# A declarative approach to data narration

Patrick Marcel, Verónika Peralta, Faten El Outa  University of Tours
<EMAIL_ADDRESS>and  Panos Vassiliadis  University of
Ioannina<EMAIL_ADDRESS>
###### Abstract.

This vision paper lays the preliminary foundations for Data Narrative
Management Systems (DNMS), systems that enable the storage, sharing, and
manipulation of data narratives. We motivate the need for such formal
foundations and introduce a simple logical framework inspired by the
relational model. The core of this framework is a Data Narrative Manipulation
Language inspired by the extended relational algebra. We illustrate its use
via examples and discuss the main challenges for the implementation of this
vision.

††copyright: rightsretained††doi: ††isbn: 978-3-89318-088-2††conference: 26th
International Conference on Extending Database Technology (EDBT); 28th
March-31st March, 2023; Ioannina, Greece††journalyear: 2023

##  1\. Introduction

A data narrative (DN) is a structured composition of messages that (a) convey
findings over the data, and, (b) are typically delivered via visual means in
order to facilitate their reception by an intended audience (Outa et al.,
2020). Data narration (Segel and Heer, 2010; Carpendale et al., 2016) refers
to the notoriously tedious process of crafting a DN by extracting insights
from data and telling stories with the goal of "exposing the unanticipated"
(Tukey, 1977) and facilitating the understanding of insights. Data narration
is practiced in many domains and by various domain experts, ranging from data
journalists to public authorities.

Consider the two infograhpics of Figure 1111American Heart Association: ”Women
and Risk of Stroke Infographic”, https://www.goredforwomen.org/en/know-your-
risk/risk-factors/risk-of-stroke-in-women-infographic,222Good: ”Infographic:
Stroke, a Silent Killer of Women, Facts About Women and Strokes”,
https://www.good.is/infographics/facts-about-women-and-strokes. These two
infographics can be seen as "physical" representations of the DNs. In (Outa et
al., 2020), a conceptual model for DNs was proposed. The goal of the present
paper is to propose a logical representation of DNs, to bridge the gap between
the physical and conceptual representations.

Figure 1. Examples of data narratives

Let us first briefly review the proposed conceptual model for DNs (Outa et
al., 2020). This model is based on 4 layers following Chatman’s organisation
(Chatman, 1980), who defined narrative as a pair of (a) story (content of the
narrative), and, (b) discourse (expression of it). In the conceptual model,
the factual layer handles the exploration of facts (i.e., the underlying data)
for fetching findings while the intentional layer models the subjective
substance of the story, identifying the messages, characters and measures the
narrator intends to communicate. As to the discourse, the structural layer
models the structure of the DN, its plot being organized in terms of episodes,
while the presentational layer deals with its rendering, that is communicated
to the audience through visual artifacts named dashboard components). The
interested reader is redirected to (Outa et al., 2020) for a deeper
presentation of the model.

For instance, in the DN of Figure 1 bottom, an episode of the DN is rendered
in the upper right dashboard component, with the message indicating that
measure ‘stroke deaths’ is 57/100000 for character ‘black women’.

Figure 2 is an excerpt of this conceptual model considered in the present
paper. Indeed, in this vision paper, our goal is to show the benefit of
manipulating data narratives declaratively, i.e., with a formal logical data
model and a manipulation language. We voluntarily keep this model and language
simple, mostly inspired by the relational data model and extended relational
algebra (Garcia-Molina et al., 2009). As will be discussed in Section 7,
taking into account all the concepts of the domain will need to revisit the
model and language, while keeping the flavor of the manipulation described
here. These manipulations rely on the concept of message which is the
conceptual model’s corner stone. A message is rooted in the facts analyzed,
conveying essential findings that can be related to one another. The message
allows introducing episodes, the building blocks of the discourse. Each
episode of the discourse is specifically tied to a message which it aims to
convey, with dashboard components being their presentational counterparts.

Figure 2. Selection of concepts used for the logical layer

While the Web abounds with DNs, manipulating them in a declarative way using
the concepts of this model has not yet been proposed, to our knowledge. This
paper aims at filling this gap, by envisioning a DN Management System (DNMS),
the foundations of which should include a logical layer enabling the
declarative manipulation of DNs.

The outline of the paper is the following. Section 2 motivates the need for a
logical layer. Section 3 introduce the logical model for DNs and Section 4 the
algebra for manipulating DNs. Section 5 illustrates the languages while
Section 6 presents related work and Section 7 concludes this vision paper by
discussing the main challenges for the implementation of DNMS.

##  2\. Motivation

As indicated above, the corner stone of a DN is a message, that associates
characters with measures. Intuitively, a DN is an ordered set of messages and
will be manipulated based on the characters and measures they deal with.

We list below simple queries that should be expressed over DNs. Each one
corresponds to an operation of the algebraic language introduced in Section 4.

  * •

Find DNs concerning some characters or measures, e.g., DNs about stroke
deaths. This selection operation allows to find DNs that satisfy a given
condition.

  * •

Retain from DNs only the messages about some characters or measures, e.g.,
messages concerning Hispanic and Native American women. This projection-like
operation produces new DNs keeping only a subset of messages.

  * •

Concatenate messages of several DNs. For example, produce a DN with all
messages of the DNs of Figure 1. This concatenation operation allows gathering
messages.

  * •

Remove duplicate messages in DNs. This duplicate elimina-tion operation only
keeps one occurrence of each message in a DN.

  * •

Synthesize groups of messages in DNs. For instance, produce DNs aggregating
messages about stroke cases and stroke deaths. This group-aggregate-like
operation groups messages in each DN using grouping conditions and merges them
using built-in merging functions.

  * •

Manipulate sets of DNs using the classical set operations (cross-product,
intersection, union and difference).

  * •

Change the plot of DNs, for example, arranging messages according to some
measures. This order by-like operation allows to modify the order of messages
in the DNs.

Composing these operations enables to devise complex query expressions as will
be seen in more details in Section 5. We simply mention here a few complex
operations that we postulate will be very useful in practice:

  * •

Connect the dots between DNs, i.e., merge DNs that have characters, or
characters and measures in common. This join-like operation can be expressed
with a combination of cross product, group-aggregate and selection.

  * •

Summarize a narrative for a particular character (e.g., from women stroke to
stroke, France to Europe, etc.). This roll-up-like operation supposes to find
in the DNDB narratives with messages having characters that generalize the
particular character. It can be expressed using selection, cross product,
projection, and group-aggregate.

  * •

Detail DNs for a particular character. This drill-down-like operation can be
seen as the inverse of the previous one and can be expressed with the same
combination of selection, cross product, projection, and group-aggregate.

##  3\. Logical data model

This section presents the data model of the logical framework. Again, in this
vision paper, the goal is not to define a thorough logical layer for DNs, but
instead to give the flavor of what this logical layer should be for an end-to-
end DNMS.

###  3.1. Data narrative components

#### Atomic concepts and relations

The atomic components of the model are characters and measures. For instance,
the DN of Figure 1 (bottom) includes character ‘black women’ and measure
‘stroke death’. To keep things simple in this preliminary version of the
model, measure values and units (e.g., 57/100000) are not part of the present
preliminary model. The semantics of the message is given by a predicate
connecting characters and measures, in the spirit of semantic triples.

We also assume binary relations between characters as the bases for the
relations between messages. Relations between characters are at the core of
relations between findings (see Figure 2). In data narration, it is common to
use these relations as transitions between episodes of the narrative’s plot,
and it has been seen that most transitions are of the following nature:
specialisation, temporal or spatial (cf. e.g., (Hullman et al., 2017)). More
precisely,

  * •

a specialization relation, noted $c\prec c^{\prime}$, allows to classify
characters in a hierarchy, for instance to indicate that black women is more
specific than women,

  * •

a spatial relation, noted $c\vdash c^{\prime}$ to indicate that $c$ is in a
spatial relation with $c^{\prime}$. For instance Greece $\vdash$ France.

  * •

a temporal relation, noted $c\dashv c^{\prime}$ to indicate that $c$ is in
temporal relation with $c^{\prime}$. For instance, in Europe, Spring $\dashv$
2nd quarter.

  * •

we also assume a general similarity relation noted $c\approx c^{\prime}$ to
indicate that character $c$ is similar to character $c^{\prime}$, for instance
birth control pills is similar to abortion pills.

#### Complex concepts

A message is a tuple associating characters with measures. Formally, a message
$m$ is a tuple $\langle C,V,P\rangle$ where $C$ is a set of characters, $V$ is
a set of measures and $P$ is a predicate333For the sake of consistency, $P$ is
a singleton.. The simplest message is the empty message,
$\langle\emptyset,\emptyset,\emptyset\rangle$.

#### Running example

Consider the messages of Figure 3. Messages $m_{1}$ to $m_{5}$ and $m_{6}$ to
$m_{8}$ are inspired from those of the DNs of Figure 1, restricting to a
subset of messages and simplifying many characters. Message $m_{9}$ is
inspired from a DN about covid. $\Box$

Figure 3. Messages of the running example

Since findings can be related to one another (e.g. findings about black women
are more specific than those concerning all women), we consider that relations
over characters also applies to messages. For example, among messages of
Figure 3, $m_{3}$ is more general than $m_{1}$ regarding characters ‘black
women’ and ‘women’. Given two messages $m,m^{\prime}$ we consider the
transition relation between them which can be one of: spatial, temporal,
generalization, similarity.

Formally, for $m=\langle C,V,P\rangle$ and $m^{\prime}=\langle
C^{\prime},V^{\prime},P^{\prime}\rangle$ it is $mRm^{\prime}$ if $\exists c\in
C,c^{\prime}\in C^{\prime},cRc^{\prime}$ and $R$ is one of
$\prec,\approx,\vdash,\dashv$.

###  3.2. Data model

To be consistent with the conceptual description of DNs of Figure 2, we define
a DN as a sequence of episodes. Each episode narrating a message, i.e., a
tuple, a DN is formally defined as a tuple of tuples. We distinguish its
schema, which consists in the number of messages (remember that each message
has the same structure), from its instance.

Unless otherwise specified, all sets are infinite and countable. Let
$\mathcal{C}$ be a set of characters, $\mathcal{V}$ a set of measures,
$\mathcal{P}$ a set of predicates and $\mathcal{M}$ the set of messages
$2^{\mathcal{C}}\times 2^{\mathcal{V}}\times\mathcal{P}$. Let $\mathcal{H}$ be
a set of DN names.

#### Data narrative

The schema of a DN of length $k$ (i.e., with $k$ messages) is a couple
$\langle h,k\rangle$ where $h\in\mathcal{H}$ is the DN name. A DN instance is
a tuple of messages.

For the sake of readability, in what follows we will consider a DN of length
$k$ as an injective function from $\mathcal{M}$ to $2^{\mathbb{N}}$. For
instance, the DN $n=\langle m,m,m^{\prime}\rangle$ can be seen as the function
$n$ where $n(m)=\\{1,2\\}$ and $n(m^{\prime})=\\{3\\}$. We abuse notations and
note $m\in n$ if a message $m$ appears in the DN $n$, and $messages(n)$ the
set of messages of DN $n$.

#### Data Narrative Database (DNDB)

A DNDB schema is a set of DN schemas and a DNDB instance is a set of DN
instances.

#### Example

Continuing the running example, $I=\\{n_{1},n_{2},n_{3}\\}$, is a DNDB
instance that organizes the messages of Figure 3:

$n_{1}=\langle m_{1},m_{2},m_{3},m_{4},m5\rangle$; $n_{2}=\langle
m_{6},m_{7},m_{8}\rangle$; $n_{3}=\langle m_{9}\rangle$. $\Box$

##  4\. Data Narrative Manipulation Language (DNML)

This section presents an algebra for manipulating DNs. The focus is the
description of the operators in their general form; more user-friendly
constructors, specially for expressing conditions, will be discussed in
Section 7. All operators have the same signature in the sense that they are
applied over a DNDB instance and output a DNDB instance. Finally, note that
DNs can be manipulated using their schemas, i.e., their names and lengths.
However, it is preferable to also manipulate DNs with conditions over their
messages and this is why most operators rely on logical formulas for
expressing these conditions. In what follows, let $D$ be a DNDB and $I$,
$I_{1}$ and $I_{2}$ be instances of $D$.

###  4.1. Constants

The first operation is the constant DN. Given a message $\langle
C,V,P\rangle$, it is simply: $\\{\langle C,V,P\rangle\\}$

###  4.2. Unary operators

#### Selection

Selects the DNs in instance $I$ that satisfy a given condition.

$\sigma_{\varphi}(I)=\\{n\in I|\varphi\\}$

where $\varphi$ is a logical formula to express selection conditions, for
instance:

  * •

$\exists m\in n,m=\langle C,V,P\rangle,c\in C$ (has character $c$)

  * •

$\exists m\in n,m=\langle C,V,P\rangle,v\in V$ (has measure $v$)

  * •

$\exists m\in n,m=\langle C,V,P\rangle,p\in P$ (has predicate $p$)

  * •

$\exists m\in n,m=\langle C,V,P\rangle,\exists c^{\prime}\in C,c^{\prime}Rc$
(has character in relation $R$ with $c$, $R$ being one of the relations over
characters)

  * •

$\exists m,m^{\prime}\in n,mRm^{\prime}$ (has messages that are in relation
$R$ where $R$ is one of the relations over messages)

  * •

$\forall m\in n,m=\langle\emptyset,\emptyset,\emptyset\rangle$ (has only empty
messages)

#### Example

The operation $\sigma_{\exists m\in n,m=\langle
C,V,P\rangle,`strokeDeaths^{\prime}\in V}(I)$
looks for DNs about stroke deaths in instance $I$ of the running example. Its
output is $\\{n_{1},n_{2}\\}$. $\Box$

#### Projection

For each DN in I, keeps only the messages satisfying a condition.

$\pi_{\varphi}(I)=\\{n|_{\\{m\in n|\varphi\\}}|n\in I\\}$

where $\varphi$ is a logical formula, similar to those used for the selection
and $n|_{\\{m\in n|\varphi\\}}$ is the restriction of DN $n$ to the set of
messages satisfying $\varphi$.

#### Example

The operation $\pi_{\exists m\in n,m=\langle
C,V,P\rangle,`BlackWomen^{\prime}\in C}(I)$ produce DNs $n_{4},n_{5}$ and
$n_{6}$, projecting a subset of messages of DNs in instance $I$, i.e., those
containing Black women as characters.

$n_{4}=\langle m_{3}\rangle$; $n_{5}=\langle m_{6},m_{7}\rangle$;
$n_{6}=\langle\rangle$. $\Box$

#### Duplicate-elimination

For each DN in $I$, keeps only one occurrence of each message.

$\delta(I)=\\{n^{\prime}|n\in I,\forall m\in n,n^{\prime}(m)=min(n(m))\\}$

#### Group-aggregate

For each DN in $I$, groups the messages using a grouping condition and
aggregates each group using a specific aggregation function.

$\gamma_{\varphi_{1},\ldots,\varphi_{j},agg_{1},\ldots,agg_{j}}(I)=$

$\\{\langle agg_{1}(\\{m\in n|\varphi_{1}\\}),\ldots,agg_{j}(\\{m\in
n|\varphi_{j}\\})\rangle|n\in I\\}$

where the $\varphi_{i}$ are grouping conditions on the set of messages of the
DN. It is not requested that the grouping conditions partition the set of
messages, which allows one message to appear in several groups. The $agg_{i}$
are aggregation functions for merging a set of messages into one message.

#### Example

The operation $\gamma_{BW,WW,A,A}(I)$, where $BW$ and $WW$ are conditions
about characters ‘Black women’ and ‘White women’ and $A$ is an aggregation
function computing the union of characters and measures in the input messages.
The output DNs contain two messages, the former concerning Black women and the
latter White women: $n_{7}=\langle
m_{3},\langle\emptyset,\emptyset,\emptyset\rangle\rangle$; $n_{8}=\langle
A(m_{6},m_{7}),A(m_{6},m_{7})\rangle$;
$n_{9}=\langle\langle\emptyset,\emptyset,\emptyset\rangle,\langle\emptyset,\emptyset,\emptyset\rangle\rangle$.
$\Box$

#### Full group-aggregate

Another group-aggregate operation allows to merge messages across DNs into one
DN.

$\gamma^{across}_{\varphi_{1},\ldots,\varphi_{j},agg_{1},\ldots,agg_{j}}(I)=$

$\\{\langle
agg_{1}(\\{m|\varphi_{1}\\}),\ldots,agg_{j}(\\{m|\varphi_{j}\\})\rangle|m\in\bigcup_{n\in
I}messsages(n)\\}$

#### Example

The operation $\gamma^{across}_{BW,WW,A,A}(I)$, with $BW$, $WW$ and $A$ as in
previous example, outputs

$n_{10}=\langle A(m_{3},m_{6},m_{7}),A(m_{6},m_{7})\rangle$ $\Box$

#### Order by

Allows to change the order of messages in the DNs.

$\tau_{\varphi_{1},\ldots,\varphi_{J},sort_{1},\ldots,sort_{J}}(I)=$

$\\{\langle sort_{1}(\\{m\in n|\varphi_{1}\\}),\ldots,sort_{J}(\\{m\in
n|\varphi_{J}\\})\rangle|n\in I\\}$

where the $\varphi_{i}$ are selection conditions over messages (like the ones
used e.g., for the selection operation) and the $sort_{i}$ are sorting
functions, mapping a set of messages to a tuple of messages.

#### Concatenation

Flattens all DNs in I into one DN, by concatenating all their messages.

$\chi(I)=\\{\langle n_{1},...n_{k}\rangle|n_{i}\in I\\}$

###  4.3. Binary operators

The classical relational binary operations $\times,\cup,\cap,\setminus$ have
their standard set theoretic meaning.

###  4.4. Properties

We give here a few insights on the properties of the algebra, a complete study
thereof is part of our future work.

#### Closure

: By definition, each operator defines a set of DNs from a set of DNs.

#### Completeness

From a given set of messages, all narratives can be obtained using the
constant and cross-product operators, which together form the core of the
algebra.

#### Minimality

The minimal set of operators includes constant, cross product, both group-
aggregates, order by, union and difference. All other operations can be
expressed from this set. We can anticipate that some expressions will be
popular and deserve to be promoted as operators, like the join in the
relational algebra that is a shortcut from cross product followed by
selection.

#### Some properties of the operators

Like in the case of the relational algebra, the cross product is non
commutative, admits one absorbing element (the empty set) and one neutral
element (the empty DN, i.e., $\langle\rangle$). Set operations keep their
usual properties. Note that, because DNDB instances are sets of DNs of
different length, intersection cannot be expressed by combining cross product,
selection and projection, while it is the case for the relational algebra.

##  5\. Example

In this section, we give some examples of useful and easy to express queries
in DNML. They are based on the running example.

#### Comparing messages

We first illustrate how to "join" DNs having contradictory messages for
characters women and stroke,

  1. (1)

select DNs about women and stroke:
$I_{1}=\sigma_{\exists m\in n,m=\langle
C,V,P\rangle,\\{stroke,women\\}\subseteq C}(I)$

  2. (2)

group by messages with women and stroke, aggregate by keeping messages that
are contradictory:
$I_{2}=\gamma^{across}_{\varphi,\neg\varphi,check,drop}(I_{1})$, where
$\varphi=\exists m\in n,m=\langle C,V,P\rangle,\\{stroke,women\\}\subseteq C$,
$check$ is a function merging messages that are contradictory otherwise
producing an empty message, and $drop$ is a function producing an empty
message.

  3. (3)

project out empty messages:
$\pi_{m\not=\langle\emptyset,\emptyset,\emptyset\rangle}(I_{2})$

#### Roll-up and drill-down

Assume we want to see DNs about ’black women, stroke’ and then to "roll-up"
from ’black women’.

  1. (1)

select the DNs with characters ’black women’,’stroke’
$I_{1}=\sigma_{\exists m\in n,m=\langle
C,V,P\rangle,\\{^{\prime}black~{}women^{\prime},^{\prime}stroke^{\prime}\\}\subseteq
C}(I)$

  2. (2)

select the DNs with characters more general than ’black women’
$I_{2}=\sigma_{\exists m\in n,m=\langle
C,V,P\rangle,^{\prime}black~{}women^{\prime}\prec x,x\in C}(I)$

  3. (3)

compute the cross product of found DNs:
$I_{3}=I_{1}\times I_{2}$

  4. (4)

group by messages with black women and one more general character, aggregate
by merging messages:
$I_{4}=\gamma_{\varphi,\neg\varphi,merge,drop}(I_{3})$, where
$\varphi=\exists m\in n,m=\langle
C,V,P\rangle,\\{^{\prime}black~{}women^{\prime}\\}\in
C\vee^{\prime}black~{}women^{\prime}\prec x,x\in C$, $merge$ is a function
merging messages, and $drop$ is a function producing an empty message.

  5. (5)

project out empty messages:
$\pi_{m\not=\langle\emptyset,\emptyset,\emptyset\rangle}(I_{4})$

##  6\. Related work

#### Data narrative modeling

Calegari et al. (Calegari, 2022) proposed a narrative metamodel based on the
conceptual model of (Outa et al., 2020), to provide abstract models to data
narratives. They explored the definition of model transformation for
converting narrative models into HTML or a Jupyter computational notebook.
Zhang et al.(Zhang et al., 2022) proposed a framework for creating data
storytelling applications from three major perspectives: concept, component,
and procedure with the absence of any logical means. Bach et al. (Bach et al.,
2018) introduces narrative design patterns, defined as "a low-level narrative
device that serves a specific intent". A pattern can be used individually or
in combination with others to give form to a story. Five major patterns of
group are identified: argumentation, flow, framing, emotion, engagement. For
example, if the intent of the data narrator is to persuade and convince
audience, he can use one of the following patterns: compare, concretize, and
repetition. Importantly, these patterns are not specifically related to a
visualization or interaction medium.

Many approaches exist for describing DN crafting, mostly describing an
essentially manual process (Chen and et al., 2018; Lee et al., 2015;
Duangphummet and Ruchikachorn, 2021), while others proposing approaches for
automatically generating simple data narratives (Wang et al., 2020; Shi et
al., 2021b, a). In all cases, no manipulation language was specifically
proposed for manipulating DNs.

#### Languages for DN

A DNDB is a set of tuples of tuples of sets, i.e., a form of nested relation,
albeit with tuples of different lengths. This means that the proposed
language, DNML, is likely to be expressed in the nested relational algebra
(NRA) (Abiteboul et al., 1995). However, due to the relatively simple
structure of DNs, some of NRA’s operations are not needed (e.g.,
nesting/unnesting, powerset).

Many languages or primitives were proposed to express data exploration
sessions (Vassiliadis et al., 2019; El et al., 2020; Youngmann et al., 2022).
While relevant for understanding the logic under the discoveries of finding,
these languages are not adapted the manipulation of messages and are not
devised as an algebra. Some are not even meant to be used by humans, since,
e.g., in (El et al., 2020), primitives are used to generate exploration
sessions through reinformcement learning.

##  7\. Discussion

We close this paper by discussing some of the main challenges raised by the
development of end-to-end DNMS.

#### Modeling the complexity of data narration

The logical layer proposed in this paper only covers a small portion of data
narration, i.e., the complex process that goes from data exploration to the
visual presentation of messages. In particular, to account for the complexity
of the process, all the concepts and relations present in the model of (Outa
et al., 2020) should have a counterpart in the logical layer.

For instance, the provenance of messages (how findings in the dataset were
discovered), and the meaning carried to the reader should be logically
modeled. For provenance, messages should be linked to a collection of
findings, independently of the form of findings, as well as to the queries the
findings are results of. For the semantics, one can assume extending the
message predicates with user-intuitive semantics, which requires including
measure values, units, quantification, etc. A challenge will be to cover all
the data narration process while keeping the model and language simple enough.

#### Data model

As noted above, the data model proposed above is very close to RA. In fact, if
the order of messages is not included in the data model, DNs could simply be
defined as sets of messages, i.e., relations of arity 2. However, this
simplicity would oblige to code the complexity of DN, leading to unnatural
queries. Besides, as explained in the previous paragraph, this model is meant
to be extended to cover the complete conceptual model of (Outa et al., 2020).

More semantics should be added in the different layers of the conceptual model
of (Outa et al., 2020). For instance, at the message level, this can be
achieved by using predicates having semantics known to the reader, like in
(Wang et al., 2020), or by modeling the relations between characters using
e.g., property graphs. At the data exploration layer, modeling findings can be
done by relating characters and measures in the spirit of (Anadiotis et al.,
2022). Semantics can also be added by modeling the intentions of the narrator
using abstract primitives like the ones of (Vassiliadis et al., 2019) or
narrative patterns of Bach et al. (Bach et al., 2018).

#### Manipulation language

As mentioned above, extending DNML will be needed to account for the
complexity of the data narration process. An eye should be kept on query
languages for sequences (Mecca and Bonner, 2001) and query languages for the
Semantic Web (Arenas et al., 2018). A challenge will be to keep the formalism
simple enough for ensuring its adoption by the narrators, analysts or data
enthusiasts.

This can be achieved by, e.g., (i) devising operations summarizing complex
expressions that are useful in practice, in the spirit of join for RA, (ii)
devising a SQL-like language for DNs, (iii) using built in predicates for the
logical formulas used in the operations (e.g., selection, projection, group-
aggregate). In this last case, for instance, predicate $hasChar(c)$ could be
used to express that DN $n$ has character $c$, i.e., logical formula $\exists
m\in n,m=\langle C,V,P\rangle,c\in C$.

#### RDBMS like stack

On the long term, the challenge will be to implement a DNMS on the model of
the successful achievements of RDBMSs, notably, a clear distinction of
conceptual, logical, physical layers with intuitive mappings between the
objects of different layers, loading facilities for populating the DNDB from
existing DNs, data organization at the physical layer, including specific
index mechanisms (e.g., inspired by information retrieval techniques),
optimizations at the logical and physical layer, etc.

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# Compact gradient $\rho$-Einstein soliton is isometric to the Euclidean
sphere

Absos Ali Shaikh1, Chandan Kumar Mondal2, Prosenjit Mandal3
Department of Mathematics,
University of Burdwan, Golapbag,
Burdwan-713104,
West Bengal, India<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this paper we have investigated some aspects of gradient $\rho$-Einstein
Ricci soliton in a complete Riemannian manifold. First, we have proved that
the compact gradient $\rho$-Einstein soliton is isometric to the Euclidean
sphere by showing that the scalar curvature becomes constant. Second, we have
showed that in a non-compact gradient $\rho$-Einstein soliton satisfying some
integral condition, the scalar curvature vanishes.

††footnotetext: $\mathbf{2020}$ Mathematics Subject Classification: 53C20;
53C21; 53C44.
Key words and phrases: Gradient $\rho$-Einstein Ricci soliton; scalar
curvature; Riemannian manifold.

##  1\. Introduction and preliminaries

A $1$-parameter family of metrics $\\{g(t)\\}$ on a Riemannian manifold $M$,
defined on some time interval $I\subset\mathbb{R}$ is said to satisfy Ricci
flow if it satisfies

$\frac{\partial}{\partial t}g_{ij}=-2R_{ij},$

where $R_{ij}$ is the Ricci curvature with respect to the metric $g_{ij}$.
Hamilton [9] proved that for any smooth initial metric $g(0)=g_{0}$ on a
closed manifold, there exists a unique solution $g(t)$, $t\in[0,\epsilon)$, to
the Ricci flow equation for some $\epsilon>0$. A solution $g(t)$ of the Ricci
flow of the form

$g(t)=\sigma(t)\varphi(t)^{*}g(0),$

where $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ is a positive function and
$\varphi(t):M\rightarrow M$ is a 1-parameter family of diffeomorphisms, is
called a Ricci soliton. It is known that if the initial metric $g_{0}$
satisfies the equation

(1) $Ric(g_{0})+\frac{1}{2}\pounds_{X}g_{0}=\lambda g_{0},$

where $\lambda$ is a constant and $X$ is a smooth vector field on $M$, then
the manifold $M$ admits Ricci soliton. Therefore, the equation (1), in
general, is known as Ricci soliton. If $X$ is the gradient of some smooth
function, then it is called gradient Ricci soliton. For more results of Ricci
soliton see [2, 7, 8]. In 1979, Bourguignon [1] introduced the notion of
Ricci-Bourguignon flow, where the metrics $g(t)$ is evolving according to the
flow equation

$\frac{\partial}{\partial t}g_{ij}=-2R_{ij}+2\rho Rg_{ij},$

where $\rho$ is a non-zero scalar constant and $R$ is the scalar curvature of
the metric $g(t)$. Following the Ricci soliton, Catino and Mazzier [4] gave
the definition of gradient $\rho$-Einstein soliton, which is the self-similar
solution of Ricci-Bourguignon flow. This soliton is also called gradient
Ricci-Bourguignon soliton by some authors.

######  Definition 1.1.

[4] Let $(M,g)$ be a Riemannian manifold of dimension $n$, $(n\geq 3)$, and
let $\rho\in\mathbb{R}$, $\rho\neq 0$. Then $M$ is called gradient
$\rho$-Einstein soliton, denoted by $(M,g,f,\rho)$, if there is a smooth
function $f:M\rightarrow\mathbb{R}$ such that

(2) $Ric+\nabla^{2}f=\lambda g+\rho Rg,$

for some constant $\lambda$.

The soliton is trivial if $\nabla f$ is a parallel vector field. The function
$f$ is known as $\rho$-Einstein potential function. If $\lambda>0$
$(\text{resp.}=0,<0)$, then the gradient $\rho$-Einstein soliton
$(M,g,f,\rho)$ is said to be shrinking (resp. steady or expanding) . On the
other hand, the $\rho$-Einstein soliton is called gradient Einstein soliton,
gradient traceless Ricci soliton or gradient Schouten soliton if
$\rho=1/2,1/n$ or $1/2(n-1)$. Later, this notion has been generalized in
various directions such as $m$-quasi Einstein manifold [11], $(m,\rho)$-quasi
Einstein manifold [12], Ricci-Bourguignon almost soliton [13].
Catino and Mazzier [4] showed that compact gradient Einstein, Schouten or
traceless Ricci soliton is trivial. They classified three-dimensional gradient
shrinking Schouten soliton and proved that it is isometric to a finite
quotient of either $\mathbb{S}^{3}$ or $\mathbb{R}^{3}$ or
$\mathbb{R}\times\mathbb{S}^{2}$. Huang [10] deduced a sufficient condition
for the compact gradient shrinking $\rho$-Einstein soliton to be isometric to
a quotient of the round sphere $\mathbb{S}^{n}$.

######  Theorem 1.1.

[10] Let $(M,g,f,\rho)$ be an $n$-dimensional $(4\leq n\leq 5)$ compact
gradient shrinking $\rho$-Einstein soliton with $\rho<0$. If the following
condition holds

$\displaystyle\Big{(}\int_{M}|W+\frac{\sqrt{2}}{\sqrt{n}(n-2)}Z\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}g|^{2}\Big{)}^{\frac{2}{n}}$
$\displaystyle+\sqrt{\frac{(n-4)^{2}(n-1)}{8(n-2)}}\lambda
vol(M)^{\frac{2}{n}}$ $\displaystyle\leq\sqrt{\frac{n-2}{32(n-1)}}Y(M,[g]),$

where $Z=Ric-\frac{R}{n}g$ is the trace-less Ricci tensor, $W$ is the Weyl
tensor and $Y(M,[g])$ is the Yamabe invariant associated to $(M,g)$, then $M$
is isometric to a quotient of the round sphere $\mathbb{S}^{n}$.

In 2019, Mondal and Shaikh [14] proved the isometry theorem for gradient
$\rho$-Einstein soliton in case of conformal vector field. In particular, they
proved the following result:

######  Theorem 1.2.

[14] Let $(M,g,f,\rho)$ be a compact gradient $\rho$-Einstein soliton. If
$\nabla f$ is a non-trivial conformal vector field, then $M$ is isometric to
the Euclidean sphere $\mathbb{S}^{n}$.

Dwivedi [13] proved an isometry theorem for gradient Ricci-Bourguignon
soliton.

######  Theorem 1.3.

[13] A non-trivial compact gradient Ricci-Bourguignon soliton is isometric to
an Euclidean sphere if any one of the following holds
(1) $M$ has constant scalar curvature.
(2) $\int_{M}g(\nabla R,\nabla f)\leq 0$.
(3) $M$ is a homogeneous manifold.

We note that Catino et. al. [5] proved many results for gradient
$\rho$-Einstein soliton in non-compact manifold.

######  Theorem 1.4.

Let $(M,g,f,\rho)$ be a complete non-compact gradient shrinking
$\rho$-Einstein soliton with $0<\rho<1/2(n-1)$ bounded curvature, non-negative
radial sectional curvature, and non-negative Ricci curvature. Then the scalar
curvature is constant.

In this paper, we have showed that a non-trivial compact gradient
$\rho$-Einstein soliton is isometric to an Euclidean sphere. The main results
of this paper are as follows:

######  Theorem 1.5.

A nontrivial compact gradient $\rho$-Einstein soliton has constant scalar
curvature and therefore $M$ is isometric to an Euclidean sphere.

We have also showed that in a non-compact gradient $\rho$-Einstein soliton
satisfying some conditions the scalar curvature vanishes.

######  Theorem 1.6.

Suppose $(M,g,f,\rho)$ is a non-compact gradient non-expanding $\rho$-Einstein
soliton with non-negative scalar curvature. If $\rho>1/n$ and the
$\rho$-Einstein potential function satisfies

(3) $\int_{M-B(p,r)}d(x,p)^{-2}f<\infty,$

then the scalar curvature vanishes in $M$.

##  2\. Proof of the results

###### Proof of the Theorem 1.5.

Since the gradient $\rho$-Einstein soliton is non-trivial, it follows that
$\rho\neq 1/n$, see [4]. Taking the trace of (2) we get

(4) $R+\Delta f=\lambda n+\rho Rn.$

From the commutative equation, we obtain

(5) $\Delta\nabla_{i}f=\nabla_{i}\Delta f+R_{ij}\nabla_{j}f.$

By using contracted second Bianchi identity, we have

$\displaystyle\Delta\nabla_{i}f=\nabla_{j}\nabla_{j}\nabla_{i}f$
$\displaystyle=$ $\displaystyle\nabla_{j}(\lambda g_{ij}+\rho Rg_{ij}-R_{ij})$
$\displaystyle=$ $\displaystyle\nabla_{i}(\rho R-\frac{1}{2}R).$

and

$\nabla_{i}\Delta f=\nabla_{i}(\lambda n+\rho Rn-R)=\nabla_{i}(\rho Rn-R).$

Therefore, (5) yields

(6) $(n-1)\rho\nabla_{i}R-\frac{1}{2}\nabla_{i}R+R_{ij}\nabla_{j}f=0,$

Taking covariant derivative $\nabla_{l}$, we get

$(n-1)\rho\nabla_{l}\nabla_{i}R-\frac{1}{2}\nabla_{l}\nabla_{i}R+\nabla_{l}R_{ij}\nabla_{j}f+R_{ij}\nabla_{l}\nabla_{j}f=0.$

Taking trace in both sides, we obtain

(7) $((n-1)\rho-\frac{1}{2})\Delta R+\frac{1}{2}g(\nabla R,\nabla f)+R(\lambda
n+\rho Rn-R)=0.$

Now integrating using divergence theorem we get

$\displaystyle\int_{M}R(\lambda n+\rho Rn-R)$ $\displaystyle=$
$\displaystyle-\int_{M}((n-1)\rho-\frac{1}{2})\Delta
R-\frac{1}{2}\int_{M}g(\nabla R,\nabla f)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{M}R\Delta f=\frac{1}{2}\int_{M}R(\lambda n+\rho
Rn-R).$

The above equation is true only if

(8) $\int_{M}R(\lambda n+\rho Rn-R)=0,$

which implies

(9) $\int_{M}R\Big{(}R+\frac{\lambda n}{n\rho-1}\Big{)}=0,$

Again integrating (4), we obtain

(10) $\int_{M}\Big{(}R+\frac{\lambda n}{n\rho-1}\Big{)}=0.$

Therefore, (9) and (10) together imply that

$\int_{M}\Big{(}R+\frac{\lambda n}{n\rho-1}\Big{)}^{2}=0.$

Hence, $R=\lambda n/(1-\rho n)$. Then from Theorem 1.3 we can conclude our
result. ∎

###### Proof of the Theorem 1.6.

From (4) we get

$(n\rho-1)R=\Delta f-\lambda n.$

Since $\lambda\geq 0$, the above equation implies that

(11) $(n\rho-1)R\leq\Delta f.$

Now, we consider the cut-off function, introduced in [6], $\varphi_{r}\in
C^{2}_{0}(B(p,2r))$ for $r>0$ such that

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

where $C>0$ is a constant. Then for $r\rightarrow\infty$, we have
$\Delta\varphi^{2}_{r}\rightarrow 0$ as
$\Delta\varphi^{2}_{r}\leq\frac{C}{r^{2}}$. Then we calculate

(12)
$\displaystyle(n\rho-1)\int_{M}R\varphi^{2}_{r}\leq\int_{M}\varphi^{2}_{r}\Delta
f$ $\displaystyle=$ $\displaystyle\int_{B(p,2r)-B(p,r)}f\Delta\varphi_{r}^{2}$
(13) $\displaystyle\leq$
$\displaystyle\int_{B(p,2r)-B(p,r)}f\frac{C}{r^{2}}\rightarrow 0,$

as $r\rightarrow\infty$. Hence, we obtain

(14) $(n\rho-1)\lim_{r\rightarrow\infty}\int_{B(p,r)}R\leq 0.$

Since $\rho>1/n$, it follows that

$\lim_{r\rightarrow\infty}\int_{B(p,r)}R\leq 0.$

But $R$ is non-negative everywhere in $M$. Therefore, $R\equiv 0$ in $M$. ∎

##  3\. acknowledgment

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

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