# Leveraging Deep Representations of Radiology Reports in Survival Analysis
for Predicting Heart Failure Patient Mortality

Hyun Gi Lee1 Evan Sholle2 Ashley Beecy3 Subhi Al’Aref4 Yifan Peng1
1Department of Population Health Sciences, Weill Cornell Medicine
2Information Technologies and Services, Weill Cornell Medicine
3Department of Medicine, Weill Cornell Medicine
4Department of Internal Medicine, UAMS College of Medicine

###### Abstract

Utilizing clinical texts in survival analysis is difficult because they are
largely unstructured. Current automatic extraction models fail to capture
textual information comprehensively since their labels are limited in scope.
Furthermore, they typically require a large amount of data and high-quality
expert annotations for training. In this work, we present a novel method of
using BERT-based hidden layer representations of clinical texts as covariates
for proportional hazards models to predict patient survival outcomes. We show
that hidden layers yield notably more accurate predictions than predefined
features, outperforming the previous baseline model by 5.7% on average across
C-index and time-dependent AUC. We make our work publicly available at
https://github.com/bionlplab/heart_failure_mortality.

##  1 Introduction

Survival analysis estimates the expected time until an event of interest
occurs Ranganath et al. (2016). In clinical research, it is used to understand
the relationship between prognostic covariates (e.g., age and treatment) and
patient survival time for important use cases such as predicting mortality of
heart failure patients and providing management recommendations for intensive
care units during a public health crisis like the COVID-19 pandemic Pandey et
al. (2020); Sprung et al. (2020); Nielsen et al. (2019).

Clinical texts such as radiology reports contain rich information about
patients that is used to diagnose disease, plan treatments and monitor
progress. It also contains the high-level reasoning of human experts that
requires years of knowledge accumulation and professional training Langlotz
(2015). Despite their clinical relevance, it is challenging to use them in
survival analysis since they are largely unstructured. Automatic labelers are
often unable to capture detailed information to distinguish between patients,
especially ones with similar conditions, because they mostly rely on a small
set of manually selected labels Lao et al. (2017). Developing methods for
accessing the critical information embedded in unstructured clinical texts
holds the potential to meaningfully benefit clinical research.

To bridge this gap, we propose a deep learning method to predict the survival
probability of heart failure (HF) patients based on the high-dimensional
feature representations of their radiology reports. Concretely, we extract
hidden features from the texts with BERT-based (Devlin et al., 2019) models
and apply a recurrent neural network (RNN) to model sequences of reports and
estimate the log-risk function for the overall mortality prediction. This
approach can encapsulate more textual information than hand-crafted features
and incorporate higher-order temporal information from report sequences. We
find that our model improves on average 5.7% in both C-index and time-
dependent AUC without requiring additional expert annotations.

We make three contributions through this work: (1) present a novel survival
analysis model to leverage feature representations from clinical texts, (2)
demonstrate that our model outperforms the ones dependent on predefined expert
features and that this approach can generalize across various biomedical and
clinical BERT models, and (3) make our work publicly available for
reproduction by others.

##  2 Related Work

Due to the lack of expert annotations, earlier automatic labelers mostly use
predefined linguistic patterns to extract relevant information. NegEx (Chapman
et al., 2001) is a regular expression algorithm that identifies observations
based on specified phrases. NegBio (Peng et al., 2018) uses universal
dependencies and subgraph matching in addition to regular expressions.
CheXpert Irvin et al. (2019) extends NegBio by adding rules to extract,
classify, and aggregate mentions to improve performance. While they typically
achieve a high precision, they suffer from a low recall because of their
limited rules.

BERT (Devlin et al., 2019) is a transformer-based method that extracts feature
representations of unlabeled text that are effective for transfer learning
across various NLP tasks. It is adapted to a wide range of domains, including
biomedical and clinical domains (Lee et al., 2020; Alsentzer et al., 2019;
Peng et al., 2019). Recently, BERT models have been applied to labeling
radiology reports. CheXbert (Smit et al., 2020) and CheXpert++ (McDermott et
al., 2020) train on silver-standard datasets created with a rule-based
labeler, CheXpert. Although they outperform rule-based labelers, these
approaches need a curated training corpus which can be costly to obtain and
error-prone. Furthermore, their labels are still limited and can miss critical
information from the reports.

Regarding survival analysis, the Cox proportional hazards model (CPH) is
widely adopted as it can deal with censored data and evaluate the prognostic
values of covariates simultaneously (Cox, 1972). DeepSurv (Katzman et al.,
2018) and DeepHit (Lee et al., 2018) are more contemporary methods that use
deep neural networks to model more complex, nonlinear relationships of
predictor variables. RNN-SURV (Giunchiglia et al., 2018) and DRSA (Ren et al.,
2019) model time-variant, sequential patterns from predictor variables. To the
best of our knowledge, the compatibility of these models and high-dimensional
features as covariates has not been tested.

Automatic extraction tools enable survival analysis to incorporate textual
information from clinical texts. Pandey et al. (2020) used a convolutional
neural network to extract findings from radiology reports of heart failure
patients and predict all-cause mortality with CPH. Heo et al. (2020) performed
stroke prognosis based on the document-level and sentence-level
representations of MRI records. Our work extends this line of research by
using contextual deep representations of clinical texts to perform survival
analysis.

##  3 Methods

###  3.1 Task

We first formulate the survival analysis problem. In the discrete context, we
divide the continuous time into disjoint intervals $V=(t_{l-1},t_{l}]$ where
$t_{0}$ and $t_{T}$ are the first and last observation interval boundaries. At
time $t_{u}$, the model predicts the survival probability in the prediction
window $(t_{u},t_{T}]$ with longitudinal features in the observation window
$(t_{0},t_{u}]$ (Figure 1).

For each participant $i$, the survival probability at each time $t_{l}$
($l>u$) is $S_{i}(t_{l})=Pr(z>t_{l})$, where $z$ is the time-to-event, time
until death in our case. The hazard rate of the survival probability is
$\lambda_{i}(t_{l})=\frac{S_{i}(t_{l-1})-S_{i}(t_{l})}{S_{i}(t_{l})}$.

###  3.2 Model

Our framework consists of two stages: feature extraction and survival analysis
(Figure 1).

Figure 1: Model Architecture.

####  3.2.1 Feature Extraction

The input of each time $t_{l}$ is given by the features extracted from the
reports of each patient $i$. In this work, we evaluate two sets of predefined
features and hidden features of the reports.

The first feature set consists of 14 common radiographic findings in computed
tomography (CT) imaging reports (aortic aneurysm, ascites, atelectasis,
atherosclerosis, cardiomegaly, enlarged liver, gall bladder wall thickening,
hernia, hydronephrosis, lymphadenopathy, pleural effusion, pneumonia, previous
surgery, and pulmonary edema). The findings are extracted using the
convolutional neural network provided by Pandey et al. (2020) which had the
reported performance of 0.90 F1 in average.

The second feature set consists of 14 predefined findings in CheXpert (Irvin
et al., 2019) which are commonly found in radiology reports (atelectasis,
cardiomegaly, consolidation, edema, enlarged cardiomediastinum, fracture, lung
lesion, lung opacity, pleural effusion, pleural other, pneumonia,
pneumothorax, support devices, and normal). These features are extracted using
CheXbert (Smit et al., 2020) with the reported performance of 0.80 F1 in
average.

For deep representations, we use CheXbert’s final hidden layer features of the
reports. More specifically, we extract the information before it passes on to
the output layer that consists of 14 linear heads. The representations are
vectors of size 768.

Lastly, we construct sequential deep representations by creating arrays of up
to three most recent reports of each patient. As the reports can change over
time based on the patient’s condition, these features are time-variant and
contain temporal information that cannot be obtained by a single report. In
addition to CheXbert, we apply BERT variations – BERT, BioBert, ClinicalBert
and BlueBert.

####  3.2.2 Deep survival analysis

In this study, the hazard rate has the form

$\lambda_{i}(t_{l}\mid X_{i})=\lambda_{u}(t_{l})e^{\psi(X_{i})}$ (1)

$\psi$ is a patient’s log-risk of failure, $X_{i}$ are covariates representing
a patient’s variables up to $t_{u}$, and $\lambda_{u}$ baseline hazard at
$t_{u}$.

For the standard Cox Proportional-Hazards (CPH) model (Cox, 1972),
$\psi(X_{i})$ has the form of a linear combination of $p$ covariates
$\beta_{1}X_{i1}+\cdots+\beta_{p}X_{ip}$. In our experiments, the covariates
are the features extracted from the reports.

$\psi$ can also be a non-linear risk function of a multilayer perceptron
(MLP). To this end, our model is the same as DeepSurv (Katzman et al., 2018).

Both CPH and DeepSurv cannot incorporate the higher-order temporal information
from report sequences. To solve this problem, we define $\psi=LSTM(X_{i})$ to
model the possible time-variant effects of the covariates leading up to
$t_{u}$ (Figure 1). Our model is similar to RNN-SURV (Giunchiglia et al.,
2018) and DRSA (Ren et al., 2019). The main difference is that the objective
function is the average partial log-likelihood Kvamme et al. (2019):

$-\frac{1}{N}{\sum_{i\in U_{l}}\left(\tilde{\psi}(x_{i})-\log\sum_{j\in
R_{l}}{e^{\tilde{\psi}(x_{j})}}\right)}$ (2)

$U_{l}$ is the set of patients that are deceased or last known to be alive
(censored) by time point $t_{l}$. $R_{l}$ is the set of all live and
uncensored patients before $t_{l}$. $N$ is the total number of deceased
patients in the dataset.

##  4 Experiments

###  4.1 Data

The dataset (Pandey et al., 2020) is a collection of thoracoabdominal CT
reports in English for heart failure patients from the New York-
Presbyterian/Weill Cornell Medical Center who were admitted and discharged
with billing codes ICD-9 Code 428 or ICD-10 Code I50 from January 2008 and
July 2018 (Table 1). It was reviewed by the institutional board and de-
identified. We use each patient’s three most recent reports or zero vectors
for any missing ones. Their time-to-event is calculated as the number of days
between the most recent report date and death date if deceased or the last
follow-up date if censored. We perform simple preprocessing steps to confirm
each patient has at least one report and nonnegative time-to-event.

Characteristics | $n$
---|---
Number of Patients | 11,971
30 days mortality | 1,209
365 days mortality | 2,062
Total mortality | 2,602
Number of Reports | 39,752
Avg number of words in reports | 306
Table 1: Dataset Overview. Model | C-index | C-index@30 | AUC@30 | AUC@365
---|---|---|---|---
CPH + Feature Set 1 | 0.499 $\pm$ 0.025 | 0.504 $\pm$ 0.032 | 0.503 $\pm$ 0.037 | 0.501 $\pm$ 0.028
CPH + Feature Set 2 | 0.621 $\pm$ 0.014 | 0.632 $\pm$ 0.033 | 0.642 $\pm$ 0.030 | 0.642 $\pm$ 0.030
CPH + Hidden Features | 0.674 $\pm$ 0.023 | 0.696 $\pm$ 0.022 | 0.710 $\pm$ 0.022 | 0.697 $\pm$ 0.026
MLP + Feature Set 1 | 0.502 $\pm$ 0.023 | 0.509 $\pm$ 0.026 | 0.509 $\pm$ 0.030 | 0.501 $\pm$ 0.032
MLP + Feature Set 2 | 0.658 $\pm$ 0.010 | 0.671 $\pm$ 0.025 | 0.685 $\pm$ 0.023 | 0.683 $\pm$ 0.008
MLP + Hidden Features | 0.704 $\pm$ 0.017 | 0.726 $\pm$ 0.020 | 0.744 $\pm$ 0.019 | 0.734 $\pm$ 0.018
LSTM + Sequential HF | 0.709 $\pm$ 0.022 | 0.733 $\pm$ 0.031 | 0.752 $\pm$ 0.033 | 0.742 $\pm$ 0.023
Table 2:  Evaluation results. Feature Set 1 - (Pandey et al., 2020), Feature Set 2 - (Irvin et al., 2019), Sequential HF - sequential hidden features  Model | C-index | C-index@30 | AUC@30 | AUC@365
---|---|---|---|---
BERT-Base Devlin et al. (2019) | 0.603 $\pm$ 0.115 | 0.611 $\pm$ 0.123 | 0.618 $\pm$ 0.134 | 0.620 $\pm$ 0.136
BioBert Lee et al. (2020) | 0.701 $\pm$ 0.021 | 0.714 $\pm$ 0.027 | 0.734 $\pm$ 0.029 | 0.739 $\pm$ 0.028
ClinicalBert Alsentzer et al. (2019) | 0.692 $\pm$ 0.019 | 0.705 $\pm$ 0.023 | 0.723 $\pm$ 0.025 | 0.727 $\pm$ 0.029
BlueBert Peng et al. (2019) | 0.713 $\pm$ 0.019 | 0.735 $\pm$ 0.024 | 0.755 $\pm$ 0.024 | 0.756 $\pm$ 0.021
CheXbert Smit et al. (2020) | 0.709 $\pm$ 0.022 | 0.733 $\pm$ 0.031 | 0.752 $\pm$ 0.033 | 0.742 $\pm$ 0.023
Table 3:  Evaluation results of LSTM + sequential hidden features using
different BERT models.

###  4.2 Metrics

To assess the discriminative accuracies of our models, we use the C-index
(Harrell et al., 1982) and time-dependent area-under-the-curve (AUC) (Heagerty
and Zheng, 2005), some of the most commonly used evaluation metrics in
clinical research (Kamarudin et al., 2017; Pencina and D’Agostino, 2004; Uno
et al., 2011). Intuitively, the C-index measures the extent to which the model
is able to assign logical risk scores. An individual with shorter time-to-
event $T$ should have a higher risk score $R$ than the ones with longer time-
to-event. Formally, it is defined as:

$C=\frac{\sum\limits_{i,j}I(T_{i}>T_{j})\cdot I(R_{i}<R_{j})\cdot
d_{j}}{\sum\limits_{i,j}I(T_{i}>T_{j})\cdot d_{j}}$ (3) $\displaystyle
I(c)=\begin{cases}1\>\text{if}\>c\>\text{is true}\\\
0\>\text{else}\end{cases}d_{j}=\begin{cases}1\>\text{if}\>T_{j}\>\text{exists}\\\
0\>\text{else}\end{cases}$

Both C-index and AUC assign a random model 0.5 and a perfect model 1. We
measure all-time C-index, C-index at 30 days (C-index@30), and AUC at 30 days
and 365 days (AUC@30 and AUC@365) to show the models’ performances dealing
with different time-to-events111https://github.com/sebp/scikit-survival.

###  4.3 Training

We perform a grid search to find the optimal hyperparameters based on the
metrics and use them for all configurations. The learning rate is set to
0.0001 with an Adam optimizer. We iterate the training process for 100 epochs
with batch size 256 and early stop if the validation loss does not decrease.
The dropout rate is 0.6. We perform five-fold cross-validation to produce 95%
confidence intervals for each metric. The training, validation and test splits
are 70%, 10%, 20%, respectively. We use pycox and PyTorch to implement the
framework222https://github.com/havakv/pycox. The end-to-end training takes
about 30 minutes with NVIDIA Tesla P100 16 GB GPU, mainly due to feature
extraction.

###  4.4 Results & Discussions

Table 2 shows our experimental results with variations in covariates and
survival analysis models.

Our LSTM model with hidden features (LSTM + Hidden Features) achieves the best
results (0.709 in C-index), 3.5% and 0.5% improvements over CPH + Hidden
Features and MLP + Hidden Features. In contrast to the MLP, its data included
the reports from patients’ prior visits with more textual and higher-order
temporal information. Nonetheless, the improvements are stll marginal,
suggesting that the evaluation of the effectiveness of LSTM in survival
analysis in this context would require more empirical evidence, particularly
with more longitudinal text data.

We observe that the hidden features provide at least 5% improvements over the
other feature sets with both CPH and MLP. This indicates that the hidden
features capture textual information more thoroughly than the predefined
features for survival analysis.

We find that Feature Set 2, obtained with CheXbert (Smit et al., 2020),
performs significantly better ($>10\%$ C-index) than Feature Set 1, obtained
with the CNN model (Pandey et al., 2020). With both CPH and MLP, Feature Set 1
yields around 0.5 in C-index and AUC, whereas Feature Set 2 shows prognostic
value in the 0.62-0.69 range. The difference of the feature sets directly
results in the performance difference. While Feature Set 1 and Feature Set 2
have overlapping features (atelectasis, cardiomegaly, pleural effusion, and
pneumonia), Feature Set 1 is not as discriminatory as Feature Set 2. This
observation informs us that much important textual information with prognostic
value is likely lost between the feature sets.

Finally, we compare our model on BERT-Base variants. BERT-Base, CheXbert and
BlueBert used “uncased” text. BioBert and ClinicalBert used “cased” text.
BioBert was pretrained on PubMed abstracts. ClinicalBert was initialized with
BioBert’s weights and further trained on MIMIC-III clinical notes. BlueBert
was pretrained on both datasets altogether. Table 3 shows that all BERT
variants (except the original BERT) capture the textual information more
comprehensively than the predefined features and yield significantly more
accurate predictions. Further, the models with more pertinence to radiology
reports perform incrementally better. BlueBert outperforms others and improves
on CheXbert slightly. This observation is consistent with the findings in Peng
et al. (2019). These results illustrate that using hidden layer
representations in survival analysis can generalize across deep learning
models based on their areas of focus.

###  4.5 Conclusion & Future Work

Incorporating the textual information of clinical texts in survival analysis
is challenging because of their unstructured format. Automatic extraction
tools have a small set of features selected by experts and fail to capture the
information fully and precisely. We show a novel method of using hidden layer
representations of clinical texts as covariates for proportional hazards
models. When applied to predicting all-cause mortality of heart failure
patients, the results indicate that hidden features encapsulate more
comprehensive and effective textual information than predefined features.

We plan to explore the use of the attention mechanism to the input sequence
and test the generalizability of this method with more datasets. In addition,
we plan to gain more insights on how hidden features are influenced (e.g. word
choice, text length, etc.) and add value for better prediction as
interpretability is highly important in the medical domain. We hope our small
contribution provides assistance in the scalable development of accurate
predictive models that harness clinical text information.

## Acknowledgment

Research reported in this publication was supported by National Library of
Medicine - National Institutes of Health (NIH) under award number R00LM013001.
It was also supported by National Heart, Lung, and Blood Institute of NIH
under award number R01HL1276610. This study also received support from
NewYork-Presbyterian Hospital (NYPH) and Weill Cornell Medical College (WCMC),
including the Clinical and Translational Science Center (CTSC) (UL1TR002384)
and Joint Clinical Trials Office (JCTO). Additionally, we would like to thank
Dr. Pranav Rajpurkar and Dr. Matthew P Lungren for providing the ChexBert
model.

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# Spatial profile of plasma temperature generated by discharge in 3D printed
capillary

Niv Barkai  Racah Institute of Physics, Hebrew University, 91904 Jerusalem,
Israel  Noa kliss  Racah Institute of Physics, Hebrew University, 91904
Jerusalem, Israel  Yair Ferber  National Laser Facility (NLF), Soreq NRC,
Yavne 81800, Israel  Yoav Raz  Racah Institute of Physics, Hebrew University,
91904 Jerusalem, Israel  Rotem Liran  Racah Institute of Physics, Hebrew
University, 91904 Jerusalem, Israel  Mordechai Botton  Racah Institute of
Physics, Hebrew University, 91904 Jerusalem, Israel  Arie Zigler  Racah
Institute of Physics, Hebrew University, 91904 Jerusalem, Israel

(September 2023)

###### Abstract

A method for evaluating the radial temperature profile of a plasma channel is
presented. The radiation spectrum of mixture of Hydrogen Nitrogen plasma in
capillary discharge is collected. Assuming a local thermal equilibrium, the
temperature is derived from the ratio of two emission lines
($N^{+}$,$N^{2+}$). The method presented here does not rely on Stark
broadening of the emitted $H_{\alpha}$ line hence can be utilized in a wide
range of plasma constituents and densities, leading to an improved control of
the plasma channel parameters as required by future laser wakefield
acceleration schemes.

##  1 Introduction

Plasma channels generated in capillaries discharges are considered to be the
backbone in promising next generation high-energy particle accelerators [1].
In the laser wakefield acceleration (LWFA) process [2, 3], the laser pulse is
focused onto a gas or plasma target, creating within the plasma a highly non-
uniform electron density moving closely behind the laser pulse. This density
non-uniformity is accompanied by a high-intensity localized electric field in
the plasma, known as a wakefield. Injecting electron bunches synchronized with
the wakefield the former can be trapped by the latter and hence be accelerated
by it to significantly higher energy. Record level of multi-GeV electron
bunches using LWFA was demonstrated in various experiments [4, 5, 6].
Underlying all these Innovative technologies is the requirement that the laser
pulse propagates in the interaction region for lengths many times longer than
Rayleigh lengths of the focused laser pulse. Such is the case in plasma
channels formed in gas-filled capillaries. Nonetheless, a particular
transverse profile of the plasma channel, more specifically, a minimum value
on the axis and increasing towards the wall is required [7]. Accordingly, an
extensive effort was and still is being given to the diagnostics of the plasma
channel parameters. One prominent method relies on the spectroscopic
properties of the plasma, more specifically on the Stark broadening of the
emitted $H_{\alpha}$ line [8, 9, 10]. Other methods used the interferometric
approach, in one or more “colors” [11, 12]. Other methods utilized the
dependence of the group velocity of a propagating electromagnetic pulse in the
plasma channel on the local density [13, 14]. Combinations of the above-
mentioned methods were employed as well [15, 16]. Spectroscopy-based
measurements albeit detailed are correlated to Hydrogen or Hydrogen like
cases. Furthermore, direct measurement of density is limited to values above
$10^{17}cm^{-3}$. As the requirements of the plasma channel may vary in
parameters (contents and density), a more robust diagnostics method may prove
useful.

In this paper, we present a utilization of such a method where the temperature
profile of the plasma channel is measured. The method presented here is
facilitated by the conditions of local thermal equilibrium (LTE). The
temperature measurement approach is based on comparing the intensities of two
separate emission lines. These lines can be transition in two sequential
ionization stages by different ionization levels, or alternatively by two
different excitation of the same ion. Thus it can be applied to the broad
range of gases in capillary discharge and need not include Hydrogen.

Subsequently, the density can inferred by relying on the reasonable assumption
of local thermal equilibrium, which guarantees a unique relation between the
temperature and density by means of the equation of state (that more often
than not is of the ideal gas). For the purpose of demonstrating this method,
we present spatial and time-dependent measurements of the plasma temperature.
accompanied by direct measurement of the density using the conventional Stark
broadening.

##  2 Experimental setup

The experiments were carried out at the High Intensity Laser Lab at the Hebrew
University of Jerusalem. The experimental setup is depicted in figure 1. A
3D-printed PolyGet capillary is positioned inside a vacuum chamber and a
controlled flow of Hydrogen-Nitrogen gases was flowing through it. The
capillary size is of [length $5\text{\,}\mathrm{c}\mathrm{m}$, diameter
$500\text{\,}\mu\mathrm{m}$]. Discharge in the gas-filled capillary is
initiated by a laser pulse injected into the capillary, followed by a high-
voltage breakdown along the capillary that produces the actual plasma channel.

The triggering laser is a Q-switched Nd:Yag laser ($30\text{\,}\mathrm{mJ}$,
$1064\text{\,}\mathrm{nm}$, $10\text{\,}\mathrm{ns}$) and is directed by a
two-mirror setup into the capillary entrance (figure1). Part of its energy is
used as a triggering signal to both the high-voltage circuit and the measuring
system. The discharge gas consists of a nitrogen-hydrogen mixture (95-5
percent) at a pressure of 1.5 $bar$, with gas flow maintained for 35 $ms$.
Synchronously following this trigger[Nd:Yag] is a high-voltage pulse (up to
$15\text{\,}\mathrm{k}\mathrm{V}$, $0.6\text{\,}\mathrm{\SIUnitSymbolMicro
s}$) applied on the capillary between two electrodes mounted at its ends. The
high voltage pulse generates a breakdown in the plasma and hence the creation
of a plasma discharge in the capillary, which effectively closes an electrical
circuit [17]. Prior to the temperature measurements and in order to establish
the proper time delay gating, a pulse train from a Ti-Sa laser (84MHz 800nm)
was injected into the generated channel (see Fig. 1). The output intensity was
collected and measured. During the propagation of the laser inside the
capillary, some part of the light is lost to the walls, when conditions of the
plasma channel are met there is internal reflection inside the capillary and
more light is being measured with P.D2

The radiation emitted from the plasma discharge was measured using a
spectrometer (Acton Research Corp spectrometer coupled to Andor ICCD camera).
The discharge and the spectrometer are synchronized to collect the radiation
by a Stanford gating device. The spectrometer is tuned to the specific lines
of the plasma involved here, The $N^{2+}$ and $N^{+}$ ions. In order to
establish a direct connection to the conventional methods, the average density
was measured by Stark-broadening of the $H_{\alpha}$ line. As will be shown
later an exact knowledge of the averaged density is not required in our
method. The spectrometer’s gating time was 50 to 20 $ns$ by width and the time
delay was interchanged regarding the guiding measurements. The temperature
profile measurement scheme utilizes the Acton SP-300i spectrometer and Andor
new-iStar ICCD camera.

Figure 1: The entire system scheme is comprised by parts : The ND:Yag laser is
used for the triggering scheme with the high voltage generator, The Ti-Sa is
used as a time measurement of the channel using the P.D2 photodiode. The
Spectrometer collects the blue light from the discharge using L3 lens.

##  3 Results and discussion

In order to relate the method presented here to previously published method,
we begin by presenting the electron density measurements carried out by the
conventional Stark-broadening of the H$\alpha$ Balmer series (the wavelength
is $656.26\text{\,}\mathrm{nm}$). The derived average density as a function of
the high voltage is shown in Fig. 2. These values can be used in the analysis
of the temperature measurements as will be demonstrated shortly.

Figure 2: Average electron density as a function of breakdown Voltage.

Temperature measurements which are based on the intensity ratio of two
separate emission lines require that the plasma would be in a local thermal
equilibrium (LTE) or near-LTE, and that the radiation loss rate would be small
compared to the collision rate [18]. Both conditions are met in this case when
the measurements take place sometime after the initiation of the plasma
discharge when the ionization fraction is high (close to one) and the primary
heating process is Ohmic [19, 20]. Accordingly, emission line measurements
were done after the discharge current reached its peak value. Out of the
multitude of possible emission lines we chose to concentrate our attention on
two which are expected to be dominant given the expected temperature scale
($\lesssim 10eV$). The lines of two distinct ionization levels of Nitrogen,
namely $N^{+}$ and $N^{2+}$ are relatively close in wavelength so the
spectrometer can be tuned to collect both of them (see Table 1 for details).

wavelength[$\lambda$] | $f_{ik}$ | $g_{k}A_{ki}$ | ${}^{J}L_{S}$ | Ion
---|---|---|---|---
571.077nm | $5.74e{-2}$ | 5.85$e7$ | ${}^{3}D\rightarrow\ ^{3}P^{0}$ | $N^{+}$
581.779nm | 8.94$e{-3}$ | 7.04$e6$ | ${}^{2}D\rightarrow^{2}P^{0}$ | $N^{2+}$
Table 1: The parameters of the two measured lines $N^{2+}$ and $N^{+}$ [21].
$f_{ik}$ is the oscillator strength which corresponds to the probability for
the transition in certain ion, the parameter $g_{k}A_{ki}$ is the degeneracy
and Einstein coefficient which relates to spontaneous emission, and the
${}^{J}L_{S}$ is the spectroscopic symbol which gives the total angular
momentum of the certain state $J=L+S$

A typical result of the radiation emitted in these lines is shown in Fig. 3
for two values of breakdown voltage. From these figures, it is clearly evident
that the emission intensity of the $N^{+}$ is approximately unchanged while
the emission intensity of the $N^{2+}$ significantly increased when the
breakdown voltage is changed from 7kV to 9kV

Figure 3:  (a.)7kV spectrum of different ionization degrees of nitrogen
(b.) 9kV spectrum of different ionization degrees of nitrogen

Relating the emission intensity from a line to its occupation fraction, and
utilizing the well know Saha-Boltzmann equation,[8] we obtain the following
expression for the intensity ratio $R$:

$R=\frac{i^{\prime}}{i}=\frac{w^{\prime}A^{\prime}g^{\prime}}{\sqrt{\pi}wAg}(4\pi
a_{0}^{3}N_{e})^{-1}(\frac{k_{B}T}{E_{H}})^{3/2}\times
exp(-\frac{E^{\prime}+E_{\infty}-E-\Delta E_{\infty}}{k_{B}T})$ (1)

Here $\omega,\omega^{\prime}$ are the frequencies of the two levels (related
to $E,E^{\prime}$ for each ion respectively), $A,A^{\prime}$ are the Einstein
coefficients, $g,g^{\prime}$ are the degeneracy levels, $a_{0}$ is the Bohr
radius $E_{H}$ is ionization energy of hydrogen $E_{\infty}$ is the difference
in ionization energy between the certain ions, $\Delta E_{\infty}$ is the
reduction to the ionization energy due to charge loss, and $N_{e}$ is the
electron density. For the calculations of the temperature from the intensity
ratio measured for each value of the breakdown voltage we use the averaged
electron densities, $N_{e}$, presented in Fig. 4. The measurements are being
averaged spatially by the ICCD, and the gating time is 50ns, so there is an
average over space and over time. Noting that the dependence of the above
ratio on the density is linear while on the temperature it is exponential, we
expect that the temperature values will only be slightly affected by
systematic errors in the density values. This effect is demonstrated in Fig.
4, where the temperature is calculated with the lowest and highest values of
the electrons density. The exact values of the derived temperature slightly
change, however, the tendency of the graph remains the same. Following [8],
the error is estimated to be 10 percent.

Figure 4: Temperature dependence on Voltage in our capillary, each point is
calculated with the appropriated density using figure 2, The multiple points
correspond to changing the density. One can see that only the scaling changes
not the trend of the points.

Following previous work the current-voltage relation is expected to be given
by Eq. 39 in [20]

$T(0)\approx 5.7{\left(\frac{I[kA]}{R_{0}[mm]}^{2/5}\right)}eV$ (2)

Here $R_{0}$ is the radius of the capillary. A plot of the inferred
temperature as a function of the measured current is presented in 5 and
compared to the theoretical fit. The correspondence is very good.

Figure 5: Temperature dependence on the current with a theoretical fit [20]

Next we address the spatial profile of the temperature. The spectrum is imaged
to a slit in the spectrometer entrance, and two images of the capillary cross
section are obtained for the two lines. The ICCD is not averaging with respect
to space, accordingly, each pixel correspond to a pixel of the cross section
of the capillary. Relating two corresponding images of the cross section of
the capillary and obtaining ratio of lines per pixel we got the ratio with
respect to the location of the cross section of the capillary. Following that,
the temperature spatial profile is obtained by repeating the ratio
calculations for the same lines for each point. The results of these
calculations are presented in Fig. 7. In order to look for the radial
temperature profile we needed to reduce the gating time from 50ns to 20ns. To
achieve that, a detailed knowledge of the time profile of the channel is
required. For that purpose, we employed a Ti:Si laser (84MHz-10ns) focused
into the capillary. A photodiode collected the laser pulse at the capillary
exit.

Figure 6: guiding of the oscillator inside our capillary the orange represents
the current discharge, and the blue is the local oscillator being amplified
and guided inside our capillary

Before the discharge, one can look in figure 6 and spot the Ti:Si with laser
pulse every 10$ns$ (voltage pulses in time before the discharge, $t<0$ in Fig.
6). Following the discharge, two distinct pulse trains with higher voltage are
observed (around $0.3\mu s$ and $0.5\mu s$). As the laser energy collected by
the photodiode is unchanged, the increase in the voltage output is related to
a corresponding increase of the collected intensity. Accordingly, the laser
light is being guided inside the capillary, so there is no loss to the walls
of the capillary, which is the result of a higher output in the photodiode.
This measurement gives us the required time-dependent data of the lifetime of
the channel with a resolution of 10$ns$. We observed two gains after the
ignition of plasma. The first one after $\simeq$230ns, was used for the
temperature profile measurements. A second channel after $\simeq$500ns, was
observed only for higher breakdown voltages. It is probably related to the
longer lifetime of the channel in higher breakdown voltages. As it did not
appear in all voltages we did not use it for the temperature profile.

The radial profile of the electron density was measured in [22] demonstrating
the conjugate profile of the temperature profile presented here. The
multiplication of the two shows that the pressure is almost constant across
the capillary cross-section.

Figure 7: Temperature profile measurements, measured with a gating of 20ns at
280ns delay from the Nd-Yag laser starting the breakdown.

Viewing figure 7 one can spot the almost parabolic appearance of the
temperature profile in the radii between values of $-50:50\mu m$. This profile
represents the core of the plasma channel. For larger radii values the profile
changes as we move closer to capillary walls.

##  4 Conclusion

In conclusion, the spectroscopic measurements of capillary discharge were
presented. Using a 3D-printed capillary filled with nitrogen/hydrogen gas at a
pressure of $1*10^{-5}$ Torr. The results showed that the emission from the
plasma was due to excited atomic states of nitrogen, and an excitation
temperature of around $3.6eV$ was calculated and correlated to theory with
[20]. This method provided insight into the plasma characteristics within the
discharge and was found to be a useful way of determining the existence of a
plasma channel within the capillary. The spatial profile of the plasma lens
was obtained and demonstrated. The method is expected to be useful for any
NLTE LTE plasma. Future work may involve improving the lifetime of the
discharge by using stronger capacitors and investigating the physical
mechanism of the appearance of the second channel we observed, possibly
improving the model for capillary discharge.

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#  On equivalence of two non-Riemannian curvatures in warped product Finsler
metrics

Ranadip Gangopadhyay, Anjali Shriwastawa, Bankteshwar Tiwari
DST-CIMS, Banaras Hindu University, Varanasi-221005, India

###### Abstract

In this paper we study warped product Finsler metrics and show that the notion
of isotropic $E$-curvature and isotropic $S$-curvature are equivalent for this
class of metrics.

##  1 Introduction

The warped product Riemannian manifold was introduced by R. L. Bishop and B.
O’Neill in 1964 [1], to construct a class of complete Riemannian manifolds of
negative curvature. Warped product Riemannian manifolds are the most natural
and fruitful generalization of Riemannian products of two manifolds. The
notion of warped products plays very important roles not only in geometry but
also in mathematical physics, especially in general relativity. In fact, many
basic solutions of the Einstein field equations, including the Schwarzschild
solution and the Robertson-Walker models, are warped product manifolds. The
famous John Nash’s embedding theorem published in 1956 implies that every
warped product Riemannian manifold manifold can be realized as a warped
product submanifold in a suitable Euclidean space ([2], [3] ).

Later on, the warped product metric was extended to the case of Finsler
manifolds by the work of Chen etal. and Kozma et al. ([4],[5] ). Recently,
some significant progress has been made in the study of Finsler warped product
metrics ( [4], [5], [6]). It has been observed in ([4]) that spherically
symmetric Finsler metrics have warped product structure. However, there are
lot of Finsler warped product metrics that are not spherically symmetric and
several authors studied those metrics ([7], [8], [9], [10]). In ([11]), the
authors obtain the differential equation that characterizes the spherically
symmetric Finsler metrics with vanishing Douglas curvature. Furthermore they
obtain all the spherically symmetric Douglas metrics by solving this equation.

A Finsler metric on a smooth manifold is a smoothly varying family of
Minkowski norms, one on each tangent space, rather than a family of inner
products one on each tangent space, as in the case of Riemannian metrics. It
turns out that every Finsler metric induces an inner product in each direction
of a tangent space at each point of the manifold. However, in sharp contrast
to the Riemannian case, these Finsler-inner products do not only depend on
where we are, but also in which direction we are looking.
In this paper we consider the warped product Finsler metrics on the manifold
$M=I\times\bar{M}$ which is a simple generalization of the Riemannian version,
where $I$ is an open interval of $\mathbb{R}$ and $(\bar{M},\bar{\alpha})$ is
an $(n-1)$ dimensional Riemannian manifold.

There are several non-Riemannian quantities in the Finsler literature, for
instance, Cartan torsion, $S$-curvature, $E$-curvature, $\Xi$-curvature,
$H$-curvature etc. These quantities become zero for a Riemannian metric. The
concept of $S$-curvature was introduced by Z. Shen [12] and it becomes a very
important quantity in both local and global Finsler geometry ([13], [14]). The
$E$-curvature is another quantity in Finsler geometry which is associated to
the $S$-curvature.
The Randers metric is one of the most simplest classes of non-Riemannian
Finsler metric that can be written in the form $F=\alpha+\beta$, where
$\alpha$ is a Riemannian metric and $\beta$ is a one-form. X. Cheng and Z.
Shen first calculated the $S$-curvature of the Randers metrics and proved that
for the Randers metrics the isotropic $S$-curvature is equivalent to the
isotropic $E$-curvature [15]. Therefore, it is very natural to ask, is there
any relation between isotropic $S$-curvature and isotropic $E$-curvature for
the other classes of Finsler metrics such as spherically symmetric Finsler
metrics and more generally for the warped product Finsler metrics $?$ In this
paper we prove the following result:

###### Theorem 1.1

Any warped product Finsler metric $F=\bar{\alpha}\phi(r,s)$ has isotropic
$S$-curvature if and only if it has isotropic $E$-curvature.

##  2 Preliminaries

Let $M$ be an $n$-dimensional smooth manifold. $T_{u}M$ denotes the tangent
space of $M$ at $u$. The tangent bundle of $M$ is the disjoint union of
tangent spaces $TM:=\sqcup_{u\in M}T_{u}M$. We denote the elements of $TM$ by
$(u,v)$ where $v\in T_{u}M$ and $TM_{0}:=TM\setminus\left\\{0\right\\}$.

###### Definition 2.1

[16] A Finsler metric on $M$ is a function $F:TM\to[0,\infty)$ satisfying the
following conditions:
(i) $F$ is smooth on $TM_{0}$,
(ii) $F$ is a positively 1-homogeneous on the fibers of tangent bundle $TM$,
(iii) The Hessian of $\frac{F^{2}}{2}$ with element
$g_{ij}=\frac{1}{2}\frac{\partial^{2}F^{2}}{\partial v^{i}\partial v^{j}}$ is
positive definite on $TM_{0}$.
The pair $(M,F)$ is called a Finsler space and $g_{ij}$ is called the
fundamental tensor.

Let us consider the product manifold $M=I\times\bar{M}$, where $I$ is an
interval of $\mathbb{R}$ and $\left(\bar{M},\bar{\alpha}\right)$ is an $(n-1)$
dimensional Riemannian manifold. Let $\left\\{{\theta^{a}}\right\\}_{a=2}^{n}$
be a local coordinate system on $\bar{M}$. Then
$\left\\{{u^{i}}\right\\}_{i=1}^{n}$ gives us a local coordinate on $M$ by
setting $u^{1}=r$ and $u^{a}=\theta^{a}$. The indices $i,j,k,...$ are ranging
from $1$ to $n$ while $a,b,c,...$ are ranging from $2$ to $n$. A vector $v$ on
$M$ can be written as $v=v^{i}\partial/\partial u^{i}$ and its projection on
$\bar{M}$ is denoted by $\bar{v}=v^{a}\partial/\partial
u^{a}=v^{a}\partial/\partial\theta^{a}$. A warped product Finsler metric can
be written in the form

$F=\bar{\alpha}\sqrt{w(s,r)},$ (1)

where $w$ is a suitable function defined on an open subset of $\mathbb{R}^{2}$
and $s=v^{1}/\bar{\alpha}$. It can be rewritten as [4]

$F=\bar{\alpha}\phi(s,r),\qquad\textnormal{where}\quad\phi(s,r)=\sqrt{w(s,r)}.$
(2)

In [4] Shen et al. also showed that warped product Finsler metrics contain the
class of spherically symmetric Finsler metrics.
The coefficients of fundamental metric tensor of the warped product Finsler
metrics are given by

$\begin{pmatrix}g_{11}&g_{1j}\\\
g_{i1}&g_{ij}\end{pmatrix}=\begin{pmatrix}\frac{1}{2}w_{ss}&\frac{1}{2}\chi_{s}\bar{\alpha}_{v^{j}}\\\
\frac{1}{2}\chi_{s}\bar{\alpha}_{v^{i}}&\frac{1}{2}\chi\bar{a}_{ij}-\frac{1}{2}s\chi_{s}\bar{\alpha}_{v^{i}}\bar{\alpha}_{v^{j}}\end{pmatrix}$

where, $\chi:=2w-sw_{s}$ and $\chi_{s}:=w_{s}-sw_{ss}$.
By some simple calculation we have,
$\det(g_{ij})=\frac{1}{2^{n-1}}\chi^{n-2}\Lambda$, where
$\Lambda=2ww_{ss}-w_{s}^{2}$.
Which can be rewritten as

$\det(g_{ij})=\phi^{n+1}\phi_{ss}(\phi-s\phi_{s})^{n-2}$ (3)

The spray coefficients of the Finsler metric $F$ are defined by

$G^{i}=\frac{1}{4}g^{il}\left\\{[F^{2}]_{v^{m}v^{l}}v^{m}-[F^{2}]_{u^{l}}\right\\},$
(4)

######  Lemma 2.1 ([4])

The spray coefficients of warped product Finsler metrics $F$ are given by

$G^{1}=\Phi\bar{\alpha}^{2},\quad G^{k}=\bar{G}^{k}+\Psi\bar{\alpha}v^{k}$ (5)

where

$\Phi=\frac{1}{4}\left\\{\left(W_{r}-\chi_{r}\right)U+s\chi_{r}V\right\\},\quad\Psi=\frac{1}{4}\left\\{\left(W_{r}-\chi_{r}\right)V+s\chi_{r}\left(W+X\right)\right\\}$
(6)

and

$\chi=2w-sw_{s},\quad\Lambda=2ww_{ss}-w_{s}^{2},\quad
U=\frac{2\chi-2s\chi_{s}}{\Lambda},\quad V=-\frac{2\chi_{s}}{\Lambda},\quad
W=\frac{2}{\Lambda},\quad X=\frac{2w_{s}\chi_{s}}{\chi\Lambda}.$ (7)

###### Definition 2.2

The $E$-curvature of a Finsler metric $F$ is defined as

$E_{ij}:=\frac{1}{2}S_{v^{i}v^{j}}(u,v)=\frac{1}{2}\frac{\partial^{2}}{\partial
v^{i}\partial v^{j}}\left(\frac{\partial G^{m}}{\partial v^{m}}\right).$ (8)

where $G^{i}$ are the spray coefficients of the Finsler metric $F$.
The Finsler metric $F$ is said to be of isotropic $E$-curvature if there
exists a scalar function $c=c(u)$ on $M$ such that

$E_{ij}=\frac{n+1}{2}c(u)F_{v^{i}v^{j}}.$ (9)

##  3 Isotropic $E$-curvature and Isotropic $S$-curvature of the warped
product Finsler metrics

In Finsler geometry two volume forms are well known, namely Busemann-Hausdorff
volume form ([12], [16]) and Holmes-Thompson volume form. In a local
coordinate system $(u^{i},v^{i})$, the Busemann-Hausdorff volume form is
defined as $dV_{BH}=\sigma_{BH}(u)du$, where

$\sigma_{BH}(u)=\dfrac{Vol(B^{n}(1))}{Vol\left\\{(v^{i})\in\mathbb{R}^{n}:F(u,v^{i}\frac{\partial}{\partial
u^{i}})<1\right\\}}.$ (10)

In a local coordinate system $(u^{i},v^{i})$, the Holmes-Thompson volume form
is defined as $dV_{HT}=\sigma_{HT}(u)du$, where

$\sigma_{HT}(u)=\dfrac{1}{Vol(B^{n}(1)}\int\limits_{F(u,v^{i}\frac{\partial}{\partial
u^{i}})<1}det(g_{ij}(u,v)dv).$ (11)

###### Definition 3.1

For a vector $v\in T_{u}M\setminus\left\\{0\right\\}$, let $\gamma=\gamma(t)$
be the geodesic with $\gamma(0)=u$ and $\dot{\gamma}(0)=v$. The $S$-curvature
of the Finsler metric $F$ with volume form $dV=\sigma_{F}(u)du$ is defined by

$S(u,v)=\frac{d}{dt}\left[\tau_{F}(\gamma(t),\dot{\gamma}(t)\right]|_{t=0},$

where $\tau_{F}$ is called distortion of the Finsler metric $F$ and defined by
$\tau_{F}=\log\frac{\sqrt{det(g_{ij})}}{\sigma_{F}}$.

###### Proposition 3.1

[16] In a standard local coordinate system in $TM$, the $S$-curvature of a
Finsler metric $F$ can also be written as

$S=\frac{\partial G^{m}}{\partial
v^{m}}(u,v)-v^{m}\frac{\partial\left(\log\sigma_{F}(u)\right)}{\partial
u^{m}}.$ (12)

###### Definition 3.2

A Finsler metric $F$ is said to be of isotropic $S$-curvature if

$S=(n+1)c(u)F$ (13)

Before proving the results we need the followings:

$\frac{\partial{\bar{\alpha}}^{2}}{\partial
v^{1}}=0,\quad\frac{\partial{\bar{\alpha}}^{2}}{\partial
v^{j}}=2v_{j}\quad\forall j=2,3,...,n.$ (14)
$s_{v^{1}}=\frac{1}{\bar{\alpha}},\quad s_{v^{1}}v^{1}=s,\quad
s_{v^{j}}=-\frac{sv_{j}}{{\bar{\alpha}}^{2}},\quad
s_{v^{j}}v^{j}=0,\quad\forall j=2,3,...,n.$ (15)
${\bar{\alpha}}_{v^{1}v^{j}}=0,\quad{\bar{\alpha}}_{v^{i}v^{j}}=\frac{{\bar{\alpha}}^{2}a_{ij}-v_{i}v_{j}}{{\bar{\alpha}}^{3}},\quad
s_{v^{1}v^{j}}=-\frac{v_{j}}{{\bar{\alpha}}^{3}},\quad
s_{v^{i}v^{j}}=\frac{3sv_{i}v_{j}-s{\bar{\alpha}}^{2}a_{ij}}{{\bar{\alpha}}^{4}}.$
(16)

###### Proposition 3.2

Let $F=\bar{\alpha}\phi(r,s)$ be a warped product Finsler metric on an
$n$-dimensional manifold $M=I\times\bar{M}$. Then the $E$-curvature of $F$ is
given by

$E_{11}=\frac{1}{\bar{\alpha}}\left[\Phi_{sss}+n\Psi_{ss}\right],\quad
E_{1j}=E_{j1}=-\frac{s}{\bar{\alpha}}v_{j}[\Phi_{sss}+n\Psi_{ss}],$ (17)
$E_{ij}=\frac{a_{ij}}{\bar{\alpha}}\left\\{(\Phi_{s}-s\Phi_{ss})+n(\Psi-s\Psi_{s})\right\\}+\frac{v_{i}v_{j}}{\bar{\alpha}^{3}}\left[ns^{2}\Psi_{ss}+s^{2}\Phi_{sss}+s(\Phi_{ss}+n\Psi_{s})-(\phi_{s}+n\Psi)\right],$
(18)

where $i,j\neq 1$.

### Proof:

From (5) we have

$\frac{\partial G^{1}}{\partial v^{1}}=\bar{\alpha}\Phi_{s}$ (19)

and for $k,m\neq 1$,

$\frac{\partial}{\partial
v^{m}}(\Psi\bar{\alpha}v^{k})=\Psi_{s}s_{v^{m}}\bar{\alpha}v^{k}+\Psi\bar{\alpha}_{v^{m}}v^{m}+(n-1)\Psi\bar{\alpha}\delta_{m}^{k}.$
(20)

Now putting $k=m$ in (20) and taking summation over $m$ we get

$\frac{\partial}{\partial
v^{m}}(\Psi\bar{\alpha}v^{m})=\Psi\bar{\alpha}_{v^{m}}v^{m}+(n-1)\Psi\bar{\alpha}=n\Psi\bar{\alpha}$

and therefore, we have

$\sum\frac{\partial G^{m}}{\partial v^{m}}=\bar{\alpha}[\Phi_{s}+n\Psi].$ (21)

Differentiating (21) with respect to $v^{i}$ gives

$\frac{\partial}{\partial v^{i}}\left(\frac{\partial G^{m}}{\partial
v^{m}}\right)=\bar{\alpha}_{v^{i}}[\Phi_{s}+n\Psi]+\bar{\alpha}[\Phi_{ss}s_{v^{i}}+n\Psi_{s}s_{v^{i}}].$
(22)

Again differentiating (22) with respect to $v^{j}$ gives

$\begin{split}\frac{\partial}{\partial v^{j}}\frac{\partial}{\partial
v^{i}}\left(\frac{\partial G^{m}}{\partial
v^{m}}\right)=\bar{\alpha}_{v^{i}}\bar{\alpha}_{v^{j}}[\Phi_{s}+n\Psi]+\left(\bar{\alpha}_{v^{i}}s_{v^{j}}+\bar{\alpha}_{v^{j}}s_{v^{i}}\right)\left[\Phi_{ss}+n\Psi_{s}\right]\\\
+\bar{\alpha}s_{v^{i}}s_{v^{j}}\left[\Phi_{sss}+n\Psi_{ss}\right]+\bar{\alpha}s_{v^{i}v^{j}}\left[\Phi_{ss}+n\Psi_{s}\right].\end{split}$
(23)

Putting $i=j=1$ and from (14), (15), (16)

$E_{11}=\frac{1}{\bar{\alpha}}\left[\Phi_{sss}+n\Psi_{ss}\right]$ (24)

and for $j\neq 1$

$E_{1j}=E_{j1}=\bar{\alpha}s_{v^{j}}\frac{1}{\bar{\alpha}}\left[\Phi_{sss}+n\Psi_{ss}\right]+\bar{\alpha}\frac{\partial}{\partial
v^{j}}\frac{1}{\bar{\alpha}}\left[\Phi_{ss}+n\Psi_{s}\right]$ (25)

and for $i,j\neq 1$ we have

$E_{ij}=\frac{a_{ij}}{\bar{\alpha}}\left[\left(\Phi_{s}-s\Phi_{ss}\right)+n(\Psi-s\Psi_{s})\right]+\frac{v_{i}v_{j}}{\bar{\alpha}}\left[ns^{2}\Psi_{ss}+s^{2}\Phi_{sss}+s(\Phi_{s}s+n\Psi_{s})-(\Phi_{s}+n\Psi)\right].$
(26)

$\square$

###### Theorem 3.1

The warped product Finsler metric $F=\bar{\alpha}\phi(r,s)$ has isotropic
$E$-curvature if and only if

$\left(\Phi_{s}-s\Phi_{ss}\right)+n\left(\Psi-s\Psi_{s}\right)=\kappa\left(\phi-s\phi_{s}\right),$
(27)

where $\Phi$, $\Psi$ are given in (6) and $\kappa\neq 0$ is a scalar function
on $M$.

### Proof:

Differentiating (2) with respect to $v^{i}$, we have

$F_{v^{i}}={\bar{\alpha}}_{v^{i}}\phi+{\bar{\alpha}}\phi_{s}s_{v^{i}}.$ (28)

Differentiating again (28) with respect to $v^{j}$ yields

$F_{v^{i}v^{j}}={\bar{\alpha}}_{v^{i}v^{j}}\phi+({\bar{\alpha}}_{v^{i}}s_{v^{j}}+{\bar{\alpha}}_{v^{j}}s_{v^{i}})\phi_{s}+{\bar{\alpha}}s_{v^{i}}s_{v^{j}}\phi_{ss}+{\bar{\alpha}}s_{v^{i}v^{j}}\phi_{s}.$
(29)

If $i=j=1$ then we have,

$F_{v^{1}v^{1}}=\frac{\phi_{ss}}{{\bar{\alpha}}}$ (30)

and if $i=1,j\neq 1$, or, $i\neq 1,j=1$ then

$F_{v^{1}v^{j}}=-\frac{v_{j}}{\bar{\alpha}^{2}}s\phi_{ss}\quad\textnormal{and}\quad
F_{v^{j}v^{1}}=-\frac{v_{j}}{\bar{\alpha}^{2}}s\phi_{ss}.$ (31)

And for $i,j\neq 1$ plugging (14), (16) into (29) yields

$F_{v^{i}v^{j}}=\frac{1}{{\bar{\alpha}}^{3}}\left[(\phi-s\phi_{s}){\bar{\alpha}}^{2}a_{ij}-(\phi-s\phi_{s}-s^{2}\phi_{ss})v_{i}v_{j}\right].$
(32)

In the view of equations (9), (17) and (18) the warped product Finsler metric
is of isotropic $E$-curvature if and only if we have,

$\Phi_{sss}+n\Psi_{ss}=\frac{n+1}{2}c(x)\phi_{ss},$ (33)
$\left(\Phi_{s}-s\Phi_{ss}\right)+n\left(\Psi-s\Psi_{s}\right)=\frac{n+1}{2}c(x)(\phi-s\phi_{s}),$
(34)
$ns^{2}\Psi_{ss}+s^{2}\Phi_{sss}+s(\Phi_{s}s+n\Psi_{s})-(\Phi_{s}+n\Psi)=\frac{n+1}{2}c(x)(\phi-s\phi_{s}-s^{2}\phi_{ss}).$
(35)

Now we will show that (33) and (35) can be obtained from (34). Differentiating
(34) with respect to $s$ we get (33). Again multiplying (33) by $s^{2}$ and
substract it from (34) we obtain (35). Therefore, $F$ has isotropic
$E$-curvature if and only if (27) holds. $\square$

###### Lemma 3.1

Let $F=\bar{\alpha}\phi(s,r)$, be a warped product Finsler metric on an
$n$-dimensional manifold $M=I\times\bar{M}$. Then the volume form $dV$ is
given by $dV=k(r)dV_{\alpha}$ where

$k(r)=\begin{cases}\dfrac{\int\limits_{-1}^{1}\frac{(1-s^{2})^{\frac{n-3}{2}}}{\phi^{n}(r,s)}ds}{\int\limits_{0}^{\pi}\sin^{n-2}(t)dt}&if,\quad
dV=dV_{BH}\\\ \dfrac{\int\limits_{0}^{\pi}(\sin^{n-2}t)\upsilon(r^{2},r\cos
t)dt}{\int\limits_{0}^{\pi}\sin^{n-2}(t)dt}&if,\quad dV=dV_{HT}\\\
\end{cases}$ (36)

and $dV_{\alpha}=\sqrt{det(a_{ij})}dx$ denotes the Riemannian volume form of
$\alpha$.

### Proof:

Fix an arbitrary point $x_{0}\in U\subset\mathbb{R}^{n},$ consider an
orthogonal basis at $x_{0}$ with respect to the Riemannian metric $\alpha$ so
that

$\alpha=\sqrt{\sum_{i=1}^{n}(v^{i})^{2}}.$

Then the volume form $dV_{\alpha}=\sigma_{\alpha}du$ at the point $x_{0}$ is
given by $dV_{\alpha}=\sigma_{\alpha}$ $du$ at $x_{0}$.

$\sigma_{\alpha}=\sqrt{det(a_{ij})}=1.$

Let us consider the coordinate transformation: $\mu:(s,z^{a})\to(v^{i})$ such
that

$v^{1}=\frac{s}{\sqrt{1-s^{2}}}\bar{\alpha}\quad\textnormal{and}\quad
v^{a}=z^{a},$ (37)

where $\bar{\alpha}=\sqrt{\sum\limits_{a=2}^{n}(z^{a})^{2}}$, $1\leq i,j,k\leq
n$, and, $2\leq a,b,c\leq n$.
Then $\alpha=\frac{1}{\sqrt{1-s^{2}}}\bar{\alpha}$.
Therefore,

$F=\alpha\phi(r,s)=\frac{\bar{\alpha}\phi}{\sqrt{1-s^{2}}}$

and the Jacobian of the transformation $\mu:(s,z^{a})\to(v^{i})$ is given by
$\dfrac{\bar{\alpha}}{(1-s^{2})^{3/2}}$.
Then

$\begin{split}Vol\left\\{(v^{i})\in\mathbb{R}^{n}:F(u,v^{i}\frac{\partial}{\partial
u^{i}})<1\right\\}&=\int_{F(u,v)<1}dv\\\ &=\int_{\alpha\phi(r,s)<1}dv\\\
&=\int_{\frac{\bar{\alpha}\phi(r,s)}{\sqrt{1-s^{2}}}<1}\frac{1}{(1-s^{2})^{3/2}}\bar{\alpha}dsdu\\\
&=\int\limits_{-1}^{1}\frac{1}{(1-s^{2})^{\frac{3}{2}}}\left[\int_{\bar{\alpha}<\frac{\sqrt{1-s^{2}}}{\phi(r,s)}}\alpha
du\right]ds\\\
&=\frac{1}{n}Vol(S^{n-2})\int\limits_{-1}^{1}\frac{1}{(1-s^{2})^{\frac{3}{2}}}\left(\frac{\sqrt{1-s^{2}}}{\phi(r,s)}\right)^{n}ds\\\
&=\frac{1}{n}Vol(S^{n-2})\int\limits_{-1}^{1}\frac{(1-s^{2})^{\frac{n-3}{2}}}{\phi^{n}(r,s)}ds\\\
&=\frac{1}{n}Vol(S^{n-2})k(r)\end{split}$

where

$k(r)=\int\limits_{-1}^{1}\frac{(1-s^{2})^{\frac{n-3}{2}}}{\phi^{n}(r,s)}ds.$

Since,
$Vol(B^{n}(1))=\frac{1}{n}Vol(S^{n-2})\int\limits_{0}^{\pi}\sin^{n-2}(t)dt$,
we have

$\sigma_{BH}=\frac{\int\limits_{-1}^{1}\frac{(1-s^{2})^{\frac{n-3}{2}}}{\phi^{n}(r,s)}ds}{\int\limits_{0}^{\pi}\sin^{n-2}(t)dt}.$
(38)

Let us consider $\upsilon=\phi\phi_{ss}(\phi-s\phi_{s})^{n-2}$. Hence, from
(11) and (3) we have,

$\begin{split}\int_{F(u,v^{i}\frac{\partial}{\partial
u^{i}}<1)}det(g_{ij}(u,v))dv&=\int_{F(u,v)<1}\phi^{n}(r,s)\upsilon(r,s)ds\\\
&=\frac{1}{n}Vol(S^{n-2})\int\limits_{-1}^{1}(1-s^{2})^{(n-3)/2}\upsilon(r,s)ds\\\
&=\frac{1}{n}Vol(S^{n-2})\int\limits_{0}^{\pi}(\sin^{n-2}t)\upsilon(r^{2},r\cos
t)dt.\end{split}$ (39)

Therefore,

$\sigma_{HT}=\frac{\int\limits_{0}^{\pi}(\sin^{n-2}t)\upsilon(r^{2},r\cos
t)dt}{\int\limits_{0}^{\pi}\sin^{n-2}(t)dt}.$ (40)

Hence we have the result. $\square$

###### Theorem 3.2

A warped product Finsler metric $F=\bar{\alpha}\phi(r,s)$ on a manifold
$M=I\times\bar{M}$ has isotropic $S$-curvature with respect to volume form
$dV_{BH}$ or $dV_{HT}$ if and only if

$\Phi_{s}+n\Psi+g(r)s=(n+1)\phi c(u),$ (41)

where $c(u)$ is a scalar function on $M$ and
$g(r)=-r\frac{k^{\prime}(r)}{k(r)}$.

### Proof:

Since, $\bar{G}^{k}=\frac{1}{2}\Gamma^{k}_{ij}v^{i}v^{j}$, we have

$[\bar{G}^{k}]_{v^{k}}=\Gamma^{k}_{kj}v^{j}=v^{m}\frac{\partial}{\partial
u^{m}}(\log\sigma_{\bar{\alpha}}).$ (42)

By Lemma (3.1), we have $dV=\sigma du=K(r)\sigma_{\bar{\alpha}}du$. Hence,

$v^{m}\frac{\partial}{\partial
u^{m}}(\log\sigma)=\dfrac{k^{\prime}(r)}{k(r)}v^{m}\dfrac{\partial r}{\partial
u^{m}}+v^{m}\dfrac{\partial}{\partial u^{m}}(\log\sigma_{\bar{\alpha}}).$ (43)

The $S$-curvature of a warped product Finsler metric $F=\bar{\alpha}\phi(r,s)$
is given by

$S=v^{m}\frac{\partial}{\partial
u^{m}}(\log\sigma_{\bar{\alpha}})+\bar{\alpha}\left[\Phi_{s}+n\Psi\right]-v^{m}\dfrac{\partial}{\partial
u^{m}}(\log\sigma).$ (44)

Again

$v^{m}\frac{\partial}{\partial
u^{m}}(\log\sigma_{\bar{\alpha}})=\bar{\alpha}sr.$ (45)

Now, from (12), (44) and (45) we obtain

$S=\bar{\alpha}\left[\Phi_{s}+n\Psi+g(r)s\right],$ (46)

where

$g(r)=-r\dfrac{k^{\prime}(r)}{k(r)}.$ (47)

Therefore, $F$ has isotropic $S$-curvature if and only if (41) holds.
$\square$

Proof of Theorem 1.1

Let the Finsler metric $F$ has isotropic $S$-curvature. Then (41) holds.
Therefore, differentiating (41) with respect to $s$ we have,

$\Phi_{ss}+n\Psi_{s}+g(r)=(n+1)c(u)\phi_{s}.$ (48)

Now multiplying (48) by $s$ and substracting it from (41) gives (27). Hence,
$F$ has isotropic $E$-curvature.
Conversely, suppose $F$ has isotropic $E$-curvature. Then (27) holds. As
$s\neq 0$, it can be rewritten as

$\frac{s\Phi_{ss}-\Phi_{s}}{s^{2}}+n\frac{s\Psi_{s}-\Psi}{s^{2}}=(n+1)c(u)\frac{\phi-s\phi_{s}}{s^{2}},$
(49)

which implies

$\frac{\partial}{\partial
s}\left(\frac{\Phi_{s}}{s}\right)+n\frac{\partial}{\partial
s}\left(\dfrac{\Psi}{s}\right)=(n+1)c(u)\frac{\partial}{\partial
s}\left(\frac{\phi_{s}}{s}\right).$ (50)

Integrating we have,

$\frac{\Phi_{s}}{s}+n\frac{\Psi}{s}+g(r)=(n+1)c(u)\frac{\phi_{s}}{s},$ (51)

where $g(r)$ is the constant of integration.
Now in particular, if we choose $g(r)=-r\dfrac{k^{\prime}(r)}{k(r)}$ and
multiplying (51) by $s$ we get (41). Therefore, $F$ has isotropic
$S$-curvature.

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