehat{\hbox{\rm
Ric}\,}$ is semi-positive definite on $M$. Then each harmonic $1$-form $\tau$
on $M$ is $\hat{\nabla}$-parallel and $\widehat{\hbox{\rm Ric}}\,(E,E)=0$ on
$M$. If, moreover, $\widehat{\hbox{\rm Ric}}\,$ is positive definite at some
point of $M$, then $\tau=0$ on $M$.

We can apply the above theorems to the Czebyshev form of a statistical
structure. For instance, we have

######  Corollary 4.5.

Let $(g,\nabla)$ be a statistical structure on a manifold $M$ and $\tau$ be
its Czebyshev form. Assume that $\hat{\nabla}^{2}\tau=0$, $\|\tau\|$ is
constant or $M$ is compact and, moreover, the Ricci tensor of the metric $g$
is non-degenerate at a point of $M$. Then the structure is trace-free.

Recall that if $\dim M=2$, then the Ricci tensor (at a point of $M$) of a
Levi-Civita connection is either non-degenerate or vanishes. In the last case,
the curvature tensor also vanishes at the point. It provides additional
information in the above theorems in the case where $\dim M=2$. We have, for
example,

######  Corollary 4.6.

Let $\tau$ be a $1$-form on a $2$-dimensional manifold $M$ and
$\hat{\nabla}^{2}\tau=0$. At each point of $M$ we have $\tau=0$ or
$\hat{R}=0$. In the case where $M$ is compact, we have $\tau=0$ or $\hat{R}=0$
on $M$.

###  4.2. For tensor fields of type $(0,2)$

If $\beta$ is a symmetric bilinear form on a Riemannian manifold $(M,g)$, then
we have

$(\hat{R}(X,Y)\cdot\beta)(U,V)=-\beta(\hat{R}(X,Y)U,V)-\beta(U,\hat{R}(X,Y)V).$

It follows that if $e_{1},...,e_{n}$ is an orthonormal basis diagonalizing
$\beta$ and $\lambda_{1},...,\lambda_{n}$ are the corresponding eigenvalues,
then

(65)
$(\hat{R}(X,Y)\cdot\beta)(e_{i},e_{j})=g(\hat{R}(X,Y)e_{i},e_{j})(\lambda_{i}-\lambda_{j})=g(\hat{R}(e_{i},e_{j})Y,X)(\lambda_{i}-\lambda_{j}).$

In particular, we have

######  Proposition 4.7.

If $\hat{\nabla}^{2}\beta$ is symmetric and the sectional curvature of $g$ is
nonzero for every plane, then $\beta=\lambda g$.

Simons’ formula for symmetric tensor fields of type $(0,2)$ can be formulated
as follows

######  Theorem 4.8.

Let $\beta$ be a symmetric tensor field of type $(0,2)$ on a Riemannian
manifold $M$. If $\hat{\nabla}\beta$ is symmetric, then we have

(66)
$\begin{array}[]{rcl}&&\frac{1}{2}\Delta\|\beta\|^{2}=\|\hat{\nabla}\beta\|^{2}+\sum_{ijk}\hat{\nabla}^{2}\beta(e_{j},e_{k},e_{i},e_{i})\beta_{jk}\\\
&&\ \ \ \ \ \ \ \ \ \ \ \ +\sum_{i<k}\hat{k}(e_{i}\wedge
e_{k})(\lambda_{i}-\lambda_{k})^{2},\end{array}$

where $\beta_{jk}=\beta(e_{j},e_{k})$, $\beta_{jk}=\delta_{jk}\lambda_{k}$ for
some orthonormal basis $e_{1},...,e_{n}$ of $T_{x}M$, $x\in M$ and
$\hat{k}(e_{i}\wedge e_{k})$ is the sectional curvature of $g$ by the plane
spanned by $e_{i},e_{k}$.

Proof. First we have

$\displaystyle\sum_{i}\hat{\nabla}^{2}\beta(e_{i},e_{i},X,Y)=\sum_{i}[\hat{\nabla}^{2}\beta(X,Y,e_{i},e_{i})+\beta(\hat{R}(X,e_{i}))Y,e_{i})+\beta(Y,\hat{R}(X,e_{i})e_{i})]$

for any orhonormal basis $e_{1},...,e_{n}$ and any $X,Y$. Hence

$\displaystyle
g(\Delta\beta,\beta)=\sum_{kli}\hat{\nabla}^{2}\beta(e_{k},e_{l},e_{i},e_{i})\beta_{kl}$
$\displaystyle\ \ \ \ \ \ \ \ \ \
+\sum_{kils}\hat{R}_{kils}\beta_{is}\beta_{kl}+\sum_{ksl}\widehat{\hbox{\rm
Ric}\,}_{ks}\beta_{kl}\beta_{ls}.$

If $e_{1},...,e_{n}$ is a basis diagonalizing $\beta$ with eigenvalues
$\lambda_{1},...,\lambda_{n}$, then we get

$\displaystyle\sum_{kils}\hat{R}_{kils}\beta_{is}\beta_{kl}+\sum_{ksl}\widehat{\hbox{\rm
Ric}\,}_{ks}\beta_{kl}\beta_{ls}=-\sum_{ik}\hat{R}_{ikki}\lambda_{i}\lambda_{k}+\sum_{k}\widehat{\hbox{\rm
Ric}\,}_{kk}\lambda_{k}^{2}$ $\displaystyle=\sum_{i<k}[-2\hat{k}(e_{i}\wedge
e_{k})\lambda_{i}\lambda_{k}+\hat{k}(e_{i}\wedge
e_{k})\lambda_{i}^{2}+\hat{k}(e_{i}\wedge e_{k})\lambda_{k}^{2}]$
$\displaystyle\ \ \ \ \ \ \ \ \ =\sum_{i<k}\hat{k}(e_{i}\wedge
e_{k})(\lambda_{i}-\lambda_{k})^{2}.$

It is now sufficient to use (60).

2

From the above formula, we immediately get

######  Theorem 4.9.

Let $(M,g)$ be a compact Riemannian manifold and $\beta$ be a symmetric
$(0,2)$-tensor field such that $\hat{\nabla}\beta$ is symmetric and the
function $\lambda=\hbox{\rm tr}\,_{g}\beta$ is constant on $M$. If the
sectional curvature of $g$ is nonnegative on $M$ and positive at some point of
$M$, then $\beta=\frac{\lambda}{n}g$.

Proof. If the sectional curvature of $g$ is everywhere nonnegative, then by
(66) we know that $\hat{\nabla}\beta=0$. The formula (66) implies also that
for every index $i\neq k$ $\hat{k}(e_{i}\wedge
e_{k})(\lambda_{i}-\lambda_{k})^{2}=0$. If at some point of $M$ all sectional
curvatures $\hat{k}$ are positive, we see that $\beta=\frac{\lambda}{n}g$ at
this point and hence everywhere on $M$. 2

In the above theorem, instead of assuming that $M$ is compact, one can assume
that $\|\beta\|$ is constant on $M$.

######  Theorem 4.10.

let $(M,g)$ be a connected Riemannian manifold whose sectional curvature is
positive (or negative) at some point of $M$. If $\beta$ is a symmetric 2-form
and $\hat{\nabla}\beta=0$, then $\beta=cg$ for some constant $c$.

######  Corollary 4.11.

Let $(M,g)$ be a compact Riemannian manifold whose sectional curvature is
nonnegative on $M$ and positive at some point of $M$. If
$\hat{\nabla}\widehat{\hbox{\rm Ric}}\,$ is symmetric and the scalar curvature
$\hat{\rho}$ is constant, then the Riemannian structure is Einstein.

######  Corollary 4.12.

Let $(M,g)$ be a Riemannian manifold whose sectional curvature is positive at
some point of $M$ (or negative at some point of $M$). If
$\hat{\nabla}\widehat{\hbox{\rm Ric}}\,=0$, then the Riemannian structure is
Einstein.

In the geometry of statistical structures, we have a few symmetric bilinear
forms, for instance, $\tau\circ K$, $g(K_{\cdot},K_{\cdot})$. The Czebyshev
form is closed if and only if the bilinear forms $\nabla\tau$, Ric ,
$\overline{\hbox{\rm Ric}}\,$ are also symmetric. In particular, we can
formulate the following

######  Corollary 4.13.

Let $(M,g,\nabla)$ be a connected statistical manifold. If Ric is symmetric,
$\hat{\nabla}\hbox{\rm Ric}\,=0$ and the sectional curvature of $g$ is
positive (or negative) at some point of $M$, then $\hbox{\rm Ric}\,=\lambda g$
for some constant $\lambda$.

######  Corollary 4.14.

Let $(M,g,\nabla)$ be a Hessian manifold and $\beta=\nabla\tau$ its Koszul
form. If $\hat{\nabla}\beta=0$ and the sectional curvature for $g$ is positive
(or negative) at some point of $M$, then the structure is Einstein-Hessian.

###  4.3. For cubic forms

We shall first compute a Simons’ type formula for symmetric cubic forms on an
$n$-dimensional Riemannian manifold $(M,g)$. Let $A$ be a symmetric cubic form
and $K=A^{\sharp}$. Assume that $\hat{\nabla}A$ is symmetric. Thus we have a
conjugate symmetric statistical structure. Let $e_{1},...,e_{n}$ be an
orthonormal basis of a tangent space $T_{x}M$. We have

$\displaystyle\begin{array}[]{rcl}&&\hat{\nabla}^{2}A(e_{i},e_{i},X,Y,Z)=\hat{\nabla}^{2}A(X,Y,Z,e_{i},e_{i})-A(\hat{R}(e_{i},X)Y,Z,e_{i})\\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-A(Y,\hat{R}(e_{i},X)Z,e_{i})-A(Y,Z,\hat{R}(e_{i},X)e_{i})\end{array}$

for every $i$. Hence

$\displaystyle\begin{array}[]{rcl}&&\hat{\nabla}A(e_{i},e_{i},e_{j},e_{k},e_{l})=\hat{\nabla}A(e_{j},e_{k},e_{l},e_{i},e_{i})-g(K(e_{l},e_{i}),\hat{R}(e_{i},e_{j})e_{k})\\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-g(K(e_{k},e_{i}),\hat{R}(e_{i},e_{j})e_{l})-g(K(e_{k},e_{l}),\hat{R}(e_{i},e_{j})e_{i})\\\
&&\ \ \ \ \ \ \
=\hat{\nabla}A(e_{j},e_{k},e_{l},e_{i},e_{i})-\sum_{r}A_{lir}\hat{R}_{ijkr}-\sum_{r}A_{kir}\hat{R}_{ijlr}-\sum_{r}A_{klr}\hat{R}_{ijir}\end{array}$

for every $i,j,k,l$. We also have

(69)
$\hat{\nabla}^{2}\tau(X,Y,Z)=\sum_{i}\hat{\nabla}^{2}A(X,Y,Z,e_{i},e_{i}).$

We now compute

$\displaystyle\begin{array}[]{rcl}&&\sum_{ijkl}\hat{\nabla}^{2}A(e_{i},e_{i},e_{j},e_{k},e_{l})A_{jkl}=\sum_{ijkl}\hat{\nabla}^{2}A(e_{j},e_{k},e_{l},e_{i},e_{i})A_{jkl}\\\
&&\ \ \ \ \ \
-\sum_{ijklr}[A_{lir}A_{jkl}\hat{R}_{ijkr}+A_{kir}A_{jkl}\hat{R}_{ijlr}+A_{klr}A_{jkl}\hat{R}_{ijir}]\\\
&&\ \ \ \ \ \ =g(\hat{\nabla}^{2}\tau,A)+\sum_{kljr}\widehat{\hbox{\rm
Ric}\,}_{jr}A_{jkl}A_{rkl}\\\ &&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \
-\sum_{ijklr}A_{irl}A_{jkl}\hat{R}_{ijkr}-\sum_{ijklr}A_{irk}A_{jlk}\hat{R}_{ijlr}\\\
&&\ \ \ \ \ \ =g(\hat{\nabla}^{2}\tau,A)+\sum_{kljr}\widehat{\hbox{\rm
Ric}\,}_{jr}A_{jkl}A_{rkl}+\sum_{ijklr}(A_{ikr}A_{jlr}-A_{ilr}A_{jkr})\hat{R}_{ijkl}\\\
&&\ \ \ \ \ \ =g(\hat{\nabla}^{2}\tau,A)+g(\widehat{\hbox{\rm
Ric}\,},g(K_{\cdot},K_{\cdot}))-\sum_{ijkl}g([K_{e_{i}},K_{e_{j}}]e_{k},e_{l})\hat{R}_{ijkl}.\end{array}$

Since for a conjugate symmetric statistical structure $R=\hat{R}+[K,K]$ and
$\hbox{\rm Ric}\,=\widehat{\hbox{\rm Ric}}\,+\tau\circ
K-g(K_{\cdot},K_{\cdot})$ (see Preliminaries), by using (60), we obtain

######  Theorem 4.15.

For any conjugate symmetric statistical structure, we have

(71)
$\frac{1}{2}\Delta(\|A\|^{2})=\|\hat{\nabla}A\|^{2}+g(\hat{\nabla}^{2}\tau,A)-g([K,K],\hat{R})+g(\widehat{\hbox{\rm
Ric}\,},g(K_{\cdot},K_{\cdot})),$ (72)
$\frac{1}{2}\Delta(\|A\|^{2})=\|\hat{\nabla}A\|^{2}+g(\hat{\nabla}^{2}\tau,A)+g(\hat{R}-R,\hat{R})+g(\widehat{\hbox{\rm
Ric}\,},g(K_{\cdot},K_{\cdot})),$

(73)
$\begin{array}[]{rcl}&&\frac{1}{2}\Delta(\|A\|^{2})=\|\hat{\nabla}A\|^{2}+g(\hat{\nabla}^{2}\tau,A)\\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\hat{R}^{2}+\widehat{\hbox{\rm
Ric}\,}^{2}-g(R,\hat{R})-g(\hbox{\rm Ric}\,,\widehat{\hbox{\rm Ric}\,})\\\ &&\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +g(\widehat{\hbox{\rm
Ric}\,},\tau\circ K).\end{array}$

Consider now the case where the sectional $K$-curvature is constant. In [14]
the following result was proved

######  Theorem 4.16.

Let $(g,K)$ be a conjugate symmetric trace-free statistical structure on a
manifold $M$. If the sectional $K$-curvature is constant (automatically
nonpositive, because of the trace-freeness), then either $K=0$ or: $\hat{R}=0$
and $\hat{\nabla}K=0$.

In a more general situation, we get

######  Theorem 4.17.

Let $(g,K)$ be a conjugate symmetric statistical structure with
$\hat{\nabla}$-parallel Czebyshev form $\tau$ and $\widehat{\hbox{\rm
Ric}\,}\geq 0$. If the sectional $K$-curvature is constant and nonpositive,
then $\hat{\nabla}K=0$. If the sectional $K$-curvature is a negative constant,
then $\widehat{\hbox{\rm Ric}\,}=0$.

Proof. If $[K,K]=\kappa R_{0}$, then (71) becomes

(74)
$\frac{1}{2}\Delta(\|A\|^{2})=\|\hat{\nabla}A\|^{2}+g(\hat{\nabla}^{2}\tau,A)-2\kappa\hat{\rho}+g(\widehat{\hbox{\rm
Ric}\,},g(K_{\cdot},K_{\cdot})).$

Since $\rho^{K}=\|E\|^{2}-\|A\|^{2}$ (see (20)) and $\hat{\nabla}E=0$, we have
that $\|A\|$ is constant. Since $\widehat{\hbox{\rm Ric}\,}\geq 0$, we have
$g(\widehat{\hbox{\rm Ric}\,},g(K_{\cdot},K_{\cdot}))\geq 0$. The first
assertion now follows from (74). If $\kappa<0$, then $\hat{\rho}=0$. Since
$\widehat{\hbox{\rm Ric}\,}\geq 0$, we have $\widehat{\hbox{\rm Ric}\,}=0$. 2

For trace-free statistical structures (73) becomes

(75)
$\frac{1}{2}\Delta(\|A\|^{2})=\|\hat{\nabla}A\|^{2}+\hat{R}^{2}+\widehat{\hbox{\rm
Ric}\,}^{2}-g(R,\hat{R})-g(\hbox{\rm Ric}\,,\widehat{\hbox{\rm Ric}\,}).$

The last formula was proved by An-Min Li for hyperbolic affine Blaschke
spheres, [8]. In that case, $\tau=0$, $R=HR_{0}$, where $H$ is a constant and
$\hbox{\rm Ric}\,=(n-1)Hg$. Hence (75) can be written as follows

(76)
$\frac{1}{2}\Delta(\|A\|^{2})=\|\hat{\nabla}A\|^{2}+\hat{R}^{2}+\widehat{\hbox{\rm
Ric}}\,^{2}-(n+1)H\hat{\rho}.$

Moreover, $\|A\|^{2}=\hat{\rho}-n(n-1)H$, see (21). Hence $\|A\|$ is constant
if $\hat{\rho}$ is constant. If $\hat{\rho}=0$, $H$ must be nonpositive. Thus
we have

######  Theorem 4.18.

[10] For a Blaschke affine sphere if $\hat{\rho}=0$, then $\hat{R}=0$ and
$\hat{\nabla}A=0$. The sphere is either a part of a trivial paraboloid (an
elliptic paraboloid with its trivial affine structure) or the sphere is
hyperbolic.

The hyperbolic sphere from the last theorem must be given by the equation
$x_{1}\cdot...\cdot x_{n}=c$, where $c$ is a positive constant, see [10].

For equiaffine spheres, we can prove

######  Theorem 4.19.

For an equiaffine hyperbolic or parabolic sphere, if $\hat{\nabla}\tau=0$,
$\widehat{\hbox{\rm Ric}\,}\geq 0$ and $\hat{\rho}$ is constant, then
$\hat{R}=0$ and $\hat{\nabla}A=0$.

Proof. Since $R=HR_{0}$ and $\hat{\nabla}\tau=0$, the formula (72) can be
written as follows

(77)
$\frac{1}{2}\Delta(\|A\|^{2})=\|\hat{\nabla}A\|^{2}+\hat{R}^{2}-2H\hat{\rho}+g(\widehat{\hbox{\rm
Ric}\,},g(K_{\cdot},K_{\cdot})).$

Moreover, the function $\|E\|$ is constant. Since the functions $\hat{\rho}$
and $\rho$ are constant, (21) implies that $\|A\|$ is also constant. The
assertion now follows from (77). 2

From (71), we immediately get

######  Proposition 4.20.

Let $(M,g,\nabla)$ be a conjugate symmetric statistical manifold such that
$\hat{\nabla}^{2}\tau=0$, $\hat{R}=0$, and $\|A\|$ is constant (or $M$ is
compact). Then $\hat{\nabla}A=0$.

From (77) one gets

######  Proposition 4.21.

For a Hessian structure $(g,A)$, if $\hat{\nabla}A=0$ and $\widehat{\hbox{\rm
Ric}}\,\geq 0$, then $\hat{R}=0$.

For a Lagrangian submanifold in a complex space form $\tilde{M}(4c)$, we have
$-[K,K]=cR_{0}-\hat{R}$, see (28). Using now (71) we obtain

(78)
$\Delta\|A\|^{2}=\|\hat{\nabla}A\|^{2}+g(\hat{\nabla}^{2}\tau,A)-\|\hat{R}\|^{2}+2c\hat{\rho}+g(\widehat{\hbox{\rm
Ric}}\,,g(K_{\cdot},K_{\cdot})).$

######  Theorem 4.22.

Let $(g,K)$ be the statistical structure on a Lagrangian submanifold in a
complex space form $\tilde{M}(4c)$ whose second fundamental tensor $K$ is
$\hat{\nabla}$-parallel. If $\widehat{\hbox{\rm Ric}}\,=0$, then $\hat{R}=0$.
If $\widehat{\hbox{\rm Ric}}\,\leq 0$ and $c\geq 0$ then $\hat{R}=0$.

If $g$ has constant curvature $\hat{k}$, then using (30) and (78) we get

(79)
$\Delta\|A\|^{2}=\|\hat{\nabla}A\|^{2}+g(\hat{\nabla}^{2}\tau,A)+\hat{k}[2(\|A\|^{2}-\|E\|^{2})+(n-1)\|A\|^{2}].$

Hence, if $\hat{k}\geq 0$, then (55) yields

(80)
$\Delta\|A\|^{2}\geq\|\hat{\nabla}A\|^{2}+g(\hat{\nabla}^{2}\tau,A)+\frac{n-1}{3}\hat{k}\|A\|^{2}.$

This formula immediately yields a theorem proved by Chen and Ogiue in [2] and
saying that a minimal Lagrangian submanifold of constant positive curvature in
a complex space form must be totally geodesic. Indeed, it is sufficient to
note that if $\tau=0$, then, by (30), $\|A\|$ is constant. Using (30) and
Theorem 4.3, we also get

######  Theorem 4.23.

Let $M$ be a flat Lagrangian submanifold in a complex space form and
$\hat{\nabla}^{2}\tau=0$. If $M$ is compact or the mean curvature is constant,
then the submanifold has the parallel second fundamental tensor.

##  5\. Using a maximum principle

In this section, we set $u=\|A\|^{2}$ and we assume that $(g,\nabla)$ is a
trace-free statistical structure on an $n$-dimensional connected manifold $M$
such that $R=HR_{0}$. It is automatically conjugate symmetric and therefore,
if $n>2$, then $H$ is constant. If $n=2$, $H$ can be a function. In this case,
the assumption $R=HR_{0}$ is equivalent to the assumption that $(g,\nabla)$ is
conjugate symmetric. If $H$ is constant, such a statistical structure can be
automatically realized (locally) on an affine Blaschke sphere.

We shall now employ two inequalities due to Calabi and An-Min Li. For detailed
proofs, we refer to [12] and [9]. Here we give a sketch of the proof. First
define tensor fields $L$, $P$ and $Q$ as follows

(81) $L(X,Y,W,Z)=g(K(X,Y),K(W,Z)),$

(82) $P(X,Y,W,Z)=L(X,Y,W,Z)-L(W,Y,X,Z),$ (83) $Q(Y,W,Z)=\hbox{\rm
tr}\,_{g}([K_{\cdot},K_{Y}]\cdot A)(\cdot,W,Z).$

In the case under consideration, we have $\hat{R}=HR_{0}-[K,K]$ and

$\Delta A(Y,W,Z)=\hbox{\rm tr}\,_{g}(\hat{R}(\cdot,Y)\cdot
A)(\cdot,W,Z)=H(n+1)A-Q(Y,W,Z).$

We have

(84) $\|L\|^{2}=\sum_{ij}a_{ij}^{2},\ \ \ \ \ \ \
\|P\|^{2}=\sum_{ijkl}b_{ij:kl}^{2},$

where

(85) $\begin{array}[]{rcl}&&a_{ij}=\sum_{kl}A_{ikl}A_{jkl},\\\
&&b_{ij;kl}=\sum_{m}(A_{ijm}A_{klm}-A_{kjm}A_{ilm}),\end{array}$

The notation in (85) is similar to that in [12] on p. 84, where it was
introduced for the cubic form $C=-2A$. It was proved there that
$-g(Q,A)=\|L\|^{2}+\|P\|^{2}$, that is,

(86) $g(\Delta A,A)=(n+1)Hu+\|L\|^{2}+\|P\|^{2}.$

Moreover, see [12],

(87) $\|L\|^{2}+\|P\|^{2}\geq\frac{n+1}{n(n-1)}u^{2},$

where $u=\|A\|^{2}$. In [12] it was assumed that $H$ is constant, but this
assumption is not necessary (although automatically satisfied if $n>2$). The
inequality actually is due to Blaschke (for $n=2$) and Calabi (for $n>2$). In
[9] A-M. Li proved the following algebraic inequality (Theorem 1 in [9])

(88) $\|L\|^{2}+\|P\|^{2}\leq\frac{3}{2}u^{2}.$

In particular, in the case where $n=2$, by comparing (87) and (88), we get

(89) $\|L\|^{2}+\|P\|^{2}=\frac{3}{2}u^{2}.$

The last equality was also directly proved in [12]. Using the above facts and
Simons’ formula (60), one can formulate

######  Theorem 5.1.

Let $(g,\nabla)$ be a trace-free statistical structure on an n-dimensional
manifold $M$ and $R=HR_{0}$. We have

(90)
$(n+1)Hu+\frac{n+1}{n(n-1)}u^{2}+\|\hat{\nabla}A\|^{2}\leq\frac{1}{2}\Delta
u\leq(n+1)Hu+\frac{3}{2}u^{2}+\|\hat{\nabla}A\|^{2}.$

If $n=2$, the equalities hold.

As an immediate consequence of this theorem, we have

######  Corollary 5.2.

Let $(g,\nabla)$ be a trace-free statistical structure on an n-dimensional
manifold $M$ and $R=HR_{0}$. Assume that $\hat{\nabla}A=0$ on $M$.
1) If $H\geq 0$, then $A=0$ on $M$.
2) If $H<0$, then either $A=0$ on $M$ or

(91) $\frac{2}{3}(n+1)(-H)\leq u\leq n(n-1)(-H).$

If $n=2$ and $\hbox{\rm sup}\,H=0$, then $A=0$.

For treating the case where $g$ is complete, we shall use the following
maximum principle

######  Lemma 5.3.

[18]. Let $(M,g)$ be a complete Riemannian manifold, whose Ricci tensor is
bounded from below and let $f$ be a $\mathcal{C}^{2}$-function bounded from
above on $M$. For every $\varepsilon>0$, there exists a point $x\in M$ at
which
i) $f(x)>\hbox{\rm sup}f-\varepsilon$
ii)$\Delta f(x)<\varepsilon$.

The results contained in the following theorem and their generalizations can
be found in [1], [10], [15].

######  Theorem 5.4.

Let $(g,\nabla)$ be a trace-free statistical structure with $R=HR_{0}$ and
complete $g$ on a manifold $M$. If $H\geq 0$, then the structure is trivial.
If $H$ is constant and negative, then the Ricci tensor of $g$ is nonpositive
and consequently

(92) $u\leq n(n-1)(-H).$

We shall now give a more delicate estimation for $u$ by looking more carefully
at the function $\|\hat{\nabla}A\|$. Recall here that for a trace-free
conjugate symmetric statistical structure we have $\widehat{\hbox{\rm
Ric}}\,>\hbox{\rm Ric}\,$, see (19). Hence, in all theorems below, the Ricci
tensor of $g$ is automatically bounded from below.

######  Theorem 5.5.

Let $(g,\nabla)$ be a trace-free statistical structure on an $n$-dimensional
manifold $M$ and $R=HR_{0}$, where $H$ is a negative constant. If $g$ is
complete and

(93) $\hbox{\rm sup}\,\|\hat{\nabla}A\|^{2}<\frac{H^{2}(n+1)^{2}}{6},$

then

(94) $\inf u\geq\frac{(n+1)(-H)+\sqrt{(n+1)^{2}H^{2}-6N_{2}}}{3},$

or

(95) $\inf u\leq\frac{(n+1)(-H)-\sqrt{(n+1)^{2}H^{2}-6N_{2}}}{3},$

where $u=\|A\|^{2}$ and $N_{2}=\hbox{\rm sup}\|\hat{\nabla}A\|^{2}$.

Proof. By Theorem 5.1, we have

(96) $\frac{1}{2}\Delta u\leq(n+1)Hu+\frac{3}{2}u^{2}+N_{2}.$

Set $N_{1}=\inf u$ and $\tilde{u}=-u$. The function $\tilde{u}$ is bounded
from above by $0$ and $-N_{1}=\hbox{\rm sup}\,\tilde{u}$. Take sequences
$\varepsilon_{i}$ and $x_{i}$ satisfying $i)$ from Lemma 5.3 applied to the
function $\tilde{u}$. Assume that the sequence $\varepsilon_{i}$ is decreasing
and tending to $0$. We have
$\tilde{u}(x_{i})>-N_{1}-\varepsilon_{i}>-N_{1}-\varepsilon_{1}$ and
consequently $u(x_{i})<N_{1}+\varepsilon_{1}$ and
$-u^{2}(x_{i})>-(N_{1}+\varepsilon_{1})^{2}$. By (96), we now have

(97)
$\frac{1}{2}(\Delta\tilde{u})(x_{i})\geq-(n+1)HN_{1}-\frac{3}{2}(N_{1}+\varepsilon_{1})^{2}-N_{2}.$

Consider the following polynomial of degree 2

(98) $\begin{array}[]{rcl}&&p(t)=-\frac{3}{2}(t+\varepsilon_{1})^{2}-(n+1)Ht-
N_{2}\\\
&&=-\frac{3}{2}t^{2}+[-3\varepsilon_{1}-(n+1)H]t-[\frac{3}{2}\varepsilon_{1}^{2}+N_{2}].\end{array}$

Denote by $\delta$ its discriminant. We have

(99) $\delta=N^{2}+6\varepsilon_{1}(n+1)H<N^{2},$

where $N=\sqrt{(n+1)^{2}H^{2}-6N_{2}}$. Assume that $\varepsilon_{1}$ is so
small that $\delta>0$. Then $\delta=N^{2}q^{2}$ for some $0<q<1$. The relation
between $q$ and $\varepsilon_{1}$ is the following

(100) $\varepsilon_{1}=\frac{N^{2}(1-q^{2})}{6(n+1)(-H)}.$

Instead of choosing $\varepsilon_{1}$, we can choose $q$, and
$\varepsilon_{1}$ tends to $0$ if and only if $q$ tends to $1$. Let $t_{1}$,
$t_{2}$ be the roots of the polynomial $p(t)$ depending also on $q$. We have

(101)
$\begin{array}[]{rcl}&&t_{1}(q)=\frac{(n+1)(-H)-q\sqrt{(n+1)^{2}H^{2}-6N_{2}}}{3}-\frac{N^{2}(1-q^{2})}{6(n+1)(-H)}\\\
&&t_{2}(q)=\frac{(n+1)(-H)+q\sqrt{(n+1)^{2}H^{2}-6N_{2}}}{3}-\frac{N^{2}(1-q^{2})}{6(n+1)(-H)}\end{array}$

One sees that

$t_{1}(q)\to\frac{(n+1)(-H)-\sqrt{(n+1)^{2}H^{2}-6N_{2}}}{3},$
$t_{2}(q)\to\frac{(n+1)(-H)+\sqrt{(n+1)^{2}H^{2}-6N_{2}}}{3}$

if $q$ tends to $1$. We shall now use the maximum principle. Suppose that

(102)
$\begin{array}[]{rcl}&&\frac{(n+1)(-H)-q\sqrt{(n+1)^{2}H^{2}-6N_{2}}}{3}<N_{1}\\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
<\frac{(n+1)(-H)+q\sqrt{(n+1)^{2}H^{2}-6N_{2}}}{3}.\end{array}$

There is $q$ (sufficiently close to $1$) such that $t_{1}(q)<N_{1}<t_{2}(q)$.
Hence $p(N_{1})$, depending only on $n$, $H$, $N_{1}$, $N_{2}$, and
$\varepsilon_{1}$ (equivalently $q$), is positive and, by (97), we cannot have
$\frac{1}{2}\Delta\tilde{u}(x_{i})<\varepsilon_{i}$ for $\varepsilon_{i}$
tending to $0$. We have got a contradiction. 2

######  Corollary 5.6.

Let $(g,\nabla)$ be a trace-free statistical structure on an $n$-dimensional
manifold $M$ and $R=HR_{0}$, where $H$ is a negative number. If $g$ is
complete,

(103) $N_{2}<\frac{H^{2}(n+1)^{2}}{6}$

and

(104) $\inf u>\frac{(n+1)(-H)-\sqrt{(n+1)^{2}H^{2}-6N_{2}}}{3},$

then

(105) $u\geq\frac{(n+1)(-H)+\sqrt{(n+1)^{2}H^{2}-6N_{2}}}{3}$

on $M$.

######  Theorem 5.7.

Let $(g,\nabla)$ be a trace-free statistical structure on an $n$-dimensional
manifold $M$ and $R=HR_{0}$, where $H$ is a negative number. If $g$ is
complete, then

(106) $\inf\,\|\hat{\nabla}A\|^{2}\leq\frac{n(n^{2}-1)H^{2}}{4}$

and the following inequalities hold

(107) $\hbox{\rm
sup}\,u\leq\frac{n(n-1)(-H)+\sqrt{n^{2}(n-1)^{2}H^{2}-4N_{4}}}{2},$ (108)
$\hbox{\rm sup}\,u\geq\frac{n(n-1)(-H)-\sqrt{n^{2}(n-1)^{2}H^{2}-4N_{4}}}{2},$

where $u=\|A\|^{2}$ and $N_{4}=\frac{n(n-1)}{n+1}\inf\,\|\hat{\nabla}A\|^{2}$.

Proof. By Theorem 5.1, we have

(109) $\frac{1}{2}\Delta
u\geq(n+1)Hu+\frac{n+1}{n(n-1)}u^{2}+\|\hat{\nabla}A\|^{2}.$

Let $N_{3}=\hbox{\rm sup}\,u$. By Theorem 5.4, $N_{3}$ is finite. Take
sequences $\varepsilon_{i}$ and $x_{i}$ satisfying $i)$ from Lemma 5.3 applied
to the function $u$. Assume that $\varepsilon_{i}$ is decreasing and tending
to $0$, and $N_{3}-\varepsilon_{1}>0$. By (109), we have

(110) $\frac{n(n-1)}{2(n+1)}\Delta u(x_{i})\geq
n(n-1)HN_{3}+(N_{3}-\varepsilon_{1})^{2}+N_{4}.$

Consider the following polynomial of degree 2

(111)
$\begin{array}[]{rcl}&&p(t)=-n(n-1)(-H)t+(t-\varepsilon_{1})^{2}+N_{4}\\\ &&\
\ \ \ \ \ \
=t^{2}-[2\varepsilon_{1}+n(n-1)(-H)]t+(\varepsilon_{1}^{2}+N_{4}).\end{array}$

Denote by $\delta$ its discriminant. We have

(112) $\delta=n^{2}(n-1)^{2}H^{2}-4N_{4}+4n(n-1)(-H)\varepsilon_{1}.$

If $n^{2}(n-1)^{2}H^{2}-4N_{4}<0$, then, by taking $\varepsilon_{1}$
sufficiently small, we get $\delta<0$ and hence $\Delta u(x_{i})$ will be
positive and bounded from zero by a positive number. This will give a
contradiction to the maximum principle. We have proved (106), that is,
$n^{2}(n-1)^{2}H^{2}-4N_{4}\geq 0$.

Set $N^{2}=n^{2}(n-1)^{2}H^{2}-4N_{4}$. Assume first that $N=0$. Then $\delta$
is positive and the roots of the polynomial (111) are

(113)
$\frac{n(n-1)(-H)}{2}+\varepsilon_{1}\pm\sqrt{n(n-1)(-H)\varepsilon_{1}}.$

If $\varepsilon_{1}$ tends to $0$, both roots tend to $\frac{n(n-1)(-H)}{2}$.
Suppose that $N_{3}>\frac{n(n-1)(-H)}{2}$. There is $\varepsilon_{1}$ such
that
$N_{3}>\frac{n(n-1)(-H)}{2}+\varepsilon_{1}\pm\sqrt{n(n-1)(-H)\varepsilon_{1}}$
and then all $\Delta u(x_{i})$ are positive and bounded from zero by a
positive number. This gives a contradiction with the maximum principle.
Similar arguments work in the case where $N_{3}<\frac{n(n-1)(-H)}{2}$.

Assume now that $N^{2}>0$. There is $Q>1$ such that $\delta=N^{2}Q^{2}$. The
relation between $\varepsilon_{1}$ and $Q$ is the following

(114) $\varepsilon_{1}=\frac{N^{2}(Q^{2}-1)}{4n(n-1)(-H)}.$

The roots of the polynomial (111) are

(115)
$\begin{array}[]{rcl}&&t_{1}(Q)=\frac{n(n-1)(-H)(Q^{2}+1)-2Q\sqrt{n^{2}(n-1)^{2}H^{2}-4N_{4}}}{4}-\frac{N_{4}(Q^{2}-1)}{n(n-1)(-H)},\\\
&&t_{2}(Q)=\frac{n(n-1)(-H)(Q^{2}+1)+2Q\sqrt{n^{2}(n-1)^{2}H^{2}-4N_{4}}}{4}-\frac{N_{4}(Q^{2}-1)}{n(n-1)(-H)}.\end{array}$

If $Q\to 1$, then

$t_{1}(Q)\to L_{1}=\frac{n(n-1)(-H)-\sqrt{n^{2}(n-1)^{2}H^{2}-4N_{4}}}{2}\geq
0$

and

$t_{2}(Q)\to P_{1}=\frac{n(n-1)(-H)+\sqrt{n^{2}(n-1)^{2}H^{2}-4N_{4}}}{2}\leq
n(n-1)(-H).$

Suppose now that $N_{3}<L_{1}$ or $N_{3}>P_{1}$. If $N_{3}<L_{1}$, then there
is $Q$ (equivalently the corresponding $\varepsilon_{1})$ such that
$N_{3}<t_{1}(Q)$, which means that $p(N_{3})$ is positive and depends only on
$n$, $H$, $N_{4}$ and $Q$. Hence (110) gives a contradiction with the maximum
principle. We argue similarly if $N_{3}>P_{1}$. 2

The case where $n=2$ should be treated separately because in this case $H$ may
be a function. Note that $\hbox{\rm Ric}\,=Hg$, hence $\widehat{\hbox{\rm
Ric}}\,$ is bounded from below, if $H$ is bounded from below. The same
arguments as those used in the the above proofs yield

######  Theorem 5.8.

Let $(g,\nabla)$ be a conjugate symmetric trace-free statistical structure
with complete $g$ on a 2-dimensional manifold $M$. Then $R=HR_{0}$ for some
function $H$ and if $-\infty<H_{2}\leq H\leq 0$ for some number $H_{2}$, then

(116) $\inf\hat{u}\leq\frac{3}{2}H_{2}^{2}$

and

(117) $-H_{2}-\sqrt{H_{2}^{2}-\frac{2}{3}\inf\hat{u}}\leq\hbox{\rm
sup}\,u\leq-H_{2}+\sqrt{H_{2}^{2}-\frac{2}{3}\inf\hat{u}},$

where $\hat{u}=\|\hat{\nabla}A\|^{2}$.
If $-\infty<H_{2}\leq H\leq H_{1}\leq 0$ for some numbers $H_{1}$, $H_{2}$ and

(118) $\hbox{\rm sup}\,\hat{u}\leq\frac{3}{2}H_{1}^{2},$

then

(119) $\inf\,u\geq-H_{1}+\sqrt{H_{1}^{2}-\frac{2}{3}\hbox{\rm sup}\,\hat{u}}$

or

(120) $\inf\,u\leq-H_{1}-\sqrt{H_{1}^{2}-\frac{2}{3}\hbox{\rm sup}\,\hat{u}}.$

##  6\. Using Ros’ integral formula

Let $s$ be a tensor of type $(0,k)$, $k\geq 2$ and $g_{0}$ – the standard
scalar product on $\mathbf{R}^{n}$. Define the following $1$-form on
$S^{n-1}=\\{V\in\mathbf{R}^{n};\|V\|=1\\}$:

$\alpha_{V}(e)=s(V,...,V,e,V,...,V),$

where $e$ stays at a fixed place $i_{0}$. Let $\delta$ denote the
codifferential relative to $g_{0}$ restricted to $S^{n-1}$. By a
straightforward computation, one gets

(121) $\begin{array}[]{rcl}&&\delta\alpha=-(n+k-2)s(V,...,V)\\\ &&\ \ \ \ \ \
\ +\hbox{\rm tr}\,_{g}s(\cdot,V,...,V,\cdot,V,...,V)+...+\hbox{\rm
tr}\,_{g}s(V,...,V,\cdot,V,...,V,\cdot),\end{array}$

where one of the dots ”$\cdot$ ” stays at the fixed $i_{0}$-th place.

Let now $g$ be a positive-definite metric tensor field on a manifold $M$. For
simplicity, we shall assume that $M$ is connected and oriented. Let $UM$
denote the unit sphere bundle over $M$. By (121), we have

######  Proposition 6.1.

For a covariant tensor field $s$ of degree $k\geq 2$ we have

(122) $\begin{array}[]{rcl}&&(n+k-2)\int_{U_{x}M}s(V,...,V)\\\ &&\ \ \ \ \ \ \
\ \ \ =\int_{U_{x}M}\hbox{\rm
tr}\,_{g}s(\cdot,V,...,V,\cdot,V,...,V)+...+\int_{U_{x}M}\hbox{\rm
tr}\,_{g}s(V,...,V,\cdot,V,...,V,\cdot)\end{array}$

for every $x\in M$.

We have the following Ros’ integral formula

(123) $\int_{UM}\hbox{\rm tr}\,_{g}(\hat{\nabla}s)(\cdot,\cdot,V,...,V)=0,$

where $s$ is a covariant tensor field of degree greater than 1 on a compact
manifold $M$ and $\int_{UM}=\int_{x\in M}\int_{U_{X}M}$, see [16].

######  Theorem 6.2.

Let $(g,A)$ be a conjugate symmetric statistical structure on a compact
manifold $M$. If the sectional curvature $\hat{k}\geq 0$ on $M$ and
$\hat{\nabla}^{2}\tau=0$, then $\hat{\nabla}A=0$ on $M$. If moreover,
$\hat{k}>0$ at some point $p$ of $M$, then the statistical structure is
trivial.

Proof. Consider the following tensor field $s$ on $M$

(124) $s(X_{1},...,X_{7})=\hat{\nabla}A(X_{1},...,X_{4})A(X_{5},X_{6},X_{7}).$

By Theorem 4.3, we have $\hat{\nabla}\tau=0$, which implies that
$\sum_{i}\hat{\nabla}A(X,Y,e_{i},e_{i})=0$, where, as usual, $e_{1},...,e_{n}$
stands for an orthonormal basis of a tangent space. Using (61), we obtain

(125) $\begin{array}[]{rcl}&&\sum_{i}\hat{\nabla}s(e_{i},e_{i},V,...,V)\\\ &&\
\ \ \ \ \ \ \ \ \ \ \ \
=\sum_{i}\hat{\nabla}^{2}A(e_{i},e_{i},V,V,V)A(V,V,V)+\sum_{i}\hat{\nabla}A(e_{i},V,V,V)^{2}\\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \
=\|(\hat{\nabla}K)(V,V,V)\|^{2}+\sum_{i}((\hat{R}(e_{i},V)A)(e_{i},V,V)A(V,V,V)\\\
&&\ \ \ \ \ \
=\|(\hat{\nabla}K)(V,V,V)\|^{2}-\sum_{i}A(\hat{R}(e_{i},V)e_{i},V,V)A(V,V,V)\\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-\sum_{i}2A(e_{i},\hat{R}(e_{i},V)V,V)A(V,V,V).\end{array}$

If we define a 1-form $\alpha$ on $U_{p}M$ by
$\alpha_{V}(e)=A(\hat{R}(e,V)V,V,V)A(V,V,V)$, then, by (121), we have

$\displaystyle\delta\alpha(V)=\sum_{i}A(\hat{R}(e_{i},V)e_{i},V,V)A(V,V,V)$
$\displaystyle\ \ \ \ \ \ \ \ \ \
+2\sum_{i}A(\hat{R}(e_{i},V)V,e_{i},V)A(V,V,V)$ $\displaystyle\ \ \ \ \ \ \ \
\ \ +3\sum_{i}A(\hat{R}(e_{i},V)V,V,V)A(e_{i},V,V),$

and consequently

$\displaystyle\int_{U_{p}M}\left[-\sum_{i}A(\hat{R}(e_{i},V)e_{i},V,V)A(V,V,V)-\sum_{i}2A(e_{i},\hat{R}(e_{i},V)V,V)A(V,V,V)\right]$
$\displaystyle\ \ \ \ \ \ \ \ \ \
=3\int_{U_{p}M}g(\hat{R}(K(V,V),V)V,K(V,V)).$

By (123) and (125), one gets

(126)
$0=\int_{UM}\|(\hat{\nabla}K)(V,V,V)\|^{2}+3\int_{UM}g(\hat{R}(K(V,V),V)V,K(V,V)).$

This implies the first assertion. To prove the second one, suppose that
$\hat{k}>0$ at $p\in M$, $K\neq 0$ (automatically at every point, in
particular at $p$). By (126), $K(V,V)$ is parallel to $V$ for each $V\in
U_{p}M$. By Theorem 4.3, we know that $E$ vanishes on $M$. Consider the
function $U_{p}M\ni V\to A(V,V,V)\in\mathbf{R}$. It attains a maximum at some
vector, say $e_{1}$. It is an eigenvector of $K_{e_{1}}$, that is,
$K(e_{1},e_{1})=\lambda_{1}e_{1}$ and $\lambda_{1}>0$. Let $e_{1},...,e_{n}$
be an orthonormal eigenbasis of $K_{e_{1}}$. Hence
$K(e_{1},e_{i})=\lambda_{i}e_{i}$ for some numbers $\lambda_{i}$. Since $E=0$,
we have $\sum_{i}\lambda_{i}=0$. Since $K(e_{i},e_{i})$ is parallel to $e_{i}$
and $K(e_{1},e_{i})=\lambda_{i}e_{i}$, we have
$0=g(K(e_{i},e_{i}),e_{1})=g(K(e_{1},e_{i}),e_{i})=\lambda_{i}$ for every
$i=2,...,n$. It follows that $\lambda_{1}=0$, which gives a contradiction. 2

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