11institutetext: Center for the Internet of Things, ITER, Siksha ‘O’
Anusandhan (Deemed to be University), Bhubaneswar-751030, Odisha, India
22institutetext: Department of Electronics and Communication Engineering,
ITER, Siksha ‘O’ Anusandhan (Deemed to be University), Bhubaneswar-751030,
Odisha, India
22email<EMAIL_ADDRESS><EMAIL_ADDRESS>
https://erkundanec.github.io/

# Parametric Scaling of Preprocessing assisted U-net Architecture for
Improvised Retinal Vessel Segmentation

Kundan Kumar1 corresponding author Sumanshu Agarwal2

###### Abstract

Extracting blood vessels from retinal fundus images plays a decisive role in
diagnosing the progression in pertinent diseases. In medical image analysis,
vessel extraction is a semantic binary segmentation problem, where blood
vasculature needs to be extracted from the background. Here, we present an
image enhancement technique based on the morphological preprocessing coupled
with a scaled U-net architecture. Despite a relatively less number of
trainable network parameters, the scaled version of U-net architecture
provides better performance compare to other methods in the domain. We
validated the proposed method on retinal fundus images from the DRIVE
database. A significant improvement as compared to the other algorithms in the
domain, in terms of the area under ROC curve ($>0.9762$) and classification
accuracy ($>95.47\%$) are evident from the results. Furthermore, the proposed
method is resistant to the central vessel reflex while sensitive to detect
blood vessels in the presence of background items viz. exudates, optic disc,
and fovea.

###### Keywords:

Deep neural network, U-net, Top-hat transform, Retinal vessel extraction,
Blood vasculature, Medical image segmentation.

##  1 Introduction

Retinal fundus imaging is a pathological procedure used by medical
professionals to diagnose various diseases. Here, the posterior portion of the
human eye that contains blood vasculature map (tree-like structure) is
captured using ophthalmic camera [1]. Retinal vasculature map helps an
ophthalmologist to detect and diagnose numerous diseases, such as diabetic
retinopathy [2, 3], hypertension [4], glaucoma [5], age-related macular
degeneration, which have direct or indirect relation with the brain and heart
diseases [6]. By tracking the growth and change in vascular map of retinal
fundus images, timely treatment for the progression of pertinent diseases
could be identified. The extraction of retinal blood vessels from the
background in presence of other structures, such as exudates, optic disk,
hemorrhages, microaneurysms, etc. (shown in Fig. 1), makes it more challenging
and difficult.

Figure 1: Labelled diagram of Retinal fundus image

The extraction of retinal blood vessels (RBV) can be done by an image
processing tool that can handle the poor contrast between vessel and non-
vessel pixels, central vessel reflex, uneven illuminance, noise, etc. [7].
Many RBV extraction algorithms have been proposed in the literature to
overcome these challenges. Nevertheless, an algorithm that could address all
of these challenges is yet to be reported. Further, the features are manually
extracted before the classification, and those features are not self-
sufficient to handle all the challenges together [8]. Conventional methods are
broadly classified as (a) filtering-based methods, (b) mathematical
morphology, (c) trace-based methods, (d) multiscale based approaches, (e)
machine learning-based methods. Here, the first four methods are purely based
on image processing tools, whereas the methods belonging to the last class,
i.e. machine learning based methods, recognize the underlying patterns using
man-made features dependent classification algorithms. Machine learning
methods can be further sub-categorized as supervised and unsupervised
learning. While supervised methods learn the classification criteria by using
ground truth labelled image of retinal vessels, unsupervised methods segment
the blood vessels without knowing the class information of blood vessel and
background pixels.

Among the various machine learning techniques, deep learning (DL) has proven
their capability in many application such as image classification, medical
image analysis, scene understanding, object detection, pose estimation, etc.
[9]. The recent development in the field of deep learning algorithms are
encouraging, and the technique has become very popular to solve many real-life
problems. For example, convolutional deep neural network (CNN) has been
endorsed in medical image processing to solve most of the segmentation
problems and thereby make the system automated by eliminating the step of
manual feature extraction using the conventional methods. Researchers have
reviewed state-of-art automated retinal analysis based on DL [10, 11].

Figure 2: Scaled U-net architecture

Various convolutional neural network architectures have been introduced to
handle the automated retinal analysis problem to achieve the state-of-the-art
findings. The following architectures have been used to automate the vessel
detection in fundus images, but not limited to, deformable U-Net (DUNet) [12],
DeepVessel [13], IterNet [14]. Furthermore, fully connected convolutional
neural network (FCN) [15], SegNet [16], U-net [17], Recurrent Residual U-net
(R2U-net) [18] are well-accepted methods to segment the medical images because
the architectures are based on end-to-end training. Among all of these, U-net
is widely used in many applications like scene understanding, self-driving
car, vegetation monitoring, medical image analysis, etc. Though U-net
architecture is advantageous due to the requirement of a minimal amount of
data for training, it needs excessive data augmentation to train the model.

That said, here, we present a scaled version of U-net model (see Fig. 2) for
RBV segmentation by using comparatively less number of learning parameters. We
find that a reduction in the number of trainable network parameters leads to
the complexity reduction without compromising the performance. For the study,
patches extracted from the preprocessed retinal images are used. In the
preprocessing step, we perform following in the order: a) grayscale
conversion, b) top-hat transformation, and c) contrast limited adaptive
histogram equalization (CLAHE), to enhance the contrast and uneven illuminance
due to background structures like, exudates, optic disc, fovea, etc.
Thereafter, a large number of image patches are randomly extracted from the
preprocessed images to train the model. The major contribution of the paper
can be summarized as follows:

  * •

White top-hat transformation based preprocessing is adopted to enhance the
blood vessel pixel along with suppressing the other structures such as fovea,
optic disk, exudates, etc.

  * •

Introduced parametric scaling of U-net architecture (scaled down U-net
architecture) to reduce the number of learning parameters to fasten the
learning process.

The model suggested in the paper is validated on widely adopted retinal fundus
images for segmentation from DRIVE database [19], which is briefly discussed
in the next section, where preprocessing, augmentation, and scaled version of
the U-net model are also described in detail. Section 3 discusses the
experimental results and the conclusions are summarized in section 4.

##  2 Materials and Method

###  2.1 Datasets and material

To assess different models, the experiments were performed on DRIVE [19]
dataset (a publicly available dataset). The DRIVE (Digital Retinal Images for
Vessel Extraction) dataset contains 40 color images which includes 20 training
and 20 test image having a plane resolution of $565\times 584$. Each retinal
image is an 8-bit color image. For the test set, two subsets of manually
segmented ground truth images (from two different observers) are provided. The
manually segmented images from the first observer are regarded as the ground
truth or gold standard to assess the performance.

The segmentation is carried out in three steps: a) preprocessing and data
augmentation, b) network architecture design and training, and c)
postprocessing. However, in the present paper, we discuss the first two
aspects of the segmentation. It should be noted that the postprocessing can
further improve the performance and the work on the same is under progress.

###  2.2 Preprocessing and data augmentation

We considered single-channel retinal images for vessel segmentation.
Accordingly, the color retinal images are converted to grayscale by using the
following conversion rule ($value=0.30*R+0.59*G+0.11*B$), where $R$, $G$, and
$B$ represent red, green, and blue components of the color images,
respectively. Here, it is worth to mention that the green channel image can
also be used instead of grayscale conversion due to its high contrast. The
grayscale conversion was followed by a negative operation. Thereafter, a
morphological operation (white top-hat transformation [20]) was applied to
reduce the non-uniform illumination from the image. It has been well
established that morphological operations reduce the contrast of the whole
image [20]. Therefore, for image contrast enhancement, CLAHE was used.

The training processes of the CNN model (discussed in the next subsection)
takes a considerable amount of time for large images. Additionally, the CNN
model needs plenty of images to be trained. Given that, after preprocessing,
$9500$ patches of size $48\times 48$ were extracted randomly from the FOV of
each image, resulting in a total collection of $9500\times 20=190,000$ image
patches. Few sample image patches are shown in Fig. 3.

Figure 3: Few sample of preprocessed patches with its corresponding ground
truth

10% of the total patches were used for validation, and rest image patches are
used for training. Validation set is used to retain the best learned model
weight during training process. For the test image, overlapped image patches
of the same resolution $48\times 48$ were extracted at a stride of 5 pixels.

${\rm
No.~{}of~{}test~{}patch}=\left({\frac{{H-h}}{{s}}+1}\right)\times\left({\frac{{W-w}}{{s}}+1}\right)$
(1)

Here, $H\times W$ is the retinal image resolution, $h\times w$ is the patch
size, and $s$ is the stride. For the DRIVE dataset image, $109\times
105=11445$ image patches were extracted for each test images. Note that,
before extracting image patches, the test images were passed through the same
preprocessing step as applied to train images.

###  2.3 Architecture design and training

In this section, we discuss the proposed CNN model as shown in Fig. 2, which
is a scaled version of the U-net architecture developed by Olaf Ronneberger et
al. for biomedical image segmentation [17]. Similar to the full-size U-net
architecture, the proposed architecture consists of two paths: encoder (named
as contraction path) and decoder (known as expanding path). The encoder is
very similar to the conventional stacked encoder which contracts as the
network go deeper and deeper. It consists of fully connected convolutional
layers and max-pooling layers for better understanding of the larger receptive
field. The receptive field is a small portion of the input to produce only one
node in a feature map. Before the down-sampling, response of the previous
layer is processed using two convolution operation with $3\times 3$ kernels
with 1 padding to the operated image. There is a rectified linear unit (ReLU)
after each convolution layer. A $2\times 2$ max pooling operation with stride
2 is also used. Here, only 3 depth level architecture is modeled to design a
less complex architecture with less number of parameters to be learned. At
level 3, the lowest resolution of $12\times 12$ pixels is obtained to learn
the local information in the image patches. The first red box at level 1 is
the input layer which takes the single-channel (grayscale) image as an input.
Remaining all boxes refers to multi-scale feature maps. The size of each
feature map is given on the left side of the box (see Fig. 2), and the number
of channel is shown on the top of the boxes. In the decoder path at each
label, white boxes and up-sampled blue boxes are concatenated to regenerate
the feature map at each level. To project the feature map to the number of
classes, we use $1\times 1$ convolutional kernel at the last layer. After all
the processing, the segmentation map is obtained at the output. It is worth to
mention here that the size of image patches should be chosen in such a way
that all $2\times 2$ max-pooling operations can be applied with an even x- and
y-size.

In the present study, the network was trained using image patches and
corresponding segmentation map (ground truth) of size $48\times 48$. We train
the network from scratch starting from weight parameter initialization. The
initialization of weights in a deep neural network is extremely important to
avoid the excessive activation problem [17]. Therefore, initial weights are
drawn randomly from a Gaussian distribution, $\mathcal{N}(0,\sqrt{2/N})$,
where $N$ denotes the number of incoming nodes of one neuron [21]. ADAM
optimizer is used to update the weights in each iteration. At the beginning of
training, the learning rate is initialized with 0.001 and after every 20
epochs learning rate is reduced at 0.9 decay factor.

##  3 Experimental Results and discussion

The proposed model was trained using extracted image patches from the training
set of DRIVE dataset. The experiments were performed under Tensorflow [22] and
Keras [23] framework on the hardware configuration NVIDIA GeForce GTX 1080Ti
GPU. In this work, the random weights were initialized in each layer. The
preprocessed patches were normalized and used to train the network for 100
epochs. During training, 32 image patches were used in a single batch to
compute the average error at the output, and that error is backpropagated in
the network to update the weights. Accuracy and loss curves are shown in Fig.
4. It shows that preprocessing assisted U-net architecture achieves more than
95% accuracy in just 15 epochs.

Figure 4: Accuracy curve (left) and loss curve (right) of the model for
training and testing

Segmented binary maps of blood vessels and background for the retinal images
taken from DRIVE dataset are shown in Fig. 5.

Figure 5: Segmented result of test images from DRIVE dataset: (a) Preprocessed
original image (first row), (b) Ground truth (second row), (c) Output response
of proposed scaled U-net architecture (third row)

The first row is the preprocessed retinal images, and the corresponding gold
standard ground truth is shown in the second row. The final response of the
proposed model is shown in the third row of the figure. The fourth column of
the figure corresponds to the pathological image; however, the first three
columns correspond to normal retinal images. The figure indicates that the
proposed model segment the blood vessels with high accuracy, even for
pathological images. Furthermore, the proposed method is resistant to the
central vessel reflex. After performing many experiments, we found that use of
top-hat transformation as morphological preprocessing reduces the computation
time (takes less number of iterations) to train the proposed U-net
architecture for the same findings as performed by many state-of-art
algorithms on the DRIVE database.

We performed the quantitative analysis in terms of area under the curve (AUC)
of receiver operating characteristic (ROC), accuracy (Acc), and sensitivity
(Sen). Table 1 compares results obtained from the scaled U-net and state-of-
art vessel segmentation methods evaluated on DRIVE dataset. It shows that the
proposed method has better AUC, accuracy, and sensitivity compare to many
algorithms in the domain.

Table 1: Experimental results of proposed method for RBV segmentation and comparison with other conventional and Deep Learning based approaches Method |  AUC |  Acc |  Sen
---|---|---|---
2nd Observer |  - |  0.9473 |  -
Soares et al. (2006) [2] |  0.9614 |  0.9466 |  -
Ricci et al. (2007) [24] |  0.9558 |  0.9563 |  -
Azzopardi et al. (2015) [25] |  0.9614 |  0.9442 |  0.7526
Liskowski et al. (2016) [26] |  0.9720 |  0.9495 |  0.7763
Fu et al.  (2016) [13] |  - |  0.9523 |  0.7603
Alom et. al. (2018) (U-net) [18] |  0.9755 |  0.9531 |  0.7537
Proposed model (scaled U-net) |  0.9762 |  0.9547 |  0.7830

##  4 Conclusion

To summarize, here, a scaled version of U-net architecture has been proposed
for retinal blood vessel segmentation. Morphological processing based
preprocessing has been introduced to enhance the segmentation performance and
computational burden. The proposed model was evaluated using DRIVE database.
The experimental results manifest that scaled version of U-net performed
better segmentation with less number of learning parameters compare to
existing methods applied on retinal fundus images; however, morphological
preprocessing reduces the run-time for the network training process.
Furthermore, the proposed method is resistant to the central vessel reflex
while sensitive to detect blood vessels even if background structures viz.
exudates, optic disc, fovea, etc. are present.

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The functions $\mathbf{F}_{\phi_{q,2}}$ and $\mathbf{F}_{\phi_{q,3}}$ satisfy
the following quasi-homogeneity property:

$\displaystyle\Big{(}\frac{x_{q,1}}{2}\partial_{x_{q,1}}+x_{q,2}\partial_{x_{q,2}}+\frac{3}{2}x_{q,3}\partial_{x_{q,3}}\Big{)}.\mathbf{F}_{\phi_{q,2}}=3\mathbf{F}_{\phi_{q,2}};$
$\displaystyle\Big{(}y_{q,1}\partial_{y_{q,1}}+\frac{3}{2}y_{q,2}\partial_{y_{q,2}}+2y_{q,3}\partial_{y_{q,3}}\Big{)}.\mathbf{F}_{\phi_{q,3}}=4\mathbf{F}_{\phi_{q,3}}.$

##  8 Appendix

###  8.1 Appendix 1: Diagonal flat metrics

The goal of this Appendix is to prove the flatness property of the metric
$\mathbf{ds^{2}_{\beta_{1},\beta_{2}}}(\omega)$ given by (3.4).

###### Theorem 8.1

Let $(\beta_{1},\beta_{2})\neq(0,0)$ and $\omega$ be a differential satisfying
(3.1) and (3.3). Then the diagonal metric
$\mathbf{ds^{2}_{\beta_{1},\beta_{2}}}(\omega)$ defined in (3.4) is flat.

We need the following lemma which investigates some useful relations involving
the Christoffel symbols (3.6) of the diagonal metric (3.4).

###### Lemma 8.1

Let $i,j,s\in\big{\\{}1,\dots,N\big{\\}}$ be distinct. Then the Christoffel
symbols of the metric $\mathbf{ds^{2}_{\beta_{1},\beta_{2}}}(\omega)$ satisfy
the following relations:

$\displaystyle\partial_{\lambda_{s}}\Gamma_{ij}^{i}=\Gamma_{ij}^{i}\Gamma_{js}^{j}-\Gamma_{ij}^{i}\Gamma_{is}^{i}+\Gamma_{is}^{i}\Gamma_{sj}^{s},$
(8.1)
$\displaystyle\partial_{\lambda_{s}}\Gamma_{ii}^{j}=\Gamma_{ii}^{j}\Gamma_{is}^{i}-\Gamma_{ii}^{j}\Gamma_{js}^{j}-\Gamma_{ii}^{s}\Gamma_{ss}^{j},$
(8.2)
$\displaystyle\partial_{\lambda_{i}}\Gamma_{ij}^{i}=\frac{\omega(P_{j})}{2\omega(P_{i})}\partial_{\lambda_{i}}W(P_{i},P_{j})+\Gamma_{ij}^{i}\Gamma_{ij}^{j}-\frac{\partial_{\lambda_{i}}\omega(P_{i})}{\omega(P_{i})}\Gamma_{ij}^{i},$
(8.3)
$\displaystyle\partial_{\lambda_{j}}\Gamma_{ij}^{i}=\frac{\omega(P_{j})}{2\omega(P_{i})}\partial_{\lambda_{j}}W(P_{i},P_{j})-(\Gamma_{ij}^{i})^{2}+\frac{\partial_{\lambda_{j}}\omega(P_{j})}{\omega(P_{j})}\Gamma_{ij}^{i},$
(8.4)
$\displaystyle\partial_{\lambda_{j}}\Gamma_{ii}^{i}=\partial_{\lambda_{i}}\Gamma_{ij}^{i},$
(8.5)
$\displaystyle\begin{split}\begin{array}[]{ll}\displaystyle\sum_{\underset{s\neq
i,j}{s=1,}}^{N}\Gamma_{jj}^{s}\Gamma_{is}^{i}&=\displaystyle\frac{\beta_{1}}{\beta_{1}\lambda_{j}+\beta+2}\Gamma_{ij}^{i}+\frac{\omega(P_{j})}{2\omega(P_{i})}\partial_{\lambda_{j}}W(P_{i},P_{j})\\\
&\quad\displaystyle+\frac{\beta_{1}\lambda_{i}+\beta_{2}}{\beta_{1}\lambda_{j}+\beta_{2}}\times\frac{\omega(P_{j})}{2\omega(P_{i})}\partial_{\lambda_{i}}W(P_{i},P_{j}).\end{array}\end{split}$
(8.6)

_Proof:_ Taking the partial derivative with respect to a branch point
$\lambda_{s}$, $s\neq i,j$, in the first relation in (3.6) and using Rauch
formulas (2.15) and (3.2), we obtain (8.1). Relation (8.2) follows immediately
from the second equality in (3.6) and (8.1). We get the two equalities (8.3)
and (8.4) using (3.6) and (3.2).
By virtue of Rauch formulas (3.2), expressions (3.6) for the Christoffel
symbols and (8.4) we obtain

$\displaystyle\partial_{\lambda_{j}}\Gamma_{ii}^{i}$
$\displaystyle=\frac{\partial_{\lambda_{i}}\partial_{\lambda_{j}}\omega(P_{i})}{\omega(P_{i})}-\frac{\partial_{\lambda_{i}}\omega(P_{i})}{(\omega(P_{i}))^{2}}\partial_{\lambda_{j}}\omega(P_{i})$
$\displaystyle=\frac{\omega(P_{j})}{2\omega(P_{i})}\partial_{\lambda_{i}}W(P_{i},P_{j})+\frac{1}{4}W(P_{i},P_{j})^{2}-\frac{\partial_{\lambda_{i}}\omega(P_{i})}{\omega(P_{i})}\Gamma_{ij}^{i}$
$\displaystyle=\partial_{\lambda_{i}}\Gamma_{ij}^{i},$

which establishes (8.5).
Finally, note that from (3.6) and the Rauch formulas, we can write

$\forall\ s\neq
i,j,\quad\Gamma_{jj}^{s}\Gamma_{is}^{i}=-\frac{\beta_{1}\lambda_{s}+\beta_{2}}{\beta_{1}\lambda_{j}+\beta_{2}}\frac{\omega(P_{j})}{2\omega(P_{i})}\partial_{\lambda_{s}}W(P_{i},P_{j}).$

We obtain (8.6) using this relation together with the action (3.7) of the
vector field $\sum_{k}(\beta_{1}\lambda_{k}+\beta_{2})\partial_{\lambda_{k}}$
on $W(P_{i},P_{j})$.

$\square$

We now move on to prove Theorem 8.1.

_Proof of Theorem 8.1:_ We are going to show that the Riemann curvature tensor
of the metric $\mathbf{ds^{2}_{\beta_{1},\beta_{2}}}(\omega)$ vanishes
identically. Recall that the Riemann curvature tensor is defined by

$R_{ijk}^{t}:=\partial_{j}\Gamma_{ik}^{t}-\partial_{k}\Gamma_{ij}^{t}+\textstyle\sum_{s}\big{(}\Gamma_{ik}^{s}\Gamma_{sj}^{t}-\Gamma_{ij}^{s}\Gamma_{sk}^{t}\big{)}.$

We distinguish four cases.

_First case:_ $i,j,k\in\\{1,\dots,N\\}$ are distinct. We have

$\displaystyle R_{ijk}^{t}$
$\displaystyle=\partial_{j}\Gamma_{ik}^{t}-\partial_{k}\Gamma_{ij}^{t}+\Gamma_{ik}^{i}\Gamma_{ij}^{t}+\Gamma_{ik}^{k}\Gamma_{kj}^{t}-\Gamma_{ij}^{i}\Gamma_{ik}^{t}-\Gamma_{ij}^{j}\Gamma_{jk}^{t}.$

If $t\notin\\{i,j,k\\}$, clearly we have $R_{ijk}^{t}=0$. By (8.1) we deduce
that if $t=i$,

$\displaystyle R_{ijk}^{i}$
$\displaystyle=\partial_{j}\Gamma_{ik}^{i}-\partial_{k}\Gamma_{ij}^{i}=\Gamma_{ik}^{i}\Gamma_{kj}^{k}-\Gamma_{ik}^{i}\Gamma_{ij}^{i}+\Gamma_{jk}^{j}\Gamma_{ij}^{i}-\Gamma_{ij}^{i}\Gamma_{jk}^{j}+\Gamma_{ij}^{i}\Gamma_{ik}^{i}-\Gamma_{kj}^{k}\Gamma_{ik}^{i}=0$

and if $t=j$,

$\displaystyle R_{ijk}^{j}$
$\displaystyle=-\partial_{k}\Gamma_{ij}^{j}+\Gamma_{ik}^{i}\Gamma_{ij}^{j}+\Gamma_{ik}^{k}\Gamma_{kj}^{j}-\Gamma_{ij}^{i}\Gamma_{ik}^{j}-\Gamma_{ij}^{j}\Gamma_{jk}^{j}$
$\displaystyle=-\Gamma_{ji}^{j}\Gamma_{ik}^{i}+\Gamma_{ji}^{j}\Gamma_{jk}^{j}-\Gamma_{jk}^{j}\Gamma_{ki}^{j}+\Gamma_{ik}^{i}\Gamma_{ij}^{j}+\Gamma_{ik}^{k}\Gamma_{kj}^{j}-\Gamma_{ij}^{j}\Gamma_{jk}^{j}=0.$

Again (8.1) implies the vanishing of $R_{ijk}^{k}$. Indeed,

$\displaystyle R_{ijk}^{k}$
$\displaystyle=\partial_{j}\Gamma_{ik}^{k}-\partial_{k}\Gamma_{ij}^{k}+\Gamma_{ik}^{i}\Gamma_{ij}^{k}+\Gamma_{ik}^{k}\Gamma_{kj}^{k}-\Gamma_{ij}^{i}\Gamma_{ik}^{k}-\Gamma_{ij}^{j}\Gamma_{jk}^{k}$
$\displaystyle=\Gamma_{ik}^{k}\Gamma_{ij}^{i}-\Gamma_{ik}^{k}\Gamma_{kj}^{k}+\Gamma_{kj}^{k}\Gamma_{ji}^{i}+\Gamma_{ik}^{k}\Gamma_{kj}^{k}-\Gamma_{ij}^{i}\Gamma_{ik}^{k}-\Gamma_{ij}^{j}\Gamma_{jk}^{k}=0.$

_Second case:_ $i=j\neq k$. In this case, the Riemann curvature tensor takes
the following form

$\displaystyle R_{iik}^{t}$
$\displaystyle=\partial_{i}\Gamma_{ik}^{t}-\partial_{k}\Gamma_{ii}^{t}+\Gamma_{ik}^{i}\Gamma_{ii}^{t}+\Gamma_{ik}^{k}\Gamma_{ki}^{t}-\sum_{s}\Gamma_{ii}^{s}\Gamma_{sk}^{t}.$

If $t\notin\\{i,k\\}$ then by (8.2), we get

$\displaystyle R_{iik}^{t}$
$\displaystyle=-\partial_{k}\Gamma_{ii}^{t}+\Gamma_{ik}^{i}\Gamma_{ii}^{t}+\Gamma_{ik}^{k}\Gamma_{ki}^{t}-\Gamma_{ii}^{k}\Gamma_{kk}^{t}-\Gamma_{ii}^{t}\Gamma_{tk}^{t}$
$\displaystyle=-\Gamma_{ii}^{t}\Gamma_{ik}^{i}+\Gamma_{ii}^{t}\Gamma_{kt}^{t}+\Gamma_{ii}^{k}\Gamma_{kk}^{t}+\Gamma_{ik}^{i}\Gamma_{ii}^{t}-\Gamma_{ii}^{k}\Gamma_{kk}^{t}-\Gamma_{ii}^{t}\Gamma_{tk}^{t}=0.$

If $t=i$, then (8.5) implies that

$\displaystyle R_{iik}^{i}$
$\displaystyle=\partial_{i}\Gamma_{ik}^{i}-\partial_{k}\Gamma_{ii}^{i}+\Gamma_{ik}^{i}\Gamma_{ii}^{i}+\Gamma_{ik}^{k}\Gamma_{ki}^{i}-\sum_{s}\Gamma_{ii}^{s}\Gamma_{sk}^{i}$
$\displaystyle=\Gamma_{ik}^{i}\Gamma_{ii}^{i}+\Gamma_{ik}^{k}\Gamma_{ki}^{i}-\Gamma_{ii}^{i}\Gamma_{ik}^{i}-\Gamma_{ii}^{k}\Gamma_{kk}^{i}$
$\displaystyle=\Gamma_{ik}^{k}\Gamma_{ki}^{i}-\Gamma_{ii}^{k}\Gamma_{kk}^{i}=0.$

If $t=k$, according to (3.6), (8.3), (8.4) and (8.6) we can write

$\displaystyle R_{iik}^{k}$
$\displaystyle=\partial_{i}\Gamma_{ik}^{k}-\partial_{k}\Gamma_{ii}^{k}+\Gamma_{ik}^{i}\Gamma_{ii}^{k}+(\Gamma_{ik}^{k})^{2}-\sum_{s}\Gamma_{ii}^{s}\Gamma_{sk}^{k}$
$\displaystyle=\partial_{i}\Gamma_{ik}^{k}+\frac{\beta_{1}}{\beta_{1}\lambda_{i}+\beta_{2}}\Gamma_{ik}^{k}+\frac{\beta_{1}\lambda_{k}+\beta_{2}}{\beta_{1}\lambda_{i}+\beta_{2}}\partial_{k}\Gamma_{ik}^{k}+\Gamma_{ik}^{i}\Gamma_{ii}^{k}+(\Gamma_{ik}^{k})^{2}-\Gamma_{ii}^{i}\Gamma_{ik}^{k}-\Gamma_{ii}^{k}\Gamma_{kk}^{k}-\sum_{s\neq
i,k}\Gamma_{ii}^{s}\Gamma_{sk}^{k}$
$\displaystyle=\frac{\omega(P_{i})}{2\omega(P_{k})}\partial_{\lambda_{i}}W(P_{k},P_{i})-(\Gamma_{ik}^{k})^{2}+\frac{\partial_{\lambda_{i}}\omega(P_{i})}{\omega(P_{i})}\Gamma_{ik}^{k}+\frac{\beta_{1}}{\beta_{1}\lambda_{i}+\beta_{2}}\Gamma_{ik}^{k}$
$\displaystyle\quad+\frac{\beta_{1}\lambda_{k}+\beta_{2}}{\beta_{1}\lambda_{i}+\beta_{2}}\bigg{(}\frac{\omega(P_{i})}{2\omega(P_{k})}\partial_{\lambda_{k}}W(P_{k},P_{i})+\Gamma_{ik}^{k}\Gamma_{ik}^{i}-\frac{\partial_{\lambda_{k}}\omega(P_{k})}{\omega(P_{k})}\Gamma_{ik}^{k}\bigg{)}$
$\displaystyle\quad-\frac{\beta_{1}\lambda_{k}+\beta_{2}}{\beta_{1}\lambda_{i}+\beta_{2}}\Gamma_{ik}^{k}\Gamma_{ik}^{i}+(\Gamma_{ik}^{k})^{2}-\bigg{(}-\frac{\beta_{1}}{2\beta_{1}\lambda_{i}+2\beta_{2}}+\frac{\partial_{\lambda_{i}}\omega(P_{i})}{\omega(P_{i})}\bigg{)}\Gamma_{ik}^{k}$
$\displaystyle\quad+\frac{\beta_{1}\lambda_{k}+\beta_{2}}{\beta_{1}\lambda_{i}+\beta_{2}}\bigg{(}-\frac{\beta_{1}}{2\beta_{1}\lambda_{k}+2\beta_{2}}+\frac{\partial_{\lambda_{k}}\omega(P_{k})}{\omega(P_{k})}\bigg{)}\Gamma_{ik}^{k}-\sum_{s\neq
i,k}\Gamma_{ii}^{s}\Gamma_{sk}^{k}$
$\displaystyle=\frac{\omega(P_{i})}{2\omega(P_{k})}\partial_{\lambda_{i}}W(P_{k},P_{i})+\frac{\beta_{1}\lambda_{k}+\beta_{2}}{\beta_{1}\lambda_{i}+\beta_{2}}\frac{\omega(P_{i})}{2\omega(P_{k})}\partial_{\lambda_{k}}W(P_{k},P_{i})+\frac{\beta_{1}}{\beta_{1}\lambda_{i}+\beta_{2}}\Gamma_{ik}^{k}-\sum_{s\neq
i,k}\Gamma_{ii}^{s}\Gamma_{sk}^{k}$ $\displaystyle=0.$

_Third case:_ $i=k\neq j$. The vanishing of $R_{iji}^{t}$ follows from the
second case and the properties of the Riemann tensor:

$\displaystyle\forall\ t,\quad R_{iji}^{t}=-R_{iij}^{t}=0.$

_Fourth case:_ If $j=k$, then we always have $R_{ijj}^{t}=R_{iii}^{t}=0$.

$\square$

###  8.2 Appendix 2: Multivalued differentials and quasi-homogeneity
properties

In this appendix we explore some results related to multivalued differentials
defined on surface $C_{g}$ of genus $g\geq 1$, where
$\big{\\{}(C_{g},\lambda),\\{a_{k},b_{k}\\}_{k}\big{\\}}$ is a fixed point in
the simple Hurwitz space $\widehat{\mathcal{H}}_{g,L}(n_{0},\dots,n_{m})$.
Let $\sigma$ and $\nu$ be two positive integers. For $k=1,\dots,g$, consider
the multivalued differentials

$\phi_{u^{k,\sigma}}(P):=\oint_{a_{k}}\big{(}\lambda(Q)\big{)}^{\sigma}W(P,Q),\quad\quad\phi_{\widetilde{u}^{k,\nu}}(P):=\oint_{b_{k}}\big{(}\lambda(Q)\big{)}^{\nu}W(P,Q).$
(8.7)

###### Proposition 8.1

Let $j=1,\dots,g$. The differentials $\phi_{u^{k,\sigma}}$ and
$\phi_{\widetilde{u}^{k,\nu}}$ (8.7) have the following jumps:

$\begin{split}&\phi_{u^{k,\sigma}}(P+b_{j})-\phi_{u^{k,\sigma}}(P)=-2{\rm
i}\pi\delta_{jk}d\big{(}\lambda(P)\big{)}^{\sigma};\\\
&\phi_{u^{k,\sigma}}(P+a_{j})-\phi_{u^{k,\sigma}}(P)=0;\\\
&\phi_{\widetilde{u}^{k,\nu}}(P+a_{j})-\phi_{\widetilde{u}^{k,\nu}}(P)=2{\rm
i}\pi\delta_{jk}d\big{(}\lambda(P)\big{)}^{\nu};\\\
&\phi_{\widetilde{u}^{k,\nu}}(P+b_{j})-\phi_{\widetilde{u}^{k,\nu}}(P)=0.\end{split}$
(8.8)

Moreover, the periods of the differentials $\phi_{u^{k,\sigma}}$ and
$\phi_{\widetilde{u}^{k,\nu}}$ are as follows:

$\begin{split}&\oint_{a_{j}}\phi_{u^{k,\sigma}}=0;\\\
&\oint_{b_{j}}\phi_{u^{k,\sigma}}=-2{\rm{i}}\pi\big{(}\lambda(P_{0})\big{)}^{\sigma}\delta_{jk}+2{\rm{i}}\pi\oint_{a_{k}}\big{(}\lambda(P)\big{)}^{\sigma}\omega_{j}(P);\\\
&\oint_{a_{j}}\phi_{\widetilde{u}^{k,\nu}}=2{\rm
i}\pi\delta_{jk}\big{(}\lambda(P_{0})\big{)}^{\nu};\\\
&\oint_{b_{j}}\phi_{\widetilde{u}^{k,\nu}}=2{\rm
i}\pi\oint_{b_{k}}\big{(}\lambda(P)\big{)}^{\nu}\omega_{j}(P).\end{split}$
(8.9)

Here $P_{0}$ denotes an arbitrary starting point of the cycles $a_{k},b_{k}$.

_Proof:_ We obtain the jumps and the periods of the multivalued differential
$\phi_{u^{k,\sigma}}$ by a direct adaptation of the proofs of (3.29) and
(3.30).
Let us deal with the differential $\phi_{\widetilde{u}^{k,\nu}}$. We have

$\displaystyle\big{(}\lambda(Q)\big{)}^{\nu}W(P,Q)=\big{(}\lambda(Q)\big{)}^{\nu}d_{P}d_{Q}\log\Big{(}\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(Q)\big{)}\Big{)}$
$\displaystyle=d_{Q}\Big{[}\big{(}\lambda(Q)\big{)}^{\nu}d_{P}\log\Big{(}\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(Q)\big{)}\Big{)}\Big{]}-\Big{[}d_{P}\log\Big{(}\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(Q)\big{)}\Big{)}\Big{]}d\big{(}\lambda(Q)\big{)}^{\nu}.$

Integrating both sides with respect to $Q$ along the contour $b_{k}$ while
choosing a starting point $P_{0}$ of the contour $b_{k}$ and keeping in mind
the quasi-periodicity properties (2.9) of the Riemann Theta function, we see
that the differential $\phi_{\widetilde{u}^{k,\nu}}$ can be represented by

$\phi_{\widetilde{u}^{k,\nu}}(P)=2{\rm
i}\pi\big{(}\lambda(P_{0})\big{)}^{\nu}\omega_{k}(P)-\oint_{b_{k}}d_{P}\log\Big{(}\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(Q)\big{)}\Big{)}d\big{(}\lambda(Q)\big{)}^{\nu}.$

Let us define

$F_{k,\nu}(P):=\int^{P}\phi_{\widetilde{u}^{k,\nu}}=2{\rm
i}\pi\big{(}\lambda(P_{0})\big{)}^{\nu}\int^{P}\omega_{k}-\oint_{b_{k}}\log\Big{(}\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(Q)\big{)}\Big{)}d\big{(}\lambda(Q)\big{)}^{\nu}.$

Then, due to the quasi-periodicity property (2.9) of the Riemann Theta
function, we see that $F_{k,\nu}$ is a multivalued function with

$\displaystyle F_{k,\nu}(P+a_{j})-F_{k,\nu}(P)$
$\displaystyle=0=\oint_{a_{j}}\phi_{\widetilde{u}^{k,\nu}},\quad\forall\ j\neq
k;$ $\displaystyle F_{k,\nu}(P+a_{k})-F_{k,\nu}(P)$ $\displaystyle=2{\rm
i}\pi\big{(}\lambda(P_{0})\big{)}^{\nu}=\oint_{a_{k}}\phi_{\widetilde{u}^{k,\nu}};$
$\displaystyle F_{k,\nu}(P+b_{j})-F_{k,\nu}(P)$ $\displaystyle=2{\rm
i}\pi\big{(}\lambda(P_{0})\big{)}^{\nu}\mathbb{B}_{jk}-\oint_{b_{k}}\Big{(}-{\rm
i}\pi\mathbb{B}_{jj}-2{\rm{i}}\pi\mathop{\langle\beta,e_{j}\rangle}-2{\rm
i}\pi\int_{Q}^{P}\omega_{j}\Big{)}d\big{(}\lambda(Q)\big{)}^{\nu}$
$\displaystyle=2{\rm
i}\pi\big{(}\lambda(P_{0})\big{)}^{\nu}\mathbb{B}_{jk}+2{\rm
i}\pi\oint_{b_{k}}\Big{(}\int_{Q}^{P}\omega_{j}\Big{)}d\big{(}\lambda(Q)\big{)}^{\nu}$
$\displaystyle=2{\rm
i}\pi\big{(}\lambda(P_{0})\big{)}^{\nu}\mathbb{B}_{jk}+2{\rm
i}\pi\Big{(}\big{(}\lambda(Q)\big{)}^{\nu}\int_{Q}^{P}\omega_{j}\Big{)}\bigg{|}_{Q=P_{0}}^{Q=P_{0}+b_{k}}+2{\rm
i}\pi\oint_{b_{k}}\big{(}\lambda(Q)\big{)}^{\nu}\omega_{j}(Q)$
$\displaystyle=2{\rm
i}\pi\oint_{b_{k}}\big{(}\lambda(Q)\big{)}^{\nu}\omega_{j}(Q).$

This, given that $\phi_{\widetilde{u}^{k,\nu}}(P)=dF_{k,\nu}(P)$, yields

$\phi_{\widetilde{u}^{k,\nu}}(P+b_{j})-\phi_{\widetilde{u}^{k,\nu}}(P)=0=\phi_{\widetilde{u}^{k,\nu}}(P+a_{l})-\phi_{\widetilde{u}^{k,\nu}}(P)$

for all $j,l=1,\dots,g$ and $l\neq k$.
On the other hand, the expression $F_{k,\nu}(P+a_{k})-F_{k,\nu}(P)$ assumes
that the point $P$ is considered as the starting point of the contour $a_{k}$.
Therefore, since the point $P_{0}$ can be chosen as the intersection point of
the contours $a_{k}$ and $b_{k}$, i.e. $P_{0}=P$, we get

$\phi_{\widetilde{u}^{k,\nu}}(P+a_{k})-\phi_{\widetilde{u}^{k,\nu}}(P)=dF_{k,\nu}(P+a_{k})-dF_{k,\nu}(P)=2\rm{i}\pi
d\big{(}\lambda(P)\big{)}^{\nu}.$

$\square$

###### Proposition 8.2

Let $\phi_{u^{k,\sigma}}$ and $\phi_{\widetilde{u}^{k,\nu}}$ be the
multivalued differentials defined by (8.7) and $t^{A}$ be an operation from
the set $\big{\\{}t^{i,\alpha},\ v^{i},\ s^{i},\ \rho^{k},\ u^{k}\Big{\\}}$
described in (3.21)-(3.22).
Then the action of the vector field $e=\sum_{j=1}^{N}\partial_{\lambda_{j}}$
on the functions $t^{A}\big{(}\phi_{u^{k,\sigma}}\big{)}$ and
$t^{A}\big{(}\phi_{\widetilde{u}^{k,\nu}}\big{)}$ is given by

$\begin{split}&e.t^{A}\big{(}\phi_{u^{k,\sigma}}\big{)}=\sigma\oint_{a_{k}}\lambda(P)^{\sigma-1}\phi_{t^{A}}(P);\\\
&e.t^{A}\big{(}\phi_{\widetilde{u}^{k,\nu}}\big{)}=2{\rm
i}\pi\big{(}\lambda(P_{0})\big{)}^{\nu}+\nu\oint_{b_{k}}\big{(}\lambda(P)\big{)}^{\nu-1}\phi_{t^{A}}(P).\end{split}$
(8.10)

_Proof:_ Let us denote by $F_{g}$ the fundamental polygon associated with the
compact surface $C_{g}$. The differential

$\Phi_{A,k,\sigma}(P):=\frac{\phi_{t^{A}}(P)}{d\lambda(P)}\phi_{u^{k,\sigma}}(P)$
(8.11)

is multivalued and, by (8.8), its jumps are such that
$\Phi_{A,k,\sigma}(P+a_{j})-\Phi_{A,k,\sigma}(P)=0$ and

$\Phi_{A,k,\sigma}(P+b_{j})-\Phi_{A,k,\sigma}(P)=-2{\rm
i}\pi\Big{(}\delta_{jk}\sigma\big{(}\lambda(P)\big{)}^{\sigma-1}\phi_{t^{A}}(P)+\delta_{\phi_{t^{A}},\phi_{u^{j}}}\phi_{u^{k,\sigma}}(P)-2{\rm
i}\pi\delta_{jk}\delta_{\phi_{t^{A}},\phi_{u^{j}}}d\big{(}\lambda(P)\big{)}^{\sigma}\Big{)}.$

Therefore, using Rauch formulas (3.15) and the residue theorem we conclude
that

$\displaystyle e.t^{A}\big{(}\phi_{u^{k,\sigma}}\big{)}$
$\displaystyle=\frac{1}{2}\sum_{j=1}^{N}\phi_{t^{A}}(P_{j})\phi_{u^{k,\sigma}}(P_{j})=\sum_{j=1}^{N}\underset{P_{j}}{{\rm
res}}\Phi_{A,k,\sigma}(P)=\frac{1}{2{\rm
i}\pi}\oint_{F_{g}}\Phi_{A,k,\sigma}(P)$ $\displaystyle=\frac{1}{2{\rm
i}\pi}\sum_{j=1}^{g}\bigg{(}\oint_{a_{j}}\Phi_{A,k,\sigma}(P)-\oint_{a_{j}}\Phi_{A,k,\sigma}(P+b_{j})\bigg{)}$
$\displaystyle=\sigma\oint_{a_{k}}\lambda(P)^{\sigma-1}\phi_{t^{A}}(P).$

Now, due to (3.29) and (8.8), the jumps of the multivalued differential

$\widetilde{\Phi}_{A,k,\nu}(P):=\frac{\phi_{t^{A}}(P)}{d\lambda(P)}\phi_{\widetilde{u}^{k,\nu}}(P)$

are as follows:

$\displaystyle\widetilde{\Phi}_{A,k,\nu}(P+a_{j})-\widetilde{\Phi}_{A,k,\nu}(P)=2{\rm
i}\pi\delta_{jk}\nu\big{(}\lambda(P)\big{)}^{\nu-1}\phi_{t^{A}}(P);$
$\displaystyle\widetilde{\Phi}_{A,k,\nu}(P+b_{j})-\widetilde{\Phi}_{A,k,\nu}(P)=-2{\rm
i}\pi\delta_{\phi_{t^{A}},\phi_{u^{j}}}\phi_{\widetilde{u}^{k,\nu}}(P).$

Using this and proceeding as in the previous case, we deduce that

$\displaystyle
e.t^{A}\big{(}\phi_{\widetilde{u}^{k,\nu}}\big{)}=\sum_{j=1}^{N}\underset{P_{j}}{{\rm
res}}\widetilde{\Phi}_{A,k,\nu}(P)=\frac{1}{2{\rm
i}\pi}\oint_{F_{g}}\widetilde{\Phi}_{A,k,\nu}(P)$
$\displaystyle=\frac{1}{2{\rm
i}\pi}\sum_{j=1}^{g}\bigg{(}\oint_{a_{j}}\widetilde{\Phi}_{A,k,\nu}(P)-\oint_{a_{j}}\widetilde{\Phi}_{A,k,\nu}(P+b_{j})\bigg{)}+\frac{1}{2{\rm
i}\pi}\sum_{j=1}^{g}\bigg{(}\oint_{b_{j}}\widetilde{\Phi}_{A,k,\nu}(P)-\oint_{b_{j}}\widetilde{\Phi}_{A,k,\nu}(P+a_{j}^{-1})\bigg{)}$
$\displaystyle=\sum_{j=1}^{g}\bigg{(}\delta_{\phi_{t^{A}},\phi_{u^{j}}}\oint_{a_{j}}\phi_{\widetilde{u}^{k,\nu}}(P)\bigg{)}+\nu\oint_{b_{k}}\big{(}\lambda(P)\big{)}^{\nu-1}\phi_{t^{A}}(P)$
$\displaystyle=2{\rm
i}\pi\big{(}\lambda(P_{0})\big{)}^{\nu}+\nu\oint_{b_{k}}\big{(}\lambda(P)\big{)}^{\nu-1}\phi_{t^{A}}(P).$

$\square$

###### Proposition 8.3

Consider the multivalued differentials $\phi_{u^{k,\sigma}}$ and
$\phi_{\widetilde{u}^{k,\nu}}$ defined by (8.7) and assume that all the cycles
$\\{a_{k},b_{k}\\}$ start at a marked point $P_{0}$ whose projection by
$\lambda$ is zero. Then, for any operation $t^{A}$ among those defining flat
coordinates in (3.21)-(3.22), the functions
$t^{A}\big{(}\phi_{u^{k,\sigma}}\big{)}$ and
$t^{A}\big{(}\phi_{\widetilde{u}^{k,\nu}}\big{)}$ are eigenfunctions of the
vector field $E=\sum_{s=1}^{N}\lambda_{s}\partial_{\lambda_{s}}$ with

$\displaystyle
E.t^{A}\big{(}\phi_{u^{k,\sigma}}\big{)}=(d_{A}+\sigma)t^{A}\big{(}\phi_{u^{k,\sigma}}\big{)};$
(8.12) $\displaystyle
E.t^{A}\big{(}\phi_{\widetilde{u}^{k,\nu}}\big{)}=(d_{A}+\nu)t^{A}\big{(}\phi_{\widetilde{u}^{k,\nu}}\big{)}.$
(8.13)

_Proof:_ Let $(C_{g},\lambda)$ be a generic branched covering representing an
equivalence class in the simple Hurwitz space
$\widehat{\mathcal{H}}_{g,L}(n_{0},\dots,n_{m})$. Denote by $F_{g}$ the
fundamental polygon associated with the compact surface $C_{g}$.
Using the properties of the multivalued differential $\Phi_{A,k,\sigma}$
defined by (8.11), we deduce that the differential
$\lambda(P)\Phi_{A,k,\sigma}(P)$ has no jumps along the cycles $a_{j}$ and its
jumps along the cycle $b_{j}$ are as follows:

$\displaystyle\lambda(P)\Phi_{A,k,\sigma}(P+b_{j})-\lambda(P)\Phi_{A,k,\sigma}(P)$
$\displaystyle=-2{\rm
i}\pi\Big{(}\delta_{jk}\sigma\big{(}\lambda(P)\big{)}^{\sigma}\phi_{t^{A}}(P)+\delta_{\phi_{t^{A}},\phi_{u^{j}}}\lambda(P)\phi_{u^{k,\sigma}}(P)-2{\rm
i}\pi\delta_{jk}\delta_{\phi_{t^{A}},\phi_{u^{j}}}\sigma\big{(}\lambda(P)\big{)}^{\sigma}d\lambda(P)\Big{)}.$

Moreover, the restriction of the differential $\phi_{u^{k,\sigma}}$ to the
fundamental polygon $F_{g}$ is holomorphic inside $F_{g}$. This enables us to
apply identity (3.58) with $\phi_{0}=\phi_{u^{k,\sigma}}$. Therefore, using
Rauch formulas (3.15) and formula (3.58) while bearing in mind the jumps of
$\lambda(P)\Phi_{A,k,\sigma}$, we get

$\displaystyle
E.t^{A}\big{(}\phi_{u^{k,\sigma}}\big{)}=\frac{1}{2}\sum_{j=1}^{N}\lambda_{j}\phi_{t^{A}}(P_{j})\phi_{u^{k,\sigma}}(P_{j})=\sum_{j=1}^{N}\underset{P_{j}}{{\rm
res}}\bigg{(}\frac{\lambda(P)\phi_{t^{A}}(P)}{d\lambda(P)}\phi_{u^{k,\sigma}}(P)\bigg{)}$
$\displaystyle=\frac{1}{2{\rm{i}}\pi}\oint_{\partial{F_{g}}}\lambda(P)\Phi_{A,k,\sigma}(P)-\sum_{i=0}^{m}\sum_{\alpha=1}^{n_{i}}\delta_{\phi_{t^{A}},\phi_{t^{i,\alpha}}}\underset{\infty^{i}}{{\rm
res}}\bigg{(}\frac{\lambda(P)\phi_{t^{A}}(P)}{d\lambda(P)}\phi_{u^{k,\sigma}}(P)\bigg{)}$
$\displaystyle\quad-\sum_{i=1}^{m}\delta_{\phi_{t^{A}},\phi_{v^{i}}}\underset{\infty^{i}}{{\rm
res}}\bigg{(}\frac{\lambda(P)\phi_{t^{A}}(P)}{d\lambda(P)}\phi_{u^{k,\sigma}}(P)\bigg{)}$
$\displaystyle=\frac{1}{2{\rm{i}}\pi}\sum_{j=1}^{g}\oint_{a_{j}}\bigg{(}\Phi_{A,k,\sigma}(P)-\Phi_{A,k,\sigma}(P+b_{j})\bigg{)}+\sum_{i=0}^{m}\sum_{\alpha=1}^{n_{i}}\delta_{\phi_{t^{A}},\phi_{t^{i,\alpha}}}\frac{\alpha}{n_{i}+1}t^{i,\alpha}\big{(}\phi_{u^{k,\sigma}}\big{)}+\sum_{i=1}^{m}\delta_{\phi_{t^{A}},\phi_{v^{i}}}v^{i}\big{(}\phi_{u^{k,\sigma}}\big{)}$
$\displaystyle=\sum_{j=1}^{g}\oint_{a_{j}}\bigg{(}\delta_{jk}\sigma\big{(}\lambda(P)\big{)}^{\sigma}\phi_{t^{A}}(P)+\delta_{\phi_{t^{A}},\phi_{u^{j}}}\lambda(P)\phi_{u^{k,\sigma}}(P)-2{\rm
i}\pi\delta_{jk}\delta_{\phi_{t^{A}},\phi_{u^{j}}}\sigma\big{(}\lambda(P)\big{)}^{\sigma}d\lambda(P)\bigg{)}$
$\displaystyle\quad+\sum_{i=0}^{m}\sum_{\alpha=1}^{n_{i}}\delta_{\phi_{t^{A}},\phi_{t^{i,\alpha}}}\frac{\alpha}{n_{i}+1}t^{i,\alpha}\big{(}\phi_{u^{k,\sigma}}\big{)}+\sum_{i=1}^{m}\delta_{\phi_{t^{A}},\phi_{v^{i}}}v^{i}\big{(}\phi_{u^{k,\sigma}}\big{)}$
$\displaystyle=\sigma\oint_{a_{k}}\big{(}\lambda(P)\big{)}^{\sigma}\phi_{t^{A}}(P)+\sum_{j=1}^{g}\delta_{\phi_{t^{A}},\phi_{u^{j}}}\oint_{a_{j}}\lambda(P)\phi_{u^{k,\sigma}}(P)+\sum_{i=0}^{m}\sum_{\alpha=1}^{n_{i}}\delta_{\phi_{t^{A}},\phi_{t^{i,\alpha}}}\frac{\alpha}{n_{i}+1}t^{i,\alpha}\big{(}\phi_{u^{k,\sigma}}\big{)}+\sum_{i=1}^{m}\delta_{\phi_{t^{A}},\phi_{v^{i}}}v^{i}\big{(}\phi_{u^{k,\sigma}}\big{)}$
$\displaystyle=\sigma
t^{A}\big{(}\phi_{u^{k,\sigma}}\big{)}+\sum_{j=1}^{g}\delta_{\phi_{t^{A}},\phi_{u^{j}}}u^{j}\big{(}\phi_{\sigma,k}\big{)}+\sum_{i=0}^{m}\sum_{\alpha=1}^{n_{i}}\delta_{\phi,\phi_{t^{i,\alpha}}}\frac{\alpha}{n_{i}+1}t^{i,\alpha}\big{(}\phi_{\sigma,k}\big{)}+\sum_{i=1}^{m}\delta_{\phi,\phi_{v^{i}}}v^{i}\big{(}\phi_{\sigma,k}\big{)}.$

In the last equality we have interchanged the order of integration over the
cycles $a_{k}$ and $b_{k}$ when $\phi_{t^{A}}=\phi_{\rho^{k}}=\omega_{k}$.
More precisely, we have used the following relation:

$\oint_{a_{k}}\big{(}\lambda(P)\big{)}^{\sigma}\omega_{k}(P)=\big{(}\lambda(P_{0})\big{)}^{\sigma}+\frac{1}{2{\rm{i}}\pi}\oint_{b_{j}}\phi_{u^{k,\sigma}}=\frac{1}{2{\rm{i}}\pi}\oint_{b_{j}}\phi_{u^{k,\sigma}},$

where the first equality follows from (8.9) and the second is an immediate
consequence of the fact that the cycles $a_{k}$ and $b_{k}$ start at a point
$P_{0}$ such that $\lambda(P_{0})=0$.
Formula (8.13) can be obtained in a similar way. $\square$

The next proposition addresses the quasi-homogeneity property of the functions

$q^{j}\big{(}\phi_{u^{k,\sigma}}\big{)}:=\oint_{a_{j}}\log\big{(}\lambda(P)\big{)}\phi_{u^{k,\sigma}}(P),\quad
j=1,\dots,g,$ (8.14)

with $\phi_{u^{k,\sigma}}$ being the differential defined by (8.7). Note that
these functions are among the family of flat coordinates (3.47) of the
intersection form $\mathbf{g}\big{(}\phi_{u^{k,\sigma}}\big{)}$ (3.46).

###### Proposition 8.4

For each $j=1,\dots,g$, the function $q^{j}\big{(}\phi_{u^{k,\sigma}}\big{)}$
(8.14) is an eigenfunction of the vector field
$E=\sum_{s=1}^{N}\lambda_{s}\partial_{\lambda_{s}}$ with

$E.q^{j}\big{(}\phi_{u^{k,\sigma}}\big{)}=\sigma
q^{j}\big{(}\phi_{u^{k,\sigma}}\big{)}.$ (8.15)

_Proof:_ Consider the multivalued differential

$\Psi_{u^{k,\sigma},q^{j}}(P):=\frac{\lambda(P)\psi_{q^{j}}(P)}{d\lambda(P)}\phi_{u^{k,\sigma}}(P),\quad\text{with}\quad\psi_{q^{k}}(P):=\oint_{a_{k}}\log(\lambda(Q))W(P,Q).$

Due to (3.51) and (8.8), the jumps of $\Psi_{u^{k,\sigma},q^{j}}(P)$ are as
follows:

$\displaystyle\Psi_{u^{k,\sigma},q^{j}}(P+a_{r})-\Psi_{u^{k,\sigma},q^{j}}(P)=0;$
$\displaystyle\Psi_{u^{k,\sigma},q^{j}}(P+b_{r})-\Psi_{u^{k,\sigma},q^{j}}(P)$
$\displaystyle=-2{\rm i}\pi\delta_{jr}\phi_{u^{k,\sigma}}(P)-2{\rm
i}\pi\sigma\delta_{kr}\big{(}\lambda(P)\big{)}^{\sigma}\psi_{q^{j}}(P)+(2{\rm
i}\pi)^{2}\sigma\delta_{jr}\delta_{kr}d\big{(}\lambda(P)\big{)}^{\sigma}.$

Thus, observing that the ramification points $P_{1},\dots,P_{N}$ are the only
simple poles of the meromorphic differential $\Psi_{u^{k,\sigma},q^{j}}$
within a fixed fundamental polygon $F_{g}$ and using its jumps, we get

$\displaystyle
E.q^{j}\big{(}\phi_{u^{k,\sigma}}\big{)}=\sum_{s=1}^{N}\underset{P_{s}}{{\rm
res}}\Psi_{u^{k,\sigma},q^{j}}(P)=\frac{1}{2{\rm{i}}\pi}\oint_{\partial{F_{g}}}\Psi_{u^{k,\sigma},q^{j}}(P)$
$\displaystyle=\frac{1}{2{\rm{i}}\pi}\sum_{r=1}^{g}\oint_{a_{r}}\bigg{(}\Psi_{u^{k,\sigma},q^{j}}(P)-\Psi_{u^{k,\sigma},q^{j}}(P+b_{r})\bigg{)}$
$\displaystyle=\oint_{a_{j}}\phi_{u^{k,\sigma}}(P)+\sigma\oint_{a_{k}}\big{(}\lambda(P)\big{)}^{\sigma}\psi_{q^{j}}(P)+2{\rm
i}\pi\sigma\sum_{r=1}^{g}\delta_{jr}\delta_{kr}\oint_{a_{r}}d\big{(}\lambda(P)\big{)}^{\sigma}$
$\displaystyle=\sigma q^{j}\big{(}\phi_{u^{k,\sigma}}\big{)}.$

$\square$

###  8.3 Appendix 3: Dubrovin’s bilinear pairing and prepotentials

Let $\omega^{(1)}$ and $\omega^{(2)}$ be two differentials on the surface
$C_{g}$ holomorphic outside the points $\infty^{0},\dots,\infty^{m}$ with the
following properties:

$\displaystyle\omega^{(\alpha)}(P)\underset{P\sim\infty^{i}}{=}\sum_{n=-n^{(\alpha)}}^{\infty}c_{n,i}^{(\alpha)}z_{i}(P)^{n}dz_{i}(P)+\frac{1}{n_{i}+1}d\left(\sum_{n=1}^{\infty}r_{n,i}^{(\alpha)}\big{(}\lambda(P)\big{)}^{n}\log\big{(}\lambda(P)\big{)}\right);$
$\displaystyle\oint_{a_{k}}\omega^{(\alpha)}=A_{k}^{(\alpha)};$
$\displaystyle\omega^{(\alpha)}(P+a_{k})-\omega^{(\alpha)}(P)=dp_{k}^{(\alpha)}\big{(}\lambda(P)\big{)},\quad\quad
p_{k}^{(\alpha)}=\sum_{j>0}p_{kj}^{(\alpha)}\big{(}\lambda(P)\big{)}^{j};$
$\displaystyle\omega^{(\alpha)}(P+b_{k})-\omega^{(\alpha)}(P)=dq_{k}^{(\alpha)}\big{(}\lambda(P)\big{)},\quad\quad
q_{k}^{(\alpha)}=\sum_{j>0}q_{kj}^{(\alpha)}\big{(}\lambda(P)\big{)}^{j},$

where $z_{i}(P)$ is the local parameter near the pole $\infty^{i}$ defined by
$\lambda(P)=z_{i}(P)^{-n_{i}-1}$, $n^{(\alpha)}\in\mathbb{Z}$ and
$c_{n,i}^{(\alpha)}$, $r_{n,i}^{(\alpha)}$, $p_{kj}^{(\alpha)}$ and
$q_{kj}^{(\alpha)}$ are some coefficients.
The Dubrovin bilinear pairing of $\omega^{(1)}$ and $\omega^{(2)}$ is defined
by

$\displaystyle\mathfrak{F}[\omega^{(1)},\omega^{(2)}]$
$\displaystyle=\sum_{i=0}^{m}\left(\sum_{n=0}^{\infty}\frac{c_{-n-2,i}^{(1)}}{n+1}c_{n,i}^{(2)}+c_{-1,i}^{(1)}p.v.\int_{P_{0}}^{\infty^{i}}\omega^{(2)}-p.v.\int_{P_{0}}^{\infty^{i}}\sum_{n=1}^{\infty}r_{n,i}^{(1)}\big{(}\lambda(P)\big{)}^{n}\omega^{(2)}\right)$
$\displaystyle\quad+\frac{1}{2{\rm
i}\pi}\sum_{k=1}^{g}\left(-\oint_{a_{k}}q_{k}^{(1)}(\lambda)\omega^{(2)}+\oint_{b_{k}}p_{k}^{(1)}(\lambda)\omega^{(2)}+A_{k}^{(1)}\oint_{b_{k}}\omega^{(2)}\right),$

where $P_{0}$ is a marked point on $C_{g}$ such that $\lambda(P_{0})=0$.

###### Theorem 8.2

Let $\phi$ be one of the primary differential (3.23) and $\Psi_{\phi}$ be the
following multivalued differential

$\Psi_{\phi}(P):=\left(p.v.\int_{\infty^{0}}^{P}\phi\right)d\lambda(P).$

Then the Dubrovin prepotential associated with the Frobenius manifold
structure determined by $\phi$ is given by means of the bilinear pairing
$\mathfrak{F}[\cdot,\cdot]$ as follows:

$F_{\phi}=\frac{1}{2}\mathfrak{F}[\Psi_{\phi},\Psi_{\phi}].$ (8.16)

Acknowledgements: The author is grateful to Vasilisa Shramchenko for several
valuable comments and discussions during the realization of this work. The
author thanks Université de Sherbrooke and Institut des Sciences mathématiques
(ISM) for their financial support.

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