# Unveiling Global Narratives: A Multilingual Twitter Dataset of News Media on
the Russo-Ukrainian Conflict

Sherzod Hakimov<EMAIL_ADDRESS>Computational Linguistics,
Department of Linguistics
University of PotsdamGermany and  Gullal S. Cheema<EMAIL_ADDRESS>L3S
Research Center
Leibniz University, HannoverGermany

###### Abstract.

The ongoing Russo-Ukrainian conflict has been a subject of intense media
coverage worldwide. Understanding the global narrative surrounding this topic
is crucial for researchers that aim to gain insights into its multifaceted
dimensions. In this paper, we present a novel multimedia dataset that focuses
on this topic by collecting and processing tweets posted by news or media
companies on social media across the globe. We collected tweets from February
2022 to May 2023 to acquire approximately 1.5 million tweets in 60 different
languages along with their images. Each entry in the dataset is accompanied by
processed tags, allowing for the identification of entities, stances, textual
or visual concepts, and sentiment. The availability of this multimedia dataset
serves as a valuable resource for researchers aiming to investigate the global
narrative surrounding the ongoing conflict from various aspects such as who
are the prominent entities involved, what stances are taken, where do these
stances originate from, how are the different textual and visual concepts
related to the event portrayed.

multimedia news discourse, multilingual twitter narrative, russo-ukrainian
conflict

††ccs: Computing methodologies Information extraction††ccs: Information
systems Summarization††ccs: Applied computing Document analysis††ccs:
Information systems Digital libraries and archives

##  1\. Introduction

The Russo-Ukrainian conflict, which began in February 2022, has been a focal
point of global attention. The conflict has been extensively covered by news
media channels worldwide, each presenting a unique perspective shaped by their
geographical location, political stance, and cultural context. This extensive
coverage has resulted in a wealth of data that, if properly harnessed, can
provide valuable insights into the global perception and narrative of the
conflict. However, to date, no comprehensive study has been conducted to
analyze the coverage of the Russo-Ukrainian conflict by news media channels on
social media, specifically focusing on multimedia data that does not only
include text but also images. This gap in the literature motivates the need
for a dataset that encompasses tweets and their images from news media
channels across the globe, focusing on the topic of the Russo-Ukrainian war.
Specifically, the analysis of discourse on social media from various
perspectives such as within a certain country and news media companies.
Currently, existing datasets about the conflict focused either more on the
collection of language-specific subsets (Park et al., 2022b; Vahdat-Nejad et
al., 2023; Toraman et al., 2022), applying only sentiment analysis methods
(Caprolu et al., 2023; Shevtsov et al., 2022; Xu et al., 2023; Džubur et al.,
2022), or use multimedia data (tweet text and image) for down-stream tasks
such as hate speech detection (Thapa et al., 2022; Bhandari et al., 2023).
However, none of these previous studies focused specifically on the coverage
of the event from news or media companies’ perspectives.

Figure 1. Distribution of tweets across countries

In response to this need, we present a multimedia dataset composed of tweets
with images from news media channels worldwide that pertain to the Russo-
Ukrainian war. This dataset spans a period of February 2022 - May 2023. The
dataset is unique in its global scope, encompassing tweets in $60$ languages
and from different parts of the world (see Figure 1). We downloaded the images
for tweets that include them. Additionally, we extracted information about the
stance, sentiment, prominent entities & concepts that occur in tweets, and
classified visual concepts in images of tweets to be able to answer questions
about the discourse on the ongoing event: who says what (prominent entities),
who stands (stance) where on what aspect (prominent concepts), how are the
aspects portrayed (sentiment), and finally what is visually portrayed. This
comprehensive collection of data allows for a nuanced analysis of the global
coverage of the Russo-Ukrainian war, providing insights into how different
regions and cultures perceive and report on the conflict. The dataset will
serve as a valuable resource for researchers interested in media studies,
conflict analysis, and international relations, facilitating a deeper
understanding of the global narrative surrounding the Russo-Ukrainian
conflict.

##  2\. Related Work

The Russo-Ukrainian conflict has been the subject of extensive research in the
realm of social media analysis. Several datasets have been curated to study
various aspects of this conflict. Park et al. (2022a) introduced the VoynaSlov
dataset, a collection of over $38$ million posts from Russian media outlets on
Twitter and VKontakte, to analyze information manipulation. The dataset is
used to analyze media effects and to discuss challenges and opportunities in
NLP research on information manipulation campaigns. Similarly, Alyukov et al.
(2023) studied the manipulation of information shared on Russian social media
platforms. Geissler et al. (2023) focused on analyzing tweets shared to
support of Russia’s stance on the conflict and concluded that many bot
accounts were deployed to disseminate propaganda on Twitter. Similar study by
Pierri et al. (2023) also focused on analyzing shared propaganda and
misinformation on Facebook and Twitter. Vahdat-Nejad et al. (2023)
investigated English tweets on the Russia-Ukraine war to analyze trends
reflecting users’ opinions and sentiments regarding the conflict. Caprolu et
al. (2023) built a dataset for the conflict by collecting more than $5.5$
million tweets related to the subject and performed Aspect-Based Sentiment
Analysis (ABSA) to characterize the sentiment about the conflict shared on
Twitter in the English-speaking world. Similarly, Shevtsov et al. (2022)
provided a dataset of $57.3$ million tweets from $7.7$ million users and
provided a glimpse of volume and sentiment trends in the data. Xu et al.
(2023) conducted a study on sentiment analysis using Long Short-Term Memory
(LSTM) and Sastrawi on an Indonesian Twitter dataset. Džubur et al. (2022)
focused on tweets related to the Russo-Ukrainian conflict and combined
sentiment and network analysis approaches to produce various important
insights into the discussion of the conflict. Chen and Ferrara (2023) have
collected more than 600 million tweets between Feb 2022-Feb 2023 while Zhu et
al. (2022) analyzed communities (subreddits) on Reddit related to the event.

Different from these aforementioned datasets, our focus lies on extracting the
global news reporting and discourse on the topic from the perspective of news
or media companies around the world. We collected tweets only from user
accounts that are tied to a specific media company. Expanding on existing
datasets and approaches on the same event, we additionally included both
textual and visual concepts that are essential for discourse analysis on
multimedia data such as stance, sentiment, and prominent entities as well as
concepts in the text and image content.

##  3\. Dataset

Table 1. Distribution of collected tweets across $60$ languages | Language | Count
---|---
English | 581469
Arabic | 169682
Spanish | 163558
French | 111041
German | 74116
Italian | 54336
Turkish | 51274
Dari | 42138
Ukrainan | 41482
Russian | 38198
Portuguese | 38180
Hebrew | 21133
Hindi | 16681
Polish | 16427
Catalan | 16068
Indonesian | 12853
Dutch | 12737
Greek | 11460
Korean | 8274
Japanese | 6143
Urdu | 5263
Romanian | 4944
Denmark | 4103
Norwegian | 3643
Swedish | 2674
Finnish | 2646
Czech | 2226
Bengali | 2116
Gujarati | 1358
Thai | 1262
| Language | Count
---|---
Malay | 1007
Telugu | 900
Tamil | 832
Oriya | 761
Tagalog | 613
Basque | 540
Slovakian | 517
Vietnamese | 368
Estonian | 274
Belarussian | 254
Marathi | 213
Nepali | 191
Azeri | 160
Bulgarian | 98
Chinese | 90
Assamese | 78
Pashto | 72
Hungarian | 67
Croatian | 59
Bosnian | 58
Luxembourgish | 44
Lithuanian | 33
Slovenian | 20
Swahili | 17
Welsh | 15
Tajik | 15
Ganda | 14
Kazakh | 13
Latvian | 12
Irish | 12

In this section, we provide details about the collection, filtering and
processing steps. Our dataset covers $60$ languages with total of
$1\,524\,826$ tweets, out of which $306\,295$ have images. These tweets were
posted by news or media companies around the world. The source code and the
dataset are publicly available 111 https://github.com/sherzod-hakimov/ru-ua-
news-discourse-twitter .

###  3.1. Data Collection

Extraction of Twitter Handles: We have used Wikidata (Vrandecic and Krötzsch,
2014) to query for the news companies, their countries, and Twitter handles
222Wikidata query for news companies’ Twitter handles and countries:
https://t.ly/XEp6. The query returned $14\,587$ news/media companies and their
respective countries.

User Account Verification: We used Twitter API to check which of the returned
user accounts are verified, which was done in March 2022. We kept verified
accounts with at least $100\,000$ and non-verified accounts with at least
$5000$ followers. In total, we ended up with $1795$ verified and $1343$ non-
verified accounts.

Querying of Tweets: Using the extracted and filtered Twitter handles of news
companies from the previous steps, we have queried all tweets posted on these
accounts between Feb 1st, 2022 - May 31st, 2023. In total, we have collected
around $47$ million tweets.

###  3.2. Data Filtering

Keywords: We manually curated a list of keywords in English that identify with
the Russo-Ukrainian conflict (similar to Chen and Ferrara (2023)). Later, we
translated the keywords into other languages, which are spoken in countries
where news companies are located, using Google Translate and DeepL. The full
list is provided333https://github.com/sherzod-hakimov/ru-ua-news-discourse-
twitter/blob/main/resources/language_resources.jsonl.

Filtering: To remove irrelevant tweets, we applied two filtering steps. The
first step is based on removing tweets that do not include the target keywords
in the respective languages. The second step is based on prompting
Flan-T5-large model (Chung et al., 2022) to check whether the tweet text is
related with the topic of interest. It was done by prompting the language
model to output yes/no by answering the following question “Does the following
tweet mention or relate to Russia-Ukraine conflict?”. After applying these
steps, we ended up with a total of $1\,524\,826$ tweets for $60$ languages.
The distribution of tweets across languages is available in Table 1.

Figure 2. Distribution of tweets across languages over timeline. Certain
prominent events are added manually (dashed lines). Figure 3. Distribution of
stance in tweets across top 50 countries Figure 4. Distribution of sentiment
in tweets across top 50 countries

(a) Word cloud of concepts extracted from images

(b) Word cloud of concepts extracted from tweet text (English translations)

Figure 5. Word cloud of concepts in text and images

###  3.3. Data Processing

The main motivation for creating this multimedia dataset is to be able to
analyse the discourse from the news or media companies’ perspectives on where
do they stand and how they reflect on which aspects. We process both textual
and visual content of each tweet. We downloaded all media links in tweets that
are of type “photo”, which resulted in $306\,295$ images. Next, we applied
pre-trained models to extract class labels of interest from both textual or
visual content.

Language detection: Each tweet by default is assigned a language tag. In cases
where the language tags were undefined, we used a language identification
model 444https://huggingface.co/papluca/xlm-roberta-base-language-detection to
classify them.

Translation: To deal with the disparity of NLP tools to apply on $60$
languages, we decided to translate all non-English tweets into English to be
able to use the same pre-trained models without having to deal with the trade-
off of having multilingual models vs. performance. We used the pre-trained
machine translation model called No Language Left Behind (NLLB Team et al.,
2022) (version nllb-200-1.3B).

Normalization: We replaced all user and URL mentions with placeholders
(¡USER_MENTION¿, ¡URL¿).

Sentiment: We used the pre-trained model (Loureiro et al., 2022) for sentiment
analysis that outputs probabilities for positive, negative, or
neutral555https://huggingface.co/cardiffnlp/twitter-roberta-base-sentiment-
latest.

Stance: Stance detection essentially reveals whether a given premise (tweet
text) is aligned with a hypothesis. Hypotheses in this study are related to
identifying whether the content is against or in favour of concepts such as
military conflict, war, Russia, Ukraine. We used following hypotheses: “This
statement is in favour of Russia”, “This statement is against Russia”, “This
statement is against Ukraine”, “This statement is in favour of Ukraine”, “This
statement is in favour of war”, “This statement is against war”, “This
statement is in favour of military conflict”, “This statement is against
military conflict”. We used a pre-trained BART model (Lewis et al., 2019) that
is fine-tuned on Multi-Genre Natural Language Inference (Williams et al.,
2018) dataset 666https://huggingface.co/facebook/bart-large-mnli.

Prominent entities, concepts: We used the Stanza model (Qi et al., 2020) to
extract Part-of-Speech (POS) tags, named entities from all tweets. The
prominent concepts can be constructed from noun POS tags whereas entities are
directly extracted by the model.

Visual concepts: We applied an image recognition model (Huang et al., 2023)
(version RAM++) on images to extract detected concepts in images.

##  4\. Analysis

###  4.1. Preliminary Findings

Languages, events & number of tweets: Figure 2 depicts the number of tweets
posted across top-5 languages (English, Arabic, Spanish, French, German) and
others combined over the timeline. We can observe that the event has attracted
similar attention across all compared languages and the peaks in a certain
time frame also correlate among the compared groups. In addition, the plot
shows some of the key events in the war that can potentially explain the
increase in tweets at a particular time.

Stance: Figure 3 shows the distribution of stance across the top-50 countries.
The stances are either pro-Russia, pro-Ukraine, or unsure. The values are
calculated by merging the stance outputs for hypotheses mentioned in Section
3.3. All confidence values equal or higher than $0.9$ for hypotheses “This
statement is in favour of Russia”, “This statement is in favour of war”, “This
statement is in favour of military conflict” are grouped under pro-Russia
category. Similarly for hypotheses “This statement is against Russia”, “This
statement is in favour of Ukraine”, “This statement is against war”, “This
statement is against military conflict” with confidence values equal or higher
than $0.9$ are grouped under pro-Ukraine. The remaining data points are simply
merged under unsure category. As a result, the tweets are categorized by
stances as follows: $89$% unsure, $7.3$% pro-Ukraine, $2.7$% pro-Russia.

Sentiment: The distribution of sentiment across the dataset is as follows:
$58$% neutral, $40$% negative, and $2$% positive (see Figure 4).

Visual concepts: Figure 5(a) shows the most prominent concepts that appear in
images. As we can observe, the concepts such as tie wear, suit, tie, person,
man being the most common on top of concepts such as damage, debris, man,
podium, give speech, army camouflage, army tank. It suggests that most images
depict either people giving speech in a formal clothing or the debris of
buildings and military vehicles and related concepts.

Textual concepts: Figure 5(b) shows word cloud generated from tweet text (or
translations for languages other than English). As we can see, the most
prominent words refer to two countries, their presidents and other related
countries.

###  4.2. Potential Use-cases

Comparative political science studies and influence analysis: Researchers can
compare and contrast global perceptions of the Russo-Ukrainian conflict with
other similar geopolitical events. This allows for a deeper understanding of
the nuances in global reactions to different conflicts and can help explain
factors that influence these reactions. Similar to the analysis of retweeted
country and retweeter country in (Chen and Ferrara, 2023), an analysis of the
country as well as media channels can be performed to identify key sources of
news and influence, and how they shape the global conversation on the
conflict. Such analysis can reveal underlying patterns of power dynamics and
geopolitical influence in shaping the narrative around international
conflicts.

Linguistic studies, journalism and media studies: The tweets in 60 different
languages can help us see how media channels talk about the conflict across
cultures and languages. It provide insights into the region and culture-
specific idioms, metaphors, and linguistic structures used in reporting and
discussing the conflict. The dataset also offers a means to examine the
framing of the conflict by different media outlets worldwide. It can
facilitate the identification of potential biases, political stances (via
stance detection), and varying reportage styles, contributing to a more
comprehensive understanding of the global media landscape and its impact on
conflict narratives.

Social media dynamics, fake news and misinformation studies: The dataset
allows for a thorough investigation of how narratives spread and evolve over
time. It offers a real-world case study on information propagation, topic
lifespan, and the dynamics of virality on social media platforms. It is also a
key resource for studying misinformation, often prevalent in conflicts,
helping identify misinformation patterns. The inclusion of verified and non-
verified accounts adds an extra dimension to fake news analysis.

##  5\. Conclusion

We present a new multimedia dataset that focuses on unveiling the global
narrative of the ongoing Russo-Ukrainian conflict by analysing the tweets
posted by news or media companies around the world. We collected the tweets
between February 2022 - May 2023 and applied certain filtering and NLP
pipelines to acquire around 1.5 million tweets with their images in $60$
different languages. Our dataset includes processed tags for each tweet to be
able to answer questions such as who says what (prominent entities), who
stands (stance) where on what aspect (prominent/visual concepts), and how are
the aspects portrayed (sentiment). The existence of such a dataset will serve
as a valuable resource for researchers aiming to study the global narrative
from various aspects.
Limitations: The main limitation of the approach is reliance on tools that are
pre-trained on English data since all tweet text were translated to English.
The main reason behind this is not having such language tools for all
languages of interest.
Ethical Considerations: The publicly shared dataset will be composed of only
tweet IDs and the processed information described in Section 3.3. The full
tweet text and images can be only be released upon formal request only for
scientific purposes.

## References

  * (1)
  * Alyukov et al. (2023) Maxim Alyukov, Maria Kunilovskaya, and Andrei Semenov. 2023\.  Wartime Media Monitor (WarMM-2022): A Study of Information Manipulation on Russian Social Media during the Russia-Ukraine War. In _Proceedings of the 7th Joint SIGHUM Workshop on Computational Linguistics for Cultural Heritage, Social Sciences, Humanities and Literature_ (Dubrovnik, Croatia). Association for Computational Linguistics, 152–161.  https://aclanthology.org/2023.latechclfl-1.17
  * Bhandari et al. (2023) Aashish Bhandari, Siddhant Bikram Shah, Surendrabikram Thapa, Usman Naseem, and Mehwish Nasim. 2023.  CrisisHateMM: Multimodal Analysis of Directed and Undirected Hate Speech in Text-Embedded Images from Russia-Ukraine Conflict. In _IEEE/CVF Conference on Computer Vision and Pattern Recognition, CVPR 2023 - Workshops, Vancouver, BC, Canada, June 17-24, 2023_. IEEE, 1994–2003.  https://doi.org/10.1109/CVPRW59228.2023.00193
  * Caprolu et al. (2023) Maurantonio Caprolu, Alireza Sadighian, and Roberto Di Pietro. 2023.  Characterizing the 2022- Russo-Ukrainian Conflict Through the Lenses of Aspect-Based Sentiment Analysis: Dataset, Methodology, and Key Findings. In _32nd International Conference on Computer Communications and Networks, ICCCN 2023, Honolulu, HI, USA, July 24-27, 2023_. IEEE, 1–10.  https://doi.org/10.1109/ICCCN58024.2023.10230192
  * Chen and Ferrara (2023) Emily Chen and Emilio Ferrara. 2023.  Tweets in time of conflict: A public dataset tracking the twitter discourse on the war between Ukraine and Russia. In _Proceedings of the International AAAI Conference on Web and Social Media_ , Vol. 17. 1006–1013.
  * Chung et al. (2022) Hyung Won Chung, Le Hou, Shayne Longpre, Barret Zoph, Yi Tay, William Fedus, Eric Li, Xuezhi Wang, Mostafa Dehghani, Siddhartha Brahma, Albert Webson, Shixiang Shane Gu, Zhuyun Dai, Mirac Suzgun, Xinyun Chen, Aakanksha Chowdhery, Sharan Narang, Gaurav Mishra, Adams Yu, Vincent Zhao, Yanping Huang, Andrew Dai, Hongkun Yu, Slav Petrov, Ed H. Chi, Jeff Dean, Jacob Devlin, Adam Roberts, Denny Zhou, Quoc V. Le, and Jason Wei. 2022.  Scaling Instruction-Finetuned Language Models.  https://doi.org/10.48550/ARXIV.2210.11416
  * Džubur et al. (2022) Benjamin Džubur, Žiga Trojer, and Urša Zrimšek. 2022.  Semantic Analysis of Russo-Ukrainian War Tweet Networks.  _SCORES: Ljubljana, Slovenia_ (2022).
  * Geissler et al. (2023) Dominique Geissler, Dominik Bär, Nicolas Pröllochs, and Stefan Feuerriegel. 2023.  Russian propaganda on social media during the 2022 invasion of Ukraine.  _EPJ Data Science_ 12, 1 (2023), 35.  https://doi.org/10.1140/epjds/s13688-023-00414-5
  * Huang et al. (2023) Xinyu Huang, Yi-Jie Huang, Youcai Zhang, Weiwei Tian, Rui Feng, Yuejie Zhang, Yanchun Xie, Yaqian Li, and Lei Zhang. 2023.  Open-Set Image Tagging with Multi-Grained Text Supervision.  arXiv:2310.15200 [cs.CV]
  * Lewis et al. (2019) Mike Lewis, Yinhan Liu, Naman Goyal, Marjan Ghazvininejad, Abdelrahman Mohamed, Omer Levy, Veselin Stoyanov, and Luke Zettlemoyer. 2019.  BART: Denoising Sequence-to-Sequence Pre-training for Natural Language Generation, Translation, and Comprehension.  _CoRR_ abs/1910.13461 (2019).  arXiv:1910.13461 http://arxiv.org/abs/1910.13461
  * Loureiro et al. (2022) Daniel Loureiro, Francesco Barbieri, Leonardo Neves, Luis Espinosa Anke, and Jose Camacho-collados. 2022\.  TimeLMs: Diachronic Language Models from Twitter. In _Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics: System Demonstrations_. Association for Computational Linguistics, Dublin, Ireland, 251–260.  https://doi.org/10.18653/v1/2022.acl-demo.25
  * NLLB Team et al. (2022) NLLB Team, Marta R. Costa-jussà, James Cross, Onur Çelebi, Maha Elbayad, Kenneth Heafield, Kevin Heffernan, Elahe Kalbassi, Janice Lam, Daniel Licht, Jean Maillard, Anna Sun, Skyler Wang, Guillaume Wenzek, Al Youngblood, Bapi Akula, Loic Barrault, Gabriel Mejia-Gonzalez, Prangthip Hansanti, John Hoffman, Semarley Jarrett, Kaushik Ram Sadagopan, Dirk Rowe, Shannon Spruit, Chau Tran, Pierre Andrews, Necip Fazil Ayan, Shruti Bhosale, Sergey Edunov, Angela Fan, Cynthia Gao, Vedanuj Goswami, Francisco Guzmán, Philipp Koehn, Alexandre Mourachko, Christophe Ropers, Safiyyah Saleem, Holger Schwenk, and Jeff Wang. 2022.  No Language Left Behind: Scaling Human-Centered Machine Translation.  (2022).
  * Park et al. (2022a) Chan Young Park, Julia Mendelsohn, Anjalie Field, and Yulia Tsvetkov. 2022a.  Challenges and Opportunities in Information Manipulation Detection: An Examination of Wartime Russian Media. In _Findings of the Association for Computational Linguistics: EMNLP 2022_. Association for Computational Linguistics, Abu Dhabi, United Arab Emirates, 5209–5235.  https://doi.org/10.18653/v1/2022.findings-emnlp.382
  * Park et al. (2022b) Chan Young Park, Julia Mendelsohn, Anjalie Field, and Yulia Tsvetkov. 2022b.  VoynaSlov: a data set of Russian social media activity during the 2022 Ukraine-Russia War.  _arXiv preprint arXiv:2205.12382_ (2022).
  * Pierri et al. (2023) Francesco Pierri, Luca Luceri, Nikhil Jindal, and Emilio Ferrara. 2023.  Propaganda and Misinformation on Facebook and Twitter during the Russian Invasion of Ukraine. In _Proceedings of the 15th ACM Web Science Conference 2023_ _(WebSci ’23)_. ACM.  https://doi.org/10.1145/3578503.3583597
  * Qi et al. (2020) Peng Qi, Yuhao Zhang, Yuhui Zhang, Jason Bolton, and Christopher D. Manning. 2020.  Stanza: A Python Natural Language Processing Toolkit for Many Human Languages. In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics: System Demonstrations_.  https://nlp.stanford.edu/pubs/qi2020stanza.pdf
  * Shevtsov et al. (2022) Alexander Shevtsov, Christos Tzagkarakis, Despoina Antonakaki, Polyvios Pratikakis, and Sotiris Ioannidis. 2022.  Twitter Dataset on the Russo-Ukrainian War.  _arXiv preprint arXiv:2204.08530_ (2022).
  * Thapa et al. (2022) Surendrabikram Thapa, Aditya Shah, Farhan Jafri, Usman Naseem, and Imran Razzak. 2022\.  A Multi-Modal Dataset for Hate Speech Detection on Social Media: Case-study of Russia-Ukraine Conflict. In _Proceedings of the 5th Workshop on Challenges and Applications of Automated Extraction of Socio-political Events from Text (CASE)_. Association for Computational Linguistics, Abu Dhabi, United Arab Emirates (Hybrid), 1–6.  https://doi.org/10.18653/v1/2022.case-1.1
  * Toraman et al. (2022) Cagri Toraman, Oguzhan Ozcelik, Furkan Şahinuç, and Fazli Can. 2022.  Not Good Times for Lies: Misinformation Detection on the Russia-Ukraine War, COVID-19, and Refugees.  arXiv:2210.05401 [cs.SI]
  * Vahdat-Nejad et al. (2023) Hamed Vahdat-Nejad, Mohammad Ghasem Akbari, Fatemeh Salmani, Faezeh Azizi, and Hamid-Reza Nili-Sani. 2023.  Russia-Ukraine war: Modeling and Clustering the Sentiments Trends of Various Countries.  _arXiv preprint arXiv:2301.00604_ (2023).
  * Vrandecic and Krötzsch (2014) Denny Vrandecic and Markus Krötzsch. 2014.  Wikidata: a free collaborative knowledgebase.  _Commun. ACM_ (2014), 78–85.  https://doi.org/10.1145/2629489
  * Williams et al. (2018) Adina Williams, Nikita Nangia, and Samuel Bowman. 2018\.  A Broad-Coverage Challenge Corpus for Sentence Understanding through Inference. In _Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long Papers)_ (New Orleans, Louisiana). Association for Computational Linguistics, 1112–1122.  http://aclweb.org/anthology/N18-1101
  * Xu et al. (2023) Anthony Xu, Matthew Evan Phanie, Allwin Simarmata, et al. 2023\.  Sentiment Analysis On Twitter Posts About The Russia and Ukraine War With Long Short-Term Memory.  _Sinkron: jurnal dan penelitian teknik informatika_ 8, 2 (2023), 789–797.
  * Zhu et al. (2022) Yiming Zhu, Ehsan ul Haq, Lik-Hang Lee, Gareth Tyson, and Pan Hui. 2022.  A Reddit Dataset for the Russo-Ukrainian Conflict in 2022\.  arXiv:2206.05107 [cs.SI]


# Some characterizations of gradient Yamabe solitons

Absos Ali Shaikh∗1, Prosenjit Mandal2
Department of Mathematics,
University of Burdwan, Golapbag,
Burdwan-713104,
West Bengal, India<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this article we have proved that a gradient Yamabe soliton satisfying some
additional conditions must be of constant scalar curvature. Later, we have
showed that in a gradient expanding or steady Yamabe soliton with non-negative
Ricci curvature if the potential function satisfies some integral condition
then it is subharmonic, in particular, for steady case the potential function
becomes harmonic. Also we have proved that, in a compact gradient Yamabe
soliton, the potential function agrees with the Hodge-de Rham potential upto a
constant.

††footnotetext:  $\mathbf{2020}$ Mathematics Subject Classification: 53C20;
53C21.
Key words and phrases: gradient Yamabe soliton; scalar curvature; harmonic
function; Hodge-de Rham; Riemannian manifold.

## Introduction

Yamabe solitons are special solutions to the Yamabe flow which is introduced
by R. S. Hamilton [6] in the late 20th century. Let $(M,g)$ be a Riemannian
manifold of dimension n, $(n\geq 2)$, such that $\\{g(t)\\}$ is the
1-parameter family of metrics and $R(t)$ is its scalar curvature. Then the
equation of Yamabe flow is given by

$\frac{\partial g(t)}{\partial t}=-R(t)g(t).$

A Riemannian manifold $(M,g,\lambda)$ of dimension n, ($n\geq 2$), which is
connected, is called a Yamabe soliton if it satisfies

(1) $\frac{1}{2}\pounds_{X}g=(R-\lambda)g,$

where $\pounds_{X}g$ is the Lie derivative of the metric g with respect to the
smooth vector field X, $\lambda$ is a constant and $R$ denotes the scalar
curvature of $M$. A Yamabe soliton $(M,g,\lambda)$ is called shrinking,
steady, and expanding Yamabe soliton if $\lambda>0$, $\lambda=0,$ and
$\lambda<0$, respectively. If there exists a smooth function
$f:M\rightarrow\mathbb{R}$ such that $X=\nabla f$, then the Yamabe soliton is
called a gradient Yamabe soliton. We write a gradient Yamabe soliton by
$(M,g,f,\lambda)$, and in this case, $(\ref{r10})$ takes the form

(2) $\nabla^{2}f=(R-\lambda)g,$

where $\nabla^{2}f$ denotes the Hessian of the potential function $f$.
Moreover, when the vector field X is trivial or the potential function $f$ is
constant, then Yamabe soliton will be called trivial, otherwise it will be a
non-trivial Yamabe soliton. Taking the trace of equation $(\ref{r11})$ we get

(3) $\Delta f=(R-\lambda)n.$

In tensors of local coordinate system, the equation $(\ref{r11})$ can be
written as

(4) $\nabla_{i}\nabla_{j}f=(R-\lambda)g_{ij}.$

For a given vector field X on a compact oriented Riemannian manifold $M$, the
Hodge-de Rham decomposition theorem, (see e.g. [10]) states that we may
decompose X as the sum of a divergence free vector field and gradient of a
function h. Hence we set

(5) $X=Y+\Delta h,$

where divY=0. Here the function $h$ is known as Hodge-de Rham potential.
Recently many authors have been studied Yamabe soliton, (see [2, 3, 5, 7, 8,
12] and also the references therein). It is proved, in [5], that a compact
gradient Yamabe soliton $(M,g,f,\lambda)$ is of constant scalar curvature.
Also in [7], Shu-Yu Hsu provides an alternative proof of this result. In the
present paper we omit the compactness condition and with the help of some
other conditions, we have proved that the scalar curvature is constant. Zhu-
Hong Zhang [11] proved a result depending on scalar curvature and potential
function for gradient Ricci solitons. Following the way of Zhu-Hong Zhang
[11], in this paper we have proved a result depending on scalar curvature and
potential function for gradient Yamabe soliton.
At first we have proved that in a gradient Yamabe soliton with potential
function bounded below and the manifold satisfying linear volume growth has
constant scalar curvature. In the last section we have showed that the
potential function in a gradient expanding or steady Yamabe soliton satisfying
some integral condition is subharmonic.

##  1\. Gradient Yamabe soliton with constant scalar curvature

######  Lemma 1.1.

[9] Let $f$ be a non-negative subharmonic function in $B(q,2r)$ then the
following inequality holds

(6) $\int_{B(q,r)}|\nabla f|^{2}\leq\frac{C}{r^{2}}\int_{B(q,2r)}f^{2},$

where $B(q,r)$ is a ball with radius $r>0$ and center at $q$ and C is a real
constant.

######  Theorem 1.2.

Let $(M,g,f,\lambda)$ be a gradient Yamabe soliton of dimension n, with the
potential function $f\geq K$ for some constant $K>0$, R$\leq\lambda$ and $M$
be of linear volume growth. Then $M$ must be of constant scalar curvature.

###### Proof.

Since $R\leq\lambda$, it follows from equation (3) that $\Delta f$ $\leq 0$.
Now

$\Big{(}\frac{1}{f}\Big{)}_{j}=-\Big{(}\frac{1}{f^{2}}\Big{)}f_{j},$

also

$\Big{(}\frac{1}{f}\Big{)}_{jj}=\Big{(}\frac{2}{f^{3}}\Big{)}f^{2}_{j}-\Big{(}\frac{1}{f^{2}}\Big{)}f_{jj}.$

Hence

(7) $\Delta\Big{(}\frac{1}{f}\Big{)}=\Big{(}\frac{2}{f^{3}}\Big{)}|\nabla
f|^{2}-\frac{\Delta f}{f^{2}}.$

As $\Delta f\leq 0$, it follows that $\Delta(\frac{1}{f})\geq 0$. Since the
manifold is of linear volume growth, so the volume of $B(q,r)$, i.e.,
$V(B(q,r))\leq C_{1}r$, for some constant $C_{1}>0$. Therefore from the
equation $(\ref{r16})$, we obtain

$\displaystyle\int_{B(q,r)}|\nabla\frac{1}{f}|^{2}$
$\displaystyle\leq\Big{(}\frac{C}{r^{2}}\Big{)}\int_{B(q,2r)}\Big{(}\frac{1}{f^{2}}\Big{)}$
$\displaystyle\leq\Big{(}\frac{C}{r^{2}K^{2}}\Big{)}V(B(q,2r))$
$\displaystyle\leq\Big{(}\frac{C}{r^{2}K^{2}}\Big{)}C_{1}2r$
$\displaystyle\leq\frac{2CC_{1}}{rK^{2}}\rightarrow 0$

as $r\rightarrow\infty$. Therefore

(8) $\int_{M}|\nabla\frac{1}{f}|^{2}=0,$

which follows that the function $\frac{1}{f}$ is constant. Consequently the
potential function $f$ is constant. Then it follows from (3) that $R=\lambda$,
which proves the result. ∎

######  Proposition 1.3.

Let $(M,g,f,\lambda)$ be a gradient Yamabe soliton of dimension n. Then the
following relation holds;

(9) $\displaystyle\Delta
R=-\frac{1}{n-1}\Big{\\{}\frac{1}{2}\nabla_{l}R\nabla_{l}f+R(R-\lambda\Big{\\}},$

where R denotes the scalar curvature of $M$.

###### Proof.

Taking the covariant derivative in $(\ref{r13})$ and using the commutating
formula for covariant derivative, we have

$g_{jk}\nabla_{i}R-g_{ik}\nabla_{j}R=\nabla_{i}\nabla_{j}\nabla_{k}f-\nabla_{j}\nabla_{i}\nabla_{k}f=-R_{ijkl}\nabla_{l}f.$

Taking the trace on $j$ and $k$, we get

$g_{jj}\nabla_{i}R-g_{ij}\nabla_{j}R=-R_{ijjl}\nabla_{l}f.$

This implies

$n\nabla_{i}R-\nabla_{i}R=-R_{il}\nabla_{l}f,$

which yields

(10) $\nabla_{i}R=-\frac{1}{(n-1)}R_{il}\nabla_{l}f.$

Now with the help of equation (10) and contracted second Bianchi identity
$\nabla_{i}R=2\nabla_{j}R_{ij}$, we get

$\displaystyle\Delta R$ $\displaystyle=$ $\displaystyle
g^{ij}\nabla_{i}\nabla_{j}R$ $\displaystyle=$
$\displaystyle-g^{ij}\nabla_{i}\Big{(}\frac{1}{n-1}R_{jl}\nabla_{l}f\Big{)}$
$\displaystyle=$
$\displaystyle-\frac{1}{n-1}g^{ij}\nabla_{i}(R_{jl}\nabla_{l}f)$
$\displaystyle=$
$\displaystyle-\frac{1}{n-1}g^{ij}\Big{\\{}\nabla_{i}(R_{jl})\nabla_{l}f+R_{jl}\nabla_{i}\nabla_{l}f\Big{\\}}$
$\displaystyle=$
$\displaystyle-\frac{1}{n-1}\Big{\\{}\frac{1}{2}\nabla_{l}R\nabla_{l}f+g^{ij}R_{jl}(Rg_{il}-\lambda
g_{il})\Big{\\}}$ $\displaystyle=$
$\displaystyle-\frac{1}{n-1}\Big{\\{}\frac{1}{2}\nabla_{l}R\nabla_{l}f+R(R-\lambda)\Big{\\}}.$

Hence, we get our result. ∎

##  2\. Yamabe soliton and potential functions

######  Theorem 2.1.

Let $(M,g,f,\lambda)$ be a gradient expanding or steady Yamabe soliton of
dimension n with non-negative scalar curvature. If the potential function $f$
satisfies the condition

(11) $\int_{M-B(q,r)}\frac{f}{d(x,q)^{2}}<\infty,$

where $B(q,r)$ is a ball with radius $r>0$ and center at $q$, then $f$ is
subharmonic.

###### Proof.

Taking trace in (2), we get

(12) $(R-\lambda)n=\Delta f.$

Let us consider the cut-off function, introduced in [4], $\psi_{r}\in
C^{\infty}_{0}(B(q,2r))$ for $r>0$ such that

$\begin{cases}0\leq\psi_{r}\leq 1&\text{ in }B(q,2r)\\\ \psi_{r}=1&\text{ in
}B(q,r)\\\ |\nabla\psi_{r}|^{2}\leq\frac{C}{r^{2}}&\text{ in }B(q,2r)\\\
\Delta\psi_{r}\leq\frac{C}{r^{2}}&\text{ in }B(q,2r),\end{cases}$

where $C>0$ is a constant. Then as $r\rightarrow\infty$, we get
$\Delta\psi_{r}\rightarrow 0$ as $\Delta\psi_{r}\leq\frac{C}{r^{2}}$. Now
using (12) and integration by parts, we obtain

$\displaystyle
0\leq\int_{B(q,2r)}\psi_{r}R=\int_{B(q,2r)}\psi_{r}(\lambda+\frac{1}{n}\Delta
f)$ $\displaystyle=$
$\displaystyle\int_{B(q,2r)}\psi_{r}\lambda+\frac{1}{n}\int_{B(q,2r)}\psi_{r}\Delta
f$ $\displaystyle\leq$
$\displaystyle\frac{1}{n}\int_{B(q,2r)-B(q,r)}f\Delta\psi_{r}$
$\displaystyle\leq$
$\displaystyle\frac{1}{n}\int_{B(q,2r)-B(q,r)}\frac{C}{r^{2}}f\rightarrow 0,$

as $r\rightarrow\infty$. Since $\psi_{r}=1$ in $B(q,r)$, so from above
inequality we have $R=0$. Then from equation (3), we get

$\Delta f\geq 0,$

This completes the proof. ∎

By the Lemma 1.1 and Theorem 2.1, we get the following corollary:

######  Corollary 2.1.1.

Let $(M,g,f,\lambda)$ be a gradient expanding or steady Yamabe soliton of
dimension n with non-negative scalar curvature. If a non-negative potential
function $f$ satisfies the condition

(13) $\int_{M-B(q,r)}\frac{f}{d(x,q)^{2}}<\infty,$

where $B(q,r)$ is a ball with radius $r>0$ and center at $q$, then $f$
satisfies the inequality

$\int_{B(q,r)}|\nabla f|^{2}\leq\frac{C}{r^{2}}\int_{B(q,2r)}f^{2}.$

######  Corollary 2.1.2.

Let $(M,g,f,\lambda)$ be a gradient steady Yamabe soliton of dimension n with
non-negative scalar curvature. If the potential function $f$ satisfies

(14) $\int_{M-B(q,r)}\frac{f}{d(x,q)^{2}}<\infty,$

where $B(q,r)$ is a ball with radius $r>0$ and center at $q$, then $f$ is
harmonic.

###### Proof.

In the above theorem we get the scalar curvature $R=0$, hence for steady
Yamabe soliton we have $\Delta f=0$, i.e., $f$ is harmonic. ∎

######  Corollary 2.1.3.

Let $(M,g,f,\lambda)$ be a gradient shrinking Yamabe soliton of dimension n
with non-negative scalar curvature and $\lambda\leq\Delta\psi$, for some
smooth function $\psi$. If the potential function $f$ satisfies

(15) $\int_{M-B(q,r)}\frac{f}{d(x,q)^{2}}<\infty,$

where $B(q,r)$ is a ball with radius $r>0$ and center at $q$, then $f$ is
superharmonic.

###### Proof.

Consider the cut-off function as in Theorem 2.1. Now using $(\ref{r7})$ and
integration by parts, we get

$\displaystyle
0\leq\int_{B(q,2r)}\psi_{r}R=\int_{B(q,2r)}\psi_{r}(\lambda+\frac{1}{n}\Delta
f)$ $\displaystyle=$
$\displaystyle\int_{B(q,2r)}\psi_{r}\lambda+\frac{1}{n}\int_{B(q,2r)}\psi_{r}\Delta
f$ $\displaystyle\leq$
$\displaystyle\int_{B(q,2r)}\psi_{r}\Delta\psi+\frac{1}{n}\int_{B(q,2r)}\psi_{r}\Delta
f$ $\displaystyle\leq$
$\displaystyle\int_{B(q,2r)-B(q,r)}\psi\Delta\psi_{r}+\frac{1}{n}\int_{B(q,2r)-B(q,r)}f\Delta\psi_{r}$
$\displaystyle\leq$
$\displaystyle\int_{B(q,2r)-B(q,r)}\frac{C}{r^{2}}\big{(}\frac{f}{n}+\psi\big{)}\rightarrow
0,$

as $r\rightarrow\infty$. Since $\psi_{r}=1$ in $B(q,r)$, so from above
inequality we have $R=0$. Then from equation (3), we get

$\Delta f\leq 0,$

which proves the result. ∎

In [1] Aquino et. al proved a result for gradient Ricci soliton. We have
proved the same result for gradient Yamabe soliton as follows:

######  Theorem 2.2.

Let $(M,g,X,\lambda)$ be a gradient Yamabe soliton of dimension n which is
compact, with potential function $f$. Then upto a constant, $f$ agrees with
the Hodge-de Rham potential $h$.

###### Proof.

For Yamabe soliton $(M,g,X,\lambda)$, it follows that

(16) $(R-\lambda)n=divX.$

Now with the help of Hodge-de Rham decomposition we get $divX=\Delta h$, and
hence

(17) $(R-\lambda)n=\Delta h.$

Again from $(\ref{r11})$ we have

(18) $\Delta f=(R-\lambda)n.$

Now, from equations $(\ref{r18})$ and $(\ref{r19})$, we get

(19) $\Delta(f-h)=0.$

Therefore, we have $f=h+C$, for some constant C, which completes the proof of
the theorem. ∎

##  3\. acknowledgment

The second author gratefully acknowledges to the CSIR(File
No.:09/025(0282)/2019-EMR-I), Govt. of India for financial assistance.

## References

  * [1] Aquino, C., Barros, A., Ribeiro, E. Jr., Some applications of the Hodge-de Rham decomposition to Ricci solitons, Results. Math. 60 (2011), 245–254.
  * [2] Calvaruso, G. and Zaeim, A., A complete classification of Ricci and Yamabe solitons of nonreductive homogeneous 4-spaces, J. Geom. Phys., 80 (2014), 15–25.
  * [3] Cao, H. D., Sun, X. and Xiaofeng, Y.S.,  On the structure of gradient Yamabe solitons, Math. Res. Lett., 19 (2012), 767–774.
  * [4] Cheeger, J. and Colding, T. H., Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. Math., 144(1) (1996), 189–237.
  * [5] Daskalopoulos, P., Sesum, N., The classification of locally conformally flat Yamabe solitons, Advances in Math., 240 (2013), 346–369.
  * [6] Hamilton, R.S., The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986), contemp. Math., 71 (1988), 237–262.
  * [7] Hsu, S. Y., A note on compact gradient Yamabe solitons, J. Math. Anal. Appl., 388 (2012), 725-–726.
  * [8] Ma, L. and Miquel, V.,  Remarks on scalar curvature of Yamabe solitons, Ann. Global Anal.Geom., 42 (2012), 195–205.
  * [9] Schoen, R., Yau, S.T., Lectures on differential geometry. Conference Proceedings and lecture notes in geometry and topology, I. International Press, Cambridge, MA, 1994.
  * [10] Warner, F., Foundations of differentiable manifolds and Lie groups, Springer- Verlag, New York, ISBN 978-0-387-90894-6, 1983.
  * [11] Zhang, H. Z., On the completeness of gradient Ricci solitons, Proc. Amer. Math. Soc., 137 (2009), 2755–2759.
  * [12] Shaikh, A. A. and Mondal, C. K., Non-existence of Riemannian metric satisfying Yamabe soliton, arXiv:1908.00334.