# On $c$-embedded subgroups of finite groups

Julian Kaspczyk111 Institut für Algebra, Technische Universität Dresden, 01069
Dresden, Germany,
Email address<EMAIL_ADDRESS>
###### Abstract

Let $G$ be a group and $H\leqslant K\leqslant G$. We say that $H$ is
$c$-embedded in $G$ with respect to $K$ if there is a subgroup $B$ of $G$ such
that $G=HB$ and $H\cap B\leqslant Z(K)$. Given a finite group $G$, a prime
number $p$ and a Sylow $p$-subgroup $P$ of $G$, we investigate the structure
of $G$ under the assumption that $N_{G}(P)$ is $p$-supersolvable or
$p$-nilpotent and that certain cyclic subgroups of $P$ with order $p$ or $4$
are $c$-embedded in $G$ with respect to $P$. New characterizations of
$p$-supersolvability and $p$-nilpotence of finite groups will be obtained.

Keywords: finite groups; $p$-supersolvable; $p$-nilpotent; $c$-embedded.
MSC (2010): 20D10, 20D20

##  1 Introduction

All groups in this paper are implicitly assumed to be finite. We use standard
notation and terminology, see for example [10] or [11]. Throughout, $p$
denotes an arbitrary but fixed prime.

Recall that a group $G$ is said to be $p$-nilpotent if $G$ has a normal Hall
$p^{\prime}$-subgroup. This concept plays an important role in local finite
group theory, and many criteria for $p$-nilpotence of finite groups can be
found in the literature. For example, a well-known theorem of Burnside asserts
that a group $G$ with Sylow $p$-subgroup $P$ is $p$-nilpotent if $P\leqslant
Z(N_{G}(P))$, or equivalently if $P$ is abelian and $N_{G}(P)$ is
$p$-nilpotent (see [11, Theorem 5.13]). This result has been generalized in
many directions, and we shall now consider some of these generalizations.

First let us introduce some notation. Let $P$ be a $p$-group and $i$ be a
positive integer. Then the subgroup of $P$ generated by all elements $x$ of
$P$ with $x^{p^{i}}=1$ is denoted by $\Omega_{i}(P)$. We set
$\Omega(P):=\Omega_{1}(P)$ if $p$ is odd and $\Omega(P):=\Omega_{2}(P)$ if
$p=2$.

In 1974, Laffey [12] published the following generalization of Burnside’s
$p$-nilpotency criterion: If $G$ is a group and $P$ is a Sylow $p$-subgroup of
$G$, then $G$ is $p$-nilpotent if $\Omega(P)\leqslant Z(P)$ and $N_{G}(P)$ is
$p$-nilpotent (see [12, p. 136]).

In 2000, Ballester-Bolinches and Guo [3] published the following more general
result: If $G$ is a group and $P$ is a Sylow $p$-subgroup of $G$, then $G$ is
$p$-nilpotent if $\Omega(P\cap G^{\prime})\leqslant Z(N_{G}(P))$ (see [3,
Theorem 1]). Moreover, they proved that a group $G$ with Sylow $2$-subgroup
$P$ is $2$-nilpotent if $\Omega_{1}(P\cap G^{\prime})\leqslant Z(P)$, $P$ is
quaternion-free and $N_{G}(P)$ is $2$-nilpotent (see [3, Theorem 2]). Here, a
group is said to be quaternion-free if it has no section isomorphic to the
quaternion group of order $8$.

For a non-empty formation $\mathfrak{F}$ and a group $G$, we use
$G^{\mathfrak{F}}$ to denote the $\mathfrak{F}$-residual of $G$, i.e.
$G^{\mathfrak{F}}$ is the smallest normal subgroup of $G$ whose quotient lies
in $\mathfrak{F}$. As usual, $\mathfrak{N}$ denotes the formation of all
nilpotent groups.

In 2004, Asaad [1] published the following result, which extends the above-
mentioned results of Ballester-Bolinches and Guo.

######  Theorem 1.

([1, Theorem 1 (a) $\Leftrightarrow$ (b)]) Let $P$ be a Sylow $p$-subgroup of
a group $G$. If $p=2$, assume that $P$ is quaternion-free. Then the following
statements are equivalent:

  1. (1)

$G$ is $p$-nilpotent.

  2. (2)

$N_{G}(P)$ is $p$-nilpotent and $\Omega_{1}(G^{\mathfrak{N}}\cap P\cap
P^{x})\leqslant Z(P)$ for all $x\in G\setminus N_{G}(P)$.

Recall that a group $G$ is said to be $p$-solvable if any chief factor of $G$
is either a $p$-group or a $p^{\prime}$-group. A $p$-solvable group $G$ is
called $p$-supersolvable if every $p$-chief factor of $G$ has order $p$. The
formation of all $p$-supersolvable groups is denoted by $\mathfrak{U}_{p}$.
Note that every $p$-nilpotent group is $p$-supersolvable. In particular, we
have $G^{\mathfrak{U}_{p}}\leqslant G^{\mathfrak{N}_{p}}$ for any group $G$.

By [13, Theorem 2.2], if $P$ is a Sylow $p$-subgroup of a group $G$, then we
have $P\cap G^{\mathfrak{N}}=P\cap G^{\mathfrak{N}_{p}}$, where
$\mathfrak{N}_{p}$ denotes the formation of all $p$-nilpotent groups.
Therefore, the nilpotent residual $G^{\mathfrak{N}}$ can be replaced by the
$p$-nilpotent residual $G^{\mathfrak{N}_{p}}$ in Theorem 1. In view of this
observation, it is natural to ask whether Theorem 1 still remains true when
the nilpotent residual $G^{\mathfrak{N}}$ is replaced by the $p$-supersolvable
residual $G^{\mathfrak{U}_{p}}$. We will show that $G^{\mathfrak{N}}$ cannot
only be replaced by $G^{\mathfrak{U}_{p}}$, but that it is enough to require
in (2) that the normalizer $N_{G}(P)$ is $p$-nilpotent and that every minimal
subgroup of $G^{\mathfrak{U}_{p}}\cap P\cap P^{x}$ is central in $P$ or
complemented in $G$ for all $x\in G\setminus N_{G}(P)$. We will also show that
Theorem 1 remains true if we additionally replace ”$p$-nilpotent” by
”$p$-supersolvable” in both statements of the theorem and furthermore assume
$G$ to be $p$-solvable.

In fact, our results are slightly more general than just stated, and they also
deal with the case that $P\in\mathrm{Syl}_{p}(G)$ is not quaternion-free when
$p=2$. In order to state our results in full generality, we introduce the
following definition.

######  Definition 2.

Let $G$ be a group and $H\leqslant K\leqslant G$. Then $H$ is said to be
$c$-embedded in $G$ with respect to $K$ if there is a subgroup $B$ of $G$ such
that $G=HB$ and $H\cap B\leqslant Z(K)$.

Let $G$ be a group and $H\leqslant K\leqslant G$ such that $H$ is $c$-embedded
in $G$ with respect to $K$, so that there exists a subgroup $B$ of $G$ with
$G=HB$ and $H\cap B\leqslant Z(K)$. If $H\cap B=1$ then $H$ is complemented in
$G$, and if $H\cap B=H$ then $H$ is central in $K$. If $1<H\cap B<H$, then $H$
can be described as being between complemented in $G$ and central in $K$. Note
that if $H$ is a minimal subgroup of $G$, then $H$ is $c$-embedded in $G$ with
respect to $K$ if and only if $H$ is complemented in $G$ or central in $K$.

Having introduced the concept of $c$-embedded subgroups, we can now state our
main results.

######  Theorem A.

Let $P$ be a Sylow $p$-subgroup of a $p$-solvable group $G$. Then $G$ is
$p$-supersolvable if and only if the following two conditions are satisfied:

  1. (1)

$N_{G}(P)$ is $p$-supersolvable.

  2. (2)

For all $x\in G\setminus N_{G}(P)$, the following hold: Any subgroup of
$G^{\mathfrak{U}_{p}}\cap P\cap P^{x}$ with order $p$ is $c$-embedded in $G$
with respect to $P$. Moreover, if $p=2$ and $G^{\mathfrak{U}_{p}}\cap P\cap
P^{x}$ is not quaternion-free, any cyclic subgroup of
$G^{\mathfrak{U}_{p}}\cap P\cap P^{x}$ with order $4$ is $c$-embedded in $G$
with respect to $P$.

The $p$-solvability condition on $G$ cannot be dropped in Theorem A. For
example, the conditions (1) and (2) from Theorem A are satisfied for $G=A_{5}$
and $p=5$ since a Sylow $5$-subgroup of $A_{5}$ is cyclic of order $5$ and has
a normalizer of order $10$, but $A_{5}$ is not $5$-supersolvable.

######  Theorem B.

Let $P$ be a Sylow $p$-subgroup of a group $G$. Then $G$ is $p$-nilpotent if
and only if the following two conditions are satisfied:

  1. (1)

$N_{G}(P)$ is $p$-nilpotent.

  2. (2)

For all $x\in G\setminus N_{G}(P)$, the following hold: Any subgroup of
$G^{\mathfrak{U}_{p}}\cap P\cap P^{x}$ with order $p$ is $c$-embedded in $G$
with respect to $P$. Moreover, if $p=2$ and $G^{\mathfrak{U}_{p}}\cap P\cap
P^{x}$ is not quaternion-free, any cyclic subgroup of
$G^{\mathfrak{U}_{p}}\cap P\cap P^{x}$ with order $4$ is $c$-embedded in $G$
with respect to $P$.

##  2 Preliminaries

In this section, we collect some lemmas needed for the proofs of our main
results. The following well-known result can be deduced from [2, Theorem 1 and
Proposition 1].

######  Lemma 3.

Let $G$ be a $p$-solvable minimal non-$p$-supersolvable group. Then the
following hold:

  1. (1)

$G^{\mathfrak{U}_{p}}$ is a $p$-group, $G^{\mathfrak{U}_{p}}$ has exponent $p$
if $p$ is odd and exponent at most $4$ if $p=2$.

  2. (2)

$G^{\mathfrak{U}_{p}}/\Phi(G^{\mathfrak{U}_{p}})$ is a chief factor of $G$.

######  Lemma 4.

Let $G$ be a $p$-solvable minimal non-$p$-supersolvable group, $H$ be a proper
subgroup of $G^{\mathfrak{U}_{p}}$ and $B$ be a subgroup of $G$ such that
$G=HB$. Then $B=G$.

[Proof]Assume for the sake of contradiction that $B$ is a proper subgroup of
$G$. Let $M$ be a maximal subgroup of $G$ such that $B\leqslant M$. Then
$G^{\mathfrak{U}_{p}}\not\leqslant M$ because otherwise $G=HB\leqslant M$.
Thus $G=MG^{\mathfrak{U}_{p}}$.

We have $\Phi(G^{\mathfrak{U}_{p}})\leqslant
G^{\mathfrak{U}_{p}}\cap\Phi(G)\leqslant G^{\mathfrak{U}_{p}}\cap M$. By Lemma
3(1), $G^{\mathfrak{U}_{p}}$ is a $p$-group. Since
$G^{\mathfrak{U}_{p}}/\Phi(G^{\mathfrak{U}_{p}})$ is elementary abelian,
$(G^{\mathfrak{U}_{p}}\cap M)/\Phi(G^{\mathfrak{U}_{p}})$ is normal in
$G^{\mathfrak{U}_{p}}/\Phi(G^{\mathfrak{U}_{p}})$. So
$G^{\mathfrak{U}_{p}}\cap M$ is normal in $G^{\mathfrak{U}_{p}}$. Since
$G^{\mathfrak{U}_{p}}\cap M$ is also normal in $M$ and since
$G=MG^{\mathfrak{U}_{p}}$, it follows that $G^{\mathfrak{U}_{p}}\cap
M\trianglelefteq G$.

As $G^{\mathfrak{U}_{p}}/\Phi(G^{\mathfrak{U}_{p}})$ is a chief factor of $G$
by Lemma 3(2), it follows that either $G^{\mathfrak{U}_{p}}\cap
M=\Phi(G^{\mathfrak{U}_{p}})$ or $G^{\mathfrak{U}_{p}}\cap
M=G^{\mathfrak{U}_{p}}$. In the latter case, we have $H\leqslant M$ and so
$G=HB\leqslant M$, a contradiction. Thus $G^{\mathfrak{U}_{p}}\cap
M=\Phi(G^{\mathfrak{U}_{p}})$.

Now we have $G^{\mathfrak{U}_{p}}=G^{\mathfrak{U}_{p}}\cap
HB=H(G^{\mathfrak{U}_{p}}\cap B)\leqslant H(G^{\mathfrak{U}_{p}}\cap
M)=H\Phi(G^{\mathfrak{U}_{p}})$. Thus
$G^{\mathfrak{U}_{p}}=H\Phi(G^{\mathfrak{U}_{p}})$ and so
$G^{\mathfrak{U}_{p}}=H$. This is a contradiction since $H$ is assumed to be a
proper subgroup of $G^{\mathfrak{U}_{p}}$. So we have $B=G$.

######  Lemma 5.

Let $G$ be a group and $L$ be a subgroup of $G$. Then
$L^{\mathfrak{U}_{p}}\leqslant G^{\mathfrak{U}_{p}}$.

[Proof]Set $L_{0}:=L\cap G^{\mathfrak{U}_{p}}$. Then $L/L_{0}\cong
LG^{\mathfrak{U}_{p}}/G^{\mathfrak{U}_{p}}\leqslant G/G^{\mathfrak{U}_{p}}$.
As subgroups of $p$-supersolvable groups are $p$-supersolvable, it follows
that $L/L_{0}$ is $p$-supersolvable. Thus $L^{\mathfrak{U}_{p}}\leqslant
L_{0}\leqslant G^{\mathfrak{U}_{p}}$.

######  Lemma 6.

Let $G$ be a $p$-supersolvable group and $P$ be a Sylow $p$-subgroup of $G$.
Suppose that $N_{G}(P)$ is $p$-nilpotent. Then $G$ is $p$-nilpotent.

[Proof]Set $\overline{G}:=G/O_{p^{\prime}}(G)$. Then $\overline{G}$ is
$p$-supersolvable and $O_{p^{\prime}}(\overline{G})=1$. So $\overline{G}$ is
$p$-closed by [5, Lemma 2.1.6]. Consequently
$\overline{P}\trianglelefteq\overline{G}$, and so
$\overline{G}=N_{\overline{G}}(\overline{P})=\overline{N_{G}(P)}$. Now, since
$N_{G}(P)$ is $p$-nilpotent, we have that $\overline{G}=\overline{N_{G}(P)}$
is $p$-nilpotent. This implies that $G$ is $p$-nilpotent.

##  3 Proofs of Theorems A and B

[Proof of Theorem A] Let $G$ and $P$ be as in the statement of Theorem A.
Assume that $G$ is $p$-supersolvable. Then, since subgroups of
$p$-supersolvable groups are $p$-supersolvable, we have that $N_{G}(P)$ is
$p$-supersolvable, whence condition (1) from Theorem A is satisfied. Also
$G^{\mathfrak{U}_{p}}=1$, so that condition (2) from Theorem A is trivially
satisfied.

Suppose now that conditions (1) and (2) from Theorem A are satisfied. We have
to show that $G$ is $p$-supersolvable. To prove this, we assume that $G$ is
not $p$-supersolvable, and we are going to derive a contradiction from this
assumption. Since $G$ is not $p$-supersolvable, $G$ has a minimal
non-$p$-supersolvable subgroup, say $L$. Without loss of generality, we assume
that $P\cap L\in\mathrm{Syl}_{p}(L)$. As $G$ is $p$-solvable, we have that $L$
is $p$-solvable, and so $L^{\mathfrak{U}_{p}}$ is a $p$-group by Lemma 3(1).
In particular $L^{\mathfrak{U}_{p}}\leqslant P\cap L$. In order to obtain the
desired contradiction, we proceed in a number of steps.

1) Any subgroup of $L^{\mathfrak{U}_{p}}$ with order $p$ is $c$-embedded in
$G$ with respect to $P$. Moreover, if $p=2$ and $L^{\mathfrak{U}_{p}}$ is not
quaternion-free, any cyclic subgroup of $L^{\mathfrak{U}_{p}}$ with order $4$
is $c$-embedded in $G$ with respect to $P$.

Since $N_{G}(P)$ is $p$-supersolvable and $L$ is not $p$-supersolvable, we
have $L\not\leqslant N_{G}(P)$. Let $x\in L\setminus N_{G}(P)$. Using Lemma 5
and the fact that $L^{\mathfrak{U}_{p}}\trianglelefteq L$, we see that
$L^{\mathfrak{U}_{p}}=L^{\mathfrak{U}_{p}}\cap(L^{\mathfrak{U}_{p}})^{x}\cap
G^{\mathfrak{U}_{p}}\leqslant P\cap P^{x}\cap G^{\mathfrak{U}_{p}}$. Since
condition (2) from Theorem A is satisfied by assumption, it follows that any
subgroup of $L^{\mathfrak{U}_{p}}$ with order $p$ is $c$-embedded in $G$ with
respect to $P$.

Assume that $p=2$ and that $L^{\mathfrak{U}_{p}}$ is not quaternion-free. Then
$P\cap P^{x}\cap G^{\mathfrak{U}_{p}}$ is not quaternion-free either, and so
the validity of condition (2) from Theorem A implies that any cyclic subgroup
of $L^{\mathfrak{U}_{p}}$ with order $4$ is $c$-embedded in $G$ with respect
to $P$.

2) If $H$ is a proper subgroup of $L^{\mathfrak{U}_{p}}$ which is $c$-embedded
in $G$ with respect to $P$, then $H\leqslant Z(P)$.

Let $H$ be a proper subgroup of $L^{\mathfrak{U}_{p}}$ such that $H$ is
$c$-embedded in $G$ with respect to $P$. Hence there is a subgroup $B$ of $G$
such that $G=HB$ and $H\cap B\leqslant Z(P)$. Set $B_{0}:=L\cap B$. Then
$L=L\cap HB=HB_{0}$. Lemma 4 implies that $L=B_{0}\leqslant B$. So it follows
that $H=H\cap B\leqslant Z(P)$.

3) $L^{\mathfrak{U}_{p}}\leqslant Z(P)$.

Suppose that $L^{\mathfrak{U}_{p}}$ has exponent $p$. Let $x\in
L^{\mathfrak{U}_{p}}$. We show that $x\in Z(P)$. Clearly, we only need to
consider the case $x\neq 1$. Then $|\langle x\rangle|=p$. We have
$|L^{\mathfrak{U}_{p}}|>p$ since $L$ would be $p$-supersolvable otherwise.
Hence $\langle x\rangle$ is a proper subgroup of $L^{\mathfrak{U}_{p}}$. So 1)
and 2) imply that $x\in Z(P)$. As $x$ was arbitrarily chosen, it follows that
$L^{\mathfrak{U}_{p}}\leqslant Z(P)$.

Suppose now that $L^{\mathfrak{U}_{p}}$ does not have exponent $p$. Then, by
Lemma 3(1), $p=2$ and $L^{\mathfrak{U}_{2}}$ has exponent $4$. Therefore $L$
is a group appearing in [4, Theorem 9] as a group of Type 3. In particular,
$L^{\mathfrak{U}_{2}}=P\cap L$ is a non-abelian special $2$-group,
$\Phi(L^{\mathfrak{U}_{2}})\leqslant Z(L)$ and
$|L^{\mathfrak{U}_{2}}/\Phi(L^{\mathfrak{U}_{2}})|=2^{2m}$,
$|\Phi(L^{\mathfrak{U}_{2}})|\leqslant 2^{m}$ for some positive integer $m$.

We claim that $L^{\mathfrak{U}_{2}}$ is not quaternion-free. Assume that
$m=1$. Then $|L^{\mathfrak{U}_{2}}|=8$, and $L^{\mathfrak{U}_{2}}$ cannot be
dihedral because then $L^{\mathfrak{U}_{2}}/\Phi(L^{\mathfrak{U}_{2}})$ would
not be a chief factor of $L$. Consequently $L^{\mathfrak{U}_{2}}$ is
isomorphic to the quaternion group of order $8$ and in particular not
quaternion-free. Assume now that $m>1$. Let $R$ be a maximal subgroup of
$\Phi(L^{\mathfrak{U}_{2}})$. Then $L^{\mathfrak{U}_{2}}/R$ is extraspecial of
order $2^{2m+1}\geqslant 2^{5}$, and so $L^{\mathfrak{U}_{2}}/R$ has a section
isomorphic to the quaternion group of order $8$ (see [8, Chapter 5, Theorem
5.2]). Hence $L^{\mathfrak{U}_{2}}$ is not quaternion-free.

Now let $x\in L^{\mathfrak{U}_{2}}$. We show that $x\in Z(P)$. Clearly, we
only need to consider the case $x\neq 1$. Then $|\langle x\rangle|=2$ or $4$.
Also $\langle x\rangle$ is a proper subgroup of $L^{\mathfrak{U}_{2}}$. So 1)
and 2) imply that $x\in Z(P)$. As $x$ was arbitrarily chosen, it follows that
$L^{\mathfrak{U}_{2}}\leqslant Z(P)$.

4) The final contradiction.

Set $N:=N_{G}(L^{\mathfrak{U}_{p}})$. By 3), we have $P\leqslant
C_{G}(L^{\mathfrak{U}_{p}})$. Since
$C_{G}(L^{\mathfrak{U}_{p}})\trianglelefteq N$, the Frattini argument implies
that $N=N_{N}(P)C_{G}(L^{\mathfrak{U}_{p}})$.

Clearly $L^{\mathfrak{U}_{p}}\trianglelefteq N_{N}(P)$. Let $1=L_{0}\leqslant
L_{1}\leqslant\cdots\leqslant L_{t}=L^{\mathfrak{U}_{p}}$ be a part of a chief
series of $N_{N}(P)$ below $L^{\mathfrak{U}_{p}}$, i.e. $L_{i}\trianglelefteq
N_{N}(P)$ for all $0\leqslant i\leqslant t$ and $L_{i+1}/L_{i}$ is a chief
factor of $N_{N}(P)$ for all $0\leqslant i<t$. Since $N_{G}(P)$ is
$p$-supersolvable and $N_{N}(P)\leqslant N_{G}(P)$, we have that $N_{N}(P)$ is
$p$-supersolvable. Consequently $|L_{i+1}/L_{i}|=p$ for all $0\leqslant i<t$.

For each $0\leqslant i\leqslant t$, we have
$C_{G}(L^{\mathfrak{U}_{p}})\leqslant C_{G}(L_{i})\leqslant N_{G}(L_{i})$ and
hence $N=N_{N}(P)C_{G}(L^{\mathfrak{U}_{p}})\leqslant N_{G}(L_{i})$. In
particular, we have $L_{i}\trianglelefteq L$ for all $0\leqslant i\leqslant
t$. Consequently, each chief factor of $L$ below $L^{\mathfrak{U}_{p}}$ has
order $p$. So it follows that $L$ is $p$-supersolvable. This contradiction
completes the proof.

[Proof of Theorem B] Let $P$ be a Sylow $p$-subgroup of a group $G$. Assume
that $G$ is $p$-nilpotent. Then, since subgroups of $p$-nilpotent groups are
$p$-nilpotent, we have that $N_{G}(P)$ is $p$-nilpotent, whence condition (1)
from Theorem B is satisfied. Also $G^{\mathfrak{U}_{p}}\leqslant
G^{\mathfrak{N}_{p}}=1$, so that condition (2) from Theorem B is trivially
satisfied.

Let $\mathfrak{Y}$ denote the class of all groups $G$ such that condition (2)
from Theorem B is satisfied for any Sylow $p$-subgroup $P$ of $G$. Note that
if condition (2) from Theorem B is satisfied for one Sylow $p$-subgroup of a
group $G$, then it is satisfied for all Sylow $p$-subgroups of $G$, so that
$G\in\mathfrak{Y}$. Let $\mathscr{Z}_{p}$ denote the class of all
$\mathfrak{N}$-groups with $p$-nilpotent normalizers of Sylow $p$-subgroups.
To complete the proof of Theorem B, we show that the class $\mathscr{Z}_{p}$
is contained in the class $\mathfrak{N}_{p}$ of all $p$-nilpotent groups.
Suppose that this is not true, and choose a non-$p$-nilpotent
$\mathscr{Z}_{p}$-group $G$ of minimal order.

Arguing similarly as in [6, Example 2], we see that $\mathfrak{Y}$ is
subgroup-closed and that $X/N\in\mathfrak{Y}$ whenever $X\in\mathfrak{Y}$ and
$N$ is a normal $p^{\prime}$-subgroup of $X$. Applying [6, Theorem A], we
conclude that $G$ is $p$-solvable.

Let $P\in\mathrm{Syl}_{p}(G)$. Then $N_{G}(P)$ is $p$-nilpotent and hence
$p$-supersolvable as $G\in\mathscr{Z}_{p}$, whence condition (1) from Theorem
A is satisfied. Also, condition (2) from Theorem A is satisfied since it is
identical to condition (2) from Theorem B, which holds as $G\in\mathfrak{Y}$.
So Theorem A implies that $G$ is $p$-supersolvable. Applying Lemma 6, we
conclude that $G$ is $p$-nilpotent. This contradiction shows that
$\mathscr{Z}_{p}$ is contained in $\mathfrak{N}_{p}$, as wanted.

##  4 Remarks and open questions

One might wonder whether Theorems A and B remain true when, in condition (2)
of the theorems, “$c$-embedded in $G$ with respect to $P$” is replaced by
“$c$-embedded in $P$ with respect to $P$”. For the case $p=2$, the answer is
negative, as the following example shows.

######  Example 7.

Let $G:=S_{4}$, the symmetric group of degree $4$, and let $P$ be a Sylow
$2$-subgroup of $G$. Then $P$ is dihedral of order $8$, and we have
$N_{G}(P)=P$. Any subgroup of $P$ with order $2$ is either complemented or
central in $P$ and thus $c$-embedded in $P$ with respect to $P$. However, $G$
is not $2$-nilpotent (or, equivalently, not $2$-supersolvable).

For the case $p=3$, the following example shows that Theorem A does not remain
true when “$c$-embedded in $G$ with respect to $P$” is replaced by
“$c$-embedded in $P$ with respect to $P$”.

######  Example 8.

Let $G$ be the group indexed in GAP [7] as SmallGroup(216,153). Then $G$ is
solvable and hence $3$-solvable. Let $P$ be a Sylow $3$-subgroup of $G$. Then
$N_{G}(P)$ is $3$-supersolvable, and any subgroup of $P$ with order $3$ is
complemented or central in $P$ and thus $c$-embedded in $P$ with respect to
$P$. However, $G$ is not $3$-supersolvable.

We were not able to answer the following question.

######  Question 9.

Suppose that $p$ is odd. Does Theorem B remain true when, in condition (2)
from Theorem B, “$c$-embedded in $G$ with respect to $P$” is replaced by
“$c$-embedded in $P$ with respect to $P$”?

Using GAP [7], we have checked that there are no counterexamples of order up
to $2000$.

Wei, Wang and Liu [13] obtained the following characterization of
$p$-nilpotent groups: A group $G$ with Sylow $p$-subgroup $P$ is $p$-nilpotent
if and only if every minimal subgroup of $P\cap G^{\mathfrak{N}_{p}}$ is
complemented in $P$ and $N_{G}(P)$ is $p$-nilpotent (see [13, Corollary 2.3]).
In a less general form, this had been proved before by Guo and Shum [9,
Theorem 2.1]. Note that if the correct answer to Question 9 is positive, then
this would generalize both Theorem B and the mentioned result of Wei, Wang and
Liu.

## Acknowledgements

I am grateful to Professor Adolfo Ballester-Bolinches for drawing my attention
to the papers [4] and [6].

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# Generalized Ricci solitons and Einstein metrics on weak $K$-contact
manifolds

Vladimir Rovenski111Department of Mathematics, University of Haifa, Israel
e-mail<EMAIL_ADDRESS>
###### Abstract

We study metric structures on a smooth manifold (introduced in our recent
works [11, 12] and called a weak contact metric structure and a weak
$K$-contact structure) which generalize the metric contact and $K$-contact
structures, and allow a new look at the classical theory. First, we
characterize weak $K$-contact manifolds among all weak contact metric
manifolds by the property well known for $K$-contact manifolds, and find when
a Riemannian manifold endowed with a unit Killing vector field forms a weak
$K$-contact structure. Second, we find sufficient conditions for a weak
$K$-contact manifold with parallel Ricci tensor or with a generalized Ricci
soliton structure to be an Einstein manifold.

Keywords: weak $K$-contact manifold, unit Killing vector field, Einstein
metric, generalized Ricci soliton, curvature

Mathematics Subject Classifications (2010) 53C15, 53C25, 53D15

##  1 Introduction

The growing interest in contact geometry is associated with its important role
in mechanics in explaining physical phenomena. In addition, many recent
articles have been motivated by the question of how interesting self-similar
solutions of the Ricci flow equation, i.e., Ricci solitons, can be for contact
metric geometry. Some of them find conditions when a contact manifold equipped
with a Ricci-type soliton structure carries a canonical (e.g., Einstein or
constant curvature) metric, e.g. [2, 6, 8, 10].

$K$-contact manifolds (i.e., contact metric manifolds whose Reeb vector field
is Killing) have been studied by several geometers, e.g., [1, 13], and it is
seen that the $K$-contact structure is intermediate between the contact and
Sasakian structures. The Reeb vector field $\xi$ of the $K$-contact structure
is a unit Killing vector field, and the influence of constant length Killing
vector fields on the Riemannian geometry has been studied by several authors
from different points of view, e.g., [3, 5, 9]. An interesting result related
to the above question is that a $K$-contact manifold equipped with generalized
Ricci soliton structure has an Einstein metric, e.g., [6].

In [12], we introduced the “weakened” metric structures on a smooth manifold
(replacing the complex structure on the characteristic distribution with a
nonsingular skew-symmetric tensor). They generalize the metric contact,
$K$-contact, Sasakian and cosymplectic structures, and allow a new look at the
classical theory. In [12], we build retraction of weak structures with
positive partial Ricci curvature onto the set of classical structures. In [11]
(where the definition of weak structures is a bit more general than in [12]),
we proved that the weak Sasakian structure is weak $K$-contact; and a weak
almost contact metric manifold is weak Sasakian if and only if it is a
Sasakian manifold. In this article we study weak $K$-contact manifolds using
their sectional and Ricci curvature in the $\xi$-direction. Our goal is to
show that the weak $K$-contact structure can be a useful tool for studying
unit Killing vector fields on Riemannian manifolds, and that some results for
$K$-contact manifolds can be extended to the case of weak $K$-contact
manifolds. For example, we answer the question of when a weak $K$-contact
manifold carries a generalized Ricci soliton structure or just an Einstein
metric.

The article is organized as follows. In Section 2, following the introductory
Section 1, we recall basics of weak contact metric manifolds. Next three
sections contain the main results. In Section 3, we characterize (in Theorem
3.1) weak $K$-contact manifolds among all weakly contact metric manifolds by
the property $\varphi=-\nabla\xi$ (well known for $K$-contact manifolds), and
find (in Theorem 3.2) when a Riemannian manifold endowed with a unit Killing
vector field forms a weak $K$-contact structure. In Section 4, for a weak
$K$-contact manifold, we calculate (in Proposition 4.1) the Ricci curvature in
the $\xi$-direction, then find (in Theorem 4.1) sufficient condition for such
a manifold with parallel Ricci tensor to be an Einstein manifold. In Section
5, we find (in Theorem 5.1) sufficient conditions for a weak $K$-contact
manifold admitting a generalized Ricci soliton structure to be an Einstein
manifold.

##  2 Preliminaries

Here, we recall basics of some metric structures that generalize the almost
contact metric structure (see [12], or, equivalently, [11] with $\nu=1$). A
weak almost contact structure on a smooth manifold $M^{2n+1}$ is a set
$(\varphi,Q,\xi,\eta)$, where $\varphi$ is a rank-(1,1) tensor, $Q$ is a
nonsingular $(1,1)$-tensor, $\xi$ is the Reeb vector field and $\eta$ is a
dual 1-form, i.e., $\eta(\xi)=1$, satisfying

$\varphi^{2}=-Q+\eta\otimes\xi,\qquad Q\,\xi=\xi.$ (1)

The form $\eta$ determines a smooth $2n$-dimensional distribution ${\cal
D}:=\ker\eta$, the collection of subspaces ${\cal D}_{m}=\\{{X}\in
T_{m}M:\eta({X})=0\\}$ for $m\in M$. We assume that ${\cal D}$ is
$\varphi$-invariant,

$\varphi{X}\in{\cal D},\quad{X}\in{\cal D},$ (2)

as in the theory of almost contact structure [1, 13], where $Q={\rm
id}_{\,TM}$. By (1) and (2), our distribution ${\cal D}$ is invariant for $Q$:
$Q({\cal D})={\cal D}$. If there is a Riemannian metric $g$ on $M$ such that

$\displaystyle
g(\varphi{X},\varphi{Y})=g({X},Q\,{Y})-\eta({X})\,\eta({Y}),\quad{X},{Y}\in\mathfrak{X}_{M},$
(3)

then $(\varphi,Q,\xi,\eta,g)$ is called a weak almost contact metric structure
on $M$, and $g$ is called a compatible metric. A weak almost contact manifold
$M(\varphi,Q,\xi,\eta)$ endowed with a compatible Riemannian metric $g$ is
called a weak almost contact metric manifold and is denoted by
$M(\varphi,Q,\xi,\eta,g)$. Some sufficient conditions for the existence of a
compatible metric on a weak almost contact manifold are given in [12].

Putting ${Y}=\xi$ in (3) and using $Q\,\xi=\xi$, we get, as in the classical
theory, $\eta({X})=g(\xi,{X})$. In particular, $\xi$ is $g$-orthogonal to
${\cal D}$ for any compatible metric $g$.

By (3), we get $g({X},Q\,{X})=g(\varphi{X},\varphi{X})>0$ for any nonzero
vector ${X}\in{\cal D}$, thus $Q$ is positive definite. For a weak almost
contact structure on a smooth manifold $M$, the tensor $\varphi$ has rank $2n$
and

$\varphi\,\xi=0,\quad\eta\circ\varphi=0,\quad\eta\circ
Q=\eta,\quad[Q,\,\varphi]=0;$

moreover, for a weak almost contact metric structure, $\varphi$ is skew-
symmetric and $Q$ is self-adjoint.

A weak contact metric structure is defined as a weak almost contact metric
structure satisfying $\Phi=d\eta$ (thus, $d\Phi=0$), where
$\Phi({X},{Y})=g({X},\varphi{Y})\ ({X},{Y}\in\mathfrak{X}_{M})$ is called the
fundamental $2$-form, and

$\displaystyle d\eta({X},{Y})$
$\displaystyle=\frac{1}{2}\,\\{{X}(\eta({Y}))-{Y}(\eta({X}))-\eta([{X},{Y}])\\},\quad{X},{Y}\in\mathfrak{X}_{M},$
(4) $\displaystyle d\Phi({X},{Y},Z)$
$\displaystyle=\frac{1}{3}\,\big{\\{}{X}\,\Phi({Y},Z)+{Y}\,\Phi(Z,{X})+Z\,\Phi({X},{Y})$
$\displaystyle-\Phi([{X},{Y}],Z)-\Phi([Z,{X}],{Y})-\Phi([{Y},Z],{X})\big{\\}},\quad{X},{Y},Z\in\mathfrak{X}_{M}.$
(5)

######  Remark 2.1.

A differential $k$-form on a manifold $M$ is a skew-symmetric tensor field
$\omega$ of type $(0,k)$. According the conventions of e.g., [7], the formula

$\displaystyle
d\omega({X}_{1},\ldots,{X}_{k+1})=\frac{1}{k+1}\sum\nolimits_{\,i=1}^{k+1}(-1)^{i+1}{X}_{i}(\omega({X}_{1},\ldots,\hat{{X}}_{i}\ldots,{X}_{k+1}))$
$\displaystyle+\sum\nolimits_{\,i<j}(-1)^{i+j}\omega([{X}_{i},{X}_{j}],{X}_{1},\ldots,\hat{{X}}_{i},\ldots,\hat{{X}}_{j},\ldots,{X}_{k+1}),$
(6)

where ${X}_{1},\ldots,{X}_{k+1}\in\mathfrak{X}_{M}$ and $\,\hat{\cdot}\,$
denotes the operator of omission, defines a $(k+1)$-form $d\omega$ called the
exterior differential of $\omega$. Note that (4) and (2), e.g., [1],
correspond to (2.1) with $k=1$ and $k=2$.

For a weak contact metric structure, the distribution ${\cal D}$ is non-
integrable (has no integral hypersurfaces), since
$g([{X},\varphi{X}],\xi)=2\,d\eta(\varphi{X},{X})=g(\varphi{X},\varphi{X})>0$
for any nonzero ${X}\in{\cal D}$.

The Nijenhuis torsion $[\varphi,\varphi]$ of $\varphi$ is given by

$\displaystyle[\varphi,\varphi]({X},{Y})=\varphi^{2}[{X},{Y}]+[\varphi{X},\varphi{Y}]-\varphi[\varphi{X},{Y}]-\varphi[{X},\varphi{Y}],\quad{X},{Y}\in\mathfrak{X}_{M}.$

A weak almost contact structure $(\varphi,Q,\xi,\eta)$ is called normal if the
following tensor is zero:

$\displaystyle
N^{\,(1)}({X},{Y})=[\varphi,\varphi]({X},{Y})+2\,d\eta({X},{Y})\,\xi,\quad{X},{Y}\in\mathfrak{X}_{M}.$
(7)

The following tensors $N^{\,(2)},N^{\,(3)}$ and $N^{\,(4)}$ are well known in
the classical theory, see [1, 13]:

$\displaystyle N^{\,(2)}({X},{Y})$
$\displaystyle=(\pounds_{\varphi{X}}\,\eta)({Y})-(\pounds_{\varphi{Y}}\,\eta)({X})\overset{\eqref{3.3A}}{=}2\,d\eta(\varphi{X},{Y})-2\,d\eta(\varphi{Y},{X}),$
$\displaystyle N^{\,(3)}({X})$
$\displaystyle=(\pounds_{\,\xi}\,\varphi){X}=[\xi,\varphi{X}]-\varphi[\xi,{X}],$
$\displaystyle N^{\,(4)}({X})$
$\displaystyle=(\pounds_{\,\xi}\,\eta)({X})=\xi(\eta({X}))-\eta([\xi,{X}])\overset{\eqref{3.3A}}{=}2\,d\eta(\xi,{X}).$

######  Remark 2.2.

Let $M(\varphi,Q,\xi,\eta)$ be a weak almost contact manifold. Consider the
product manifold $\bar{M}=M\times\mathbb{R}$, where $\mathbb{R}$ has the
Euclidean basis $\partial_{t}$, and define tensor fields $\bar{\varphi}$ and
$\bar{Q}$ on $\bar{M}$ putting

$\displaystyle\bar{\varphi}({X},\,a\,\partial_{t})$
$\displaystyle=(\varphi{X}-a\,\xi,\,\eta({X})\,\partial_{t}),$
$\displaystyle\bar{Q}({X},\,a\,\partial_{t})$
$\displaystyle=(Q{X},\,a\,\partial_{t}),$

where $a\in C^{\infty}(M)$. Thus, $\bar{\varphi}({X},0)=(\varphi{X},0)$,
$\bar{Q}({X},0)=(Q{X},0)$ for ${X}\in\ker\varphi$,
$\bar{\varphi}(\xi,0)=(0,\partial_{t})$, $\bar{Q}(\xi,0)=(\xi,0)$ and
$\bar{\varphi}(0,\partial_{t})=(-\xi,0)$,
$\bar{Q}(0,\partial_{t})=(0,\partial_{t})$. By the above, $\bar{\varphi}^{\
2}=-\bar{Q}$. The tensors $N^{\,(i)}\ (i=1,2,3,4)$ appear when we derive the
integrability condition $[\bar{\varphi},\bar{\varphi}]=0$ (vanishing of the
Nijenhuis torsion of $\bar{\varphi}$ ) and express the normality condition
$N^{\,(1)}=0$ of a $(\varphi,Q,\xi,\eta)$ on $M$.

######  Theorem 2.1 (see [11] with $\nu=1$).

a) For a weak almost contact metric structure $(\varphi,Q,\xi,\eta,g)$, the
vanishing of $N^{\,(1)}$ implies that $N^{\,(3)}$ and $N^{\,(4)}$ vanish and
$N^{\,(2)}({X},{Y})=\eta([\tilde{Q}{X},\,\varphi{Y}])$.

b) For a weak contact metric manifold, the tensors $N^{\,(2)}$ and $N^{\,(4)}$
vanish and the trajectories of $\xi$ are geodesics, i.e.,
$\nabla_{\xi}\,\xi=0$; moreover, $N^{\,(3)}\equiv 0$ if and only if $\,\xi$ is
a Killing vector field.

######  Definition 2.1 (see [11] with $\nu=1$).

Two weak almost contact structures $(\varphi,Q,\xi,\eta)$ and
$(\varphi^{\prime},Q^{\prime},\xi,\eta)$ on $M$ are said to be homothetic if
the following is valid for some real $\lambda>0$:

$\displaystyle\varphi=\sqrt{\lambda}\ \varphi^{\prime},$ (8a) $\displaystyle
Q\,|_{\,{\mathcal{D}}}=\lambda\,Q^{\prime}|_{\,\mathcal{D}}.$ (8b) Two weak
contact metric structures $(\varphi,Q,\xi,\eta,g)$ and
$(\varphi^{\prime},Q^{\prime},\xi,\eta,g^{\prime})$ on $M$ are said to be
homothetic if they satisfy conditions (8a,b) and $\displaystyle
g|_{\,{\mathcal{D}}}=\lambda^{\,-\frac{1}{2}}\,g^{\prime}|_{\,{\mathcal{D}}},\quad
g(\xi,\,\cdot)={g}^{\prime}(\xi,\,\cdot).$ (8c)

######  Lemma 2.1 (see [11] with $\nu=1$).

Let $(\varphi,Q,\xi,\eta)$ be a weak almost contact structure such that

$\displaystyle Q\,|_{\,{\mathcal{D}}}=\lambda\,{\rm id}_{\mathcal{D}},$ for
some real $\lambda>0$. Then the following is true:

$\bullet$ $(\varphi^{\prime},\xi,\eta)$ is an almost contact structure, where
$\varphi^{\prime}$ is given by (8a).

$\bullet$ If $(\varphi,Q,\xi,\eta,g)$ is a weak contact metric structure, and
$\varphi^{\prime},g^{\prime}$ satisfy (8a,c), then
$(\varphi^{\prime},\xi,\eta,{g}^{\prime})$ is a contact metric structure.

A weak $K$-contact manifold is defined as a weak contact metric manifold whose
Reeb vector field $\xi$ is Killing (or, infinitesimal isometry, e.g., [7]),
i.e.,

$\displaystyle(\pounds_{\,\xi}\,g)({X},{Y}):=\xi(g({X},{Y}))-g([\xi,{X}],{Y})-g({X},[\xi,{Y}])=g(\nabla_{X}\,\xi,{Y})+g(\nabla_{Y}\,\xi,{X})=0.$
(10)

Here $\pounds_{\,\xi}$ is the Lie derivative in the $\xi$-direction and
$\nabla$ is the Levi-Civita connection. A normal weak contact metric manifold
is called a weak Sasakian manifold. Recall that a weak Sasakian structure is
weak $K$-contact, see [11, Proposition 4.1], furthermore, a weak almost
contact metric structure is weak Sasakian if and only if it is homothetic to a
Sasakian structure, see [11, Theorem 4.1].

The relationships between the different classes of weak structures (considered
in this article) can be summarizes in the diagram (well known in the case of
classical structures):

$\left|\begin{array}[]{c}weak\\\ almost\ contact\\\
\end{array}\right|\overset{metric}{\longrightarrow}\left|\begin{array}[]{c}weak\
almost\\\ contact\ metric\\\
\end{array}\right|\overset{\Phi=d\eta}{\longrightarrow}\left|\begin{array}[]{c}weak\\\
contact\ metric\\\
\end{array}\right|\overset{\xi\,-\,Killing}{\longrightarrow}\left|\begin{array}[]{c}weak\\\
K-contact\\\ \end{array}\right|.$

A “small” (1,1)-tensor $\tilde{Q}=Q-{\rm id}$ is a measure of the difference
between a weakly contact structure and a contact one, and $\tilde{Q}=0$ means
the classical contact geometry. Note that $[\tilde{Q},\varphi]=0$ and
$\tilde{Q}\,\xi=0$.

######  Lemma 2.2 ([11] with $\nu=1$).

For a weak contact metric manifold, we get

$g((\nabla_{{X}}\varphi){Y},Z)=\frac{1}{2}g(N^{\,(1)}({Y},Z),\varphi{X})+g(\varphi{X},\varphi{Y})\,\eta(Z)-g(\varphi{X},\varphi
Z)\,\eta({Y})+\frac{1}{2}N^{\,(5)}({X},{Y},Z),$ (11)

where the skew-symmetric with respect to ${Y}$ and $Z$ tensor
$N^{\,(5)}({X},{Y},Z)$ supplements the sequence of tensors $N^{\,(i)}\
(i=1,2,3,4)$ and for a weak contact metric manifold is given by

$\displaystyle N^{\,(5)}({X},{Y},Z)=(\varphi
Z)\,(g({X},\tilde{Q}{Y}))-(\varphi{Y})\,(g({X},\tilde{Q}Z))$
$\displaystyle+\,g([{X},\varphi
Z],\tilde{Q}{Y})-g([{X},\varphi{Y}],\tilde{Q}Z)+g([{Y},\varphi
Z]-[Z,\varphi{Y}]-\varphi[{Y},Z],\ \tilde{Q}{X}).$

######  Remark 2.3.

For a contact metric structure, (11) (with $N^{\,(5)}=0$) gives the result in
[1, Corollary 6.1]. Note that only one new tensor $N^{\,(5)}$ is needed for
further study of a weak contact metric structure. In particular, by (11), we
get $g((\nabla_{\xi}\,\varphi){Y},Z)=\frac{1}{2}\,N^{\,(5)}(\xi,{Y},\,Z)$ and

$\displaystyle N^{\,(5)}({X},\xi,Z)$
$\displaystyle=g(N^{\,(3)}(Z),\,\tilde{Q}{X}),$ $\displaystyle
N^{\,(5)}(\xi,{Y},Z)$ $\displaystyle=g([\xi,\varphi
Z],\tilde{Q}{Y})-g([\xi,\varphi{Y}],\tilde{Q}Z),$ $\displaystyle
N^{\,(5)}(\xi,\xi,Z)$ $\displaystyle=N^{\,(5)}(\xi,{Y},\xi)=0.$ (12)

##  3 Unit Killing vector fields

######  Proposition 3.1.

On a weak $K$-contact manifold, we get $N^{\,(1)}(\xi,\,\cdot)=0$ and

$\displaystyle N^{\,(5)}(\xi,\,\cdot\,,\,\cdot)$ $\displaystyle=$
$\displaystyle N^{\,(5)}(\,\cdot\,,\,\xi,\,\cdot)=0,$ (13)
$\displaystyle\pounds_{\,\xi}\,{\tilde{Q}}$ $\displaystyle=$
$\displaystyle\nabla_{\xi}\,{\tilde{Q}}=0,$ (14)
$\displaystyle\nabla_{\xi}\,\varphi$ $\displaystyle=$ $\displaystyle 0.$ (15)

###### Proof.

By (7) and $d\eta(\xi,\,\cdot)=\Phi(\xi,\,\cdot)=0$ we get

$N^{\,(1)}(\xi,{X})=[\varphi,\varphi]({X},\xi)=\varphi^{2}[{X},\xi]-\varphi[\varphi{X},\xi]=\varphi
N^{\,(3)}({X})=0.$

By [11, Lemma 3.1] with $\nu=1$ and $h=\frac{1}{2}\,N^{\,(3)}=0$, we get
$N^{\,(5)}(\xi,\,\cdot\,,\,\cdot)=0$, $\pounds_{\,\xi}{\tilde{Q}}=0$ and

$g(Q\,\nabla_{{X}}\,\xi,Z)=g(\varphi
Z,Q{X})-\frac{1}{2}\,N^{\,(5)}({X},\xi,\varphi Z).$

By (2.3)1 with $N^{\,(3)}=0$, we get $N^{\,(5)}(\,\cdot\,,\xi,\,\cdot)=0$. We
use $[\varphi,{\tilde{Q}}]=0$ to obtain $\nabla_{\xi}\,{\tilde{Q}}=0$:

$(\pounds_{\,\xi}\,{\tilde{Q}}){X}=[\xi,{\tilde{Q}}{X}]-{\tilde{Q}}[\xi,{X}]=(\nabla_{\xi}{\tilde{Q}}){X}+[\varphi,{\tilde{Q}}]{X}=(\nabla_{\xi}\,{\tilde{Q}}){X}.$

This completes the proof of (13) and (14). Next, from (11) with ${X}=\xi$ we
get (15). ∎

In the next theorem, we characterize weak $K$-contact manifolds among all weak
contact metric manifolds by the following well known property of $K$-contact
manifolds, see [1]:

$\nabla\,\xi=-\varphi.$ (16)

######  Theorem 3.1.

A weak contact metric manifold is weak $K$-contact $($that is $\xi$ is a
Killing vector field$)$ if and only if (16) is valid.

###### Proof.

Let a weak contact metric manifold satisfy (16). By skew-symmetry of
$\varphi$, we get
$(\pounds_{\,\xi}\,g)({X},{Y})=g(\nabla_{X}\,\xi,{Y})+g(\nabla_{Y}\,\xi,{X})=-g(\varphi{X},{Y})-g(\varphi{Y},{X})=0$,
thus, $\xi$ is a Killing vector field.

Conversely, let our manifold be weak $K$-contact. By (11) with ${Y}=\xi$,
using $N^{\,(1)}(\xi,\,\cdot)=0$ and $N^{\,(5)}({X},\xi,Z)=0$, see (13), we
get

$\displaystyle g((\nabla_{{X}}\,\varphi)\,\xi,Z)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,g(N^{\,(1)}(\xi,Z),\varphi{X})-g(\varphi{X},\varphi
Z)+\frac{1}{2}\,N^{\,(5)}({X},\xi,Z)=g(\varphi^{2}{X},Z).$

Hence, $(\nabla_{{X}}\,\varphi)\,\xi=\varphi^{2}{X}$. From this and
$0=\nabla_{{X}}\,(\varphi\,\xi)=(\nabla_{{X}}\,\varphi)\,\xi+\varphi\nabla_{{X}}\,\xi$,
we obtain $\varphi(\nabla_{{X}}\,\xi+\varphi{X})=0$. Since
$\nabla_{{X}}\,\xi+\varphi{X}\in{\cal D}$ and $\varphi$ is invertible when
restricted on ${\cal D}$, we get that $\nabla_{{X}}\,\xi=-\varphi{X}$. ∎

If a plane contains $\xi$, then its sectional curvature is called
$\xi$-sectional curvature. It is well known that the $\xi$-sectional curvature
of a $K$-contact manifold is constant equal to 1. Recall that a Riemannian
manifold with a unit Killing vector field and the property
$R_{{X},\,{\xi}}\,{\xi}={X}\ ({X}\bot\,\xi)$ is a $K$-contact manifold, e.g.,
[13, Theorem 3.1] or [1, Proposition 7.4]. We generalize this result in the
following

######  Theorem 3.2.

A Riemannian manifold $(M^{\,2n+1},g)$ admitting a unit Killing vector field
$\xi$ with positive $\xi$-sectional curvature is a weak $K$-contact manifold
$M(\varphi,Q,\xi,\eta,g)$ with the following structural tensors:
$\eta({X})=g({X},\xi)$, $\varphi=-\nabla\,\xi$, see (16), and
$Q{X}=R_{{X},\,{\xi}}\,{\xi}$ for ${X}\in\ker\eta$.

###### Proof.

Let $\eta=g(\,\cdot\,,\xi)$ and ${\cal D}=\ker\eta$. Put
$\varphi{X}=-\nabla_{X}\,\xi$ and $Q{X}=R_{{X},\,{\xi}}\,{\xi}$ for
${X}\in{\cal D}$. Since $\xi$ is a Killing vector field, we obtain the
property $d\eta=\Phi$ for $\Phi({X},{Y})=g({X},\varphi{Y})$:

$d\eta({X},{Y})=\frac{1}{2}\,(g(\nabla_{X}\,\xi,{Y})-g(\nabla_{Y}\,\xi,{X}))=-g(\nabla_{Y}\,\xi,{X})=g({X},\varphi{Y}).$

Since $\xi$ is a unit Killing vector field, we get $\nabla_{\xi}\,\xi=0$ and
$\nabla_{X}\nabla_{Y}\,\xi-\nabla_{\nabla_{X}\,{Y}}\,\xi=R_{\,\xi,{X}}\,{Y}$.
Thus, $\varphi\,\xi=0$ and

$\varphi^{2}{X}=\nabla_{\nabla_{X}\,\xi}\,\xi=R_{\,\xi,{X}}\,\xi=-Q{X}\quad({X}\in{\cal
D}).$

Put $Q\,\xi=\xi$. Therefore, (1) is valid, and $Q$ is positive definite, that
completes the proof. ∎

######  Example 3.1.

By Theorem 3.2, we can search for examples of weak $K$-contact (not
$K$-contact) manifolds among Riemannian manifolds of positive sectional
curvature that admit unit Killing vector fields. Indeed, let $M$ be a convex
hypersurface (ellipsoid) with induced metric $g$ of the Euclidean space
$\mathbb{R}^{2n+2}$,

$M=\Big{\\{}(u_{1},\ldots,u_{2n+2})\in\mathbb{R}^{2n+2}:\sum\nolimits_{\,i=1}^{n+1}u_{i}^{2}+a\sum\nolimits_{\,i=n+2}^{2n+1}u_{i}^{2}=1\Big{\\}},$

where $0<a=const\neq 1$ and $n\geq 1$ is odd. The sectional curvature of
$(M,g)$ is positive. It follows that

$\xi=(-u_{2},u_{1},\ldots,-u_{n+1},u_{n},-\sqrt{a}\,u_{n+3},\sqrt{a}\,u_{n+2},\ldots,-\sqrt{a}\,u_{2\,n+2},\sqrt{a}\,u_{2\,n+1})$

is a Killing vector field on $\mathbb{R}^{2\,n+2}$, whose restriction to $M$
has unit length. Since $\xi$ is tangent to $M$ (i.e., $M$ is invariant under
the flow of $\xi$), so $\xi$ is a unit Killing vector field on $(M,g)$, see
[5, p. 5]. For $n=1$, we get a weak $K$-contact manifold
$M^{3}=\big{\\{}u_{1}^{2}+u_{2}^{2}+au_{3}^{2}+au_{4}^{2}=1\big{\\}}\subset\mathbb{R}^{4}$
with $\xi=(-u_{2},u_{1},-\sqrt{a}\,u_{4},\sqrt{a}\,u_{3})$.

Other examples of weak $K$-contact (not $K$-contact) manifolds are obtained
from [3, Theorem 12]: “On every sphere $S^{2n-1},\ n\geq 2$, for any
$\varepsilon>0$ there exists a (real analytic) Riemannian metric $g$ of
cohomogeneity 1 and a (real analytic) Killing vector field $\xi$ of unit
length on $(S^{2n-1},g)$ such that

1) all sectional curvatures of $(S^{2n-1},g)$ differ from 1 by at most
$\varepsilon$;

2) the vector field $\xi$ has both closed and non-closed integral
trajectories”.

######  Corollary 3.1.

A weak $K$-contact structure $(\varphi,Q,\xi,\eta,g)$ with constant positive
$\xi$-sectional curvature, $K(\xi,{X})=\lambda>0$ for some
$\lambda=const\in\mathbb{R}$ and all ${X}\in{\cal D}$, is homothetic to a
$K$-contact structure $(\varphi^{\prime},\xi,\eta,g^{\prime})$ after the
transformation (8a-c).

###### Proof.

Note that $K(\xi,{X})=\lambda\ ({X}\in{\cal D})$ if and only if
$R_{{X},\xi}\,\xi=\lambda\,{X}\ ({X}\in{\cal D})$. By $Q{X}=R_{{X},\xi}\,\xi\
({X}\in{\cal D})$, see Theorem 3.2, we get $Q{X}=\lambda{X}\ ({X}\in{\cal
D})$. By Lemma 2.1(ii), $(\varphi^{\prime},\xi,\eta,{g}^{\prime})$ is a
contact metric structure. Using (10), we get
$(\pounds_{\,\xi}\,g^{\prime})({X},{Y})=\lambda(\pounds_{\,\xi}\,g)({X},{Y})\
({X},{Y}\in{\cal D})$ and $(\pounds_{\,\xi}\,g^{\prime})(\xi,\,\cdot)=0$. By
$\pounds_{\,\xi}\,g=0$, we get $\pounds_{\,\xi}\,g^{\prime}=0$; thus
$(\varphi^{\prime},\xi,\eta,g^{\prime})$ is a $K$-contact structure. ∎

##  4 The Ricci curvature in the characteristic direction

Denote by $R_{{X},{Y}}\,Z$ the curvature tensor and by
$\operatorname{Ric}^{\sharp}$ the Ricci operator of $g$ associated with the
Ricci tensor ${\rm Ric}$ and given by ${\rm
Ric}({X},{Y})=g(\operatorname{Ric}^{\sharp}{X},{Y})$ for all
${X},{Y}\in\mathfrak{X}_{M}$. The Ricci curvature in the $\xi$-direction is
given by ${\rm
Ric}({\xi},{\xi})=\sum_{\,i=1}^{\,2n}g(R_{e_{i},\,{\xi}}\,{\xi},e_{i})$, where
$(e_{i})$ is any local orthonormal basis of ${\cal D}$.

In the next proposition, we generalize three particular properties of
$K$-contact manifolds to weak $K$-contact manifolds.

######  Proposition 4.1.

For a weak $K$-contact manifold, the following equalities are true:

$\displaystyle R_{{\xi},\,{X}}\,{Y}=(\nabla_{X}\,\varphi){Y},$ (17)
$\displaystyle R_{{X},\,{\xi}}\,{\xi}=-\varphi^{2}{X},$ (18)
$\displaystyle{\rm
Ric}({\xi},{\xi})=\operatorname{trace\,}Q=2\,n+\operatorname{trace\,}\tilde{Q}.$
(19)

###### Proof.

Using (16), we derive

$\displaystyle R_{Z,\,{X}}\,{\xi}$ $\displaystyle=$
$\displaystyle\nabla_{Z}(\nabla_{X}\,{\xi})-\nabla_{X}(\nabla_{Z}\,{\xi})-\nabla_{[Z,{X}]}\,{\xi})$
(20) $\displaystyle=$ $\displaystyle\nabla_{X}(\varphi
Z)-\nabla_{Z}(\varphi{X})+\varphi([Z,{X}])=(\nabla_{X}\,\varphi)Z-(\nabla_{Z}\,\varphi){X}.$

Note that
$(\nabla_{X}\,\Phi)({Y},Z)=g((\nabla_{X}\,\varphi)Z,{Y})=-g((\nabla_{X}\,\varphi){Y},Z)$.
Using condition $d\Phi=d^{2}\eta=0$, we get

$(\nabla_{X}\,\Phi)({Y},Z)+(\nabla_{Y}\,\Phi)(Z,{X})+(\nabla_{Z}\,\Phi)({X},{Y})=0.$
(21)

From (20), using (21) and skew-symmetry of $\Phi$, we get (17):

$\displaystyle g(R_{{\xi},\,{X}}\,{Y},Z)$ $\displaystyle=$ $\displaystyle
g(R_{{Y},\,Z}\,{\xi},{X})\overset{\eqref{Eq-R}}{=}(\nabla_{Z}\,\Phi)({X},{Y})+(\nabla_{Y}\,\Phi)(Z,{X})$
$\displaystyle\overset{\eqref{E-dPhi-three}}{=}$
$\displaystyle-(\nabla_{X}\,\Phi)({Y},Z)=g((\nabla_{X}\,\varphi){Y},Z).$

By (17) with ${Y}=\xi$, using $\varphi\,\xi=0$ and (16), we find

$\displaystyle
R_{{\xi},{X}}\,{\xi}=(\nabla_{X}\,\varphi)\,\xi=-\varphi\nabla_{X}\,{\xi}=\varphi^{2}{X}.$

This and (1)1 yield (18). By this, for any local orthonormal basis $(e_{i})$
of ${\cal D}$, we get

${\rm
Ric}({\xi},{\xi})\overset{\eqref{E-R1}}{=}-\sum\nolimits_{\,i=1}^{\,2n}g(\varphi^{2}e_{i},e_{i})\overset{\eqref{2.1}}{=}\sum\nolimits_{\,i=1}^{\,2n}g(Qe_{i},e_{i}).$

By the above and the equality
$\operatorname{trace\,}Q=2\,n+\operatorname{trace\,}\tilde{Q}$, (19) is true.
∎

Note that if a Riemannian manifold admits a unit Killing vector field $\xi$,
then $K(\xi,{X})\geq 0\ ({X}\,\perp\,\xi,\ {X}\neq 0)$, thus ${\rm
Ric}({\xi},{\xi})\geq 0$; moreover, ${\rm Ric}({\xi},{\xi})\equiv 0$ if and
only if $\xi$ is parallel: $\nabla\xi\equiv 0$, for example, [9]. In the case
of $K$-contact manifolds, $K(\xi,{X})=1$, see [1, Theorem 7.2].

######  Corollary 4.1.

For a weak $K$-contact manifold, the $\xi$-sectional curvature is positive:

$K(\xi,{X})=g(Q{X},{X})>0\quad({X}\in{\cal D},\ \|{X}\|=1),$ (22)

therefore, for the Ricci curvature we get ${\rm Ric}({\xi},{\xi})>0$.

###### Proof.

For any unit vector ${X}\in{\cal D}$, by (18), we get

$0<g(\varphi{X},\varphi{X})=-g(\varphi^{2}{X},{X})=g(Q{X},{X}),$

thus $K(\xi,{X})>0$ and $\operatorname{trace\,}Q>0$. Therefore, from (19) we
get the statement. ∎

By Theorem 3.2 and Corollary 4.1, using [12, Corollary 3], we conclude the
following.

######  Corollary 4.2.

A weak $K$-contact manifold $M(\varphi,Q,\xi,\eta,g_{0})$ admits a smooth
family of metrics $g_{t}\ (t\in\mathbb{R})$, such that
$M(\varphi_{t},Q_{t},\xi,\eta,g_{t})$ are weak $K$-contact manifolds with
certainly defined $\varphi_{t}$ and $Q_{t}$; moreover, $g_{t}$ converges
exponentially fast, as $t\to-\infty$, to a limit metric $\hat{g}$ that gives a
$K$-contact structure.

The following theorem generalizes a well known result, e.g., [13, Proposition
5.1].

######  Theorem 4.1.

A weak $K$-contact manifold with conditions
$(\nabla\operatorname{Ric})(\xi,\,\cdot)=0$ $($in particular, the Ricci tensor
is parallel$)$ and $\operatorname{trace\,}Q=const$ is an Einstein manifold of
scalar curvature $(2\,n+1)\operatorname{trace\,}Q$.

###### Proof.

Differentiating (19) and using (16) and the conditions, we get

$0=\nabla_{Y}\,({\rm Ric}({\xi},{\xi}))=(\nabla_{Y}\,{\rm
Ric})({\xi},{\xi})+2\,{\rm Ric}(\nabla_{Y}\,{\xi},{\xi}))=-2\,{\rm
Ric}(\varphi{Y},{\xi})),$

hence ${\rm Ric}({Y},{\xi})=\eta({Y})\,{\rm
Ric}(\xi,\xi)=\eta({Y})\operatorname{trace\,}Q$. Differentiating this, then
using

${X}(\eta({Y}))=g(\nabla_{X}\xi,{Y})=-g(\varphi{X},{Y})+g(\nabla_{X}{Y},\xi)$

and assuming $\nabla_{X}{Y}=0$ at $x\in M$, gives

$(\operatorname{trace\,}Q)\,g(\varphi{Y},{X})=\nabla_{X}\,({\rm
Ric}({Y},{\xi}))=(\nabla_{X}\,{\rm Ric})({Y},{\xi})+2\,{\rm
Ric}({Y},\nabla_{X}\,{\xi})=-2\,{\rm Ric}({Y},\varphi{X}),$

hence ${\rm
Ric}({Y},\varphi{X})=(\operatorname{trace\,}Q)\,g({Y},\varphi{X})$. Therefore,
we obtain

${\rm Ric}({X},{Y})=(\operatorname{trace\,}Q)\,g({X},{Y})$

for any vector fields ${X}$ and ${Y}$ on $M$, which means that $(M,g)$ is an
Einstein manifold. Using the definition of scalar curvature,
$\tau=\operatorname{trace\,}\operatorname{Ric}$, we find
$\tau=(2\,n+1)\operatorname{trace\,}Q$. ∎

######  Remark 4.1.

For a weak $K$-contact manifold, by (17) and
$\nabla_{\xi}\,\varphi=\frac{1}{2}\,N^{\,(5)}(\xi,{Y},\,Z)=0$, see Remark 2.3
and (13), we get the following equality (well known for $K$-contact manifolds,
e.g., [1]):

$\operatorname{Ric}^{\sharp}(\xi)=\sum\nolimits_{\,i=1}^{\,2n}(\nabla_{e_{i}}\,\varphi)\,e_{i},$

where $(e_{i})$ is any local orthonormal basis of ${\cal D}$. For $K$-contact
manifolds, this gives $\operatorname{Ric}^{\sharp}(\xi)=2\,n\,\xi$, see [1,
Proposition 7.2], and $\operatorname{Ric}(\xi,\xi)=2\,n$; moreover, the last
condition characterizes $K$-contact manifolds among all contact metric
manifolds.

##  5 Generalized Ricci solitons on weak $K$-contact manifolds

The generalized Ricci soliton equation in a Riemannian manifold $(M,g)$ is
defined by [10],

$\frac{1}{2}\,\pounds_{{X}}\,g=-c_{1}{X}^{\flat}\otimes{X}^{\flat}+c_{2}\operatorname{Ric}+\lambda\,g$
(23)

for some smooth vector field ${X}$ and real $c_{1},c_{2}$ and $\lambda$. If
${X}=\nabla f$ in (23) for some $f\in C^{\infty}(M)$, then by the definition
${\rm Hess}_{f}({X},{Y})=\frac{1}{2}\,(\pounds_{\nabla f}\,g)$(X,Y), we get
the generalized gradient Ricci soliton equation

${\rm Hess}_{f}=-c_{1}df\otimes df+c_{2}\operatorname{Ric}+\lambda\,g.$ (24)

Each equation above is a generalization of the Einstein metric,
$\operatorname{Ric}+\lambda\,g=0$. For different values of $c_{1},\,c_{2}$ and
$\lambda$, equation (23) is a generalization of Killing equation
($c_{1}=c_{2}=\lambda=0$), equation for homotheties ($c_{1}=c_{2}=0$), Ricci
soliton equation ($c_{1}=0,\,c_{2}=-1$), vacuum near-horizon geometry equation
($c_{1}=1,\ c_{2}=1/2$), e.g., [6].

First, we formulate some lemmas.

######  Lemma 5.1.

For a weak $K$-contact manifold we get

$(\pounds_{\,\xi}(\pounds_{{X}}\,g))({Y},\xi)=g({X},{Y})+g(\nabla_{\xi}\nabla_{\xi}\,{X},{Y})+{Y}g(\nabla_{\xi}\,{X},\xi)$

for all smooth vector fields ${X},{Y}$ with ${Y}$ orthogonal to $\xi$.

###### Proof.

This uses the equalities $\nabla_{\xi}\,\xi=0$ and (18), and is the same as
for [6, Lemma 3.1]. ∎

######  Lemma 5.2 (see, for example, [6]).

Let $(M;g)$ be a Riemannian manifold and $f\in C^{\infty}(M)$. Then the
following holds for every vector fields $\xi,{Y}$ on $M:$

$\pounds_{\,\xi}(df\otimes
df)({Y},\xi)={Y}(\xi(f))\,\xi(f)+{Y}(f)\,\xi(\xi(f)).$

Recall that the Ricci curvature of any $K$-contact manifold satisfies the
following condition:

$\operatorname{Ric}(\xi,{X})=0\quad({X}\in{\cal D}).$ (25)

######  Lemma 5.3.

Let a weak $K$-contact manifold satisfy (25) and admit the generalized
gradient Ricci soliton structure (24). Then

$\nabla_{\xi}\nabla
f=(\lambda+2\,c_{2}n+c_{2}\operatorname{trace\,}\tilde{Q})\,\xi-
c_{1}\xi(f)\,\nabla f.$

###### Proof.

This uses (19) and (24) and is analogous to the proof of [6, Lemma 3.3]. By
(19) and (25) we get

$\lambda\,\eta({Y})+c_{2}\operatorname{Ric}(\xi,{Y})=(\lambda+2\,c_{2}n+c_{2}\operatorname{trace\,}\tilde{Q})\,\eta({Y}).$
(26)

Using (24) and (26), we get

${\rm Hess}_{f}(\xi,{Y})=-c_{1}\xi(f)\,g(\nabla
f,{Y})+(\lambda+2\,c_{2}n+c_{2}\operatorname{trace\,}\tilde{Q})\,\eta({Y}).$
(27)

Thus, (27) and the condition (24) for the Hessian complete the proof. ∎

The next theorem generalizes [6, Theorem 3.1].

######  Theorem 5.1.

Let a weak $K$-contact manifold with $\operatorname{trace\,}\tilde{Q}=const$
satisfy the generalized gradient Ricci soliton equation (24) with
$c_{1}(\lambda+2\,c_{2}n+c_{2}\operatorname{trace\,}\tilde{Q})\neq-1$. Suppose
that condition (25) is true. Then $f=const$. Furthermore, if $c_{2}\neq 0$,
then the manifold is an Einstein one.

###### Proof.

Let ${Y}\in{\cal D}$. Then by Lemma 5.1 with ${X}=\nabla f$, we obtain

$2\,(\pounds_{\,\xi}({\rm
Hess}_{f}))({Y},\xi)={Y}(f)+g(\nabla_{\xi}\nabla_{\xi}\nabla
f,{Y})+{Y}g(\nabla_{\xi}\nabla f,\xi).$ (28)

Using Lemma 5.3 in (28) and the properties $\nabla_{\xi}\,\xi=0$ and
$g(\xi,\xi)=1$, yields

$\displaystyle 2\,(\pounds_{\,\xi}({\rm
Hess}_{f}))({Y},\xi)={Y}(f)+(\lambda+2\,c_{2}n+c_{2}\operatorname{trace\,}\tilde{Q})\,g(\nabla_{\xi}\,\xi,{Y})$
$\displaystyle-c_{1}g(\nabla_{\xi}(\xi(f)\nabla
f),{Y})+(\lambda+2\,c_{2}n+c_{2}\operatorname{trace\,}\tilde{Q})\,{Y}(g(\xi,\xi))-c_{1}{Y}(\xi(f)^{2})$
$\displaystyle={Y}(f)-c_{1}g(\nabla_{\xi}(\xi(f)\nabla
f),{Y})-c_{1}{Y}(\xi(f)^{2}).$ (29)

Using Lemma 5.3 with ${Y}\in{\cal D}$, from (5) it follows that

$2\,(\pounds_{\,\xi}({\rm
Hess}_{f}))({Y},\xi)={Y}(f)-c_{1}\xi(\xi(f))\,{Y}(f)+c_{1}^{2}\xi(f)^{2}{Y}(f)-c_{1}{Y}(\xi(f)^{2}).$
(30)

Since $\xi$ is a Killing vector field, thus $\pounds_{\,\xi}\,g=0$, this
implies $\pounds_{\,\xi}\operatorname{Ric}=0$. Using the above fact and
applying the Lie derivative to equation (24), gives

$2\,(\pounds_{\,\xi}({\rm
Hess}_{f}))({Y},\xi)=-2\,c_{1}(\pounds_{\,\xi}(df\otimes df))({Y},\xi).$ (31)

Using (30), (31) and Lemma 5.2, we obtain

${Y}(f)\big{(}1+c_{1}\xi(\xi(f))+c_{1}^{2}\,\xi(f)^{2}\big{)}=0.$ (32)

By Lemma 5.3, we get

$c_{1}\xi(\xi(f))=c_{1}\,\xi(g(\xi,\nabla f))=c_{1}g(\xi,\nabla_{\xi}\nabla
f)=c_{1}(\lambda+2\,c_{2}n+c_{2}\operatorname{trace\,}\tilde{Q})-c_{1}^{2}\,\xi(f)^{2}.$
(33)

Using (32) in (33), we get
${Y}(f)(c_{1}(\lambda+2\,c_{2}n+c_{2}\operatorname{trace\,}\tilde{Q})+1)=0$.
This implies ${Y}(f)=0$ provided by
$c_{1}(\lambda+2\,c_{2}n+c_{2}\operatorname{trace\,}\tilde{Q})+1\neq 0$.
Hence, $\nabla f$ is parallel to $\xi$. Taking the covariant derivative of
$\nabla f=\xi(f)\,\xi$ and using (16), we obtain

$g(\nabla_{Z}\,\nabla
f,{Y})={Z}(\xi(f))\,\eta({Y})-\xi(f)\,g(\varphi{Z},{Y}),\quad{Z},{Y}\in\mathfrak{X}_{M}.$

From this, by symmetry of ${\rm Hess}_{f}$, i.e., $g(\nabla_{Z}\,\nabla
f,{Y})=g(\nabla_{Y}\,\nabla f,{Z})$, we get $\xi(f)\,g(\varphi{Z},{Y})=0$. For
${Y}=\varphi{Z}$ for some ${Z}\neq 0$, since $g(\varphi{Z},\varphi{Z})>0$, we
get $\xi(f)=0$; so $\nabla f=0$, i.e. $f=const$. Thus, from (24) and
$c_{2}\neq 0$ we conclude that the manifold is an Einstein manifold. ∎

######  Remark 5.1.

The following generalization of gradient Ricci soliton equation was given in
[8]:

${\rm Hess}_{f_{1}}=-c_{1}df_{2}\otimes
df_{2}+c_{2}\operatorname{Ric}+\lambda\,g$ (34)

for some functions $f_{1},f_{2}\in C^{\infty}(M)$ and real $c_{1},c_{2}$ and
$\lambda$. For $f_{1}=f_{2}$, (34) reduces to (24).

Let a weak $K$-contact manifold with $\operatorname{trace\,}\tilde{Q}=const$
satisfy (25) and admit the generalized Ricci soliton structure (34) with
$c_{1}(\lambda+2\,c_{2}n+c_{2}\operatorname{trace\,}\tilde{Q})\neq-1$. Then
similarly to Lemma 5.3, we get

$\nabla_{\xi}\nabla
f_{1}=(\lambda+2\,c_{2}n+c_{2}\operatorname{trace\,}\tilde{Q})\,\xi-
c_{1}\xi(f_{2})\,\nabla f_{2}.$ (35)

Using (35) and Lemmas 5.1 and 5.2, and slightly modifying the proof of Theorem
5.1, we find that the vector field $\nabla f$ is parallel to $\xi$, where
$f=f_{1}+c_{1}(\lambda+2\,c_{2}n+c_{2}\operatorname{trace\,}\tilde{Q})\,f_{2}$.
Thus $df=0$, i.e.,

$df_{1}=-c_{1}(\lambda+2\,c_{2}n+c_{2}\operatorname{trace\,}\tilde{Q})\,df_{2}.$

Using this in (34) and denoting
$a:=\lambda+2\,c_{2}n+c_{2}\operatorname{trace\,}\tilde{Q}$, we get

$-c_{1}a\,{\rm Hess}_{f_{2}}=-c_{1}\,df_{2}\otimes
df_{2}+c_{2}\operatorname{Ric}+\lambda\,g.$ (36)

Then we obtain the following assertion (with three cases) that generalizes
Theorem 5.1.

1\. If $c_{1}a\neq 0$, then (36) reduces to

${\rm Hess}_{f_{2}}=\frac{1}{a}\,df_{2}\otimes
df_{2}-\frac{c_{2}}{c_{1}a}\,\operatorname{Ric}-\frac{\lambda}{c_{1}a}\,g.$

By Theorem 5.1, if $c_{1}a\neq-1$, then $f_{2}=const$; moreover, if $c_{2}\neq
0$, then $(M,g)$ is an Einstein manifold.

2\. If $a=0$ and $c_{1}\neq 0$, then (36) reduces to

$0=c_{2}\operatorname{Ric}-c_{1}\,df_{2}\otimes df_{2}+\lambda\,g.$

If $c_{2}\neq 0$ and $f_{2}\neq const$, then we get a gradient quasi Einstein
manifold. The concept of a quasi Einstein manifold was introduced in [4] by
the condition
$\operatorname{Ric}({X},{Y})=a\,g({X},{Y})+b\,\mu({X})\,\mu({Y})$ for all
vector fields ${X},{Y}$, where $a$ and $b\neq 0$ are real scalars, and $\mu$
is a 1-form of unit norm.

3\. If $c_{1}=0$, then (36) reduces to $0=c_{2}\operatorname{Ric}+\lambda\,g$,
and for $c_{2}\neq 0$ we get an Einstein manifold.

##  6 Conclusion

It is shown that the weak $K$-contact structure is a useful tool for studying
unit Killing vector fields on Riemannian manifolds and that some results for
$K$-contact manifolds can be extended to the case of weak $K$-contact
manifolds. Inspired by [1, Theorems 7.1 and 7.2], the following question can
be posed: is the condition (22), or the weaker condition (19), sufficient for
a weak contact metric manifold to be weak $K$-contact? To answer the question,
some well known results for contact metric manifolds, such as [1, Proposition
7.1], must be generalized for weak contact metric manifolds. In conclusion, we
pose the following questions (about “weak” analogues of results mentioned in
[6, Remark 3.2]): is a compact weak $K$-contact Einstein manifold a Sasakian
manifold? Thus, is a compact weak $K$-contact manifold admitting generalized
Ricci soliton structure a Sasakian manifold?

## References

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  * [2] D.E. Blair,  A survey of Riemannian contact geometry,  _Complex Manifolds_ , 6 (2019), 31–64.
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  * [6] G. Ghosh, and U.C. De,  Generalized Ricci soliton on $K$-contact manifolds,  _Math. Sci. Appl. E-Notes_ , 8 (2020), 165–169.
  * [7] S. Kobayashi, K. Nomizu, Foundations of differential geometry, Vols. I, II, USA, Interscience Publishers, New York–London–Sydney, 1963, 1969.
  * [8] A. Mohammed Cherif, K. Zegga and G. Beldjilali, On the generalised Ricci solitons and Sasakian manifolds. Communications in Mathematics, 30, Issue 1 (2022) 119–123.
  * [9] V. N. Berestovskij, Yu. G. Nikonorov, Killing vector fields of constant length on Riemannian manifolds. _Sib. Math. J._ 49, No. 3 (2008), 395–407.
  * [10] P. Nurowski, and M. Randall,  Generalised Ricci solitons,  _J. Geom. Anal._ , 26 (2016), 1280–1345.
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# The role of geometric phase in the formation of electronic coherences at
conical intersections

Simon P. Neville1 Albert Stolow1,2,3,4 Michael S. Schuurman1,2 1National
Research Council of Canada, 100 Sussex Drive, Ottawa, Ontario K1A 0R6, Canada
2Department of Chemistry and Biomolecular Sciences, University of Ottawa, 150
Louis Pasteur, Ottawa, Ontario, K1N 6N5, Canada  3Department of Physics,
University of Ottawa, 150 Louis Pasteur,Ottawa, ON K1N 6N5 Canada  4University
of Ottawa - National Research Council Joint Centre for Extreme Photonics,
Ottawa ON K1A 0R6, Canada

###### Abstract

The direct observation of non-adiabatic dynamics at conical intersections is a
long-standing goal of molecular physics. Novel time-resolved spectroscopies
have been proposed which are sensitive to electronic coherences induced by the
passage of an excited state wavepacket through a region of conical
intersection. Here we demonstrate that inclusion of the geometric phase
effect, and its manifestations, is essential for the correct description of
the transient electronic coherences that may or may not develop. For
electronic states of different symmetry, electronic coherences are suppressed
by the geometric phase. Conversely, for states of the same symmetry,
appreciable electronic coherences are possible, but their magnitude depends on
both the topography of and direction of approach to the conical intersection.
These general results have consequences for all studies of electronic
coherences at conical intersections.

The concept of seams of conical intersections (CIs) is central to our
understanding of ultrafast electronic relaxation from molecular excited
states. The points forming the seam correspond to degeneracies between
adiabatic electronic states, providing the pathways by which internal
conversion may occur on vibrational timescalesSchuurman and Stolow (2018). The
existence of such ultrafast non-adiabatic pathways is of immense importance in
many areas of photoinduced molecular dynamics, including vision,
photosynthesis, the photostability of biomolecules, and light harvesting
systems. There is thus global interest in developing experimental methods to
directly observe populations and coherences induced by CI mediated molecular
dynamics.

Ultrafast spectroscopic probes of dynamics at CIs can be broadly split into
two categories. The first encompasses methods which are sensitive to
electronic state population dynamics of a vibronic wavepacket as it transits
the region of strong non-adiabatic coupling surrounding a CI seam. These
include, but are not limited to, ultrafast pump-probe spectroscopies such as
transient absorption or time-resolved photoelectron spectroscopyNeville  _et
al._ (2016, 2018); Bhattacherjee and Leone (2018); Schuurman and Stolow
(2018); Stolow and Underwood (2008); Stolow  _et al._ (2004). The second
category is comprised of (generally non-linear) spectroscopic methods which
are sensitive to the transient electronic coherences induced as a wavepacket
passes through a region encompassing a CIKowalewski  _et al._ (2015, 2017);
Cho  _et al._ (2020); Makhija  _et al._ (2020); Sun  _et al._ (2020).

For a coherence to exist between two electronic states, two conditions must be
satisfied: (i) both electronic states must be appreciably populated; (ii) the
nuclear components of the wavepacket on each adiabatic state must have
appreciable overlap. Since the nonadiabatic coupling becomes large in the
region around a CI, population is readily transferred between the electronic
states, ensuring the first criterion is met. However, we show here that the
geometric phaseRyabinkin  _et al._ (2017) effect can lead to zero, or near-
zero, overlaps of the nuclear components of the wavepacket on different
electronic states. For example, the geometric phase effect is found to
suppress the formation of coherences between electronic states of different
symmetry. Conversely, for CIs between electronic states of the same symmetry,
large coherences may form. However, the magnitude of the electronic coherences
thus formed strongly depends on both the CI topography and the direction of
approach of the wavepacket.

We begin with a brief exposition of the problem using a minimal model which
captures all the fundamental physical effects involved. Consider the
excitation of a molecule from its ground electronic state to an excited state
manifold spanned by two adiabatic states $|\psi_{1}^{(a)}\rangle$ and
$|\psi_{2}^{(a)}\rangle$. We consider the case where only
$|\psi_{2}^{(a)}\rangle$ is initially populated. The non-stationary vibronic
wavepacket produced in $|\psi_{2}^{(a)}\rangle$ may evolve towards a CI
between the two states, leading to internal conversion and population transfer
to $|\psi_{1}^{(a)}\rangle$. This process is more easily modeled by switching
to a set of diabatic states, $|\psi_{1}^{(d)}\rangle$ and
$|\psi_{2}^{(d)}\rangle$, related to the adiabatic states by a unitary
transformation,

$\begin{bmatrix}|\psi_{1}^{(a)}\rangle\\\
|\psi_{2}^{(a)}\rangle\end{bmatrix}=\begin{bmatrix}\cos{\theta}&\sin{\theta}\\\
-\sin{\theta}&\cos{\theta}\\\
\end{bmatrix}\begin{bmatrix}|\psi_{1}^{(d)}\rangle\\\
|\psi_{2}^{(d)}\rangle\end{bmatrix},$ (1)

where $\theta$ is the (geometry-dependent) adiabatic-to-diabatic
transformation (ADT) angle.

In the diabatic representation, the Hamiltonian matrix reads

$\boldsymbol{H}=\hat{T}_{n}\boldsymbol{1}_{2}+\begin{bmatrix}W_{11}&W_{21}\\\
W_{12}&W_{22}\end{bmatrix},$ (2)

where $\hat{T}_{n}$ is the nuclear kinetic energy operator and $W_{ij}$ are
the elements of the diabatic potential matrix, the off-diagonal elements of
which account for the non-adiabatic coupling between the two electronic
states.

First, consider a system comprised of two electronic states of different
symmetry. The simplest possible model correctly accounting for the underlying
physics and symmetries of this problem is a two-mode linear vibronic coupling
(LVC) Hamiltonian, $\hat{H}_{LVC}$, expressed in terms of a totally symmetric
tuning mode $q_{t}$ and a coupling mode $q_{c}$:

$\displaystyle\boldsymbol{H}_{LVC}$
$\displaystyle=\left[\hat{T}_{n}+\frac{1}{2}\left(\omega_{t}q_{t}^{2}+\omega_{c}q_{c}^{2}\right)\right]\boldsymbol{1}_{2}$
(3) $\displaystyle+\begin{bmatrix}\kappa_{1}q_{t}&\lambda q_{c}\\\ \lambda
q_{c}&\Delta+\kappa_{2}q_{t}\end{bmatrix}.$

This model describes a system of shifted, coupled harmonic oscillators with
frequencies $\omega_{t/c}$ and a CI located at
$\boldsymbol{Q}_{CI}=(q_{t}=\Delta/(\kappa_{1}-\kappa_{2}),q_{c}=0)$. Note
that the coupling mode in this case is non-totally symmetric, with a symmetry
given by the direct product of the irreducible representations generated by
the two electronic states.

The quantity of interest here is the magnitude of the coherence between the
adiabatic electronic states as the excited state wavepacket passes through the
CI. Let $\Psi(t)$ denote the total vibronic wavepacket, which can be expressed
in the Born-Huang framework as

$\Psi(t)=\sum_{j=1}^{2}|\psi_{j}^{(a)}\rangle\chi_{j}^{(a)}(q_{t},q_{c},t).$
(4)

where the $\chi_{j}^{(a)}$ are the adiabatic nuclear wavepackets.

The magnitude of the coherence between the electronic states is given by the
absolute value of off-diagonal element of the reduced electronic density
matrix in the adiabatic representation,

$\rho_{12}^{(a)}(t)=\left\langle\Psi(t)\middle|\psi_{1}^{(a)}\right\rangle\left\langle\psi_{2}^{(a)}\middle|\Psi(t)\right\rangle=\left\langle\chi_{1}^{(a)}(t)\middle|\chi_{2}^{(a)}(t)\right\rangle,$
(5)

As stated above, in order for $|\rho_{12}^{(a)}(t)|$ to be large in value, two
conditions must be met: both electronic states must be populated and the
nuclear wavepackets, $\chi_{j}^{(a)}$, on each adiabatic state must have
appreciable overlap. The strong non-adiabatic coupling in the CI region
ensures that the first condition is met. The second condition, however, is
generally not satisfied when the electronic states are of different symmetry.

The reason for poor overlap between the two adiabatic nuclear wavepackets of
concern is best seen by re-writing $\rho_{12}^{(a)}(t)$ as

$\rho_{12}^{(a)}(t)=A_{12}(t)+B_{12}(t),$ (6)

$2A_{12}(t)=\left\langle\chi_{1}^{(d)}\middle|\sin
2\theta\middle|\chi_{1}^{(d)}\right\rangle+\left\langle\chi_{2}^{(d)}\middle|\sin
2\theta\middle|\chi_{2}^{(d)}\right\rangle$ (7)

$2B_{12}(t)=\left\\{\left\langle\chi_{1}^{(d)}\middle|\chi_{2}^{(d)}\right\rangle+\left\langle\chi_{1}^{(d)}\middle|\cos
2\theta\middle|\chi_{2}^{(d)}\right\rangle\right\\}+h.c.$ (8)

where $\theta$ is the ADT angle, and the $\chi_{j}^{(d)}$ are the diabatic
nuclear wavepackets. In the following, we will refer to $A_{12}$ as the “on-
diagonal” contribution to $\rho_{12}^{(a)}$, and $B_{12}$ as the “off-
diagonal” contribution.

We can show that both the $A_{12}$ and $B_{12}$ terms in Equation 7 vanish
using the following simple symmetry arguments:

  1. 1.

Because the two electronic states are of different symmetry, the coupling mode
$q_{c}$ will be non-totally symmetric. This implies that $W_{11}$ and $W_{22}$
are even functions with respect to $q_{c}$, while $W_{12}$ is an odd function
of $q_{c}$.

  2. 2.

Given the even symmetry of $W_{22}$ with respect to $q_{c}$, and assuming
vertical excitation from the ground state, $\chi_{2}^{(d)}$ will be an even
function of $q_{c}$.

  3. 3.

Given the odd symmetry of $W_{12}$ with respect to $q_{c}$ and the even
symmetry of $\chi_{2}^{(d)}$, the nuclear wavepacket $\chi_{1}^{(d)}$ formed
on the lower electronic state will have a node along the coupling mode
$q_{c}$, and thus will be an odd function of this coordinate.

  4. 4.

Using the relation

$2\theta=\arctan\frac{2W_{12}}{W_{22}-W_{11}},$ (9)

it can be shown that the functions $\sin{2\theta(q_{t},q_{c})}$ and
$\cos{2\theta(q_{t},q_{c})}$ are odd and even with respect to $q_{c}$,
respectively (see the Supplementary Information for more details).

Given these symmetries, we see that the integrands in both Equations 7 and 8
are all overall odd with respect to $q_{c}$ and vanish in this two-mode LVC
model. We conclude that electronic coherences are suppressed for CIs between
states of different symmetry.

As detailed in the Supplementary Information, the symmetries of
$\sin{2\theta(q_{t},q_{c})}$ and $\cos{2\theta(q_{t},q_{c})}$ are intimately
linked to the geometric phase effect. Further, the geometric phase effect will
exist if the leading term in the diabatic coupling is odd with respect to
$q_{c}$. This symmetry is realized in the LVC model above and is responsible
for the node in $\chi_{1}^{(d)}$. This is the reason why the geometric phase
effect is present in the case of conical intersections, but is absent for
glancing (e.g. Renner-Teller) intersections. Thus, the suppression of
coherences between electronic states of different symmetry is best understood
as simply a consequence of the geometric phase effect.

To demonstrate this, we note that by forcing the diabatic coupling to contain
only even terms with respect to the coupling mode(s), the geometric phase
effect may be “turned off” Izmaylov  _et al._ (2016). This can be achieved by
replacing $W_{12}$ in Equation 2 by its absolute value:
$W_{12}\rightarrow|W_{12}|$. In the following, we denote the physically
correct Hamiltonian as $\hat{H}_{\text{wGP}}$ (i.e. with geometric phase), and
the Hamiltonian with $W_{12}$ replaced by its absolute value by
$\hat{H}_{\text{noGP}}$ (i.e. no geometric phase). Importantly, using
$\hat{H}_{\text{noGP}}$ has no effect on the adiabatic potential energy
surfaces. However, it allows us to clearly demonstrate the role of the
geometric phase effect on the formation of electronic coherences. In the
following, we perform wavepacket propagations using both
$\hat{H}_{\text{wGP}}$ and $\hat{H}_{\text{noGP}}$ and track the resulting
changes in the values of $|\rho_{12}^{(a)}(t)|$.

As a representative system, we choose the excited state dynamics of pyrazine,
using a four-mode model to describe the coupled dynamics of the
$B_{3u}(n\pi^{*})$ and $B_{2u}(\pi\pi^{*})$ states (adapted from Reference
14). This well-known model correctly reproduces the short-time excited state
dynamics of pyrazine, including the passage of the excited state wavepacket
through the CI between the $B_{3u}(n\pi^{*})$ and $B_{2u}(\pi\pi^{*})$ states.
We begin by considering the symmetries of the nuclear wavepackets
$\chi_{1}^{(d)}$ and $\chi_{2}^{(d)}$ formed following vertical excitation to
the bright $B_{3u}(n\pi^{*})$ state. These symmetries are most readily
discerned from the phase angles $\zeta_{j}$ defined via the the polar
representation of the diabatic nuclear wavepackets,

$\chi_{j}^{(d)}(\boldsymbol{Q},t)=r_{j}(\boldsymbol{Q},t)e^{i\zeta_{j}(\boldsymbol{Q},t)}.$
(10)

A phase angle satisfying $\zeta_{j}(q_{c})=\zeta_{j}(-q_{c})$ corresponds to
an even function of $q_{c}$, while $\zeta_{j}(q_{c})=\zeta_{j}(-q_{c})\pm\pi$
implies and odd function of $q_{c}$. Shown in Figure 1 are the squared
absolute values $|\chi_{j}^{(d)}|^{2}$ of the diabatic nuclear wavepackets
colored by the phase angles $\zeta_{j}$. In these plots,
$|\chi_{j}^{(d)}|^{2}$ and $\zeta_{j}$ are shown plotted along $q_{c}$ with
all other modes set to their time-evolving centroid values.

As shown by the amplitude (i.e. the magnitude in the z direction) of the
nuclear wavepackets in Figures 1a and 1b, it is the upper diabatic state which
is primarily populated at early times. However, this population is quickly
depleted following the initial passage through the CI region at around 40 fs.
Conversely, the lower diabatic state rapidly accumulates population at this
time, as evinced by Figures 1c and 1d. These observations regarding the state
populations apply equally well to both the $\hat{H}_{\text{wGP}}$ and
$\hat{H}_{\text{noGP}}$ simulations. That is, the geometric phase is of little
consequence for the population dynamics, as shown below in Figure 2a (vide
infra).

In contrast, there are significant differences in the phase of nuclear wave
packets determined from $\hat{H}_{\text{wGP}}$ and $\hat{H}_{\text{noGP}}$.
The phase angle for $\chi_{2}^{(d)}$ is an even function of $q_{c}$ for both
models, as can be seen in Figures 1a and 1b. As discussed above, the use of
the physically correct Hamiltonian $\hat{H}_{\text{wGP}}$ causes a node to
form in $\chi_{1}^{(d)}$, resulting in it being an odd function of $q_{c}$
(see Figure 1c). Using the unphysical Hamiltonian $\hat{H}_{\text{noGP}}$
results in $\chi_{1}^{(d)}$ being an even function of $q_{c}$, as shown in
Figure 1d. Thus, when coupled with the above discussed symmetry properties of
the ADT angle (see Supplementary Information), the use of
$\hat{H}_{\text{wGP}}$ results in the suppression of the electronic coherences
in this system, whilst the use of $\hat{H}_{\text{noGP}}$ results in
(spurious) electronic coherences of large magnitude. This is clearly seen in
Figure 2b, where we show the magnitudes $|\rho_{12}^{(a)}(t)|$ of the
electronic coherences formed when using both $\hat{H}_{\text{wGP}}$ and
$\hat{H}_{\text{noGP}}$, where the latter are completely suppressed.
Significantly, although the magnitude of the electronic coherences formed
using $\hat{H}_{\text{wGP}}$ and $\hat{H}_{\text{noGP}}$ are entirely
different, the population dynamics are remarkably similar, as illustrated in
Figure 2a. This result shows that although neglecting the geometric phase
effect may have only mild consequences for the simulation of electronic state
population dynamics, it must be properly accounted for when studying
electronic coherences.

We now consider the case of CIs between two electronic states of the same
symmetry. Here, the coupling mode $q_{c}$ will generate the totally symmetric
irreducible representation of the point group of the molecule. Hence, there
can exist non-zero gradients of diabatic potentials with respect to $q_{c}$,
and the diabatic nuclear wavepackets $\chi_{1/2}^{(d)}$ will no longer
necessarily be even or odd functions of it, Accordingly, neither the on- or
off-diagonal contributions to $\rho_{12}^{(a)}$ necessarily vanish, and it is
possible for appreciable electronic coherences to form.

The magnitude of the electronic coherence formed in this same symmetry case
depends on both the topography of the CI and the direction of approach of the
wavepacket to the CI. To see this, consider the first-order expansion of the
diabatic potentials about the CI point, $\boldsymbol{X}_{CI}$, in terms of
intersection-adapted coordinates $x$ and $y$Atchity  _et al._ (1991),

$\boldsymbol{W}^{(1)}(x,y)=\left(s_{x}x+s_{y}y\right)\boldsymbol{1}_{2}+\begin{bmatrix}-gx&hy\\\
hy&gx\end{bmatrix},$ (11)

where $g$ and $h$ are the norms of the gradient difference and non-adiabatic
coupling vectors evaluated at the CI point, respectively. The terms $s_{x/y}$
are the gradients of the average energy with respect to $x$ and $y$ at the CI
point, and determine whether the CI is “sloped” or “peaked”Atchity  _et al._
(1991); Yarkony (2001). In Equation 11, the diabatic and adiabatic
representations are equal at the CI point $\boldsymbol{X}_{CI}$. In order to
apply symmetry arguments analogous to the different state symmetry case above,
we require that the linear component of the diabatic coupling introduces a
node at the centre of $\chi_{1}^{(d)}$. This is achieved by transforming to a
different, but entirely equivalent, diabatic representation in which the
diabatic and adiabatic representations are equal at the centre of the initial
wavepacket $\Psi(t=0)$. Note that this is possible because the ADT is only
defined up to a constant unitary transformation. Let $\theta_{0}$ denote the
ADT angle of the original diabatic representation evaluated at the centre of
the initial wavepacket. Then, as detailed in the Supplementary Information,
the first-order potential in the new diabatic representation takes the form

$\boldsymbol{W}^{(1)}(x,y)=\left(s_{x}x+s_{y}y\right)\boldsymbol{1}_{2}+\begin{bmatrix}-\cos{\left(2\theta_{0}\right)}gx+\sin{\left(2\theta_{0}\right)}hy&\cos{\left(2\theta_{0}\right)}hy+\sin{\left(2\theta_{0}\right)}gx\\\
\cos{\left(2\theta_{0}\right)}hy+\sin{\left(2\theta_{0}\right)}gx&\cos{\left(2\theta_{0}\right)}gx-\sin{\left(2\theta_{0}\right)}hy\end{bmatrix}.$
(12)

To form an electronic coherence in this first-order vibronic coupling model,
there must be both on- and off-diagonal elements which are non-vanishing with
respect to either $x$ or $y$. The diabatic nuclear wavepackets
$\chi_{1/2}^{(d)}$ will then be neither even nor odd with respect to both
nuclear degrees of freedom, allowing for non-zero values of both the on- and
off-diagonal contributions to $\rho_{12}^{(a)}$. Importantly, from Equation 12
we see that this criterion will be satisfied if either: (i) $s_{y}\neq 0$,
and/or; (ii) $\sin{\left(2\theta_{0}\right)}\neq 0$. The parameter $s_{y}$
determines the tilt of the CI axis along the non-adiabatic coupling
directionAtchity  _et al._ (1991), whereas $\theta_{0}$ is determined by the
position of the initial wavepacket relative to the CI point. In a trajectory-
based treatment of the nuclear dynamics, this initial position would be
analogous to the “direction of approach” to the CI. In the limiting case of a
peaked (i.e., non-tilted) CI and an initial wavepacket displaced from the CI
purely along the gradient difference direction, the dominant first-order
contributions to the electronic coherence will, again, vanish.

To illustrate the effects of CI topography on electronic coherences, we
consider the case where the centre of the initial wavepacket is displaced from
the CI point purely along the gradient difference direction. In this case, the
magnitude of the electronic coherence will be determined by the tilt of the CI
axis along the non-adiabatic coupling direction $y$, and thus, by the
parameter $s_{y}$ in Equation 12. A two-mode, two-state LVC Hamiltonian was
constructed to describe ultrafast, gradient-directed internal conversion
through a CI (see the Supplementary Information for the parameters used). The
parameter $s_{y}$ was varied to yield tilt angles, $\alpha_{y}$, of
$0.5^{\circ}$, $3^{\circ}$, $7^{\circ}$ and $10^{\circ}$ along the $y$
direction. The conical intersections for these tilt angles are shown in
Figures 3a through 3d. For $\alpha_{y}=0.5^{\circ}$, the CI is almost peaked
along $y$. Upon increasing $\alpha_{y}$, the slope of the CI along $y$
gradually increases. As illustrated in Figure 3d, the magnitude of the
electronic coherence is negligible for a nearly peaked CI
($\alpha_{y}=0.5^{\circ}$), and grows with increasing tilt angle along $y$.
Again, the adiabatic state population dynamics are found to be only very
weakly affected by the tilt of the CI along $y$. Finally, we show in Figure 3e
the same electronic state populations and coherences, but with the geometric
phase effect removed from the model Hamiltonian (via the use of
$\hat{H}_{\text{noGP}}$). Strikingly, the removal of the geometric phase
effect results - erroneously - in large magnitude electronic coherences for
all tilt angles. This again serves to highlight the importance of correctly
accounting for the geometric phase in any simulation of electronic coherences
at CIs.

In summary, we have explored the factors affecting the formation of electronic
coherences as a wavepacket passes through a CI. Specifically, we have shown
that the explicit consideration of the geometric phase effect is essential for
a qualitatively correct description of the coherences which may or may not
form in the vicinity of a CI. In the case of two electronic states of
different symmetry, geometric phase is responsible for the suppression of
electronic coherences around a CI. For the unavoidable case of a CI between
electronic states of the same symmetry, electronic coherences may in general
form. However, their magnitude depends on both the topography of the CI and
the direction of approach of the wavepacket to it. These results will help to
identify molecular systems for the experimental study of electronic coherences
in dynamics at CIs, an opportunity identified in a recent road-map on
ultrafast X-ray scienceHeinz  _et al._ . We emphasize that a proper accounting
of the geometric phase effect is required in any theoretical study of
electronic coherences induced by nuclear motion near CIs. This recognition
should result in the simulations required to guide experimental efforts to
identify unique signatures of CI dynamics.

##  I Acknowledgments

The authors thank A. F. Izmaylov, I. G. Ryabikin, and L. Joubert-Doriol for
helpful discussions.

Figure 1: Dynamics at a conical intersection in pyrazine, showing the nuclear
wavepacket evolution in the coupled $B_{3u}(n\pi^{*})$ and
$B_{2u}(\pi\pi^{*})$ states. The the magnitude in the z direction in each plot
shows the squared absolute values $|\chi_{j}^{(d)}|^{2}$ of the diabatic
nuclear wavepackets. The phase angle $\zeta_{j}$ of the wavepackets is encoded
in the color maps. (a) Upper diabatic state using $\hat{H}_{\text{wGP}}$. (b)
Upper diabatic state using $\hat{H}_{\text{noGP}}$. (c) Lower diabatic state
using $\hat{H}_{\text{wGP}}$. (d) Lower diabatic state using
$\hat{H}_{\text{noGP}}$. In all cases, the phase angle is plotted along the
coupling mode $q_{c}$ with all other nuclear degrees set to their time-
dependent centroid values.

Figure 2: Quantum dynamics of pyrazine following vertical excitation to the
optically bright $B_{2u}(\pi\pi^{*})$ using $\hat{H}_{\text{wGP}}$ (solid
lines) and $\hat{H}_{\text{noGP}}$ (dashed lines). (a) Adiabatic state
populations. (b) Electronic coherences. It can clearly be seen that omission
of the geometric phase leads, incorrectly, to large electronic coherences
which vanish when geometric phase is properly included.

Figure 3: Conical intersections, population and electronic coherence dynamics
in a two-mode, two-state model system as a function of the CI tilt angle
$\alpha_{y}$ along the non-adiabatic coupling direction $y$. In panels (a) to
(d) we show adiabatic potential surfaces as a function of increasing tilt
angle $\alpha_{y}$. (e) Populations (solid lines) and electronic coherences
(dashed lines) computed calculated using the physically correct
$\hat{H}_{\text{wGP}}$. (f) Populations and coherences computed using the
physically incorrect $\hat{H}_{\text{noGP}}$. It can be seen that the omission
of geometric phase leads to artificially larger electronic coherences for all
tilt angles. In both cases, the population and coherence dynamics correspond
to an initial wavepacket displaced from the CI point only along the gradient
difference direction, thus isolating the effects due to CI topography alone.

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