The Riemannian Bianchi identities of metric connections
with skew torsion and generalized Ricci solitons

S. Ivanov1 and N. Stanchev2

1 University of Sofia, Faculty of Mathematics and Informatics,

blvd. James Bourchier 5, 1164, Sofia, Bulgaria

and Institute of Mathematics and Informatics, Bulgarian Academy of Sciences

2 University of Sofia, Faculty of Mathematics and Informatics,

blvd. James Bourchier 5, 1164, Sofia, Bulgaria

###### Abstract

Curvature properties of a metric connection with totally skew-symmetric
torsion are investigated. It is shown that if either the 3-form $T$ is
harmonic, $dT=\delta T=0$ or the curvature of the torsion connection $R\in
S^{2}\Lambda^{2}$ then the scalar curvature of a ${\nabla}$-Einstein manifold
is determined by the norm of the torsion up to a constant. It is proved that a
compact generalized gradient Ricci soliton with closed torsion is Ricci flat
if and only if either the norm of the torsion or the Riemannian scalar
curvature are constants. In this case the torsion 3-form is harmonic and the
gradient function has to be constant.

Necessary and sufficient conditions a metric connection with skew torsion to
satisfy the Riemannian first Bianchi identity as well as the contracted
Riemannian second Binachi identity are presented. It is shown that if the
torsion connection satisfies the Riemannian first Bianchi identity then it
satisfies the contracted Riemannian second Bianchi identity. It is also proved
that a metric connection with skew torsion satisfying the curvature identity
$R(X,Y,Z,V)=R(Z,Y,X,V)$ must be flat.

AMS MSC2010: 53C55, 53C21, 53C29, 53Z05

###### Contents

  1. 1 Introduction
  2. 2 Metric connection with skew-symmetric torsion and its curvature
    1. 2.1 Proof of Theorem 1.3
  3. 3 The contracted second Bianchi identity
    1. 3.1 Proof of Theorem 1.2
  4. 4 The ${\nabla}$-Einstein condition
  5. 5 Generalized gradient Ricci solitons. Proof of Theorem 1.1

##  1 Introduction

Riemannian manifolds with metric connections having totally skew-symmetric
torsion and special holonomy received a strong interest mainly from
supersymmetric string theories and supergravity. The main interest is the
existence and properties of such a connection with holonomy inside $U(n)$
which also preserves the $U(n)$ structure i.e. metric connections with skew-
symmetric torsion on an almost hermitian manifold preserving the almost
hermitian structure.

Hermitian manifolds have widespread applications in both physics and
differential geometry. These are complex manifolds equipped with a metric $g$,
and a Kaehler form $\omega$ which are of type (1,1) with respect to the
complex structure $J$. There are many examples of Hermitian manifolds as every
complex manifold admits a Hermitian structure. In many applications, Hermitian
manifolds have additional properties which are expressed as either a condition
on $\omega$ or as a restriction on the holonomy of one of the Hermitian
connections.

Given a Hermitian manifold $(M,g,J)$ the Strominger-Bismut connection $\nabla$
is the unique connection on $M$ that is Hermitian ($\nabla g=0$ and $\nabla
J=0$) and has totally skew-symmetric torsion tensor $T$. Its existence and
explicit expression first appeared in Strominger’s seminal paper [43] in 1986
in connection with the heterotic supersymmetric string background, where he
called it the H-connection. Three years later, Bismut formally discussed and
used this connection in his local index theorem paper [6], which leads to the
name Bismut connection in literature. Note that this connection was also
called KT connection (Kähler with torsion) and characteristic connection.

If the torsion 3-form $T$ of $\nabla$ is closed, $dT=0$, which is equivalent
to the condition $\partial\bar{\partial}\omega=0$, the Hermitian metric g is
called SKT (strong Kähler with torsion) [24] or pluriclosed. The SKT
(pluriclosed) metrics has found many applications in both physics, see e.g.
[21, 25, 43, 22, 23, 10] and geometry, see e.g. [33, 27, 28, 12, 44, 17, 18,
11, 14, 40, 35]. For example in type II string theory, the torsion 3-form $T$
is identified with the 3-form field strength. This is required by construction
to satisfy $dT=0$. Streets and Tian [40] introduced a hermitian Ricci flow
under which the pluriclosed or equivalently strong KT structure is preserved.
Generalizations of the pluriclosed condition $\partial\bar{\partial}\omega=0$
on 2n dimensional Hermitian manifolds in the form
$\omega^{\ell}\wedge\partial\bar{\partial}\omega^{k}=0~{},~{}~{}~{}1\leq
k+\ell\leq n-1~{}$ has been investigated in [15, 38, 19, 29] etc.

Hermitian metrics with the Strominger-Bismut connection being Kähler-like,
namely, its curvature satisfies the Riemannian first Bianchi identity ( the
identity (2.11) below), have been studied in [4], investigating this property
on 6-dimensional solvmanifolds with holomorphically trivial canonical bundle.
It was conjectured by Angela-Otal-Ugarte-Villakampa in [4] that such metrics
should be SKT (pluriclosed).

Support to that can be derived from [Theorem 5][45] where the simply connected
hermitian manifolds with flat Strominger-Bismut connection are classified and
it easily follows that these spaces are SKT (pluriclosed) with parallel 3-form
torsion [4]. The latter conclusion also follows from the Cartan-Schouten
theorem [7], (see also [Theorem 2.2][3]).

The conjecture by Angela-Otal-Ugarte-Villakampa was proved in [47]. In the
first version of [47] in the arXive the authors stated, by a misprint, that
the Strominger-Bismut connection is Kähler like if and only if the curvature
satisfies the identity

$R(X,Y,Z,V)=R(Z,Y,X,V).$ (1.1)

It is shown in [47] that the curvature of Strominger-Bismut connection is
Kähler like then one has $dH=\nabla H=0$. In fact, in the proof, they used the
correct Riemannian first Bianchi identity (2.11). The corrected definition of
the Kähler-like condition was given in the second version of [47].

It was shown by Fino and Tardini in [13] that the curvature condition (1.1) is
strictly stronger than the Riemannian first Bianchi identity by constructing
an explicit example whose Strominger-Bismut curvature satisfies the Riemannian
first Bianchi identity but it does not obey the condition (1.1).

In general, metric connections with skew symmetric and closed torsion
$T,dT=0$, are closely connected with the generalized Ricci flow. Namely, the
fixed points of the generalized Ricci flow are Ricci flat metric connections
with harmonic torsion 3-form, $Ric=dT=\delta T=0$, we refer to the recent book
[20] and the references given there for mathematical and physical motivation.

The first goal in the paper is to investigate peoperties of a metric
connection with skew-symmetric torsion with applications to generalized Ricci
flow. We consider ${\nabla}$-Einstein spaces determined with the condition
that the symmetric part of the Ricci tensor of the torsion connection is a
scalar multiple of the metric. These spaces are introduced by Agricola and
Ferreira [1, Definition 2.2] as the critical points of the
${\nabla}$-Einstein-Hilbert functional [1, Theorem 2.1]. In the case when the
torsion 3-form is ${\nabla}$-parallel the ${\nabla}$-Einstein spaces are
investigated in [1, 2, 8, 9] and a large number of examples are given there.
The ${\nabla}$-Einstein spaces appear also in generalized geometry. In [20,
Proposition 3.47] it is shown that a Riemannian metric $g$ and a harmonic
3-form $T$ are critical point of the generalized Einstein-Hilbert functional
if and only if it is ${\nabla}$-Einstein.

Notice that in contrast to the Riemannian case, for a ${\nabla}$-Einstein
manifold the scalar curvature $Scal$ of the torsion connection is not
necessarily constant (for details see [1]). If the torsion is ${\nabla}$\-
parallel then the scalar curvature of the torsion connection and the
Riemannian scalar curvature are constants, similarly to an Einstein manifold
[1, Proposition 2.7].

We investigate the constancy of the scalar curvatures of a ${\nabla}$-Einstein
space. We show in Theorem 4.2 that if either the 3-form $T$ has zero torsion
1-forms, $\delta T\lrcorner T=T\lrcorner dT=0$, (in particular if $T$ is
harmonic, $dT=\delta T=0$) or the curvature of the torsion connection $R\in
S^{2}\Lambda^{2}$ then the scalar curvature of a ${\nabla}$-Einstein manifold
is determined by the norm of the torsion up to a constant. In particular, the
scalar curvature of the torsion connection is constant if and only if the norm
of the torsion is constant.

Observing that if the torsion is parallel, ${\nabla}T=0$, then the torsion
1-forms vanish, $\theta=\delta T\lrcorner T=0,\Theta=T\lrcorner dT=0$, we
obtain that any ${\nabla}$-Einstein manifold with parallel torsion of
dimension bigger than 2 the Riemannian scalar curvarture and the scalar
curvature of the torsion connections are constants, thus confirm the result of
Agricola and Ferreira, [1, Proposition 2.7].

A special phenomena occurs in dimednsion six, namaely (4.32) below imply that
if $(M,g,T)$ is a six dimensional Riemannian manifold with zero torsion
1-forms (in particular $T$ is harmonic 3-form) and the metric connection with
torsion $T$ is ${\nabla}$-Einstein then the Riemannian scalar curvature is
constant, Corollary 4.3.

We recall [20, Definition 4.31] that a Riemannian manifold $(M,g,T)$ with a
closed 3-form $T$ is a generalized steady Ricci soliton with $k=0$ if one has

$Ric^{g}=\frac{1}{4}T^{2}-\mathbb{L}_{X}g,\qquad\delta T=-X\lrcorner T,\qquad
dT=0.$

If the vector field $X$ is a gradient of a smooth function $f$ then we have
the notion of a generalized gradient Ricci soliton. A complete generalized
gradient Ricci solitons are constructed on complex surfaces in [42]. The
existence and classifivation of non-trivial solitons on compact (complex)
4-manifolds has been recently proved in [39, 41, 5].

Our first main observation is the following

######  Theorem 1.1.

Let $(M,g,T)$ be a compact Riemannian manifold with closed 3-form $T,dT=0$.

If $(M,g,T,f)$ is a generelized gradient Ricci soliton then the following
conditions are equivalent:

  * a).

The norm of the torsion is constant, $d||T||^{2}=0$;

  * b).

The function $f$ is constant.

  * c).

The torsion connection is Ricci flat, $Ric=0$;

  * d).

The Riemannian scalar curvature is constant, $Scal^{g}=const$.

In all four cases the 3-form $T$ is harmonic.

Examples of Ricci flat torsion connections are constructed in [37, 34].

The second aim of this note is to express necessary and sufficient conditions
the curvature of a metric connection with totally skew-symmetric torsion to
satisfy the Riemannian first Bianchi identity as well as the contracted
Riemannian second Binachi identity ((2.14) below) and equation (1.1).

Our second main result is the next

######  Theorem 1.2.

The curvature of a metric connection $\nabla$ with skew-symmetric torsion $T$
on a Riemannian manifold $(M,g)$ satisfies the Riemannian first Bianchi
identity if and only if the next identities hold

$dT=-2\nabla T=\frac{2}{3}\sigma^{T},$ (1.2)

where the four form $\sigma^{T}$ corresponding to the 3-form $T$ is defined
below with (2.5).

In this case the norm of the 3-form $T$ is a constant, $||T||^{2}=const.$ and
the curvature of the connection ${\nabla}$ satisfies the contracted Riemannian
second Bianchi identity.

Clearly, any torsion-free connection satisfying (1.1) must be flat which is a
simple consequence of the first Bianchi identity. In particular, a Riemannian
manifold satisfies (1.1) if and only if it is flat.

We show that this is valid also for metric connections with skew-symmetric
torsion which explains the reason for the existence of the example in [13]. We
derive from Theorem 1.2 the next general result

######  Theorem 1.3.

A metric connection with skew-symmetric torsion satisfies the condition (1.1)
if and only if it is flat, $R=0$.

######  Remark 1.4.

We note that Theorem 4.2 generalizes the Ricci flat case established recently
in [36, Lemma 2.21] where it was proved that if the torsion is harmonic (it is
sufficient only to be closed) and the Ricci tensor of the torsion connection
vanishes then the norm of the torsion is constant.

The converse is not true in general. Namely not any space with closed torsion
of constant norm $dT=d||T||^{2}=0$ is Ricci flat.

In some cases the converse holds true. For example in the case of compact
generalized gradient Ricci soliton the Ricci flatness is equivalent to the
constancy of the norm of the torsion due to Theorem 1.1.

Another cases occur if the torsion connection has special holonomy, contained
in the groups $SU(3)$, $G_{2}$ or $Spin(7)$. It is shown very recently in
[31], [32] and [30] that the $SU(3),G_{2},Spin(7)$-torsion connection with
closed torsion is Ricci flat on a compact manifold if and only if the norm of
the torsion is constant.

Everywhere in the paper we will make no difference between tensors and the
corresponding forms via the metric as well as we will use Einstein summation
conventions, ie repeated Latin indices are summed over up to $n$.

##  2 Metric connection with skew-symmetric torsion and its curvature

On a Riemannian manifold $(M,g)$ of dimension $n$ any metric connection
${\nabla}$ with totally skew-symmetric torsion $T$ is connected with the Levi-
Civita connection ${\nabla}^{g}$ of the metric $g$ by

${\nabla}^{g}={\nabla}-\frac{1}{2}T.$ (2.3)

The exterior derivative $dT$ has the following expression (see e.g. [26, 28,
16])

$\begin{split}dT(X,Y,Z,V)=(\nabla_{X}T)(Y,Z,V)+(\nabla_{Y}T)(Z,X,V)+(\nabla_{Z}T)(X,Y,V)\\\
+2\sigma^{T}(X,Y,Z,V)-(\nabla_{V}T)(X,Y,Z)\end{split}$ (2.4)

where the 4-form $\sigma^{T}$, introduced in [16], is defined by

$\sigma^{T}(X,Y,Z,V)=\frac{1}{2}\sum_{j=1}^{n}(e_{j}\lrcorner
T)\wedge(e_{j}\lrcorner T)(X,Y,Z,V),$ (2.5)

$(e_{j}\lrcorner T)(X,Y)=T(e_{j},X,Y)$ is the interior multiplication and
$\\{e_{1},\dots,e_{n}\\}$ is an orthonormal basis.

The properties of the 4-form $\sigma^{T}$ are studied in detail in [2] where
it is shown that $\sigma^{T}$ measures the ‘degeneracy’ of the 3-form $T$.

For the curvature of ${\nabla}$ we use the convention
$R(X,Y)Z=[\nabla_{X},\nabla_{Y}]Z-\nabla_{[X,Y]}Z$ and
$R(X,Y,Z,V)=g(R(X,Y)Z,V)$. It has the well known properties

$R(X,Y,Z,V)=-R(Y,X,Z,V)=-R(X,Y,V,Z).$ (2.6)

The Ricci tensors and scalar curvatures of the Levi-Civita connection
${\nabla^{g}}$ and the torsion connection ${\nabla}$ are related by [16,
Section 2], (see also [20, Prop. 3.18])

$\begin{split}Ric^{g}(X,Y)=Ric(X,Y)+\frac{1}{2}(\delta
T)(X,Y)+\frac{1}{4}\sum_{i=1}^{n}g(T(X,e_{i}),T(Y,e_{i});\\\
Scal^{g}=Scal+\frac{1}{4}||T||^{2},\qquad Ric(X,Y)-Ric(Y,X)=-(\delta
T)(X,Y),\end{split}$ (2.7)

where $\delta=(-1)^{np+n+1}*d*$ is the co-differential acting on $p$-forms and
$*$ is the Hodge star operator.

Following [20] we denote

$T^{2}_{ij}=T_{iab}T_{jab}:=\sum_{a,b=1}^{n}T_{iab}T_{jab}.$

Then the first equality in (2.7) takes the form

$Ric^{g}_{ij}=Ric_{ij}+\frac{1}{2}\delta T_{ij}+\frac{1}{4}T^{2}_{ij}.$

The first Bianchi identity for $\nabla$

$\begin{split}R(X,Y,Z,V)+R(Y,Z,X,V)+R(Z,X,Y,V)\\\
=(\nabla_{X}T)(Y,Z,V)+(\nabla_{Y}T)(Z,X,V)+(\nabla_{Z}T)(X,Y,V)+\sigma^{T}(X,Y,Z,V)\end{split}$

can be written in the following form (see e.g. [26, 28, 16])

$\begin{split}R(X,Y,Z,V)+R(Y,Z,X,V)+R(Z,X,Y,V)\\\
=dT(X,Y,Z,V)-\sigma^{T}(X,Y,Z,V)+(\nabla_{V}T)(X,Y,Z)\end{split}$ (2.8)

It is proved in [16, p.307] that the curvature of a metric connection
${\nabla}$ with totally skew-symmetric torsion $T$ satisfies also the next
identity

$\begin{split}R(X,Y,Z,V)+R(Y,Z,X,V)+R(Z,X,Y,V)-R(V,X,Y,Z)-R(V,Y,Z,X)-R(V,Z,X,Y)\\\
=\frac{3}{2}dT(X,Y,Z,V)-\sigma^{T}(X,Y,Z,V).\end{split}$ (2.9)

We obtain from (2.9) and (2.8)

######  Proposition 2.1.

The curvature of a metric connection with skew-symmetric torsion satisfies

$\begin{split}R(V,X,Y,Z)+R(V,Y,Z,X)+R(V,Z,X,Y)=-\frac{1}{2}dT(X,Y,Z,V)+(\nabla_{V}T)(X,Y,Z)\end{split}$
(2.10)

Following [4] we have

######  Definition 2.2.

We say that the curvature $R$ satisfies the Riemannian first Bianchi identity
if

$R(X,Y,Z,V)+R(Y,Z,X,V)+R(Z,X,Y,V)=0.$ (2.11)

It is well known algebraic fact that (2.6) and (2.11) imply $R\in
S^{2}\Lambda^{2}$, i.e it holds

$R(X,Y,Z,V)=R(Z,V,X,Y),$ (2.12)

Note that, in general, (2.6) and (2.12) do not imply (2.11).

The precise condition the curvature of a metric connection ${\nabla}$ with
totally skew-symmetric torsion $T$ to satify (2.12) is given in [26, Lemma
3.4], namely the covariant derivative of the torsion with respect to the
torsion connection ${\nabla}T$ must be a four form,

######  Lemma 2.3.

[26, Lemma 3.4]) The next equivalences hold for a metric connection with
torsion 3-form

$({\nabla}_{X}T)(Y,Z,V)=-({\nabla}_{Y}T)(X,Z,V)\Longleftrightarrow
R(X,Y,Z,V)=R(Z,V,X,Y))\Longleftrightarrow dT=4{\nabla^{g}}T.$ (2.13)

An immediate consequence of (2.13) is the next

######  Corollary 2.4.

Suppose that a metric connection with torsion 3-form $T$ has curvature $R\in
S^{2}\Lambda^{2}$.

Then the torsion 3-form is closed, $dT=0$, if and only if the torsion is
parallel with respect to the Levi-Civita connection, ${\nabla^{g}}T=0$.

######  Definition 2.5.

We say that a the curvature of a metric connection with skew-symmetric torsion
satisfies the contracted Riemannian second Bianchi identity if

$d(Scal)(X)-2\sum_{i=1}^{n}({\nabla}_{e_{i}}Ric)(X,e_{i})=0.$ (2.14)

Following [3], we consider a 1-parameter family of metric connections
$\nabla^{t}$ with torsion $tT$ defined by

$g(\nabla^{g}_{X}Y,Z)=g(\nabla^{t}_{X}Y,Z)-\frac{t}{2}T(X,Y,Z)$

yielding the equality (see e.g. [3])

$(\nabla^{g}_{X}T)(Y,Z,V)=(\nabla^{t}_{X}T)(Y,Z,V)+\frac{t}{2}\sigma^{T}(X,Y,Z,V)$
(2.15)

We continue with the following result which generalizes [Proposition 2.1][3],
[Theorems 3.1,3.2][13] and proves the first part of Theorem 1.2.

######  Theorem 2.6.

The curvature of a metric connection with skew-symmetric torsion $T$ satisfies
the Riemannian first Bianchi identity if and only if the identities (1.2)
hold.

In this case the 3-form $T$ is parallel with respect to the metric connection
$\nabla^{1/3}$ with torsion equal to $\frac{1}{3}T,\nabla^{1/3}T=0$. In
particular, the norm of the 3-form $T$ is constant, $||T||^{2}=const$.

###### Proof.

Substitute (1.2) into the right-hand side of (2.8) to get that the Riemannian
first Bianchi identity (2.11) holds.

For the converse, suppose the curvature $R$ of ${\nabla}$ satisfies the
Riemannian first Bianchi identity (2.11). Then it satisfies the identity
(2.12) and [26, Lemma 3.4] imply that $\nabla T$ is a 4-form. Then we have
from (2.8), (2.4) and (2.9)

$3\nabla T=-\sigma^{T}=-\frac{3}{2}dT$

which proofs (1.2).

One gets from (1.2) $\nabla T=-\frac{1}{3}\sigma^{T}$ and (2.15) yields
$\nabla^{1/3}T=0$, which was first obsered in [3]. ∎

###  2.1 Proof of Theorem 1.3

###### Proof.

We will show that (1.1) implies (1.2). Indeed, substitute (1.1) into (2.9) to
get

$3dT=2\sigma^{T}.$ (2.16)

Further, (1.1) yields $R(X,Y,Z,V)=R(Z,V,X,Y)$ leading by [26, Lemma 3.4] that
$\nabla T$ is a 4-form which applied to (2.4) gives

$\begin{split}dT(X,Y,Z,V)=(\nabla_{X}T)(Y,Z,V)+(\nabla_{Y}T)(Z,X,V)+(\nabla_{Z}T)(X,Y,V)-(\nabla_{V}T)(X,Y,Z)\\\
+2\sigma^{T}(X,Y,Z,V)=4\nabla
T(X,Y,Z,V)+2\sigma^{T}(X,Y,Z,V)=\frac{2}{3}\sigma^{T}(X,Y,Z,V),\end{split}$

where we have used (2.16). Hence, (1.2) holds and Theorem 1.2 shows the
validity of the Riemannian first Bianchi identity. Hence, we get

$\begin{split}0=R(X,Y,Z,V)+R(Y,Z,X,V)+R(Z,X,YV)\\\
=R(Z,Y,X,V)+R(Y,Z,X,V)+R(Z,X,Y,V)=R(Z,X,Y,V)\end{split}$

where we have applying (1.1) to get the second identity. The proof is
completed. ∎

##  3 The contracted second Bianchi identity

In this section we investigate the second Bianchi identity for the curvature
of a metric connection with skew-symmetric torsion.

First we show the validity of an algebraic identity.

######  Proposition 3.1.

For an arbitrary 3-form $T$ the next identity holds

$T_{abc}\sigma^{T}_{abci}=0$ (3.17)

###### Proof.

Using (2.5) we calculate

$T_{abc}\sigma^{T}_{abci}=T_{abc}\Big{(}T_{abs}T_{sci}+T_{bcs}T_{sai}+T_{cas}T_{sbi}\Big{)}=3T_{abc}T_{abs}T_{sci}=0$

since $T_{abc}T_{abs}$ is symmetric in $c$ and $s$ while $T_{sci}$ is skew-
symmetric in $c$ and $s$. ∎

The next observation expresses the ${\nabla}$-divergence of $\delta T$,

######  Proposition 3.2.

For a metric connection with torsion 3-form $T$ the next identity holds

$2{\nabla}_{i}\delta T_{ij}=\delta T_{ia}T_{iaj}.$ (3.18)

###### Proof.

Applying (2.3), we calculate using the Ricci identity for the Levi-Civita
connection, the symmetricity of its Ricci tensor and the first Bianchi
identity

${\nabla}_{i}\delta T_{ij}={\nabla^{g}}_{i}\delta T_{ij}-\frac{1}{2}\delta
T_{is}T_{ijs}=-\frac{1}{2}\Big{(}{\nabla^{g}}_{i}{\nabla^{g}}_{s}-{\nabla^{g}}_{s}{\nabla^{g}}_{i}\Big{)}T_{sij}+\frac{1}{2}\delta
T_{is}T_{isj}=\\\
-\frac{1}{2}\Big{(}R^{g}_{issq}T_{qij}+R^{g}_{isiq}T_{sqj}+R^{g}_{isjq}T_{isq}\Big{)}+\frac{1}{2}\delta
T_{is}T_{isj}\\\
=Ric^{g}_{iq}T_{iqj}-\frac{1}{6}\Big{(}R^{g}_{isqj}+R^{g}_{siqj}+R^{g}_{iqsj}\Big{)}T_{isq}+\frac{1}{2}\delta
T_{is}T_{isj}=\frac{1}{2}\delta T_{is}T_{isj}.$

The proof of Proposition 3.2 is completed. ∎

######  Definition 3.3.

For any 3-form $T$ we define two torsion 1-forms $\theta,\Theta$ naturally
associated to $T$ by

$\theta_{j}=\delta T_{ab}T_{jab},\quad\Theta_{j}=T_{abc}dT_{jabc}.$

Further, we have

######  Lemma 3.4.

The 1-forms $\theta$ and $\Theta$ are connected by the equality

$3\theta_{j}+\Theta_{j}=\frac{1}{2}{\nabla^{g}}_{j}||T||^{2}-3{\nabla^{g}}_{s}T^{2}_{sj}=\frac{1}{2}{\nabla}_{j}||T||^{2}-3{\nabla}_{s}T^{2}_{sj}$
(3.19)

###### Proof.

We obtain from (2.4)

$\Theta_{j}=-dT_{abcj}T_{abc}=-3{\nabla^{g}}_{a}T_{bcj}T_{abc}+\frac{1}{2}{\nabla^{g}}_{j}||T||^{2}=-3{\nabla}_{a}T_{bcj}T_{abc}+\frac{1}{2}{\nabla}_{j}||T||^{2},$
(3.20)

where we used (2.15) and (3.17).

On the other hand, we calculate in view of (3.17) and (3.20) that

$\theta_{j}=\delta
T_{ia}T_{iaj}=-{\nabla}_{s}T_{sia}T_{iaj}=-{\nabla}_{s}T^{2}_{sj}+{\nabla}_{s}T_{iaj}T_{sia}=-{\nabla^{g}}_{s}T^{2}_{sj}+{\nabla^{g}}_{s}T_{iaj}T_{sia}\\\
=-{\nabla^{g}}_{s}T^{2}_{sj}+\frac{1}{3}({\nabla^{g}}_{s}T_{iaj}+{\nabla^{g}}_{i}T_{asj}T_{sia}+{\nabla^{g}}_{a}T_{sij})T_{sia}\\\
=-{\nabla^{g}}_{s}T^{2}_{sj}-\frac{1}{3}dT_{jsia}T_{sia}+\frac{1}{6}{\nabla^{g}}_{j}||T||^{2}=-{\nabla^{g}}_{s}T^{2}_{sj}-\frac{1}{3}\Theta_{j}+\frac{1}{6}{\nabla^{g}}_{j}||T||^{2}.$
(3.21)

The lemma is proved. ∎

Note that if the torsion is ${\nabla}$-paralel then $\theta=\delta T\lrcorner
T=0$ and $\Theta=T\lrcorner dT=2T\lrcorner\sigma^{T}=0$ due to (2.4) and
(3.17).

######  Proposition 3.5.

The contracted second Bianchi identity for the curvature of the torsion
connection ${\nabla}$ reads

$d(Scal)(X)-2\sum_{i=1}^{n}({\nabla}_{e_{i}}Ric)(X,e_{i})+\frac{1}{6}d||T||^{2}(X)+\theta(X)+\frac{1}{6}\Theta(X)=0.$
(3.22)

If the torsion 1-forms satisfy the identity

$6\theta+\Theta=0$

then

$d(Scal)_{j}-2{\nabla}_{i}Ric_{ji}+\frac{1}{6}{\nabla}_{j}||T||^{2}=0.$ (3.23)

In particular, if the 3-form $T$ is harmonic, $dT=\delta T=0$ then (3.23)
holds.

###### Proof.

The second Bianchi identity for the curvature of a metric connection
${\nabla}$ with torsion $T$ is

$({\nabla}_{X}R)(V,Y,Z,W)+({\nabla}_{V}R)(Y,X,Z,W)+({\nabla}_{Y}R)(X,V,Z,W)\\\
+R(T(X,V),Y,Z,W)+R(T(V,Y),X,Z,W)+R(T(Y,X),V,Z,W)=0.$ (3.24)

Take the trace of (3.24) to get

$\begin{split}({\nabla}_{X}Ric)(Y,Z)+\sum_{i=1}^{n}({\nabla}_{e_{i}}R)(Y,X,Z,e_{i})-({\nabla}_{Y}Ric)(X,Z)\\\
\sum_{i,j=1}^{n}\Big{[}T(X,e_{i},e_{j})R(e_{j},Y,Z,e_{i})+T(e_{i},Y,e_{j})R(e_{j},X,Z,e_{i})\Big{]}-\sum_{i=1}^{n}T(Y,X,e_{i})Ric(e_{i},Z)=0.\end{split}$
(3.25)

The trace in (3.25) together with (2.7) and (2.10) yields

$\begin{split}0=d(Scal)(X)-2\sum_{i=1}^{n}({\nabla}_{e_{i}}Ric)(X,e_{i})-2\sum_{i,j=1}^{n}T(X,e_{i},e_{j})Ric(e_{i},e_{j})\\\
+\frac{1}{3}\sum_{i,j,k=1}^{n}T(e_{i},e_{j},e_{k})\Big{[}R(X,e_{i},e_{j},e_{k})+R(X,e_{j},e_{k},e_{i})+R(X,e_{k},e_{i},e_{j})\Big{]}\\\
=d(Scal)X)-2\sum_{i=1}^{n}({\nabla}_{e_{i}}Ric)(X,e_{i})+\sum_{i,j=1}^{n}T(X,e_{i},e_{j})\delta
T(e_{i},e_{j})+\frac{1}{6}{\nabla}_{X}||T||^{2}\\\
+\frac{1}{6}\sum_{i,j,k=1}^{n}T(e_{i},e_{j},e_{k})dT(X,e_{i},e_{j},e_{k}).\end{split}$
(3.26)

which proves (3.22). The proof of the Proposition 3.5 is completed. ∎

Apply (3.20) to the last term of (3.22) to get

######  Corollary 3.6.

The contracted second Bianchi identity for the curvature of the torsion
connection has also the form

$d(Scal)_{j}-2{\nabla}_{s}Ric_{js}+\frac{1}{4}{\nabla}_{j}||T||^{2}+\theta_{j}-\frac{1}{2}{\nabla}_{i}T_{jka}T_{ijk}=0.$
(3.27)

######  Corollary 3.7.

The curvature of the torsion connection satisfies the contracted Riemannian
second Bianchi identity (2.14) if and only if the next equality holds

$6\theta_{j}+\Theta_{j}+{\nabla}_{j}||T||^{2}=0\Longleftrightarrow
4\theta_{j}+{\nabla}_{j}||T||^{2}-2{\nabla}_{i}T_{jka}T_{ijk}=0$ (3.28)

In particular, the equation (3.28) holds for any Ricci flat torsion
connection.

Note that a special case of Proposition 3.5 and the above corollaries, when
the torsion is ${\nabla}$-parallel, is given in [1, Corollary 2.6].

######  Theorem 3.8.

Let the curvature $R$ of a metric connection ${\nabla}$ with skew-symmetric
torsion $T$ satisfies $R\in S^{2}\Lambda^{2}$, i.e. (2.12) holds.

Then the curvature of ${\nabla}$ satisfies the contracted Riemannian second
Bianchi identity (2.14) if and only if the norm of the torsion is a constant,
$||T||^{2}=const$.

In particular, if ${\nabla}$ is Ricci flat, $Ric=0$, then the norm of the
torsion is constant, $||T||=const.$

###### Proof.

The condition (2.12) imply the Ricci tensor is symmetric and (2.7) yields
$\delta T=0$. Now, taking into account (2.13) we obtain from (3.27)

$d(Scal)(X)-2\sum_{i=1}^{n}({\nabla}_{e_{i}}Ric)(X,e_{i})+\frac{1}{2}{\nabla}_{X}||T||^{2}=0$

which completes the proof of Theorem 3.8 since any Ricci flat connection
satisfies the contracted Rieamannian second Bianchi identity. ∎

Theorem 2.6 and Theorem 3.8 imply

######  Corollary 3.9.

If the curvature of the torsion connectin satisfies the Riemannian first
Bianchi identity (2.11) then it satisfies the contracted Riemannian second
Bianchi identity (2.14).

###  3.1 Proof of Theorem 1.2

The proof of Theorem 1.2 follows from Theorem 2.6 and Corollary 3.9.

######  Remark 3.10.

It is known that if the curvature of a metric connection satisfies the
Riemannian first Bianchi identity (2.11) then it satisfies the curvature
identity (2.12) but the converse is not true in general.

In some cases the converse is true. If the torsion connection has special
holonomy, contained for example in the group $SU(3)$ in dimension six or in
$G_{2}$ in dimension seven the first Bianchi identity implies the vanishing of
the Ricci tensor, $Ric=0$. It is shown very recently in [31] and [32] that
(2.12) together with $Ric=0$ for a torsion connection with holonomy contained
in $SU(3)$ or in $G_{2}$ imply that the Riemannian first Bianchi identity
(2.11) follows.

##  4 The ${\nabla}$-Einstein condition

Since the Ricci tensor of the torsion connection is not symmetric, the usual
Einstein condition seem to be restrictive. We consider the following weaker
condition introduced by Agricola and Ferreira [1]

######  Definition 4.1.

[1, Definition 2.2] A metric connection with skew-symmetric torsion is said to
be ${\nabla}$-Einsten if the symmetric part of the Ricci tensor is a scalar
multiple of the metric,

$\frac{Ric(X,Y)+Ric(Y,X)}{2}=\lambda g(X,Y).$

In view of (2.7) the ${\nabla}$-Einstein condition is equivalent to

$Ric(X,Y)=\frac{Scal}{n}g(X,Y)-\frac{1}{2}\delta T(X,Y).$ (4.29)

It is shown in [1, Theorem 2.1] that on a compact Riemanian manifold $(M,g,T)$
with a Riemannian metic $g$ and a 3-form $T$, the critical points of the
${\nabla}$-Einstein-Hilbert functional

$\mathcal{L}(g,T)=\int_{M}Scal.Vol_{g}$

are precisely the pairs $(g,T)$ satisfying the ${\nabla}$-Einstein condition.

We have

######  Theorem 4.2.

Let a metric connection with skew-symmetric torsion $T$ be
${\nabla}$-Einstein.

  * a)

Then the next identity holds

$\frac{n-2}{n}d(Scal)_{j}-\frac{1}{2}{\nabla^{g}}_{s}T^{2}_{sj}+\frac{1}{4}{\nabla}_{j}||T||^{2}=0$
(4.30)

  * b)

If the torsion 1-forms satisfy the identity

$3\theta+\Theta=0$ (4.31)

then the scalar curvature is determined by the norm of the torsion up to a
constant $C$ due to

$Scal=-\frac{n}{6(n-2)}||T||^{2}+C\quad and\quad
Scal^{g}=Scal+\frac{1}{4}||T||^{2}=\frac{n-6}{12(n-2)}||T||^{2}+C,$ (4.32)

In particulart, if the 3-form $T$ is harmonic, $dT=\delta T=0$ then (4.32)
holds.

  * c)

If the curvature of the torsion connection $R\in S^{2}\Lambda^{2}$ then the
scalar curvature and the norm of the torsion satisfy the next identity with a
constant $B$

$Scal=-\frac{n}{2(n-2)}||T||^{2}+B,\quad
Scal^{g}=-\frac{n+2}{4(n-2)}||T||^{2}+B,$ (4.33)

In particular, if the scalar curvature of the torsion connection is constant
in b) and c) then the norm of the torsion is constant, $||T||^{2}=const$.

###### Proof.

We have from (4.29) and (3.18) that

${\nabla}_{i}Ric_{ji}=\frac{d(Scal)_{j}}{n}-\frac{1}{2}{\nabla}_{i}\delta
T_{ji}=\frac{d(Scal)_{j}}{n}+\frac{1}{4}\delta
T_{ia}T_{iaj}=\frac{d(Scal)_{j}}{n}+\frac{1}{4}\theta_{j}$ (4.34)

Substitute (4.34) and (3.21) into (3.26) to get

$0=\frac{n-2}{n}d(Scal)_{j}+\frac{1}{2}\delta
T_{ia}T_{iaj}+\frac{1}{6}{\nabla}_{j}||T||^{2}+\frac{1}{6}T_{abc}dT_{jabc}\\\
=\frac{n-2}{n}d(Scal)_{j}-\frac{1}{2}{\nabla^{g}}_{s}T^{2}_{sj}+\frac{1}{4}{\nabla}_{j}||T||^{2}$
(4.35)

since ${\nabla}_{j}||T||^{2}={\nabla^{g}}_{j}||T||^{2}$ due to (3.17). This
proves (4.30).

If $3\theta+\Theta=0$ then (4.35) takes the form

$0=d(Scal)_{j}-2{\nabla}_{i}Ric_{ij}+\frac{1}{6}{\nabla}_{j}||T||^{2}=d(\frac{n-2}{n}Scal+\frac{1}{6}||T||^{2})_{j}.$
(4.36)

Hence, (4.32) holds.

If $R\in S^{2}\Lambda^{2}$ then ${\nabla}T$ is a four form due to [26, Lemma
3.4]. Consequently, we have $\delta T=0$ and
${\nabla}_{s}T^{2}_{sj}=-\frac{1}{2}{\nabla}_{j}||T||^{2}$ because of (3.21).
Now, (4.35) takes the form

$0=d(\frac{n-2}{n}Scal+\frac{1}{2}||T||^{2}),$

which implies (4.33). The proof is completed. ∎

We remark that due to (3.19) the condition (4.31) in Theorem 4.2 is equivalent
to the condition

$3\theta+\Theta=0\Longleftrightarrow{\nabla}_{j}||T||^{2}=6{\nabla}_{i}T^{2}_{ij}.$

The second equality in (4.32) leads to the next

######  Corollary 4.3.

Let $(M,g,T)$ be a six dimensional Riemannian manifold and the 3-form $T$
satisfies (4.31), (in particular $T$ be a harmonic 3-form). If the metric
connection with torsion $T$ is ${\nabla}$-Einstein then the Riemannian scalar
curvature is constant.

######  Remark 4.4.

We remark that if the torsion is harmonic then the last equality in (4.36)
follows from [20, Proposition 3.47].

######  Remark 4.5.

We also remark that if $(M,g,T)$ is Ricci flat with closed torsion then the
equality (4.32) yields the norm of the torsion is constant wich recovers the
recent result [36, Lemma 2.21].

##  5 Generalized gradient Ricci solitons. Proof of Theorem 1.1

One fundamental consequence of Perelman’s energy formula for Ricci flow is
that compact steady solitons for Ricci flow are automatically gradient.
Adapting these energy functionals to generalized Ricci flow it is proved in
[20, Chapter 6] that steady generalized Ricci solitons on compact manifolds
are automatically gradient, and moreover satisfy k = 0.

We recal [20, Definition 4.31] that a Riemannian manifold $(M,g,T,f)$ with a
closed 3-form $T$ and a smooth function $f$ is a generalized gradient Ricci
soliton with $k=0$ if one has

$Ric^{g}_{ij}=\frac{1}{4}T^{2}_{ij}-{\nabla^{g}}_{i}{\nabla^{g}}_{j}f,\qquad\delta
T_{ij}=-df_{s}T_{sij},\qquad dT=0.$ (5.37)

Using the torsion connection ${\nabla}$ with 3-form torsion $T$, (2.3) and the
second equation in(5.37), we have

${\nabla}_{i}{\nabla}_{j}f-{\nabla}_{j}{\nabla}_{i}f=-df_{s}T_{sij}=\delta
T_{ij}.$ (5.38)

In view of (2.7) and (5.38) we write (5.37) in the form

$Ric_{ij}=-\frac{1}{2}({\nabla}_{i}{\nabla}_{j}f+{\nabla}_{j}{\nabla}_{i}f)-\frac{1}{2}\delta
T_{ij}=-{\nabla}_{i}{\nabla}_{j}f,\quad Scal=-{\nabla}_{i}{\nabla}_{i}f=\Delta
f.$ (5.39)

The second Bianchi identity (3.26) and (5.39) yield

${\nabla}_{j}\Delta f-2{\nabla}_{i}Ric_{ji}+\delta
T_{ab}T_{abj}+\frac{1}{6}{\nabla}_{j}||T||^{2}={\nabla}_{j}\Delta
f+2{\nabla}_{i}{\nabla}_{j}{\nabla}_{i}f+\delta
T_{ab}T_{abj}+\frac{1}{6}{\nabla}_{j}||T||^{2}=0$ (5.40)

We evaluate the second term of (5.40) in two ways. First using the Ricci
identities for ${\nabla}$ and (5.38)

${\nabla}_{i}{\nabla}_{j}{\nabla}_{i}f={\nabla}_{j}{\nabla}_{i}{\nabla}_{i}f-R_{ijis}{\nabla}_{s}f-T_{ija}{\nabla}_{a}{\nabla}_{i}f=-{\nabla}_{j}\Delta
f+Ris_{js}{\nabla}_{s}f-\frac{1}{2}({\nabla}_{a}{\nabla}_{i}f-{\nabla}_{i}{\nabla}_{a}f)T_{aij}\\\
=-{\nabla}_{j}\Delta
f-{\nabla}_{j}{\nabla}_{s}f.{\nabla}_{s}f+\frac{1}{2}df_{s}T_{sai}T_{aij}=-{\nabla}_{j}\Delta
f-{\nabla}_{j}{\nabla}_{s}f.{\nabla}_{s}f-\frac{1}{2}\delta T_{ai}T_{aij}$
(5.41)

Applying (5.38) and (3.18), we obtain

${\nabla}_{i}{\nabla}_{j}{\nabla}_{i}f={\nabla}_{i}({\nabla}_{i}{\nabla}_{j}f-\delta
T_{ij})={\nabla}_{i}{\nabla}_{i}{\nabla}_{j}f-{\nabla}_{i}\delta
T_{ij}={\nabla}_{i}{\nabla}_{i}{\nabla}_{j}f-\frac{1}{2}\delta T_{ia}T_{iaj}$
(5.42)

Substitute (5.41) and (5.42) into (5.40), we get

$\begin{split}-{\nabla}_{j}\Delta
f-{\nabla}_{j}||df||^{2}+\frac{1}{6}{\nabla}_{j}||T||^{2}=-{\nabla}_{j}\Delta
f+2Ric_{js}{\nabla}_{s}f+\frac{1}{6}{\nabla}_{j}||T||^{2}=0;\\\
{\nabla}_{j}\Delta
f+2{\nabla}_{i}{\nabla}_{i}{\nabla}_{j}f+\frac{1}{6}{\nabla}_{j}||T||^{2}={\nabla}_{j}\Delta
f-2{\nabla}_{i}Ric_{ij}+\frac{1}{6}{\nabla}_{j}||T||^{2}=0.\end{split}$ (5.43)

Note that the first equality in (5.43) is precisely [20, Proposition 4.33].

Differentiate the first line in (5.43) apply (5.39) and the second equality in
(5.43) to get

$0=\Delta(\Delta
f-\frac{1}{6}||T||^{2})+2{\nabla}_{j}Ric_{js}{\nabla}_{s}f+2Ric_{js}{\nabla}_{j}{\nabla}_{s}f=\Delta(\Delta
f-\frac{1}{6}||T||^{2})+2{\nabla}_{j}Ric_{js}{\nabla}_{s}f-2||Ric||^{2}\\\
=\Delta(\Delta f-\frac{1}{6}||T||^{2})+{\nabla}_{j}(\Delta
f+\frac{1}{6}||T||^{2}){\nabla}_{j}f-2||Ric||^{2}\\\ \leq\Delta(\Delta
f-\frac{1}{6}||T||^{2})+g({\nabla}(\Delta f+\frac{1}{6}||T||^{2}),{\nabla}f)$
(5.44)

If the norm of the torsion $T$ is constant, ${\nabla}||T||^{2}=0$, then (5.44)
takes the form

$\Delta\Delta f+g({\nabla}\Delta f,{\nabla}f)\geq 0.$

If $M$ is compact, $\Delta f$ is constant due to the strong maximum principle
(see e.g. [46, 20]) which yields $f=const$. Conversely, if the function $f$ is
constant then (5.44) together with the strong maximum principle implies
$d||T||^{2}=0$ which yields the equivalence of a) and b).

Assume a) or b). Then $Ric=Scal=\delta T=0$ due to (5.38) and (5.39). If
$Ric=0$ then $\Delta f=0$ leading to $f=const$ since $M$ is compact. Hence, b)
is equivalent to c).

To show d) is equivalent to a) we use (2.7) and (5.39) to find
$Scal^{g}=\Delta f+\frac{1}{4}||T||^{2}$ and we can write (5.44) in the form

$0\leq\Delta(Scal^{g}-\frac{5}{12}||T||^{2})+g({\nabla}(Scal^{g}-\frac{1}{12}||T||^{2}),{\nabla}f).$
(5.45)

Then $Scal^{g}=const.$ if and only $d||T||^{2}=0$ by the strong maximum
principle applied to (5.45).

The proof of Theorem 1.1 is completed.

Acknowledgements We would like to thank Jeffrey Streets and Ilka Agricola for
extremly useful remarks, comments and suggestions.

The research of S.I. is partially financed by the European Union-Next
Generation EU, through the National Recovery and Resilience Plan of the
Republic of Bulgaria, project N: BG-RRP-2.004-0008-C01. The research of N.S.
is partially supported by Contract 80-10-192 / 17.5.2023 with the Sofia
University ”St.Kl.Ohridski” and the National Science Fund of Bulgaria,
National Scientific Program “VIHREN”, Project KP-06-DV-7.

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# Effects of Social Ties in Knowledge Diffusion:
case study on PLOS ONE

Felipe Eltermann1,2, Alan Godoy1,3, Fernando J. Von Zuben1

###### Abstract

In order to capture the effects of social ties in knowledge diffusion, this
paper examines the publication network that emerges from the collaboration of
researchers, using citation information as means to estimate knowledge flow.
For this purpose, we analyzed the papers published in the PLOS ONE journal
finding strong evidence to support that the closer two authors are in the co-
authorship network, the larger the probability that knowledge flow will occur
between them. Moreover, we also found that when it comes to knowledge
diffusion, strong co-authorship proximity is more determinant than geographic
proximity.

Keywords: knowledge diffusion, co-authorship network, multiplex network

##  1 Introduction

It is of primary interest of organizations and society in general to promote
knowledge diffusion. This interest is often twofold: first, knowledge
acquisition fosters best problem solving techniques and allows better decision
making; on the other hand, a knowledge producer may want to position itself as
source (or holder) of a piece of knowledge. The literature highlights two
knowledge-intensive activities: academic research111Public research,
scientific research and academic research are commonly interchangeable and in
this paper, except when explicitly put otherwise, represents research
conducted in universities and governmental laboratories. [Liberman and Wolf
1997, Barabási et al. 2002, Newman 2004] and industrial R&D [Jaffe et al.
1993, Singh 2005] and their interactions [Trajtenberg et al. 1992, Cohen et
al. 2002, Sorenson and Singh 2006].

Based on their capability to represent flow, exchange and communication,
networks are used in literature as a functional framework for the study of
organizational arrangements between economic actors [Powell 1990]. Considering
its production-consumption structure, the study of knowledge diffusion
commonly uses network-based models as a foundation [Barabási et al. 2002,
Newman 2004, Singh 2005].

Published documents (scientific papers and patents) are considered to be a
recorded track that reflects the actual occurrence of knowledge flow between
its producers and consumers [Jaffe et al. 1993]. Specifically, it has been
considered that knowledge flow occurs when authors cite previous works of
others [Jaffe et al. 1993, Singh 2005, Sorenson and Singh 2006] or when they
collaborate as joint co-authors [Sidone et al. 2016] of the same document.

In this paper, we explore the methodology used by [Singh 2005] to study how
collaboration ties in a patent network predict the occurrence of knowledge
flow between teams, applying it to a network of scientific papers. A multiplex
network is considered, in which papers are represented by nodes and two layers
are present: the citation layer – in which oriented edges represent the
citations between papers – and the co-authorship layer in which edges
represent the co-authorship relationships between papers.

We observed that social proximity is a strong determinant for the occurrence
of knowledge diffusion in scientific research.

This paper is organized as follows: Section 2 is devoted to describing the
preprocessing stages necessary to achieve to the ready-to-use dataset,
followed by a brief explanation of how to built the multiplex network directly
from the analysis of the dataset. Section 3 deals with the methodology adopted
to extract knowledge from the multiplex network. The main results are outlined
at Section 4, followed by a discussion in Section 5. Concluding remarks and
further steps of the research are presented in Section 6.

##  2 Methods

The implementation of the methods described below is fully available
online222https://bitbucket.org/eltermann/plosone-analysis.

###  2.1 Citation Network and Paper-centric Co-authorship Network

The methodology to relate social ties with knowledge flow was strongly based
on [Singh 2005], which argues that knowledge flow occurs when one team cites
work authored by another. This definition provides a reasonable intuition for
detecting the actual occurrence of knowledge flow that takes place within the
scientific community. The original definition requires that, to consider a
citation as corresponding to a flow of knowledge, the authors of the citing
paper must be all different from the authors of the cited work. We also tested
a relaxed version of this definition, considering that knowledge flow occurs
provided that the citing team has at least one different author from the cited
team. Our preliminary analysis showed that the difference considering both
definitions was not significant, so we only present results based on the
original definition.

While it is possible to map the occurrence of knowledge flow in a structure of
teams or authors, greater granularity is achieved when allowing the concept of
flow to be applied between two individual papers. Knowledge actually flows
between people, but we consider that the concrete output of a particular
occurrence of flow is embedded in the citation between two papers (provided
the constraint mentioned in the previous paragraph).

Therefore, differently from the usual _author-centric_ co-authorship network
analysis [Barabási et al. 2002, Newman 2004], it is also possible to represent
the co-authorship relationship as follows: papers are nodes and there is an
edge between any given pair of papers if, and only if, there is at least one
person that is author of both papers. This _paper-centric_ definition reflects
the social ties in the sense that the network evolves according to the pattern
of collaboration between authors.

Figure 1: Let A, B, C, D and E be scientific papers. Also, let
$a_{A}=\\{1,2\\}$, $a_{B}=\\{3\\}$, $a_{C}=\\{5\\}$, $a_{D}=\\{3,4,5\\}$, and
$a_{E}=\\{1,4\\}$ be the sets of authors of each paper. Citations are drawn as
follows: A is cited by B, C and D; B is cited by D; and D is cited by E.
Finally, A, B, C and D are published in year n and E is published in year n+1.
Given this setting, the top half of the figure presents the citation layer of
our paper-centric network: papers are represented as nodes and there is an
oriented edge from one paper to another if, and only if, the former cites the
latter. The bottom half presents the co-authorship layer of the network, in
which there is an edge between two papers if, and only if, they share at least
one common author. The social distance between two papers is defined as the
number of nodes in the shortest path between them in the co-authorship layer.
Given $d_{n}(X,Y)$ as the social distance between papers X and Y in year n, we
pick the following examples: $d_{n}(B,D)=0$, $d_{n+1}(B,E)=1$,
$d_{n}(A,B)=\infty$, and $d_{n+1}(A,B)=2$. Now, the occurrence of knowledge
flow between two papers $y(X,Y)$ is defined as a Boolean variable that is set
to 1 if, and only if, one paper cites the other and there is no common author
between them. Considering this definition, we have: $y(B,D)=0$, $y(B,E)=0$,
and $y(A,B)=1$.

From the two-layered multiplex network represented in Figure 1, it is possible
to observe that the social distance between two papers can assume an infinite
value (when the papers are in different connected components in the co-
authorship layer). It is also possible to observe that the distance between
two papers may vary with time, directly affecting the design of the
experiment: each pair of papers must be considered in its proper period (this
is detailed next).

Algorithm 1 describes how to build the paper-centric co-authorship layer of
the network and to collect data of social distance and knowledge flow for
later analysis. It is important to keep in mind that only the paper-centric
co-authorship layer is indeed built, while information from the citation layer
is implicitly used to compute the occurrence of knowledge flow.

In order to ensure that the social distances are computed correctly, the co-
authorship layer expands gradually in a discrete year-by-year fashion. For
each year, the graph is incremented with papers published that year and for
each of those papers, pairs are formed by visiting all other nodes in the
graph. By doing so, we only compute those social distances that are related to
at least one paper that was published in the most recent year available in the
graph. To shorten our notation, we define $d(X,Y)$ (without the subscript) as
the aforementioned contextualized distance between papers X and Y. Even though
it would be more precise to consider the passage of time in a more granular
fashion, the use of years is imposed by the data available. This limitation is
argued to be negligible because of the nature of the application: while recent
papers are more likely to receive new citations [Redner 2005], both research
and publishing take time, imposing a delay of at least a few months between
the publishing of a paper and the first appearance of its citation.

Algorithm 1 Extracting social distances and occurrence of knowledge flow.

1:$G\leftarrow empty\\_graph$

2:$d\leftarrow\\{\\}$

3:$y\leftarrow\\{\\}$

4:for each available year $n$ taken in order do

5: $G.vertices\leftarrow G.vertices\cup\\{paper\ with\ year==n\\}$

6: for each paper $A$ with $year=n$ do:

7: for each paper $B$ from $G.vertices$ with $B\neq A$ do:

8: if $A$ and $B$ share at least one common author then

9: $G.edges\leftarrow G.edges\cup\\{(A,B)\\}$

10: end if

11: end for

12: end for

13: for each paper $A$ with $year=n$ do:

14: for each paper $B$ from $G.vertices$ with $B\neq A$ do:

15: $d(A,B)\leftarrow num\\_intermediate\\_nodes\\_in\\_shortest\\_path(A,B)$

16: if $A$ cites $B$ and there is no person being author of both $A$ and $B$
then

17: $y(A,B)\leftarrow 1$

18: else

19: $y(A,B)\leftarrow 0$

20: end if

21: end for

22: end for

23:end for

###  2.2 Data collection and pre-processing

We decided to use data from PLOS ONE333http://journals.plos.org/plosone for
the following reasons: first, PLOS ONE content is published under the
permissive Creative Commons license which allows its usage for any
purpose444PLOS ONE also stands up for Open Access Research, a publishing
format that does not charge users for having access to the full outputs of
published research. For more information, see https://www.plos.org/open-
access.; second, its data is made available in XML files making it easy to be
gathered and processed; third, it comprises publications from different fields
providing a broad (yet limited) view on patterns of scientific publication;
and finally, its large corpus of more than 150,000 papers allows the usage of
tools that exploit statistical characteristics of data.

A preprocessing step is performed to transform the original XML data into a
database, in which it is included: all PLOS ONE papers (and their authors),
each of the cited papers (and their authors) – which included several non-PLOS
ONE papers –, and each citation made by PLOS ONE papers (information on
citations made by non-PLOS ONE papers were not available). Data was collected
on February 2016.

###  2.3 Entities Disambiguation

Precise connections are essential in our analysis: we want to be able to
properly assign authors to papers and citations between papers. Unfortunately,
no identifier is available for authors and the DOI555The digital object
identifier (DOI) is a code that uniquely identifies electronic documents, as
scientific papers. was present for only a fraction of 15% of all papers. This
led to the necessity of dealing with pure text information to establish an
approximate canonical representation of those entities.

When DOI is not available, matching of papers is solely based on the title
information. Since it is rare that two different papers have the same title,
our effort was to deal with a single paper title being spelled in slightly
different ways. The method used is described as follows: non-alphabetical
characters were removed, all letters were changed to uppercase and exceeding
blank spaces were removed. We manually checked a random sample of one hundred
papers matched by title and the result was satisfactory (at least 95% of the
matches occurred as expected).

Author names were already provided as separate parts: surname and given names.
In one hand, since it is common that two different authors have similar names,
it is possible to assign two different authors names to the same entity
(considering the task of name matching as a classification, this case would be
a false positive). On the other hand, depending on the abbreviation used, two
different representations of one author’s name may lead to duplicate entries
in the database (that being a false negative). Our approach here was the one
adopted by [Barabási et al. 2002]: surnames were changed to uppercase, and
given names were represented as initials (all characters but capital letters
were removed). The parts were concatenated into what we call canonical author
name. We manually checked a random sample of one hundred authors matched by
name and the strategy was satisfactory (at least 85% of the matches occurred
as expected).

Also, we are aware of more robust disambiguation methods that could be used.
Nevertheless, we believe that the chosen method is simple enough to be easily
replicated and that its false negatives introduce only a negligible noise on
the effect of social ties to be analyzed.

##  3 Extraction and Analysis Methodology

The following aspects of the methodology are also implemented in the
aforementioned source code we made available.

###  3.1 Geographic Localization of Knowledge Flow

Taking the geographic dimension into consideration when studying knowledge
diffusion is relevant to broaden the understanding of its patterns and to
provide richer information for strategic decisions of public and private
institutions.

A widely accepted pattern is that knowledge flow is geographically localized:
it occurs mostly between teams geographically close to each other [Singh
2005]. However, considering our definition of co-authorship distance (or
social distance) between papers, it is reasonable to state that geographically
closer papers tend to be socially closer too (for instance, in Brazil, the
increase of 100 km between two researchers reduces the probability of
collaboration by 16% on average [Sidone et al. 2016]). Therefore, there is
space to explore the joint effects of social and geographic distances in the
occurrence of knowledge flow.

Fortunately, geographic information is available in our dataset: information
of country and region666Considering the hierarchy in an address line, the
region is the highest entity after the country. In our dataset, it is commonly
a state or a city. is available in 99% of PLOS ONE papers and 83% of the
others. To be precise, the country and region of each paper were inferred from
the countries and regions of its authors. In order to decide which country and
region to assign for each paper, we tested two approaches: taking the country
and region of the first author and taking the majority of the countries and
regions between the authors. The experiment revealed insignificant difference
between these approaches and therefore the former one was chosen for further
discussions.

When comparing a pair of papers with geographic assignments, we can then
define two Boolean variables: $is\\_same\\_country$, which is 1 if, and only
if, the two papers are from the same country; and $is\\_same\\_region$, which
is 1 if, and only if, the two papers are from the same region. Even though
these variables do not accurately represent the distance between the two
addresses, it provides a good approximation.

###  3.2 Variables and Representation

Considering a pair of papers $(X,Y)$, we have so far defined three categories
of variables: occurrence of knowledge flow $y(X,Y)$, co-authorship distance
$d(X,Y)$, and geographic assignments $is\\_same\\_country(X,Y)$ and
$is\\_same\\_region(X,Y)$. Taking into consideration that the co-authorship
distances, when finite, increase much slower than the number of papers in the
network [Watts and Strogatz 1998], the distance variable was represented as a
one-hot vector, in which position $i$ is 1 only if the distance between the
pair of papers is equal to $i$ and all other positions are equal to 0.

In order to study the impact of each of these variables in the probability of
knowledge flow, we trained Logistic Regression [Bishop 2006] models and
analyzed their coefficients. The higher a coefficient, the most relevant the
variable is considered in determining the occurrence of knowledge flow.

###  3.3 Sampling Strategy and Weighted Regression Model

It is possible to note that the number of possible pairs of papers increases
quadratically as more papers are added to the network. In fact, the
straightforward approach of using all data to train regression is
unnecessarily expensive, since a large portion of this data corresponds to
pairs for which there is no occurrence of knowledge flow. This issue is
addressed next.

In a large dataset, the occurrence of knowledge flow is a rare event, i.e. the
probability of randomly choosing a pair of papers in which knowledge flow is
present is very low. In other words, it was observed that the occurrence of
knowledge flow ($y(X,Y)=1$) is a highly infrequent event in comparison with
the volume of all pairs of papers. Therefore, an efficient method is to use
all registers with $y(X,Y)=1$ and just a small sample taken from the
$y(X,Y)=0$ cases:

$num\\_samples_{y(X,Y)=1}=\alpha*num\\_records_{y(X,Y)=1}$
$num\\_samples_{y(X,Y)=0}=\beta*num\\_records_{y(X,Y)=0},$

in which $num\\_samples$ is the number of samples used to train the
regression, $num\\_records$ is the total number of samples in our database and
$\alpha$ and $\beta$ are coefficients indicating the proportion of the
database to be sampled.

Aiming at avoiding the introduction of bias because of this endogenous
stratified sampling, we associated each sample with a weight inversely
proportional to the probability of it being sampled:

$weight_{y(X,Y)=1}=1/\alpha$ $weight_{y(X,Y)=0}=1/\beta$

In other words, the unbalance between $num\\_records_{y(X,Y)=1}$ and
$num\\_records_{y(X,Y)=0}$ is equipoised by the weighted regression. In our
approach, $\alpha$ is 1 and $\beta$ is taken empirically: we take $\beta$ so
that $num\\_samples_{y(X,Y)=1}~{}num\\_samples_{y(X,Y)=0}$ and, thus,
$\beta~{}number\\_of\\_citations/number\\_of\\_papers^{2}$.

##  4 Results and Discussion

On average, each of the 152,406 PLOS ONE papers was authored by 6.8 people and
cited 43.3 papers. Given that 92% of these papers are from the major subject
Biology and Life Sciences, the high number of authors per paper is expected
[Newman 2004]. The database resulting from the preliminary step consists of
3,373,968 papers (including papers that are cited by PLOS ONE papers),
2,435,359 authors and 6,294,099 citations.

Collected PLOS ONE papers range from 2006 to 2015. Given that computing the
shortest paths in a large network is a very time-consuming task, we only
managed to build the graph up to 2009, encompassing 8,340 PLOS ONE papers (we
used the igraph777http://igraph.org library and it took 7 days of computation
on a DELL PowerEdge R430 server with Intel® Xeon® E5-2609 processor and 96GB
of RAM). Even though this corresponds to just 5% of the total PLOS ONE papers,
we were able to observe some interesting phenomena. The resulting
collaboration network consisted of 2,353,314 nodes (papers) and 313,507,251
edges.

(a) All pairs

(b) Same country

(c) Different countries

(d) Same region

(e) Different regions

Figure 2: Histograms for 5 cohorts: (a) all sampled pairs, (b) pairs of the
same country, (c) pairs of different countries, (d) pairs of the same region
and (e) pairs of different regions. For each plot, the horizontal axis
presents each variable of the one-hot distance vector $d(X,Y)$. The reason we
do not consider the case where $d(X,Y)=0$ (both here and in the regression) is
that by construction, theses cases always imply $y(X,Y)=0$. The largest
observed finite social distance between two papers was 9, which reflects the
small-world effect. It is also possible to note the effect of absence of
geographic assignment in many papers (for instance, the sum of pairs of the
same country (b) with pairs of different countries (c) is lower than the
unrestricted total (a)). Figure 3: Each set of data in Figure 2 was used to
fit the regression: the x-axis represents the binary variables of the one-hot
input vectors and the y-axis corresponds to the coefficients of the regression
in each case. It is possible to observe a clear decrease in relevance to
predict knowledge flow as the distance increases (specially until a distance
of 4 is reached). Figure 4: Data of cohorts (b) and (c) of Figure 2 were used
to generate the aforementioned Boolean variable $is\\_same\\_country$ and
jointly fit the regression. The resulting coefficients are displayed with
cross marks and the coefficient for variable $is\\_same\\_country$ is
presented as the dashed horizontal rule. The same process was performed to
data of cohorts (d) and (e) of Figure 2 and the coefficients are presented
with bullet marks. The coefficient related to variable $is\\_same\\_region$ is
presented as the dotted horizontal rule.

The coefficients obtained for the logistic regression are displayed in Figure
3. They evidence that low social distances lead to a high frequency of
knowledge flow, confirming the initial expectations. Despite it is possible to
note a clear decrease in the effect of distance, the tendency smooths at some
point. Interestingly, this point (around 3 and 4) is consonant with
[Christakis and Fowler 2013], which states that the effects of social ties
(e.g. in diffusion of behaviors) fades until a distance of approximately 3
(beyond which it becomes inconspicuous). It is important to keep in mind that
this comparison was drawn between our results from a network of produced
papers and others directly from social networks. A systematic exploration of
the effects of an author-centric network in the corresponding document-centric
network (one can name it: scientific papers, patents, software projects) would
be useful to support this kind of comparison and to deepen the understanding
of social collaboration.

We also fit a second set of logistic regressions, but using the binary
variables $is\\_same\\_country$ and $is\\_same\\_region$ in addition to the
variables indicating social distance, in order to compare the effects of
geographic and social distances. Figure 4 shows the coefficients of the
regressions. It is possible to see that the coefficients related to the social
distance are extremely close to those observed in the previous regressions.
Also, it is noticeable that coefficients related to the geographic variables
are less relevant than those related to short social distances, which
indicates that being in the same geographic region matter less than being
socially close to other teams for the occurrence of knowledge flow.

##  5 Concluding Remarks

Comparing our results to those presented by [Singh 2005] related to patent
data, geographic location is markedly less relevant in the present work. Such
finding may indicate either that academic research is globally more integrated
than industrial R&D or that citation practices are significantly different
between both communities.

Though restricted to the context of scientific production, it is possible to
generalize the main consequences of the obtained results to other types of
organizations. The results are suggesting that the promotion of information
flow is not only influenced by the geographical proximity of key players, but
also by the density of the social interactions established. Information
technology is certainly a key instrument to mitigate the effects of
geographical distance, together with initiatives to establish offices and
representatives in strategic locations. Additionally, any organization devoted
to maximizing the information flow among its peers should strongly encourage
the creation of solid professional links involving external actors. This is
particularly interesting for developing countries, like Brazil, as it
indicates that strategic foreign social connections are a valuable path for
economic and scientific growth as a result of the tapping of knowledge from
abroad.

###  5.1 Future work

A clear point for improvement in this experiment would be to enrich the
database. Since the information of citations was not available in any of the
non-PLOS ONE papers, there is space to improve the approximation of the
occurrence of knowledge flow. In that sense, co-authorship also represents
only a fraction of the actual social ties between authors, and other sources
of interactions could be incorporated.

Another useful analysis could be drawn considering the subject of papers,
which would allow the comparison between general fields.

Finally, in order to speed-up the time for computing the distances in the co-
authorship network, we plan to incorporate the usage of a GPU-based
implementation to leverage the possibility of calculating the shortest paths
in parallel.

##  6 Acknowledgements

We thank Elabora Consultoria for providing the computational structure to
collect the data and run the experiments. We also thank Leonardo Maia for the
helpful suggestions on working with data provided by PLOS ONE.

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