# Elastocapillary menisci mediate interaction of neighboring structures at the
surface of a compliant solid

Lebo Molefe  John M. Kolinski<EMAIL_ADDRESS>Department of Mechanical
Engineering (IGM), School of Engineering (STI), Ecole Polytechnique Fédérale
de Lausanne (EPFL), 1015 Lausanne, Switzerland

###### Abstract

Surface stress drives long-range elastocapillary interactions at the surface
of compliant solids, where it has been observed to mediate interparticle
interactions and to alter the transport of liquid drops. We show that such an
elastocapillary interaction arises between neighboring structures that are
simply protrusions of the compliant solid. For compliant micropillars arranged
in a square lattice with spacing $p$ less than an interaction distance
$p^{*}$, the distance of a pillar to its neighbors determines how much it
deforms due to surface stress: pillars that are close together tend to be
rounder and flatter than those that are far apart. The interaction is mediated
by the formation of an elastocapillary meniscus at the base of each pillar,
which sets the interaction distance and causes neighboring structures to
deform more than those that are relatively isolated. Neighboring pillars also
displace toward each other to form clusters, leading to the emergence of
pattern formation and ordered domains.

††preprint: APS/123-QED

Surface stress can completely change the shape of the interface of a compliant
solid at sufficiently small scales [10, 4, 8, 1, 7, 20, 12, 17, 21, 5, 19, 2,
11, 3, 14, 13, 15, 16, 9, 6, 18], just as it leads to large deformation of
small fluid volumes [23, 22]. The excess tangential stress in a liquid-air
interface promotes the instability of a cylindrical liquid jet, causing it to
break into droplets [22] and surface stress has been observed to cause an
analogous instability for cylinders of compliant solid gels [1]. Compliant
solids routinely undergo large deformation in response to stresses that they
are exposed to. However, the consequences for an interfacial structure that
deforms subject to surface stress remain poorly understood.

The typical scale at which elastocapillary phenomena are expected to be
observed is defined by the ratio of the surface stress $\Upsilon$ to the
material’s elastic modulus $E$, and is called the elastocapillary length
$\ell_{ec}=\Upsilon/E$ 111In general, $\mathbf{\Upsilon}$ is a tensor, but
constant $\Upsilon$ is used in the uniform and isotropic case.; this arises
from a comparison of capillary pressure to the elastic strain in the bulk [2].
For structures at a solid-fluid interface with radius of curvature
$\mathcal{R}$ below $\ell_{ec}$, surface stress prompts rounding and large
deformations [2, 3]. In this way, regions of a surface that have initially
small radius of curvature, such as the corners of a sharp feature, deform into
regions of large radius of curvature that extend farther from the original
feature and are in some cases reminscent of liquid menisci. It is unknown how
the deformation of a single structure influences the deformation of other
structures at the interface, and ultimately the entire interface geometry.

Here we probe the mutual interaction of deforming structures at the interface
of compliant gels by molding grids of three-dimensional (3D) cuboid
micropillar features with size similar to the material’s elastocapillary
length, and varying the center-to-center spacing $p$ between the
microtextures. Following release from the mold, the solid-air interface
deforms due to surface stress. We experimentally study the deformed
micropillar surface profiles through two metrics: their final deformed height
$h_{d}$ and mean curvature $H$. An elastocapillary interaction emerges below
an interaction distance $p^{*}$. At distances $p<p^{*}$, pillar deformation is
strongly dependent on the location of neighboring pillars.

The spacing-dependent pillar deformation generates two regimes of the system’s
behavior: above the interaction distance $p^{*}$, a pillar’s deformation
depends only on its initial shape and material properties ($\ell_{ec}$). Below
$p^{*}$, however, proximity to neighboring structures increases deformation.
The interaction is shown to be mediated by elastocapillary menisci formed at
the base of each pillar. Within the elastocapillary interaction regime,
besides the increase of individual pillar deformation, additional behaviors
such as clustering emerge. In addition to its relevance for practical
applications in microfabrication and other areas [4], our study also improves
our fundamental understanding of surface stress, including highlighting the
emergence of ordered domains.

Figure 1: Surface fabrication using a dissolvable mold. (a) Microfabrication
procedure: A silicon wafer is coated with (i) a 2 µm layer of photoresist
(AZnLOF2020, Merck), followed by (ii) a $30$ µm layer of photoresist (AZ40XT,
Merck). (iii) The top layer is patterned using photolithography. (iv) An
approximately $200$ µm thick layer of compliant gel (PDMS CY 52-276, Dowsil)
is spin coated on the mold and cured by baking. (v) A glass wafer is bonded to
the gel surface using oxygen plasma. (vi) Photoresist layers are subsquently
dissolved with acetone, after which the surface is immersed in water to rinse,
then allowed to dry in ambient conditions; thus, the textured surface is
released. (b) The mold shape is a grid of recessed rectangular prisms with
center-to-center spacing $p$. (c) Upon release from the mold, the PDMS deforms
to height $h_{d}$ and width $w_{d}$. (d–f) Optical microcoscopy images for
$p=90$ µm of the: (d) dissolvable mold; (e) stiff surface, $E\approx 190$ kPa;
(f) compliant surface, $E\approx 3$ kPa. The mold is well-replicated by the
stiff PDMS, whereas significant deformation is observed for the compliant
PDMS. Scale bars are 100 µm.

To make textured surfaces in a compliant solid material (E $\lesssim 3$ kPa),
we use the microfabrication technique shown in Fig. 1(a). Rather than
attempting to separate the surface cleanly from a mold by peeling [25, 26, 27,
6, 5], where textures are prone to breaking, we instead cast a compliant
silicone gel (PDMS CY 52-276, Dow Europe GmbH) into a dissolvable mold [Fig.
1(a)(iii–iv)]. The method of using a dissolvable mold has previously been used
to study surface stress deformation of millimeter-scale elastic rods using
styrofoam molds dissolved in toluene [1, 7] and we demonstrate that this idea
can also be applied to produce micrometer-scale patterns.

First, a mold is produced from two polymeric layers: the first is a
sacrificial layer in contact with the tops of the pillars and the second is a
photoresist patterned by photolithograpy, which shapes the pillars’ side walls
[Fig. 1(a)(i–iii)]. A schematic of the mold is shown in Fig. 1(b), where each
hole has depth $h_{0}=26$ µm and width $w_{0}=30$ µm. The holes are arranged
on a square lattice with spacing $p$ varying between $45-480$ µm. Fig. 1(d) is
an image of the mold for $p=90$ µm. The initial mold dimensions were verified
by optical microscopy (for $w_{0}$) and optical profilometry (for $h_{0}$).
After spin-coating a compliant gel into the mold and curing the material by
baking at $80^{\circ}$C for 30 minutes, the gel layer is bonded to a glass
surface for support [Fig. 1(a)(v)]. The whole ensemble is then placed in an
acetone bath, where the mold dissolves, but the cast gel is left intact [Fig.
1(a)(vi)] 222Acetone has been shown to swell PDMS by 6% [28], yet because this
is a slight effect, we assume gel deformation is not significantly affected by
the solvent, especially since acetone is volatile and does not remain within
the gel once it is dry.. After dissolution of the mold, we remove the surface
from acetone, immerse the surface in deionized water to rinse off the acetone,
and dry the PDMS by leaving it in air.

A white light interferometric profilometer (Bruker Contour X) is used to
measure the solid-air interface profile. Pillar height $h_{0}$ is small
compared to the bulk thickness, estimated to be 200 µm based on the spin-
coating speed. To vary the material stiffness, we vary the ratio of part A to
part B in PDMS CY 52-276. A 1:1 A:B ratio produces a compliant surface and a
1:10 A:B ratio produces a stiff surface. A schematic of the released textures
is shown in Fig. 1(c), where they deform to a final height $h_{d}$ and width
$w_{d}$. This technique can be used to produce 3D pillar textures [Figs.
1(e)–1(f)] as well as more complex shapes.

To characterize the elastocapillary length, we measure the gel’s shear modulus
$G$ with a rheometer (Anton Paar MCR 301). For the compliant material
$G=0.84\pm 0.07$ kPa and for the stiff material $G=63\pm 2$ kPa. The surface
stress is $\Upsilon=35\pm 5$ mN/m and was measured by imaging the wetting
ridge of a glycerol drop deposited on the PDMS with a confocal fluorescence
microscope (Leica SP8 STED 3X with a 93$\times$ glycerol immersion lens, NA =
1.3). See the Supplemental Material for fabrication and material
characterization details [34].

Figure 2: Deformed surface profiles. (a) Height profile for a stiff
micropillar, $E\approx 190$ kPa. (b) Height profile for a compliant
micropillar, $E\approx 3$ kPa. Grid spacing in both cases is $p=90$ µm. (c)
Profiles of the stiff and compliant pillars as viewed along the dashed lines
indicated in (a) and (b). Data are compared to a theoretical model for low-
aspect-ratio features [9] (lines). For the compliant pillar $h_{d}$ is less
than predicted. (d) Deformed height $h_{d}$ of stiff pillars (squares) and
compliant pillars (circles) as a function of grid spacing $p$. For compliant
pillars, $h_{d}$ decreases as $p$ becomes small. The theory (solid lines) [9]
is plotted with a shaded confidence band representing a 95% confidence
interval in the elastic modulus. The confidence band for the stiff material is
too small to be visible. Figure 3: Deformed curvature. (a–b) Mean curvature
field $H(x,y)$ for a deformed compliant micropillar ($E\approx 3$ kPa) with
spacing (a) $p=90$ µm and (b) $p=480$ µm. Negative curvature indicates that
the pillar curves toward the $-\hat{z}$ direction. (c) Histogram of $H$ for
the pillars with (c) $p=90$ µm and (d) $p=480$ µm, considering data inside the
dashed boxes in (a–b). Gaussian fits (solid lines) have peaks at
$H_{\text{peak}}=-0.010\pm 0.003$ 1/µm and $H_{\text{peak}}=-0.012\pm 0.002$
1/µm, respectively (68% confidence interval). (e–f) Circular arcs (dashed red
lines) with radius of curvature (e) $\mathcal{R}=94$ µm and (f)
$\mathcal{R}=80$ µm are superimposed on the pillar profiles (gray points). (g)
Radius of curvature $\mathcal{R}$ as a function of spacing $p$. As pillars
$p\to 0$, radius of curvature increases – in other words, pillars are flatter.
Dashed line is the elastocapillary length $\ell_{ec}$. (inset) Curvature
$|H_{\text{peak}}|$ as a function of $p$. Figure 4: Elastocapillary meniscus
scale. (a) Mean curvature field $H(x,y)$ around a pillar with $p=480$ µm. (b)
$H$ as a function of position $s$ along $y=0$ (light blue) and $x=0$ (dark
blue). The pillar profile is plotted in gray. (c) Curvature along $y=0$ for 8
different pillars with $p=480$ µm. The red dashed line is a moving average
using bin width of 3 µm. Beyond $\sim$60 µm from the pillar center, curvature
decays to zero. Figure 5: Interaction of elastocapillary menisci. (a)
Schematic of a pair of pillars depicting elastocapillary meniscus scale
$s^{*}\approx p^{*}/2$. When the distance between adjacent pillars is below
$p^{*}$, pillars interact. (b–c) Profiles of adjacent pillars (orange) at
spacings: (b) $p=60$ µm and (c) $p=45$ µm. Gray profiles in each plot are
profiles of a pillar at large spacing ($p=360$ µm) that is duplicated so we
can visualize the hypothetical overlap of its elastocapillary meniscus with
another isolated pillar. (d–f) Optical microscopy of compliant pillar clusters
at (d) $p=45$ µm, (e) $p=60$ µm, and (f) $p=90$ µm. Scale bars are 200 µm. (g)
Profilometry data depict arrested coalescence of a clustered pair for $p=90$
µm. (h) Clustering behavior, quantified by pillar centroids’ root-mean-squared
deviation (RMSD), divided by $(p-w_{0})$.

Fig. 2 compares the deformed profiles of stiff [Fig. 2(a)] and compliant [Fig.
2(b)] micropillars at $p=90$ µm. The stiff pillar has replicated the mold with
fidelity, having the mold’s original height $h_{0}=26$ µm, and also retaining
the original lateral dimensions of the mold (30 µm $\times$ 30 µm), whereas
the compliant one has become much shorter, rounding into an almost circular
profile and expanding significantly in the $xy$-plane. Fig. 2(c) displays a
side-view of the profiles. We compare the profiles to the predicted
deformation in a theory proposed for 3D structures with low aspect ratio
proposed by Hui et al. [9] and find that it models the stiff pillar well but
overpredicts the height of the compliant one. This suggests that the theory
may need to be modified for structures that have a characteristic scale
similar to $\ell_{ec}$ but do not satisfy the assumption of infinitesimal
aspect ratio. Significant rounding and flattening of compliant pillars were
similarly observed for all spacings $p=45-480$ µm.

Fig. 2(d) displays the deformed height $h_{d}$ measured at the pillar center
as a function of spacing $p$ for surfaces of different stiffness. Error bars
are 95% confidence intervals, where each data point represents the mean height
of $30-146$ pillars. The elastocapillary length of the stiff material is
$\ell_{ec}\approx 0.5$ µm, so that we expect surface stress to have a minimal
effect on structures at a 30 µm scale. Indeed, the stiff pillar heights match
the original mold design ($h_{0}=26$ µm) across all spacings $p$. By contrast,
the elastocapillary length of the compliant material is $\ell_{ec}\approx 42$
µm, so we expect surface stress to drive significant deformations. A compliant
pillar’s height $h_{d}$ is highly dependent on spacing for low values of $p$,
whereas $h_{d}$ reaches a constant value as $p$ increases, as is evident in
Fig. 2(d). We identify the length scale for the transition between these two
behaviors around $p^{*}\approx 120-150$ µm, which defines the elastocapillary
interaction distance. For $p<p^{*}$, a pillar’s deformation is affected by its
neighbors, whereas for $p>p^{*}$, deformation depends only on the pillar’s
initial shape and material properties $G$ and $\Upsilon$.

We observe a similar trend in the mean curvature $H$ as a function of $p$
333Mean curvature $H$ is computed by fitting a paraboloid to local subsets of
the pillar surface, from which we compute the curvature at each point, as
described in detail in the Supplemental Material [34].. Curvature fields for
pillars at $p=90$ µm and $p=480$ µm are shown in Figs. 3(a–b). For both
spacings $p$, the curvature across the top of the pillar is negative,
indicating that the pillar curves toward the surface. Moreover, $H$ varies
little across the pillar center, as demonstrated by the narrow spread in the
histograms of $H$ shown in Figs. 3(c)–3(d), indicating that the initial
cuboids have deformed into approximately spherical caps with radius
$\mathcal{R}=1/|H_{\text{peak}}|$, where $H_{\text{peak}}$ is the value of $H$
at the location of the Gaussian peak. The radius of curvature is
$\mathcal{R}=94$ µm for the pillar with $p=90$ µm, indicating that it is
flatter than the relatively isolated pillar with $p=480$ µm, for which
$\mathcal{R}=80$ µm. Circular arcs with these $\mathcal{R}$ values overlaid
onto the profiles are an excellent match [Figs. 3(e)–3(f)]. Fig. 3(g) displays
$\mathcal{R}$ as a function of $p$ for compliant pillars. $\mathcal{R}$ is
largest at small spacings, whereas it decreases to a constant value as $p$
increases, indicating that close pillars flatten significantly more (having
larger radius of curvature) than relatively isolated ones. Notably, the
transition to a constant curvature occurs at the same distance $p^{*}\approx
120-150$ µm identified in the analysis of $h_{d}$ [Fig. 2(d)].

The increased deformation and rounding for $p<p^{*}$ suggests an
elastocapillary interaction between pillars. To probe this interaction, we
measure the curvature field $H(x,y)$ around a relatively isolated pillar, as
shown in Fig. 4(a). In Fig. 4(b), we observe that the curvature field has
almost constant negative curvature near the top of the pillar, which becomes
positive where the pillar meets the surface, and subsequently decays to zero.
By doing the same analysis for several pillars [Fig. 5(c)], we observe that
the curvature consistently decays with a characteristic distance of
$s^{*}\approx 60$ µm. This horizontal decay length defines the scale of an
elastocapillary meniscus. Notably, the pillar shape that was originally square
in the $xy$-plane becomes nearly axisymmetric, as is evident in Fig. 4(a) and
discussed in detail in the Supplemental Material [34].

The scale of the curvature decay $s^{*}\approx 60$ µm is about half of the
scale of the interaction distance $p^{*}\approx 120-150$ µm. This relation
$s^{*}=p^{*}/2$ is a clear indication that the interaction between pillars is
mediated by their elastocapillary menisci: for a pair of pillars, the length
scale $2s^{*}$ is the scale at which the meniscus of one pillar touches that
of another pillar – schematically illustrated in Fig. 5(a). This is the same
length scale $p^{*}=2s^{*}$ at which we observe spacing-dependent deformation.

To illustrate more clearly what happens to the elastocapillary menisci in the
valley between neighboring pillars as compared to isolated ones, in Figs.
5(b)–5(c) we overlay profiles of two isolated pillars (gray) whose original
spacing was $p=480$ µm as if they were separated by a distance of $p=60$ µm or
$p=45$ µm. The superposition of two isolated pillars represents a hypothetical
scenario where elastocapillary interaction with neighboring pillars does not
affect deformation. The region where the menisci overlap near $x=0$ µm in this
hypothetical situation has a higher curvature than the tops of the pillars,
and we observe that the profile of a real interface (orange) is rounder by
comparison. Thus, as a result of rounding and lifting bulk material between
pillars, as well as decreasing pillar height, the meniscus decreases the
surface area and curvature, likely reducing the total energy of the system.

Elastocapillary interaction not only increases individual pillar deformation,
but also leads to displacement of the pillars’ centroids, such that groups of
pillars form clusters [Figs. 5(d)–5(f)]. The profile in Fig. 5(g) depicts a
pair of pillars displaced toward each other. Clustering for a group of pillars
can be quantified by the root-mean-squared deviation (RMSD) of pillar
centroids from their ideal grid positions, divided by the initial space
$(p-w_{0})$ between two pillar edges. If the deviation of a pillar and its
neighbor toward each other is equal to half the distance between two pillar
edges ($\frac{p-w_{0}}{2}$), the pillars will touch. In Fig. 5(f), we show
that the RMSD of the ensemble of compliant pillars has a value such that
$\text{RMSD}/(p-w_{0})\approx 1/2$ for the lowest spacings $p=45-75$ µm,
indicating high levels of clustering that are not observed for the stiff
material.

In this work, we demonstrate that neighboring structures at a solid-air
interface can interact through an elastocapillary meniscus, leading to
pronounced deformation and rounding. We introduce a microfabrication technique
for producing structures at the surface of a compliant solid and study how
their deformation to final height $h_{d}$ and mean curvature $H$ depends on
grid spacing $p$. Deformation depends on spacing for $p<p^{*}$ where
$p^{*}\approx 2s^{*}$ is set by the horizontal scale $s^{*}$ of the
elastocapillary meniscus around a deformed pillar. Excess deformation due to
elastocapillary interaction is significant – the most closely spaced pillars
shrink to around 20% of their initial height, $h_{d}\approx 0.2h_{0}$. This is
much less than the deformed height of a relatively isolated pillar,
$h_{d}\approx 0.5h_{0}$.

In addition to large deformations, pillar centroids displace from their
initial grid positions, similar to droplet ‘durotaxis’ or the attraction
between glass beads at a gel-air interface, which are also mediated by regions
of nonzero curvature between the objects [19, 20, 21]. Here we observe a
similar long-range elastocapillary interaction between structures that are
part of the interface itself, without introducing external objects. This
system is an attractive alternative for studying elastocapillary interaction
at interfaces, because the singularity at the contact line is not present.
Interaction between pillars also differs from the interactions between
external objects, however: first, a pillar’s displacement toward its neighbor
is limited due to the elastic energy cost. Second, the pillar’s final shape is
affected by the interaction, which does not occur for objects such as rigid
beads. Excess deformation and rounding are added mechanisms by which compliant
pillars decrease the surface area of their elastocapillary menisci.
Intriguingly, we observe the emergence of pattern formation in a system driven
solely by the solid’s surface stress; by contrast, clustering patterns
previously observed for stiff micropillar arrays are driven by liquid-air
surface tension [29, 30, 31, 32, 33]. Our observations suggest future
directions for introducing defects or varied initial conditions to better
characterize emergence of order at low $p$.

Beyond the fundamental study of surface stress, our experiments suggest new
directions toward probing self-assembly on gel and elastomer interfaces [20,
21, 24] that may open new avenues for microfabrication techniques. Future
study of the dynamic response of solid surfaces to changing outer fluids may
also lead to new directions for producing soft machines, valves, or other
tools to be used, for example, in microfluidic devices.

###### Acknowledgements.

We thank Guillermo Villanueva, Mojtaba Abdolkhani, Abigail Plummer, and Andrej
Košmrlj for insightful discussions. We gratefully acknowledge the staff of the
Center for MicroNanotechnology (CMi) at EPFL who provided valuable technical
support for the fabrication, in particular Cyrille Hibert, Joffrey Pernollet,
Niccolò Piacentini, Georges-André Racine, Gatera Kumuntu, and Vivigan
Sinnathamby. Data supporting this study are openly available from [35].

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# Polyp Segmentation Generalisability of Pretrained Backbones

Edward Sanderson ∗,1 and Bogdan J. Matuszewski 1

Keywords: polyp, segmentation, generalisability, self-supervised, pretraining,
colonoscopy

##  1 Introduction

Due to the relatively low amount of annotated data available for training
polyp segmentation models, e.g. Sanderson and Matuszewski (2022), which
typically take the form of an autoencoder with UNet-style skip connections
(Ronneberger et al., 2015), pretraining of the encoder, also known as the
backbone, is typically performed. Until very recently, this was almost
exclusively done in a supervised manner with ImageNet-1k (Deng et al., 2009).
However, we recently demonstrated that pretraining backbones in a self-
supervised manner generally provides better fine-tuned performance (Sanderson
and Matuszewski, 2024), and that models with ViT-B (Dosovitskiy et al., 2020)
backbones typically perform better than models with ResNet50 (He et al., 2016)
backbones.

In this paper, we extend this recent work to consider generalisability. I.e.,
we assess the performance of models on a different dataset to that used for
fine-tuning, accounting for variation in network architecture and pretraining
pipeline (algorithm and dataset). This reveals how well models with different
pretrained backbones generalise to data of a somewhat different distribution
to the training data, which will likely arise in deployment due to different
cameras and demographics of patients, amongst other factors.

##  2 Analysis

We consider 12 polyp segmentation models pretrained and fine-tuned in a
previous study (Sanderson and Matuszewski, 2024), specifically those fine-
tuned on Kvasir-SEG (Jha et al., 2020). Each model is either a ResNet50
encoder with a DeepLabV3+ (Chen et al., 2018) decoder, or a ViT-B encoder with
a DPT (Ranftl et al., 2021) decoder. Additionally, each model was pretrained
on either Hyperkvasir-unlabelled (Borgli et al., 2020) or ImageNet-1k in a
self-supervised manner using either MoCo v3 (Chen et al., 2021), Barlow Twins
(Zbontar et al., 2021) (ResNet50 only), or MAE (He et al., 2022) (ViT-B only),
or pretrained in a supervised manner (ImageNet-1k only), or not pretrained at
all.

We evaluate performance on the full CVC-ClinicDB dataset (Bernal et al., 2015)
with mDice, mIoU, mPrecision, and mRecall. The results are reported in Table
1, where we also specify each model’s rank on each metric, as well as any
change in rank relative to the model’s evaluation on the Kvasir-SEG test set
(Sanderson and Matuszewski, 2024). The results show that self-supervised
pretraining on ImageNet-1k generally provides the best generalisation, that
supervised pretraining on ImageNet-1k is generally better than self-supervised
pretraining on Hyperkvasir-unlabelled, and that any considered pretraining is
better than no pretraining. These findings are consistent with the evaluation
on the Kvasir-SEG test set.

Table 1: Performance of models fine-tuned on the Kvasir-SEG training set and tested on CVC-ClinicDB. In addition to reporting the value of each metric, we also indicate the rank of each model, as well as any change in this rank relative to the model’s evaluation on the Kvasir-SEG test set. For conciseness, we abbreviate Hyperkvasir-unlabelled to HK, ImageNet-1k to IN, and Barlow Twins to BT. Backbone | Pretraining | mDice | mIoU | mPrecision | mRecall
---|---|---|---|---|---
arch. | Data | Algo. | Value | Rank | Value | Rank | Value | Rank | Value | Rank
ResNet50 | HK | MoCo v3 | 0.789 | 9 ($\uparrow$1) | 0.686 | 10 | 0.785 | 10 | 0.856 | 3 ($\uparrow$7)
BT | 0.801 | 8 ($\uparrow$1) | 0.709 | 8 ($\uparrow$1) | 0.831 | 8 ($\uparrow$1) | 0.848 | 7 ($\downarrow$2)
IN | MoCo v3 | 0.843 | 1 ($\uparrow$3) | 0.760 | 1 ($\uparrow$3) | 0.867 | 3 ($\uparrow$5) | 0.874 | 1
BT | 0.826 | 5 | 0.735 | 6 ($\uparrow$1) | 0.858 | 6 | 0.854 | 4 ($\uparrow$2)
Supervised | 0.822 | 6 | 0.735 | 5 | 0.899 | 1 ($\uparrow$4) | 0.811 | 10 ($\downarrow$2)
None | None | 0.520 | 12 | 0.394 | 12 | 0.496 | 12 | 0.724 | 12
ViT-B | HK | MoCo v3 | 0.789 | 10 ($\downarrow$2) | 0.696 | 9 ($\downarrow$1) | 0.812 | 9 ($\downarrow$2) | 0.848 | 8 ($\downarrow$1)
MAE | 0.828 | 3 | 0.743 | 3 | 0.852 | 7 ($\downarrow$4) | 0.858 | 2 ($\uparrow$1)
IN | MoCo v3 | 0.830 | 2 | 0.742 | 4 ($\downarrow$2) | 0.861 | 4 ($\downarrow$2) | 0.849 | 5 ($\downarrow$3)
MAE | 0.827 | 4 ($\downarrow$3) | 0.746 | 2 ($\downarrow$1) | 0.868 | 2 ($\downarrow$1) | 0.848 | 6 ($\downarrow$2)
Supervised | 0.809 | 7 | 0.717 | 7 ($\downarrow$1) | 0.860 | 5 ($\downarrow$1) | 0.832 | 9
None | None | 0.637 | 11 | 0.519 | 11 | 0.670 | 11 | 0.759 | 11

However, the model pretrained with MAE on ImageNet-1k, which performs best on
the Kvasir-SEG test set, reduces its rank on every metric, notably dropping
from rank 1 to 4 on mDice. In contrast, models with a ResNet50 backbone
generally improve their ranking, implying greater generalisability than models
with a ViT-B backbone, which generally experience a drop in ranking, and the
best generalisation is achieved by the model with a ResNet50 backbone that was
pretrained on ImageNet-1k using MoCo v3, notably improving from rank 4 to 1 on
mDice. To better understand this, we compare the distribution of instance-wise
Dice scores from each model’s evaluation on the Kvasir-SEG test set against
the distribution from its evaluation on CVC-ClinicDB in Fig. 1. This indicates
that all models experience a drop in overall performance that primarily arises
from a higher variance. However, the portion of each distribution for the
highest Dice scores shows that most models with ResNet50 backbones achieve
better performance on some instances of CVC-ClinicDB than any in the Kvasir-
SEG test set, while models with ViT-B backbones fail to exceed their maximum
Dice score across the Kvasir-SEG test set when evaluated on CVC-ClinicDB. We
verify that all models experience a drop in performance, and quantify the
relative drop, in Fig. 1, which reveals that most models with ResNet50
backbones do indeed experience less of a drop, potentially as a result of
their improvement in maximum Dice score, explaining the improvement in
ranking.

##  3 Conclusion

In this paper, we showed that the findings of previous work, regarding
pretraining pipelines for polyp segmentation, hold true when considering
generalisability. However, our results imply that models with ResNet50
backbones typically generalise better, despite being outperformed by models
with ViT-B backbones in evaluation on the test set from the same dataset used
for fine-tuning. We expect that this is a result of the larger complexity of
the models with ViT-B backbones allowing for overfitting on the distribution
underlying the training data. However, this challenges the assumption that the
considered pretraining pipelines should help prevent this, and more work is
required to better understand the relationships between architecture,
pretraining pipeline, and performance on different distributions of data.

Figure 1: Comparison of the distribution of instance-wise Dice score from each
model’s evaluation on the Kvasir-SEG test set (blue) against the distribution
from its evaluation on CVC-ClinicDB (red). For conciseness, we denote
ResNet50s with RN, ViT-Bs with VT, Hyperkvasir-unlabelled with HK, ImageNet-1k
with IN, MoCo v3 with MC, Barlow Twins with BT, MAE with MA, supervised
pretraining with SL, and no pretraining with NA-NA. Figure 2: Relative drop in
mDice from each model’s evaluation on the Kvasir-SEG test set to its
evaluation on CVC-ClinicDB. For conciseness, we denote ResNet50s with RN, ViT-
Bs with VT, Hyperkvasir-unlabelled with HK, ImageNet-1k with IN, MoCo v3 with
MC, Barlow Twins with BT, MAE with MA, supervised pretraining with SL, and no
pretraining with NA-NA. For clarity, the results for ResNet50 models are
coloured blue and the results for ViT-B models are coloured red.

## Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.

## Author Contributions

ES: Conceptualization, Formal analysis, Visualization, Writing – original
draft, Writing – review & editing; BJM: Funding acquisition, Project
administration, Resources, Supervision, Writing – review & editing.

## Funding

This work was supported by the Science and Technology Facilities Council
[grant number ST/S005404/1].

## Data Availability Statement

This publication is supported by Kvasir-SEG and CVC-ClinicDB which are openly
available as cited in the ’References’ section of this paper.

## References

  * Bernal et al. (2015) Bernal, J., Sánchez, F. J., Fernández-Esparrach, G., Gil, D., Rodríguez, C., and Vilariño, F. (2015).  Wm-dova maps for accurate polyp highlighting in colonoscopy: Validation vs. saliency maps from physicians.  _Computerized medical imaging and graphics_ 43, 99–111
  * Borgli et al. (2020) Borgli, H., Thambawita, V., Smedsrud, P. H., Hicks, S., Jha, D., Eskeland, S. L., et al. (2020).  Hyperkvasir, a comprehensive multi-class image and video dataset for gastrointestinal endoscopy.  _Scientific data_ 7, 283
  * Chen et al. (2018) Chen, L.-C., Zhu, Y., Papandreou, G., Schroff, F., and Adam, H. (2018).  Encoder-decoder with atrous separable convolution for semantic image segmentation.  In _Proceedings of the European conference on computer vision (ECCV)_. 801–818
  * Chen et al. (2021) Chen, X., Xie, S., and He, K. (2021).  An empirical study of training self-supervised vision transformers.  _CoRR_ abs/2104.02057
  * Deng et al. (2009) Deng, J., Dong, W., Socher, R., Li, L.-J., Li, K., and Fei-Fei, L. (2009).  Imagenet: A large-scale hierarchical image database.  In _2009 IEEE conference on computer vision and pattern recognition_ (Ieee), 248–255
  * Dosovitskiy et al. (2020) Dosovitskiy, A., Beyer, L., Kolesnikov, A., Weissenborn, D., Zhai, X., Unterthiner, T., et al. (2020).  An image is worth 16x16 words: Transformers for image recognition at scale.  _arXiv preprint arXiv:2010.11929_
  * He et al. (2022) He, K., Chen, X., Xie, S., Li, Y., Dollár, P., and Girshick, R. (2022).  Masked autoencoders are scalable vision learners.  In _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_. 16000–16009
  * He et al. (2016) He, K., Zhang, X., Ren, S., and Sun, J. (2016).  Deep residual learning for image recognition.  In _Proceedings of the IEEE conference on computer vision and pattern recognition_. 770–778
  * Jha et al. (2020) Jha, D., Smedsrud, P. H., Riegler, M. A., Halvorsen, P., de Lange, T., Johansen, D., et al. (2020).  Kvasir-seg: A segmented polyp dataset.  In _MultiMedia Modeling: 26th International Conference, MMM 2020, Daejeon, South Korea, January 5–8, 2020, Proceedings, Part II 26_ (Springer), 451–462
  * Ranftl et al. (2021) Ranftl, R., Bochkovskiy, A., and Koltun, V. (2021).  Vision transformers for dense prediction.  In _Proceedings of the IEEE/CVF international conference on computer vision_. 12179–12188
  * Ronneberger et al. (2015) Ronneberger, O., Fischer, P., and Brox, T. (2015).  U-net: Convolutional networks for biomedical image segmentation.  In _Medical image computing and computer-assisted intervention–MICCAI 2015: 18th international conference, Munich, Germany, October 5-9, 2015, proceedings, part III 18_ (Springer), 234–241
  * Sanderson and Matuszewski (2022) Sanderson, E. and Matuszewski, B. J. (2022).  Fcn-transformer feature fusion for polyp segmentation.  In _Annual conference on medical image understanding and analysis_ (Springer), 892–907
  * Sanderson and Matuszewski (2024) Sanderson, E. and Matuszewski, B. J. (2024).  A study on self-supervised pretraining for vision problems in gastrointestinal endoscopy.  _IEEE Access_ 12, 46181–46201.  10.1109/ACCESS.2024.3381517
  * Zbontar et al. (2021) Zbontar, J., Jing, L., Misra, I., LeCun, Y., and Deny, S. (2021).  Barlow twins: Self-supervised learning via redundancy reduction.  In _International Conference on Machine Learning_ (PMLR), 12310–12320


# Utilizing Large Language Models for Natural Interface to Pharmacology
Databases

Hong Lu, Chuan Li, Yinheng Li, Jie Zhao  honglu, chuanl, yinhengli,
<EMAIL_ADDRESS>
(2018)

###### Abstract.

The drug development process necessitates that pharmacologists undertake
various tasks, such as reviewing literature, formulating hypotheses, designing
experiments, and interpreting results. Each stage requires accessing and
querying vast amounts of information. In this abstract, we introduce a Large
Language Model (LLM)-based Natural Language Interface designed to interact
with structured information stored in databases. Our experiments demonstrate
the feasibility and effectiveness of the proposed framework. This framework
can generalize to query a wide range of pharmaceutical data and knowledge
bases.

††copyright: acmcopyright††journalyear: 2018††doi:
XXXXXXX.XXXXXXX††conference: Make sure to enter the correct conference title
from your rights confirmation emai; June 03–05, 2018; Woodstock, NY††price:
15.00††isbn: 978-1-4503-XXXX-X/18/06

##  1\. Introduction

Numerous databases are commonly used in pharmacology (Mo et al., 2022; Chandak
et al., 2023), encompassing various properties of biological entities (such as
genes, proteins, drugs, phenotypes, and diseases) and their relationships.
Simple queries might involve retrieving GO terms for a specific protein, while
complex queries could necessitate analyzing diseases associated with the
dysregulation of a genetic pathway. However, formulating SQL queries and
comprehending these databases can be challenging and time-consuming for
pharmacologists (Pourreza and Rafiei, 2023; Huang et al., 2021). In this work,
we study employing Large Language Models (LLMs) to develop a natural language
interface that enables pharmacologists to query public or private databases.

##  2\. Method

As illustrated in Fig. 1, our end-to-end Question Answering (QA) system allows
users to ask natural language questions, which are then translated into SQL
queries by GPT-4. Using these queries, the SQL database retrieves relevant
information from the databases. Our system employs an in-house Named Entity
Recognition (NER) model, followed by an entity linker built on Cognitive
Search. Detected entities are replaced with their type-aware placeholders. The
GPT-4 powered SQL generator facilitates schema linking, question
decomposition, Chain-of-Thought, and self-correction of the generated SQL
query.

In addition to the QA system, there is a separate question generation
pipeline. This pipeline serves two purposes: generating a synthetic dataset to
evaluate the QA model and providing in-context demonstrations for few-shot QA.
This dual functionality ensures that the system is effective, accurate, and
user-friendly, streamlining the research process and enhancing data
exploration efficiency.

##  3\. Results

We evaluated our solution using both synthetic and realistic questions. Our
synthetic dataset comprises 60 single-hop questions and 204 two-hop questions,
based on the knowledge graph of entities. Domain experts reviewed these
generated questions to ensure their quality and relevance. Furthermore, our
realistic dataset includes 50 question-answer pairs curated by experts who are
unaware of the database schema. This dataset represents the system’s
effectiveness and applicability to real-world scenarios, further demonstrating
its practical utility.

We utilize PrimeKG(Chandak et al., 2023) as the knowledge graph, which
comprehensively includes 10 types of entities and 30 types of relations with
over 8 million edges, aggregated from 20 bio-medical data sources. We employ
Exact Match (EM) and F1-score as evaluation metrics. To assess the importance
of each module, we conducted an ablation test and evaluated model performance
on the realistic dataset with and without named entity recognition (NER) or
self-correction (Table1). This test highlights the significance of both
modules. Our model also demonstrated reasonable performance on synthetic
questions (Table2). The results indicate that NER is a bottleneck and confirm
that two-hop reasoning can be more challenging for LLMs.

Figure 1. The proposed framework. Table 1. Realistic dataset ablation results.
”$-$” denotes removing NER (using oracle entity names) or self-correction
(SC).

Setting | ChatGPT | GPT-4
---|---|---
EM | F1 | EM | F1
Full | 0.396 | 0.433 | 0.708 | 0.712
$-$ NER | 0.458 | 0.491 | 0.792 | 0.797
$-$ SC | 0.313 | 0.316 | 0.708 | 0.712
$-$ NER $-$ SC | 0.375 | 0.399 | 0.792 | 0.797

Table 2. Evaluation on the synthetic dataset.

LLM | Setting | single-hop | two-hop | Overall
---|---|---|---|---
EM | F1 | EM | F1 | EM | F1
GPT-4 | Full | 0.450 | 0.496 | 0.320 | 0.343 | 0.378 | 0.350
Full $-$ NER | 0.700 | 0.746 | 0.458 | 0.483 | 0.513 | 0.543

## References

  * (1)
  * Chandak et al. (2023) P. Chandak, K. Huang, and M. Zitnik. 2023.  Building a knowledge graph to enable precision medicine.  _Sci Data_ (2023).
  * Huang et al. (2021) X. Huang, J. Zhang, Z. Xu, L. Ou, and J. Tong. 2021.  A knowledge graph based question answering method for medical domain.  _PeerJ Comput Sci_ (2021).
  * Mo et al. (2022) L. Mo, Z. Wang, J. Zhao, and H. Sun. 2022\.  Knowledge Transfer between Structured and Unstructured Sources for Complex Question Answering. In _SUKI_.
  * Pourreza and Rafiei (2023) M. Pourreza and D. Rafiei. 2023.  DIN-SQL: Decomposed In-Context Learning of Text-to-SQL with Self-Correction.  arXiv:2304.11015 [cs.CL]


# Conjugate Connections and their Applications on Pure Metallic Metric
Geometries

Olgun DURMAZ  Ataturk University, Faculty of Science, Department of
Mathematics, 25240, Erzurum-Turkey<EMAIL_ADDRESS>and  Aydin
GEZER  Ataturk University, Faculty of Science, Department of Mathematics,
25240, Erzurum-Turkey<EMAIL_ADDRESS>
###### Abstract.

Let $\left(M,J,g\right)$ be a metallic pseudo-Riemannian manifold equipped
with a metallic structure $J$ and a pseudo-Riemannian metric $g$. The paper
deals with interactions of Codazzi couplings formed by conjugate connections
and tensor structures. The presence of Tachibana operator and Codazzi
couplings presented a new characterization for locally metallic pseudo-
Riemannian manifold. Also, a necessary and sufficient condition a non-
integrable metallic pseudo-Riemannian manifold is a quasi metallic pseudo
Riemannian manifold is derived. Finally, it is introduced metallic-like
pseudo-Riemannian manifolds and presented some results concerning them.

Mathematics Subject Classification: Primary 53C05, 53C55; Secondary 62B10.

Keywords : Codazzi-coupled, conjugate connection, generalized dual connection,
metallic pseudo-Riemannian manifold, statistical manifold.

##  1\.
Introduction

The concept of tensor structures on differentiable manifolds is one of the
central concepts of modern differential geometry. Among tensor structures on a
differentiable manifold, one of the best known is almost complex structure,
that is, a $(1,1)-$ tensor field $J$ whose square, at each point, is minus the
identity. The manifold must be even-dimensional, that is, $dim=2n$. Usually,
it is equipped with a Hermitian metric $g$ which is a (Riemannian or pseudo-
Riemannian) metric that preserves the almost complex structure, that is, the
almost complex structure $J$ acts as an isometry with respect to the
(pseudo-)Riemannian metric. In this case, the $(J,g)$ is called an almost
Hermitian structure. The associated $(0,2)-$tensor of the Hermitian metric $g$
is a $2-$form $\omega$ which is defined by $\omega=g(JX,Y)$ and hence the
relationship with symplectic geometry.

The other case is that the almost complex structure $J$ acts as an anti-
isometry with respect to a pseudo-Riemannian metric $g$. Such a metric is
known as an anti-Hermitian metric or a Norden metric (first studied by and
named after Norden [17]). The Norden metric is necessary pseudo-Riemannian of
neutral signature. In this case, $(J,g)$ is called an almost anti-Hermitian
structure or almost Norden structure or almost $B-$structure. The associated
$(0,2)-$tensor of any Norden metric which is defined by $G=g(JX,Y)$ is also a
Norden metric. So, in this case we dispose with a pair of mutually associated
Norden metrics, known also as twin Norden metrics.

Besides almost complex structures, other tensor structures on differentiable
manifolds which are called almost tangent, almost product structures etc.
exist and are important for modern differential geometry. All the tensor
structures we have talked about so far are actually a polynomial structure.
Polynomial structures can be considered as $(1,1)-$ tensor field which are
roots of the algebraic equation

$Q(J)=J^{n}+a_{n}J^{n-1}+...+a_{2}J+a_{1}J=0,$

where $a_{1},a_{2},...,a_{n}$ are real numbers and $I$ is the identity tensor
of type $(1,1)$. Polynomial structures on a manifold were defined in [9]. In
particular, we can say the followings

  * •

If $Q(J)=J^{2}+I=0$, its solution $J$ is called an almost complex structure.

  * •

If $Q(J)=J^{2}-I=0$, its solution $J$ is called an almost product structure.

  * •

If $Q(J)=J^{2}=0$, its solution $J$ is called an almost tangent structure.

  * •

If $Q(J)=J^{2}-J-I=0$, its solution $J$ is called an golden structure. This
tensor structure was inspired by the golden ratio, which was described by
Johannes Kepler (1571–1630). The number $\eta=\frac{1+\sqrt{5}}{2}=1,618...$,
which is a solution of the equation $x^{2}-x-1=0$, is the golden ratio.

  * •

If $Q(J)=J^{2}-pJ-qI=0$, , with $p$ and $q$ positive integers, its solution
$J$ is called a metallic structure. The name is motivated by the fact that the
metallic means family or metallic proportions introduced by by de Spinadel [6]
is the positive root of the quadratic equation $x^{2}-px-q=0$, namely
$\sigma_{p,q}=\frac{p+\sqrt{p^{2}+4q}}{2}$. If $p=0$, that is, $J^{2}=qI$,
then we call them trivial metallic structures.

We will follow the notations and definitions concerning with metallic pseudo-
Riemannian manifolds given in [3].

######  Definition 1.

[3] Let $M$ be a smooth (real) manifold and $J:TM\rightarrow TM$ be a tangent
bundle isomorphism. A $(1,1)-$tensor field $J$ on $M$ is called a metallic
structure if $J^{2}=pJ+qI$, where $I$ is the identity operator and
$p,q\in\mathbb{R}.$

The classical definition is used for integrability of a metallic structure.

######  Definition 2.

[3] For a metallic structure $J$, the integrability is equivalent to the
vanishing of the Nijenhuis tensor $N_{J}$ :

$N_{J}(X,Y)=\left[JX,JY\right]-J\left[JX,Y\right]-J\left[X,JY\right]+J^{2}\left[X,Y\right].$

Here and further, $X,Y,Z$ will stand for arbitrary differentiable vector
fields on the considered manifold (or vectors in its tangent space at an
arbitrary point of the manifold).

######  Definition 3.

A linear connection $\nabla$ on a smooth manifold $M$ is called a
$J-$connection if $\nabla J=0$.

Another equivalent definition for integrability of a polynomial structure is
given by Vanzura in [18]: in order that a metallic structure $J$ be
integrable, it is necessary and sufficient that we can introduce a torsion-
free linear connection $\nabla$ such that $\nabla J=0$, that is, the torsion-
free linear connection $\nabla$ is a $J-$connection.

######  Definition 4.

Let $\left(M,g\right)$ be a pseudo-Riemannian manifold and $J$ be a
$g-$symmetric $\left(1,1\right)-$tensor field on $M$ such that $J^{2}=pJ+qI$,
where $p$ and $q$ are real numbers. Then $\left(J,g\right)$ is called a
metallic pseudo-Riemannian structure and $\left(M,J,g\right)$ is named a
metallic pseudo-Riemannian manifold. A metallic pseudo-Riemannian manifold
$\left(M,J,g\right)$ such that the Levi-Civita connection $\nabla^{g}$ with
respect to $g$ is a $J-$connection is called a locally metallic pseudo-
Riemannian manifold.

Next, we can give the following proposition for another way of describing
locally metallic pseudo-Riemannian manifolds. The following proposition can be
proven by following the method used in [19]. It will be presented without
proof. So, we will avoid unnecessary repetition.

######  Proposition 1.

Let $\left(M,J,g\right)$ be a metallic pseudo-Riemannian manifold and
$\nabla^{g}$ be the Levi-Civita connection of $g$. Then $\Phi_{J}g=0$ is
equivalent to $\nabla^{g}J=0$, where $\Phi_{J}$ is the Tachibana operator
defined by

$\left(\Phi_{J}g\right)\left(X,Y,Z\right)=JXg\left(Y,Z\right)-X\left(g\left(JY,Z\right)\right)+g\left(\left(L_{Y}J\right)X,Z\right)+g\left(Y,\left(L_{Z}J\right)X\right).$

Here $\left(L_{X}J\right)Y=\left[X,JY\right]-J\left[X,Y\right].$

Now let’s turn our attention to the other important actor in our article, the
conjugate connections. The notion of conjugate connections with respect to a
metric tensor field was originally introduced by Norden in the context of Weyl
geometry [17]. Such linear connections were independently developed by Nagaoka
and Amari [15] under the name dual connections and used by Lauritzen in the
definition of statistical manifolds [12]. In this direction, we can define two
conjugate connection on a metallic pseudo-Riemannian manifold
$\left(M,J,g\right)$, which are respectively called $g-$conjugate connection
($\nabla^{\ast}$) and $G-$conjugate connection ($\nabla^{{\dagger}}$):

(1.1)
$Zg\left(X,Y\right)=g\left(\nabla_{Z}X,Y\right)+g\left(X,\nabla_{Z}^{\ast}Y\right)$

and

(1.2)
$ZG\left(X,Y\right)=G\left(\nabla_{Z}X,Y\right)+G\left(X,\nabla_{Z}^{{\dagger}}Y\right),$

where $G=g(JX,Y)$ is the twin metric on $\left(M,J,g\right)$ and $\nabla$ is a
linear connection. It is easy to see that $\nabla^{\ast}$ and
$\nabla^{{\dagger}}$ are indeed connections such that
$\left(\nabla^{\ast}\right)^{\ast}=\left(\nabla^{{\dagger}}\right)^{{\dagger}}=\nabla$.
Another type of conjugate connections are those concerning with tensor
structures (for details, [1, 2, 4]). In our setting, this connection is
defined by

$\nabla_{X}^{J}Y=J^{-1}\left(\nabla_{X}\left(JY\right)\right)$

which is called $J-$conjugate connection ($\nabla^{J}$).

##  2\. Codazzi Coupling of $\nabla$ with $J$

Let $\nabla$ be an arbitrary linear connection on a pseudo-Riemannian manifold
$(M,g)$. A symmetric $(0,2)-$tensor field $\rho$ is Codazzi if it satisfies
the symmetry property

$(\nabla_{X}\rho)(Y,Z)=(\nabla_{Z}\rho)(X,Y).$

Alternatively, a $(1,1)-$tensor field $J$ is Codazzi if it is self-adjoint and

$(\nabla_{X}J)Y=(\nabla_{Y}J)X.$

We call the pairs $(\nabla,\rho)$ and $(\nabla,J)$, respectively, a Codazzi-
coupled.

Let $\left(M,J,g\right)$ be a metallic pseudo-Riemannian manifold. The inverse
of the metallic structure $J$ is as follow: $J^{-1}=\frac{1}{q}(J-pI)$
$\left(q\neq 0\right)$. The following proposition is analogue to the known
result given by Fei and Zhang [7] for Hermitian setting.

######  Proposition 2.

Let $\nabla$ be a linear connection and $J$ be a metallic structure on $M$.
Then the following situations are equivalent

$1)\ \left(\nabla,J\right)$ is Codazzi-coupled;

$2)\ \nabla$ and $\nabla^{J}$ have equal torsions;

$3)\ \left(\nabla^{J},J^{-1}\right)$ is Codazzi-coupled.

###### Proof.

$1)\Rightarrow 2):\ $Assume that $\left(\nabla,J\right)$ is Codazzi-coupled.
Then we have

$\displaystyle
J^{-1}\left(\left(\nabla_{X}J\right)Y-\left(\nabla_{Y}J\right)X\right)$
$\displaystyle=$ $\displaystyle
J^{-1}\left(\nabla_{X}JY-J\nabla_{X}Y-\nabla_{Y}JX+J\nabla_{Y}X\right)$
$\displaystyle=$ $\displaystyle
J^{-1}\left(\nabla_{X}JY\right)-J^{-1}\left(\nabla_{Y}JX\right)-\left(\nabla_{X}Y-\nabla_{Y}X\right)$
$\displaystyle=$
$\displaystyle\nabla_{X}^{J}Y-\nabla_{Y}^{J}X-\left(\nabla_{X}Y-\nabla_{Y}X\right)$
$\displaystyle=$ $\displaystyle
T^{\nabla^{J}}\left(X,Y\right)-T^{\nabla}\left(X,Y\right).$

$2)\Rightarrow 3):\ $Suppose that we have
$T^{\nabla^{J}}\left(X,Y\right)=T^{\nabla}\left(X,Y\right)$. Since

$\displaystyle
J\left(\left(\nabla_{X}^{J}J^{-1}\right)Y-\left(\nabla_{Y}^{J}J^{-1}\right)X\right)$
$\displaystyle=$ $\displaystyle
J\left(\nabla_{X}^{J}J^{-1}Y\right)-\nabla_{X}^{J}Y-J\left(\nabla_{Y}^{J}J^{-1}X\right)+\nabla_{Y}^{J}X$
$\displaystyle=$
$\displaystyle\nabla_{X}Y-\nabla_{Y}X-\left(\nabla_{X}^{J}Y-\nabla_{Y}^{J}X\right)$
$\displaystyle=$ $\displaystyle
T^{\nabla}\left(X,Y\right)-T^{\nabla^{J}}\left(X,Y\right),$

from which it is clear that $\left(\nabla^{J},J^{-1}\right)$ is Codazzi-
coupled.

$3)\Rightarrow 1):$ The result immediately follows from (2) and (2). ∎

Fei and Zhang [7] proved that if $J$ is either an almost complex structure or
an almost para-complex structure, then $\left(\nabla^{J}\right)^{J}=\nabla$.
For the metallic structure $J$, we have the following proposition.

######  Proposition 3.

Let $J$ be a metallic structure on $M$ and $\nabla$ be a linear connection. If
$J$ is a trivial metallic structure $(p=0)$, then we get
$\left(\nabla^{J}\right)^{J}=\nabla$.

###### Proof.

For a linear connection $\nabla$ and a metallic structure $J$, we know that

$\nabla_{X}^{J}Y=J^{-1}\left(\nabla_{X}\left(JY\right)\right).$

Using the definition of metallic structure $J$, we can rewrite it as

$\displaystyle\left(\nabla^{J}\right)_{X}^{J}Y$ $\displaystyle=$
$\displaystyle J^{-1}\left(\nabla_{X}^{J}\left(JY\right)\right)$
$\displaystyle=$ $\displaystyle
J^{-2}\left(\left(\nabla_{X}J^{2}\right)Y+J^{2}\nabla_{X}Y\right)$
$\displaystyle=$ $\displaystyle
pJ^{-2}\left(\left(\nabla_{X}J\right)Y\right)+qJ^{-2}\left(\left(\nabla_{X}I\right)Y\right)+\nabla_{X}Y$
$\displaystyle=$ $\displaystyle
pJ^{-2}\left(\left(\nabla_{X}J\right)Y\right)+\nabla_{X}Y,$

Thus, if $p=0,$ then we have $\left(\nabla^{J}\right)_{X}^{J}Y=\nabla_{X}Y.$ ∎

######  Remark 1.

Let $M$ be an almost (para)complex manifold equipped a $(1,1)-$tensor field
$J$ such that $J^{2}=-I$ (or $J^{2}=I)$ where $I$ is the identity operation.
Fei and Zhang obtained a nice result stating that $g-$conjugation,
$\omega-$conjugation and $J-$conjugation (along with an identity operation)
together form a $4-$element Klein group on the space of linear connections,
where $\omega$ is the fundemantal $2-$form (see Theorem 2.13 in [7]).
Unfortunately, for metallic structure $J$, $J-$conjugation is not involutive,
that is, $\left(\nabla^{J}\right)^{J}\neq\nabla$. So, in our setting, we
cannot create a $4-$element Klein group with our arguments. But, this shows
the difference of our structure from structures used in [7].

######  Proposition 4.

Let $J$ be a metallic structure on $M$ and $\nabla$ be a linear connection on
$M$ . If $\left(\nabla,J\right)$ is a Codazzi-coupled, then$\
\left(\nabla^{J},J\right)$ is so.

###### Proof.

Assume that $\left(\nabla,J\right)$ is a Codazzi-coupled. Then, standard
calculations give

$\displaystyle\left(\nabla_{X}^{J}J\right)Y-\left(\nabla_{Y}^{J}J\right)X$
$\displaystyle=$
$\displaystyle\nabla_{X}^{J}\left(JY\right)-J\nabla_{X}^{J}Y-\nabla_{Y}^{J}\left(JX\right)+J\nabla_{Y}^{J}X$
$\displaystyle=$ $\displaystyle
J^{-1}\left(\left(\nabla_{X}J^{2}\right)Y+J^{2}\left(\nabla_{X}Y\right)\right)-\nabla_{X}\left(JY\right)$
$\displaystyle-J^{-1}\left(\left(\nabla_{Y}J^{2}\right)X+J^{2}\left(\nabla_{Y}X\right)\right)+\nabla_{Y}\left(JX\right)$
$\displaystyle=$ $\displaystyle
pJ^{-1}\left(\left(\nabla_{X}J\right)Y-\left(\nabla_{Y}J\right)X\right)$
$\displaystyle+qJ^{-1}\left(\left(\nabla_{X}I\right)Y-\left(\nabla_{Y}I\right)X\right)$
$\displaystyle=$ $\displaystyle 0.$

∎

Let $J$ be either an almost complex structure or an almost para-complex
structure. Then $\left(\nabla,J\right)$ is Codazzi-coupled if and only if$\
\left(\nabla^{J},J\right)$ is Codazzi-coupled [7]. For the metallic structure
$J$, the reverse of the Proposition 4 is only true for trivial metallic
structure $(p=0)$, which is meaningless.

######  Lemma 1.

Let $J$ be a metallic structure on $M$ and $\nabla$ be a linear connection on
$M$. If $\left(\nabla,J\right)$ is a Codazzi-coupled, then

$N_{J}\left(X,Y\right)=-J^{2}T^{\nabla}\left(X,Y\right)+T^{\nabla}\left(X,JY\right)+JT^{\nabla}\left(JX,Y\right)-T^{\nabla}\left(JX,JY\right),$

where $T^{\nabla}\left(X,Y\right)=\nabla_{X}Y-\nabla_{Y}X-\left[X,Y\right]$
(also, see [7]).

For the integrability of a metallic structure $J$, the following proposition
immediately follows from Lemma 1.

######  Proposition 5.

Let $J$ be a metallic structure on $M$ and $\nabla$ be a linear connection on
$M$. If $\left(\nabla,J\right)$ is Codazzi-coupled and
$T^{\nabla}\left(X,Y\right)=0$, then the metallic structure $J$ is integrable.

##  3\. Codazzi Coupling of $\nabla$ with $g$ and $G$

Given a pseudo-Riemannian manifold $(M,g)$ endowed with a metallic structure
$J$, then the triple $(M,J,g)$ is called a metallic pseudo-Riemannian manifold
if [3]

$g(JX,Y)=g(X,JY).$

Also, the twin metallic pseudo-Riemannian metric $G$ is defined by

$G(X,Y)=g(JX,Y).$

From the equalities of the above, it is easy to see the following result.

######  Proposition 6.

Let $(M,J,g)$ be metallic pseudo-Riemannian manifold and $G$ be the twin
pseudo-Riemannian metric. Then the following equalities hold:

$1)\ G\left(X,Y\right)=G\left(Y,X\right)$;

$2)\ G\left(JX,Y\right)=G\left(X,JY\right)$, that is, $(M,J,G)$ is a metallic
pseudo-Riemannian manifold;

$3)\ g\left(J^{-1}X,Y\right)=g\left(X,J^{-1}Y\right)$, that is, $(M,J^{-1},g)$
is a metallic pseudo-Riemannian manifold;

$4)\ G\left(J^{-1}X,Y\right)=G\left(X,J^{-1}Y\right)$, that is, $(M,J^{-1},G)$
is a metallic pseudo-Riemannian manifold.

Let $C$ and $\Gamma$ be the $\left(0,3\right)$-tensor defined by

(3.1)
$C\left(X,Y,Z\right)\equiv\left(\nabla_{Z}g\right)\left(X,Y\right)=Zg\left(X,Y\right)-g\left(\nabla_{Z}X,Y\right)-g\left(X,\nabla_{Z}Y\right)$

and

(3.2)
$\Gamma\left(X,Y,Z\right)\equiv\left(\nabla_{Z}G\right)\left(X,Y\right)=ZG\left(X,Y\right)-G\left(\nabla_{Z}X,Y\right)-G\left(X,\nabla_{Z}Y\right).$

Due to symmetry of $g$ and $G$, it is clear that
$C\left(X,Y,Z\right)=C\left(Y,X,Z\right)$ and
$\Gamma\left(X,Y,Z\right)=\Gamma\left(Y,X,Z\right)$. From (1.1) and (3.1)
(resp., (1.2) and (3.2)), we get

$C\left(X,Y,Z\right)=g\left(X,\left(\nabla^{\ast}-\nabla\right)_{Z}Y\right)$

and

$\Gamma\left(X,Y,Z\right)=G\left(X,\left(\nabla^{{\dagger}}-\nabla\right)_{Z}Y\right).$

Also, it is easy to see that

(3.3)
$C^{\ast}\left(X,Y,Z\right)\equiv\left(\nabla_{Z}^{\ast}g\right)\left(X,Y\right)=-C\left(X,Y,Z\right)$

and

(3.4)
$\Gamma^{{\dagger}}\left(X,Y,Z\right)\equiv\left(\nabla_{Z}^{{\dagger}}G\right)\left(X,Y\right)=-\Gamma\left(X,Y,Z\right).$

For the metallic structure $J$, the relationship between $C=\nabla g$ and
$\Gamma=\nabla G$ is as follow

(3.5)
$\Gamma\left(X,Y,Z\right)=C\left(X,JY,Z\right)+g\left(X,\left(\nabla_{Z}J\right)Y\right).$

From (3.5), if $\nabla$ is a $J-$connection, then we get
$\Gamma\left(X,Y,Z\right)=C\left(X,JY,Z\right)$.

Next, we give the following proposition which is analogue to result given in
[7] for Hermitian setting.

######  Proposition 7.

Let $G$ be the twin metallic pseudo-Riemannian metric, $\nabla$ be a linear
connection and $\nabla^{{\dagger}}$ be the $G-$conjugate connection of
$\nabla$. Then the following conditions are equivalent

$1)\ \left(\nabla,G\right)$ is Codazzi-coupled,

$2)\ \left(\nabla^{{\dagger}},G\right)$ is Codazzi-coupled,

$3)\ \Gamma$ is totally symmetric,

$4)\ \Gamma^{{\dagger}}$ is totally symmetric,

$5)\ T^{\nabla}=T^{\nabla^{{\dagger}}}$,
where $\Gamma=\nabla G$ and $\Gamma^{{\dagger}}=\nabla^{{\dagger}}G$.

###### Proof.

The assertion can be easily proved by following the method in [7, 8]. ∎

######  Proposition 8.

Let $(M,J,g)$ be a metallic pseudo-Riemannian manifold, $G$ be the twin
metallic pseudo-Riemannian metric and $\nabla$ be a linear connection. If both
$\left(\nabla,J\right)$ and $\left(\nabla,G\right)$ are Codazzi-coupled, then

$G\left(\left(\nabla_{X}^{{\dagger}}J\right)Y-\left(\nabla_{Y}^{{\dagger}}J\right)X,Z\right)=G\left(Y,\left(\nabla_{Z}J\right)X-\left(\nabla_{Z}^{{\dagger}}J\right)X\right),$

where $\nabla^{{\dagger}}$ is the $G-$conjugate connection of $\nabla$.

###### Proof.

If both $\left(\nabla,J\right)$ and $\left(\nabla,G\right)$ are Codazzi-
coupled, then

$\displaystyle
G\left(\left(\nabla_{X}^{{\dagger}}J\right)Y-\left(\nabla_{Y}^{{\dagger}}J\right)X,Z\right)$
$\displaystyle=$ $\displaystyle
G\left(\nabla_{X}^{{\dagger}}JY-J\nabla_{X}^{{\dagger}}Y,Z\right)-G\left(\nabla_{Y}^{{\dagger}}JX-J\nabla_{Y}^{{\dagger}}X,Z\right)$
$\displaystyle=$ $\displaystyle
XG\left(Z,JY\right)-G\left(\nabla_{X}Z,JY\right)-YG\left(Z,JX\right)+G\left(\nabla_{Y}Z,JX\right)$
$\displaystyle+G\left(J\nabla_{Y}^{{\dagger}}X,Z\right)-G\left(J\nabla_{X}^{{\dagger}}Y,Z\right)$
$\displaystyle=$
$\displaystyle\Gamma\left(Z,JY,X\right)+G\left(Z,\nabla_{X}JY\right)-\Gamma\left(Z,JX,Y\right)-G\left(\nabla_{Y}JX,Z\right)$
$\displaystyle+G\left(JZ,\nabla_{Y}^{{\dagger}}X-\nabla_{X}^{{\dagger}}Y\right)$
$\displaystyle=$
$\displaystyle\Gamma\left(Z,JY,X\right)-\Gamma\left(Z,JX,Y\right)+G\left(JZ,T^{\nabla}\left(X,Y\right)-T^{\nabla^{{\dagger}}}\left(X,Y\right)\right)$
$\displaystyle=$
$\displaystyle\Gamma\left(X,JY,Z\right)-\Gamma\left(Y,JX,Z\right)$
$\displaystyle=$ $\displaystyle
G\left(Y,\left(\nabla_{Z}J\right)X-\left(\nabla_{Z}^{{\dagger}}J\right)X\right),$

which completes the proof. ∎

######  Proposition 9.

Let $(M,J,g)$ be a metallic pseudo-Riemannian manifold, $G$ be the twin
metallic pseudo-Riemannian metric and $\nabla$ be a linear connection. If both
$\left(\nabla,g\right)$ and $\left(\nabla,G\right)$ are Codazzi-coupled, then
$\left(\nabla^{\ast},J\right)$ is Codazzi-coupled, where $\nabla^{\ast}$ is
the $g-$conjugate connection of $\nabla$.

###### Proof.

From (3.5), we have

$C\left(X,JY,Z\right)=\Gamma\left(X,Y,Z\right)-g\left(X,\left(\nabla_{Z}J\right)Y\right)$

and

$C\left(Z,JY,X\right)=\Gamma\left(Z,Y,X\right)-g\left(Z,\left(\nabla_{X}J\right)Y\right).$

Since both $\left(\nabla,g\right)$ and $\left(\nabla,G\right)$ are Codazzi-
coupled, we can write

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
g\left(Z,\left(\nabla_{X}J\right)Y\right)-g\left(X,\left(\nabla_{Z}J\right)Y\right)$
$\displaystyle=$ $\displaystyle
g\left(Y,\left(\nabla_{X}^{\ast}J\right)Z\right)-g\left(Y,\left(\nabla_{Z}^{\ast}J\right)X\right)$
$\displaystyle=$ $\displaystyle
g\left(Y,\left(\nabla_{X}^{\ast}J\right)Z-\left(\nabla_{Z}^{\ast}J\right)X\right),$

which gives that $\left(\nabla^{\ast},J\right)$ is Codazzi-coupled. ∎

######  Proposition 10.

Let $(M,J,g)$ be a metallic pseudo-Riemannian manifold and $G$ be the twin
metallic pseudo-Riemannian metric. Then

1) If both $\left(\nabla,J\right)$ and $\left(\nabla^{\ast},J\right)$ are
Codazzi-coupled, then we have $\nabla=\nabla^{\ast}$, where $\nabla^{\ast}$ is
the $g-$conjugate connection of $\nabla$;

2) If both $\left(\nabla,J\right)$ and $\left(\nabla^{{\dagger}},J\right)$ are
Codazzi-coupled, then we have $\nabla=\nabla^{{\dagger}}$, where
$\nabla^{{\dagger}}$ is the $G-$conjugate connection of $\nabla$.

###### Proof.

Here we will prove only 1). Similarly, 2) can be proven. Assume that both
$\left(\nabla,J\right)$ and $\left(\nabla^{\ast},J\right)$ are Codazzi-
coupled. From (1.1), it is easy to see that

$g\left(\left(\nabla_{X}J\right)Y,Z\right))=g\left(Y,\left(\nabla_{X}^{\ast}J\right)Z\right).$

Thus, we can write

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
g\left(\left(\nabla_{X}J\right)Y-\left(\nabla_{Y}J\right)X,Z\right)$
$\displaystyle=$ $\displaystyle
g\left(Y,\left(\nabla_{Z}^{\ast}J\right)X\right)-g\left(Y,\left(\nabla_{Z}J\right)X\right)$
$\displaystyle=$ $\displaystyle
g\left(Y,\left(\nabla_{Z}^{\ast}J\right)X-\left(\nabla_{Z}J\right)X\right)$

that implies $\nabla^{\ast}J=\nabla J$. ∎

######  Proposition 11.

Let $(M,J,g)$ be a metallic pseudo-Riemannian manifold and $G$ be the twin
metallic pseudo-Riemannian metric. If $\nabla$ is a $J-$connection, then both
$\nabla^{\ast}$ and $\nabla^{{\dagger}}$ are $J-$connections.

###### Proof.

Assume that $\nabla$ is a $J-$connection. Then, we have

$\displaystyle g\left(Y,\nabla_{X}^{\ast}JZ-J\nabla_{X}^{\ast}Z\right)$
$\displaystyle=$ $\displaystyle
g\left(Y,\nabla_{X}^{\ast}JZ\right)-g\left(Y,J\nabla_{X}^{\ast}Z\right)$
$\displaystyle=$ $\displaystyle
X\left(g\left(Y,JZ\right)\right)-g\left(\nabla_{X}Y,JZ\right)-g\left(Y,J\nabla_{X}^{\ast}Z\right)$
$\displaystyle=$ $\displaystyle
X\left(g\left(Y,JZ\right)\right)-g\left(\nabla_{X}Y,JZ\right)-X\left(g\left(JY,Z\right)\right)+g\left(\nabla_{X}JY,Z\right)$
$\displaystyle=$ $\displaystyle 0$

such that $\nabla_{X}^{\ast}JZ=J\nabla_{X}^{\ast}Z$. Similarly, using the twin
metallic pseudo-Riemannian metric $G$, it can be proven that
$\nabla^{{\dagger}}$ is a $J-$connection. ∎

######  Proposition 12.

Let $(M,J,g)$ be a metallic pseudo-Riemannian manifold, $G$ be the twin
metallic pseudo-Riemannian metric, and $\nabla^{\ast}$ and
$\nabla^{{\dagger}}$ respectively denote the $g-$conjugate and $G-$conjugate
connection of a linear connection $\nabla$. If $\nabla$ is a $J-$connection,
then

$\displaystyle C\left(X,JY,Z\right)$ $\displaystyle=$
$\displaystyle-C^{\ast}\left(X,JY,Z\right)=-C^{{\dagger}}\left(X,JY,Z\right)$
$\displaystyle=$
$\displaystyle\Gamma\left(X,Y,Z\right)=-\Gamma^{{\dagger}}\left(X,Y,Z\right)=-\Gamma^{\ast}\left(X,Y,Z\right),$

where $C^{{\dagger}}=\nabla^{{\dagger}}g$ and $\Gamma^{\ast}=\nabla^{\ast}G$.

###### Proof.

Suppose that $\nabla$ is a $J-$connection. From (3.3), (3.4) and (12), we have

$\displaystyle C\left(X,JY,Z\right)$ $\displaystyle=$
$\displaystyle-C^{\ast}\left(X,JY,Z\right)$
$\displaystyle\Gamma\left(X,Y,Z\right)$ $\displaystyle=$
$\displaystyle-\Gamma^{{\dagger}}\left(X,Y,Z\right)$

and

$C\left(X,JY,Z\right)=\Gamma\left(X,Y,Z\right).$

Moreover, since

$\displaystyle\Gamma^{\ast}\left(X,Y,Z\right)$ $\displaystyle\equiv$
$\displaystyle\left(\nabla_{Z}^{\ast}G\right)\left(X,Y\right)=ZG\left(X,Y\right)-G\left(\nabla_{Z}^{\ast}X,Y\right)-G\left(X,\nabla_{Z}^{\ast}Y\right)$
$\displaystyle=$ $\displaystyle
Zg\left(JX,Y\right)-g\left(\nabla_{Z}^{\ast}X,JY\right)-g\left(JX,\nabla_{Z}^{\ast}Y\right)$
$\displaystyle=$ $\displaystyle
g\left(J\nabla_{Z}Y,X\right)-g\left(X,J\nabla_{Z}^{\ast}Y\right)$
$\displaystyle=$ $\displaystyle-C\left(X,JY,Z\right)$

and

$\displaystyle C^{{\dagger}}\left(X,JY,Z\right)$ $\displaystyle\equiv$
$\displaystyle\left(\nabla_{Z}^{{\dagger}}g\right)\left(X,JY\right)=Zg\left(X,JY\right)-g\left(\nabla_{Z}^{{\dagger}}X,JY\right)-g\left(JX,\nabla_{Z}^{{\dagger}}Y\right)$
$\displaystyle=$ $\displaystyle
ZG\left(X,Y\right)-G\left(\nabla_{Z}^{{\dagger}}X,Y\right)-G\left(X,\nabla_{Z}^{{\dagger}}Y\right)$
$\displaystyle=$ $\displaystyle\Gamma^{{\dagger}}\left(X,Y,Z\right),$

we obtain (12) which ends the proof. ∎

As a direct result of Proposition 12, we have the following proposition.

######  Proposition 13.

Let $(M,J,g)$ be a metallic pseudo-Riemannian manifold, $G$ be the twin
metallic pseudo-Riemannian metric, and $\nabla^{\ast}$ and
$\nabla^{{\dagger}}$ respectively denote the $g-$conjugate and $G-$conjugate
connection of a linear connection $\nabla$. Under the assumption that $\nabla$
is a $J-$connection, If at least one of the pairs $\left(\nabla,g\right),$
$\left(\nabla^{\ast},g\right),$ $\left(\nabla^{{\dagger}},g\right),$
$\left(\nabla,G\right),$ $\left(\nabla^{\ast},G\right)$ and
$\left(\nabla^{{\dagger}},G\right)$ is Codazzi-coupled, then all the remaining
pairs are Codazzi-coupled.

######  Proposition 14.

Let $(M,J,g)$ be a metallic pseudo-Riemannian manifold, $G$ be the twin
metallic pseudo-Riemannian metric, and $\nabla^{\ast}$ and
$\nabla^{{\dagger}}$ respectively denote the $g-$conjugate and $G-$conjugate
connection of a linear connection $\nabla$. If $\nabla$ is a $J-$connection,
then

$\begin{array}[]{cc}C\left(JX,Y,Z\right)=C\left(X,JY,Z\right),&\Gamma\left(JX,Y,Z\right)=\Gamma\left(X,JY,Z\right),\\\
C^{\ast}\left(JX,Y,Z\right)=C^{\ast}\left(X,JY,Z\right),&\Gamma^{\ast}\left(JX,Y,Z\right)=\Gamma^{\ast}\left(X,JY,Z\right),\\\
C^{{\dagger}}\left(JX,Y,Z\right)=C^{{\dagger}}\left(X,JY,Z\right),&\Gamma^{{\dagger}}\left(JX,Y,Z\right)=\Gamma^{{\dagger}}\left(X,JY,Z\right).\end{array}$

###### Proof.

If $\nabla$ is a $J-$connection, then

$\displaystyle C\left(JX,Y,Z\right)$ $\displaystyle=$ $\displaystyle
Zg\left(JX,Y\right)-g\left(\nabla_{Z}JX,Y\right)-g\left(JX,\nabla_{Z}Y\right)$
$\displaystyle=$ $\displaystyle
Zg\left(X,JY\right)-g\left(\nabla_{Z}X,JY\right)-g\left(X,\nabla_{Z}JY\right)$
$\displaystyle=$ $\displaystyle C\left(X,JY,Z\right)$

and

$\displaystyle\Gamma\left(JX,Y,Z\right)$ $\displaystyle=$ $\displaystyle
ZG\left(JX,Y\right)-G\left(\nabla_{Z}JX,Y\right)-G\left(JX,\nabla_{Z}Y\right)$
$\displaystyle=$ $\displaystyle
ZG\left(X,JY\right)-G\left(\nabla_{Z}X,JY\right)-G\left(X,\nabla_{Z}JY\right)$
$\displaystyle=$ $\displaystyle\Gamma\left(X,JY,Z\right).$

With help of Proposition 11, the other equalities can be obtained in a similar
way. ∎

Next, let us introduce a tensor operator which is applied to pure tensor
fields. We refer to [20] for details about pure tensor fields and tensor
operators. One of the most important classes of tensor operators is the class
of Tachibana operators associated with a considering fixed $(1,1)-$tensor
field. The Tachibana operators were firstly defined and studied by Tachibana
in [24]. Lather, the Tachibana operators was applied in Norden geometry and in
theory of lifts in [19, 20, 21, 22, 10]. In here, we aims to give a new
characterization of locally metallic pseudo-Riemannian manifolds by means of
the Tachibana operator.

######  Proposition 15.

Let $(M,J,g)$ be a metallic pseudo-Riemannian manifold and $\nabla$ be a
torsion-free connection. Then

$\left(\Phi_{J}g\right)\left(X,Y,Z\right)=C\left(Y,Z,JX\right)-\Gamma\left(Y,Z,X\right)+g\left(\left(\nabla_{Y}J\right)X,Z\right)+g\left(\left(\nabla_{Z}J\right)X,Y\right),$

where $\Phi_{J}g:T_{2}^{0}\left(M\right)\rightarrow T_{3}^{0}\left(M\right)$
is the Tachibana operator applied to the pseudo-Riemannian metric $g$.

###### Proof.

Using the definition of the operator $\Phi_{J}g$, we have

$\left(\Phi_{J}g\right)\left(X,Y,Z\right)=JXg\left(Y,Z\right)-X\left(g\left(JY,Z\right)\right)+g\left(\left(L_{Y}J\right)X,Z\right)+g\left(Y,\left(L_{Z}J\right)X\right),$

where $\left(L_{X}J\right)Y=\left[X,JY\right]-J\left[X,Y\right]$. Since
$\nabla$ is a torsion-free linear connection, we can rewrite it as

$\displaystyle\left(\Phi_{J}g\right)\left(X,Y,Z\right)$ $\displaystyle=$
$\displaystyle JXg\left(Y,Z\right)-X\left(g\left(JY,Z\right)\right)$
$\displaystyle+g\left(\left(\nabla_{Y}J\right)X,Z\right)-g\left(\nabla_{JX}Y,Z\right)+g\left(J\nabla_{X}Y,Z\right)$
$\displaystyle+g\left(Y,\left(\nabla_{Z}J\right)X\right)-g\left(Y,\nabla_{JX}Z\right)+g\left(Y,J\nabla_{X}Z\right)$
$\displaystyle=$ $\displaystyle
C\left(Y,Z,JX\right)-\Gamma\left(Y,Z,X\right)+g\left(\left(\nabla_{Y}J\right)X,Z\right)+g\left(\left(\nabla_{Z}J\right)X,Y\right),$

where $\left[X,Y\right]=\nabla_{X}Y-\nabla_{Y}X$. ∎

A statistical manifold is a (pseudo-) Riemannian manifold $(M,g,\nabla)$ with
a (pseudo-) Riemannian metric $g$ and a torsion-free linear connection
$\nabla$ for which $C=\nabla g$ is totally symmetric, that is, the Codazzi
equation holds [12]

$(\nabla_{X}g)(Y,Z)=(\nabla_{Y}g)(X,Z)=(\nabla_{Z}g)(X,Y).$

The following theorem gives a new characterization of locally metallic pseudo-
Riemannian manifolds by means of the Tachibana operator and Codazzi-coupled.

######  Theorem 1.

Let $(M,J,g)$ be a metallic pseudo-Riemannian manifold and $\nabla$ be a
torsion-free connection. If $\nabla$ is a $J-$connection and
$\left(\nabla,g\right)$ is Codazzi-coupled, then $\Phi_{J}g=0$. Furthermore,
the triple $(M,J,g)$ is a locally metallic pseudo-Riemannian manifold.

###### Proof.

From Proposition 15 and $\nabla J=0$, we have

$\left(\Phi_{J}g\right)\left(X,Y,Z\right)=C\left(Y,Z,JX\right)-C\left(Y,JZ,X\right).$

Since $\left(\nabla,g\right)$ is Codazzi-coupled, from Proposition 14, we get
the following

$\displaystyle\left(\Phi_{J}g\right)\left(X,Y,Z\right)$ $\displaystyle=$
$\displaystyle C\left(JX,Z,Y\right)-C\left(X,JZ,Y\right)$ $\displaystyle=$
$\displaystyle 0.$

Because of Proposition 1, we can say that the triple $(M,J,g)$ is a locally
metallic pseudo-Riemannian manifold. ∎

In [23], for a non-integrable almost paracomplex manifold $(M,P)$ (that is,
$P^{2}=I$, where $I=identity$) with compatible metric $g$, authors give the
following

$\displaystyle\left(\Phi_{P}g\right)\left(X,Y,Z\right)+\left(\Phi_{P}g\right)\left(Y,Z,X\right)+\left(\Phi_{P}g\right)\left(Z,X,Y\right)$
$\displaystyle=$ $\displaystyle
g\left(\left(\nabla_{X}^{g}P\right)Y,Z\right)+g\left(\left(\nabla_{Y}^{g}P\right)Z,X\right)+g\left(\left(\nabla_{Z}^{g}P\right)X,Y\right),$

where $\nabla^{g}$ is the Levi-Civita connection of $g$. If
$\left(\Phi_{P}g\right)\left(X,Y,Z\right)+\left(\Phi_{P}g\right)\left(Y,Z,X\right)+\left(\Phi_{P}g\right)\left(Z,X,Y\right)=g\left(\left(\nabla_{X}^{g}P\right)Y,Z\right)+g\left(\left(\nabla_{Y}^{g}P\right)Z,X\right)+g\left(\left(\nabla_{Z}^{g}P\right)X,Y\right)=0$,
they called the triple $(M,P,g)$ a quasi (para-)Kähler manifold which is
analogue of quasi Kähler manifold in [13].

Analogously, if a non-integrable metallic pseudo Riemannian manifold $(M,J,g)$
satisfies

$\left(\Phi_{J}g\right)\left(X,Y,Z\right)+\left(\Phi_{J}g\right)\left(Y,Z,X\right)+\left(\Phi_{J}g\right)\left(Z,X,Y\right)=0,$

then we are calling the triple $(M,J,g)$ a quasi metallic pseudo-Riemannian
manifold.

######  Proposition 16.

Let $(M,J,g)$ be a non-integrable metallic pseudo-Riemannian manifold and
$\nabla$ be a torsion-free connection. If $\left(\nabla,g\right)$ is Codazzi-
coupled, then

$\displaystyle\left(\Phi_{J}g\right)\left(X,Y,Z\right)+\left(\Phi_{J}g\right)\left(Y,Z,X\right)+\left(\Phi_{J}g\right)\left(Z,X,Y\right)$
$\displaystyle=$ $\displaystyle
g\left(\left(\nabla_{X}J\right)Y,Z\right)+g\left(\left(\nabla_{Y}J\right)Z,X\right)+g\left(\left(\nabla_{Z}J\right)X,Y\right).$

###### Proof.

From Proposition 15, we know that

$\left(\Phi_{J}g\right)\left(X,Y,Z\right)=C\left(Y,Z,JX\right)-\Gamma\left(Y,Z,X\right)+g\left(\left(\nabla_{Y}J\right)X,Z\right)+g\left(\left(\nabla_{Z}J\right)X,Y\right).$

If $\left(\nabla,g\right)$ is Codazzi-coupled, then

$\displaystyle\left(\Phi_{J}g\right)\left(X,Y,Z\right)+\left(\Phi_{J}g\right)\left(Y,Z,X\right)+\left(\Phi_{J}g\right)\left(Z,X,Y\right)$
$\displaystyle=$ $\displaystyle
C\left(JX,Z,Y\right)+C\left(JY,X,Z\right)+C\left(JZ,Y,X\right)$
$\displaystyle-\left(\Gamma\left(Y,Z,X\right)+\Gamma\left(Z,X,Y\right)+\Gamma\left(X,Y,Z\right)\right)$
$\displaystyle+g\left(\left(\nabla_{Y}J\right)X,Z\right)+g\left(\left(\nabla_{Z}J\right)X,Y\right)+g\left(\left(\nabla_{Z}J\right)Y,X\right)+g\left(\left(\nabla_{X}J\right)Y,Z\right)$
$\displaystyle+g\left(\left(\nabla_{X}J\right)Z,Y\right)+g\left(\left(\nabla_{Y}J\right)Z,X\right)$

from which we can write the following

$\displaystyle\left(\Phi_{J}g\right)\left(X,Y,Z\right)+\left(\Phi_{J}g\right)\left(Y,Z,X\right)+\left(\Phi_{J}g\right)\left(Z,X,Y\right)$
$\displaystyle=$ $\displaystyle
g\left(\left(\nabla_{X}J\right)Y,Z\right)+g\left(\left(\nabla_{Y}J\right)Z,X\right)+g\left(\left(\nabla_{Z}J\right)X,Y\right).$

∎

By means of the above proposition, we get the following result.

######  Corollary 1.

Let $(M,J,g)$ be a non-integrable metallic pseudo-Riemannian manifold and
$\nabla$ be a torsion-free connection. Under the assumption that
$\left(\nabla,g\right)$ is Codazzi-coupled, the triple $(M,J,g)$ is a quasi
metallic pseudo Riemannian manifold if and only if
$g\left(\left(\nabla_{X}J\right)Y,Z\right)+g\left(\left(\nabla_{Y}J\right)Z,X\right)+g\left(\left(\nabla_{Z}J\right)X,Y\right)=0$.

##  4\. Generalized conjugate connections and their applications

Generalizations of conjugate connections are studied in this section. Similar
problems studied for conjugate connections in the previous section will be
searched for generalized connections.

######  Definition 5.

Let $(M,J,g)$ be a metallic pseudo-Riemannian manifold and $\nabla$ be a
linear connection on $M$. The generalized conjugate connection
$\nabla^{\ast\prime}$ of $\nabla$ with respect to $g$ by a $1-$form $\tau$ is
defined by [14, 16, 5]

$Xg\left(Y,Z\right)=g\left(\nabla_{X}Y,Z\right)+g\left(Y,\nabla_{X}^{\ast\prime}Z\right)-\tau\left(X\right)g\left(Y,Z\right).$

It can easily be checked that the generalized conjugate connection
$\nabla^{\ast\prime}$ of $\nabla$ with respect to $g$ is involutive, that is,
$\left(\nabla^{\ast\prime}\right)^{\ast\prime}=\nabla$.

######  Proposition 17.

Let $(M,J,g)$ be a metallic pseudo-Riemannian manifold and $\nabla$ be a
linear connection on $M$. If $\nabla^{\ast\prime}$is a generalized conjugate
connection of $\nabla$ with respect to $g$ by $\tau$, then we have

$\nabla\text{ is a }J-\text{connection if and only if
}\nabla^{\ast\prime}\text{ is a }J-\text{connection.}$

###### Proof.

Suppose that $\nabla$ is a $J-$connection. Using the definition of the
generalized conjugate connection, it is possible to write the following

$\displaystyle g\left(Y,\nabla_{X}^{\ast\prime}JZ\right)$ $\displaystyle=$
$\displaystyle
Xg\left(Y,JZ\right)-g\left(\nabla_{X}Y,JZ\right)+\tau\left(X\right)g\left(Y,JZ\right)$
$\displaystyle=$ $\displaystyle
g\left(\nabla_{X}JY,Z\right)+g\left(JY,\nabla_{X}^{\ast\prime}Z\right)-g\left(\nabla_{X}Y,JZ\right)$
$\displaystyle=$ $\displaystyle g\left(Y,J\nabla_{X}^{\ast\prime}Z\right)$

such that $\nabla_{X}^{\ast\prime}JZ=J\nabla_{X}^{\ast\prime}Z$. Conversely,
let $\nabla^{\ast\prime}$ be a $J-$connection. Then

$\displaystyle g\left(Y,\nabla_{X}JZ\right)$ $\displaystyle=$ $\displaystyle
Xg\left(Y,JZ\right)-g\left(JZ,\nabla_{X}^{\ast\prime}Y\right)+\tau\left(X\right)g\left(Y,JZ\right)$
$\displaystyle=$ $\displaystyle
g\left(\nabla_{X}^{\ast\prime}JY,Z\right)+g\left(JY,\nabla_{X}Z\right)-g\left(JZ,\nabla_{X}^{\ast\prime}Y\right)$
$\displaystyle=$ $\displaystyle g\left(Y,J\nabla_{X}Z\right)$

which implies $\nabla_{X}JZ=J\nabla_{X}Z$. ∎

######  Proposition 18.

Let $(M,J,g)$ be a metallic pseudo-Riemannian manifold and $\nabla$ be a
linear connection, and $\nabla^{\ast\prime}$ and $\nabla^{{\dagger}\prime}$
denote generalized conjugate connections of $\nabla$ with respect to $g$ and
$G$ by $\tau$, respectively. Then

$\text{if }\nabla\text{ is a }J-\text{connection, then
}\nabla^{\ast\prime}=\nabla^{{\dagger}\prime}\text{. }$

where $\left(\nabla^{{\dagger}\prime}\right)^{{\dagger}\prime}=\nabla$.

###### Proof.

The definition of the generalized conjugate connection of $\nabla$ with
respect to $G$ by $\tau$ immediately gives

$\displaystyle XG\left(Y,Z\right)$ $\displaystyle=$ $\displaystyle
G\left(\nabla_{X}Y,Z\right)+G\left(Y,\nabla_{X}^{{\dagger}\prime}Z\right)-\tau\left(X\right)G\left(Y,Z\right)$
$\displaystyle=$ $\displaystyle
g\left(J\nabla_{X}Y,Z\right)+g\left(JY,\nabla_{X}^{{\dagger}\prime}Z\right)-\tau\left(X\right)g\left(JY,Z\right)$
$\displaystyle=$ $\displaystyle Xg\left(JY,Z\right).$

Since

$Xg\left(JY,Z\right)=g\left(\nabla_{X}JY,Z\right)+g\left(JY,\nabla_{X}^{\ast\prime}Z\right)-\tau\left(X\right)g\left(JY,Z\right),$

we find

$g\left(\nabla_{X}JY-J\nabla_{X}Y,Z\right)+g\left(JY,\nabla_{X}^{\ast\prime}Z-\nabla_{X}^{{\dagger}\prime}Z\right)=0$

from which we immediately obtain that if $\nabla J=0$, then
$\nabla^{\ast\prime}=\nabla^{{\dagger}\prime}$. ∎

######  Proposition 19.

Let $(M,J,g)$ be a metallic pseudo-Riemannian manifold and
$\nabla^{\ast\prime}$ be a generalized conjugate connection of $\nabla$ with
respect to $g$ by $\tau$. Then $\nabla^{\ast\prime}$ is a generalized
conjugate connection of $\nabla^{J}$ with respect to $G$ by $\tau$.

###### Proof.

We have

$\displaystyle G\left(\nabla_{X}^{J}Y,Z\right)$ $\displaystyle=$
$\displaystyle G\left(J^{-1}\nabla_{X}JY,Z\right)$ $\displaystyle=$
$\displaystyle g\left(\nabla_{X}JY,Z\right)$ $\displaystyle=$ $\displaystyle
Xg\left(JY,Z\right)-g\left(JY,\nabla_{X}^{\ast\prime}Z\right)+\tau\left(X\right)g\left(JY,Z\right)$
$\displaystyle=$ $\displaystyle
XG\left(Y,Z\right)-G\left(Y,\nabla_{X}^{\ast\prime}Z\right)+\tau\left(X\right)G\left(Y,Z\right).$

∎

Now, we consider another generalized connection which is called dual-
projectively equivalent connection.

######  Definition 6.

[5] Let $(M,J,g)$ be a metallic pseudo-Riemannian manifold and $\nabla$ and
$\nabla^{\prime}$ be linear connections on $M$. We say that $\nabla$ and
$\nabla^{\prime}$ are dual-projectively equivalent if there exists a $1$-form
$\tau$ such that

$\nabla_{X}^{\prime}Y=\nabla_{X}Y-g\left(X,Y\right)\tau^{\sharp},$

where $g\left(X,\tau^{\sharp}\right)=\tau\left(X\right)$.

The following proposition ends this section.

######  Proposition 20.

If $\left(\nabla,J\right)$ is Codazzi-coupled, then each of the pairs
$(\nabla^{\prime},J),$ $(\nabla^{\prime},J^{-1}),$
$(\left(\nabla^{J}\right)^{\prime},J)$ and
$(\left(\nabla^{J}\right)^{\prime},J^{-1})$ is Codazzi-coupled.

###### Proof.

Assume that the pair $\left(\nabla,J\right)$ is Codazzi-coupled. Then we have

$\displaystyle\left(\nabla_{X}^{\prime}J\right)Y-\left(\nabla_{Y}^{\prime}J\right)X$
$\displaystyle=$
$\displaystyle\nabla_{X}^{\prime}JY-J\nabla_{X}^{\prime}Y-\nabla_{Y}^{\prime}JX+J\nabla_{Y}^{\prime}X$
$\displaystyle=$
$\displaystyle\nabla_{X}JY-g\left(X,JY\right)\tau^{\sharp}-\nabla_{Y}JX+g\left(Y,JX\right)\tau^{\sharp}$
$\displaystyle-J\left(\nabla_{X}Y-g\left(X,Y\right)\tau^{\sharp}-\nabla_{Y}X+g\left(Y,X\right)\tau^{\sharp}\right)$
$\displaystyle=$
$\displaystyle\left(\nabla_{X}J\right)Y-\left(\nabla_{Y}J\right)X$
$\displaystyle=$ $\displaystyle 0.$

Similarly, it can be easily shown that the pairs $(\nabla^{\prime},J^{-1})$ ,
$(\left(\nabla^{J}\right)^{\prime},J)$ and
$(\left(\nabla^{J}\right)^{\prime},J^{-1})$ are Codazzi-coupled too. ∎

##  5\. Meatallic-like pseudo-Riemannian manifolds

In this section we shall present some results concerning with a metallic-like
pseudo-Riemannian manifold.

######  Definition 7.

[11] Let $\left(M,g\right)$ be a pseudo-Riemannian manifold with a metallic
structure $J$ which has another $\left(1,1\right)-$tensor field $J^{\ast}$
satisfying

$g\left(JX,Y\right)=g\left(X,J^{\ast}Y\right),$

where $\left(J^{\ast}\right)^{\ast}=J$. Then $(M,J,g)$ is called a metallic-
like pseudo-Riemannian manifold.

Standard calculations immediately give the following proposition.

######  Proposition 21.

Let $(M,J,g)$ be a metallic-like pseudo-Riemannian manifold. Then, the
following properties hold

$1)\ \left(J^{\ast}\right)^{2}=pJ^{\ast}+qI,$

$2)\ \left(J^{\ast}\right)^{-1}=\frac{1}{q}J^{\ast}-\frac{p}{q}I,$

$3)\ g\left(J^{-1}X,Y\right)=g\left(X,\left(J^{\ast}\right)^{-1}Y\right),$

$4)\
g\left(JX,J^{\ast}Y\right)=pg\left(X,J^{\ast}Y\right)+qg\left(X,Y\right),$

$5)\
g\left(\left(\nabla_{X}J^{\ast}\right)Y,Z\right)=g\left(Y,\left(\nabla_{X}^{\ast}J\right)Z\right),$

$6)\
g\left(\left(\nabla_{X}^{\ast}J^{\ast}\right)Y,Z\right)=g\left(Y,\left(\nabla_{X}J\right)Z\right).$

######  Proposition 22.

Let $(M,J,g)$ be a metallic-like pseudo-Riemannian manifold. Then we have

$1)\ \nabla$ is a $J^{\ast}-$connection if and only if $\nabla^{\ast}$ is a
$J-$connection.

$2)\ \nabla$ is a $J-$connection if and only if $\nabla^{\ast}$ is a
$J^{\ast}-$connection.

###### Proof.

$1)\ $Assume that $\nabla J^{\ast}=0$. From (1.1), we can write

$\displaystyle g\left(Y,\nabla_{X}^{\ast}JZ\right)$ $\displaystyle=$
$\displaystyle Xg\left(Y,JZ\right)-g\left(\nabla_{X}Y,JZ\right)$
$\displaystyle=$ $\displaystyle
Xg\left(J^{\ast}Y,Z\right)-g\left(\nabla_{X}J^{\ast}Y,Z\right)$
$\displaystyle=$ $\displaystyle g\left(J^{\ast}Y,\nabla_{X}^{\ast}Z\right)$
$\displaystyle=$ $\displaystyle g\left(Y,J\nabla_{X}^{\ast}Z\right)$

such that $\nabla^{\ast}JZ=J\nabla_{X}^{\ast}Z$. Conversely, if
$\nabla^{\ast}$ is $J-$connection, then

$\displaystyle g\left(Y,\nabla_{X}J^{\ast}Z\right)$ $\displaystyle=$
$\displaystyle
Xg\left(J^{\ast}Z,Y\right)-g\left(J^{\ast}Z,\nabla_{X}^{\ast}Y\right)$
$\displaystyle=$ $\displaystyle
Xg\left(Y,JY\right)-g\left(Z,\nabla^{\ast}JY\right)$ $\displaystyle=$
$\displaystyle g\left(\nabla_{X}Z,JY\right)$ $\displaystyle=$ $\displaystyle
g\left(Y,J^{\ast}\nabla_{X}Z\right)$

which implies $\nabla_{X}J^{\ast}Z=J^{\ast}\nabla_{X}Z$.

$2)$ Similarly, it can be obtained with standard calculations. ∎

######  Proposition 23.

Let $(M,J,g)$ be a metallic-like pseudo-Riemannian manifold.

$1)$ If both $\left(\nabla,J^{\ast}\right)$ and $\left(\nabla^{\ast},J\right)$
are Codazzi-coupled, then $\nabla J^{\ast}=\nabla^{\ast}J.$

$2)\ $If both $\left(\nabla^{\ast},J^{\ast}\right)$ and
$\left(\nabla,J\right)$ are Codazzi-coupled, then
$\nabla^{\ast}J^{\ast}=\nabla J.$

###### Proof.

$1)$ From $(5)$ of Proposition 21, we have

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
g\left(\left(\nabla_{X}J^{\ast}\right)Y-\left(\nabla_{Y}J^{\ast}\right)X,Z\right)$
$\displaystyle=$ $\displaystyle
g\left(Y,\left(\nabla_{X}^{\ast}J\right)Z\right)-g\left(X,\left(\nabla_{Y}^{\ast}J\right)Z\right)$
$\displaystyle=$ $\displaystyle
g\left(Y,\left(\nabla_{Z}^{\ast}J\right)X\right)-g\left(X,\left(\nabla_{Z}^{\ast}J\right)Y\right)$
$\displaystyle=$ $\displaystyle
g\left(X,\left(\nabla_{Z}J^{\ast}\right)Y\right)-g\left(X,\left(\nabla_{Z}^{\ast}J\right)Y\right)$
$\displaystyle=$ $\displaystyle
g\left(Y,\left(\nabla_{Z}J^{\ast}\right)Y-\left(\nabla_{Z}^{\ast}J\right)Y\right)$

which implies $\nabla J^{\ast}=\nabla^{\ast}J$.

$2)$ Similarly, it can be easily obtained. ∎

######  Proposition 24.

On a metallic-like pseudo-Riemannian manifold $(M,J,g)$, the following two
identities hold

$C\left(JX,Y,Z\right)=C\left(X,J^{\ast}Y,Z\right)+g\left(Y,\left(\nabla_{Z}^{\ast}J\right)X-\left(\nabla_{Z}J\right)X\right),$

where $C=\nabla g$, and

$C^{\ast}\left(JX,Y,Z\right)=C^{\ast}\left(X,J^{\ast}Y,Z\right)+g\left(X,\left(\nabla_{Z}^{\ast}J^{\ast}\right)Y-\left(\nabla_{Z}J^{\ast}\right)Y\right),$

where $C^{\ast}=\nabla^{\ast}g.$

###### Proof.

From Definition 7, we get

$\displaystyle C\left(JX,Y,Z\right)$ $\displaystyle=$ $\displaystyle
Zg\left(JX,Y\right)-g\left(\nabla_{Z}JX,Y\right)-g\left(JX,\nabla_{Z}Y\right)$
$\displaystyle=$ $\displaystyle
Zg\left(X,J^{\ast}Y\right)-g\left(\nabla_{Z}X,J^{\ast}Y\right)-g\left(X,\nabla_{Z}J^{\ast}Y\right)$
$\displaystyle+g\left(X,\left(\nabla_{Z}J^{\ast}\right)Y\right)-g\left(\left(\nabla_{Z}J\right)X,Y\right)$
$\displaystyle=$ $\displaystyle
C\left(X,J^{\ast}Y,Z\right)+g\left(Y,\left(\nabla_{Z}^{\ast}J\right)X-\left(\nabla_{Z}J\right)X\right)$

and similarly

$C^{\ast}\left(JX,Y,Z\right)=C^{\ast}\left(X,J^{\ast}Y,Z\right)+g\left(X,\left(\nabla_{Z}^{\ast}J^{\ast}\right)Y-\left(\nabla_{Z}J^{\ast}\right)Y\right).$

∎

Also note that if $\nabla J=0$ and $\nabla J^{\ast}=0,$ then we obtain

$C\left(JX,Y,Z\right)=C\left(X,J^{\ast}Y,Z\right)=-C^{\ast}\left(JX,Y,Z\right)=-C^{\ast}\left(X,J^{\ast}Y,Z\right)\text{.}$

######  Theorem 2.

On a metallic-like pseudo-Riemannian manifold $(M,J,g)$, if $\nabla J=0$,
$\nabla J^{\ast}=0$ and $\left(\nabla,g\right)$ is Codazzi-coupled, where
$\nabla$ is a torsion-free connection, then we get

$\left(\Phi_{J^{\ast}}g\right)\left(X,Y,Z\right)+\left(\Phi_{J}g\right)\left(X,Y,Z\right)=0.$

###### Proof.

Using the definition of the operator $\Phi_{J}g,$ we have

$\displaystyle\left(\Phi_{J^{\ast}}g\right)\left(X,Y,Z\right))$
$\displaystyle=$ $\displaystyle
J^{\ast}Xg\left(Y,Z\right)-g\left(\nabla_{J^{\ast}X}Y,Z\right)-g\left(Y,\nabla_{J^{\ast}X}Z\right)$
$\displaystyle-\left(Xg\left(J^{\ast}Y,Z\right)-g\left(J^{\ast}\nabla_{X}Y,Z\right)-g\left(Y,J^{\ast}\nabla_{X}Z\right)\right)$
$\displaystyle+g\left(\left(\nabla_{Y}J^{\ast}\right)X,Z\right)+g\left(Y,\left(\nabla_{Z}J^{\ast}\right)X\right)$
$\displaystyle=$ $\displaystyle
C\left(Y,Z,J^{\ast}X\right)-C\left(Y,JZ,X\right)-g\left(Y,\nabla_{X}JZ\right)+g\left(Y,J^{\ast}\nabla_{X}Z\right).$

From the hypothesis and Proposition 24, we get

$\left(\Phi_{J^{\ast}}g\right)\left(X,Y,Z\right))=g\left(Y,\left(J^{\ast}-J\right)\nabla_{X}Z\right).$

Similarly, we can calculate

$\left(\Phi_{J}g\right)\left(X,Y,Z\right)=g\left(Y,\left(J-J^{\ast}\right)\nabla_{X}Z\right).$

When we add the last two equations, we find

$\left(\Phi_{J^{\ast}}g\right)\left(X,Y,Z\right)+\left(\Phi_{J}g\right)\left(X,Y,Z\right)=0.$

∎

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