# K-contact manifolds with minimal closed Reeb orbits

Hui Li School of mathematical Sciences
Soochow University
Suzhou, 215006, China<EMAIL_ADDRESS>
###### Abstract.

We use the Boothby-Wang fibration to construct certain simply connected
K-contact manifolds and we give sufficient and necessary conditions on when
such K-contact manifolds are homeomorphic to the odd dimensional spheres. For
the symplectic base manifold of the fibration which admits a Hamiltonian torus
action, we show that the total space of the fibration admits other than the
above K-contact structures whose closed Reeb orbits correspond exactly to the
fixed points of the Hamiltonian circle action on the base manifold. Then we
give a collection of examples of compact simply connected K-contact manifolds
with minimal number of closed Reeb orbits which are not homeomorphic to the
odd dimensional spheres, while having the real cohomology ring of the spheres.
Finally, we give a family of examples of simply connected K-contact manifolds
which have one more than the minimal number of closed Reeb orbits and which do
not have the real cohomology ring of the spheres.

###### Key words and phrases:

K-contact manifold, Reeb orbit, Hamiltonian action, cohomology ring, Chern
classes

2010 classification. Primary: 53D10, 53D05, 53D20; Secondary: 55N10, 57R20

##  1\. introduction

A contact manifold is a smooth manifold $M$ of odd dimension $2n+1$ with a
contact 1-form $\alpha$ such that $\alpha\wedge(d\alpha)^{n}\neq 0$
everywhere. The Reeb vector field $R$ of $\alpha$ is the vector field
characterized by $\alpha(R)=1$ and $d\alpha(R,\cdot)=0$; the flow of $R$ is
called the contact flow or the Reeb flow. Hence the contact 1-form $\alpha$
defines a symplectic form $\omega=d\alpha$ on the $2n$-dimensional subbundle
$\ker(\alpha)$. The contact manifold $(M,\alpha)$ is called regular if the
Reeb vector field $R$ of $\alpha$ generates a free circle action. A Riemannian
metric $g$ on $M$ is said to be adapted to $\alpha$ if $g$ is preserved by the
Reeb flow of $\alpha$, and there exists an almost complex structure $J$ on
$\ker(\alpha)$ such that $g|_{\ker(\alpha)}$, $J$ and $d\alpha$ are
compatible, i.e., $g(X,Y)=d\alpha(X,JY)$ for all $X,Y$ in $\ker(\alpha)$. When
there is an adapted Riemannian metric $g$ on $(M,\alpha)$, we call
$(M,\alpha,g)$ a K-contact manifold, and call $(\alpha,g)$ a K-contact
structure on $M$. For a compact K-contact manifold $(M,\alpha,g)$, the closure
of the Reeb flow in the isometry group of $M$ is a compact abelian Lie group,
a $k$-dimensional torus $T^{k}$, with $1\leq k\leq n+1$ if $\dim(M)=2n+1$ (see
[23, 25]), and $k$ is called the rank of the K-contact manifold
$(M,\alpha,g)$. Clearly the torus $T^{k}$ acts on the K-contact manifold
$(M,\alpha,g)$ preserving the contact form $\alpha$ and the Riemannian metric
$g$. On a smooth manifold $M$, there may exist different contact forms and
K-contact structures with different ranks. For example, the sphere $S^{2n+1}$
with its standard contact form $\alpha_{0}$ induced from the standard
symplectic form on ${\mathbb{C}}^{n+1}$ is a regular contact manifold;
$\alpha_{0}$ and the standard metric $g_{0}$ on $S^{2n+1}$ give $S^{2n+1}$ a
K-contact structure $(\alpha_{0},g_{0})$ of rank 1. A perturbation of
$\alpha_{0}$ can give $S^{2n+1}$ a different contact form $\alpha$, which has
an adapted Riemannian metric $g$, a perturbation of $g_{0}$. We may have a
K-contact manifold $(S^{2n+1},\alpha,g)$ of rank $k$ for any $2\leq k\leq
n+1$, see Example 6.5.

Recall that a contact toric manifold is a contact manifold of dimension $2n+1$
with an effective contact $T^{n+1}$-action; it is called of Reeb type if the
Reeb vector field of the contact form is generated by a one parameter subgroup
action of $T^{n+1}$. We can equip contact toric manifolds of Reeb type with
$T^{n+1}$-invariant Riemannian metrics adapted to the contact form, hence
contact toric manifolds of Reeb type is a special subclass of K-contact
manifolds, which are of maximal rank.

In this paper, we are interested in working on a special class of K-contact
manifolds which are principal circle bundles over symplectic manifolds with
symplectic forms representing integral cohomology classes. The fact that these
spaces possess contact and K-contact structures is due to Boothby and Wang.
Before the K-contact structures are very involved, we first make some claims
on principal circle bundles over general compact manifolds, and over compact
symplectic manifolds with Hamiltonian circle actions.

Recall that principal circle bundles over a compact manifold $N$ are
classified by $H^{2}(N;{\mathbb{Z}})$ through a characteristic class, which we
call the Euler class of the circle bundle ([10]). We first take a general base
manifold in Theorem 1 and then take the base manifold to be a symplectic
manifold in Theorem 2.

######  Theorem 1.

Let $(N,\omega)$ be a compact smooth manifold of dimension $2n$, $x$ be a
primitive integral degree 2 cohomology class of $N$, and $\pi\colon M\to N$ be
a principal circle bundle over $N$ with Euler class $x$. Then the following
claims hold.

  1. (1)

$M$ has the real cohomology ring of $S^{2n+1}$ if and only if $N$ has the real
cohomology ring of ${\mathbb{CP}}^{n}$, i.e.,
$H^{*}(N;{\mathbb{R}})={\mathbb{R}}[x]/x^{n+1}$ as a ring.

  2. (2)

$M$ has the integral cohomology ring of $S^{2n+1}$ if and only if $N$ has the
integral cohomology ring of ${\mathbb{CP}}^{n}$, i.e.,
$H^{*}(N;{\mathbb{Z}})={\mathbb{Z}}[x]/x^{n+1}$ as a ring.

  3. (3)

$M$ is homeomorphic to a sphere $S^{2n+1}$ if and only if $\pi_{1}(N)=1$ and
$H^{*}(N;{\mathbb{Z}})={\mathbb{Z}}[x]/x^{n+1}$ as a ring.

Recall that if a compact symplectic manifold $(N,\omega)$ of dimension $2n$
admits a Hamiltonian circle action, then the action has at least $n+1$ fixed
points. This is due to the fact that the moment map is a perfect Morse
function, and $b_{2i}(N)\geq 1$ for any $0\leq 2i\leq 2n$. Now we take a
compact symplectic manifold $(N,\omega)$ of dimension $2n$ as a base manifold
in a principal circle bundle, then the Euler class of the principal circle
bundle is the same as the first Chern class of the principal circle bundle. We
introduce a Hamiltonian circle action on $N$, then a condition on the base
manifold is sufficient or sufficient and necessary for the total space to be
homeomorphic to $S^{2n+1}$, this is Theorem 2 below. In Theorem 2 (2),
$c_{1}(\cdot)$ and $c(\cdot)$ are respectively the first Chern class and total
Chern class of the corresponding space.

######  Theorem 2.

Let $(N,\omega)$ be a compact symplectic manifold of dimension $2n$ with
$[\omega]$ being a primitive integral cohomology class, and let $\pi\colon
M\to N$ be a principal circle bundle with first Chern class $x=[\omega]$.
Assume $N$ admits a Hamiltonian circle action with fixed point set consisting
of $n+1$ isolated points. Then the following claims hold.

  1. (1)

$M$ is homeomorphic to a sphere $S^{2n+1}$ if and only if
$H^{*}(N;{\mathbb{Z}})={\mathbb{Z}}[x]/x^{n+1}$ as a ring.

  2. (2)

If $c_{1}(N)=(n+1)x$, then $c(N)\cong c({\mathbb{CP}}^{n})=(1+x)^{n+1}$, and
$M$ is homeomorphic to a sphere $S^{2n+1}$.

In Theorem 2, if there is a Hamiltonian torus action on $N$, then there is a
Hamiltonian subcircle action on $N$ with the same fixed point set as the torus
action. Hence for the claims in Theorem 2, assuming a circle action on $N$ is
more general than assuming a torus action. See Remark 3.5.

In Theorem 2 (2), except in low dimensions $2n\leq 8$, the author does not
know if $c_{1}(N)=(n+1)x$ is a necessary condition for $M$ to be homeomorphic
to $S^{2n+1}$. If $H^{*}(N;{\mathbb{Z}})={\mathbb{Z}}[x]/x^{n+1}$ implied
$c_{1}(N)=(n+1)x$, then we would have the necessary condition by (1). When
there is a maximal dimensional symmetry, namely there is a Hamiltonian
$T^{n}$-action on $N$ with exactly $n+1$ isolated fixed points, then $M$ is
homeomorphic to $S^{2n+1}$ implies $\pi_{1}(N)=1$ and
$H^{*}(N;{\mathbb{Z}})={\mathbb{Z}}[x]/x^{n+1}$ as a ring by Theorem 1 (3); by
Petrie’s work ([24]), we have the total Chern class isomorphism $c(N)\cong
c({\mathbb{CP}}^{n})=(1+x)^{n+1}$ (in particular $c_{1}(N)=(n+1)x$). Or,
solely for the symplectic manifold $N$, when there is a Hamiltonian
$T^{n}$-action on $N$ with exactly $n+1$ isolated fixed points, by Jang’s work
in [6] for almost complex torus manifolds, we have $c(N)\cong
c({\mathbb{CP}}^{n})$, hence $c_{1}(N)=(n+1)x$ (independant of what $M$ is).

In Theorem 2, we assume that the fixed points of the Hamiltonian circle action
on $N$ are isolated, this is what we concern in the current work. When the
fixed points of the Hamiltonian circle action on $N$ are not isolated, we may
still have the claims (1) and (2) of the theorem, and in (2), in certain case,
$M$ is homeomorphic to $S^{2n+1}$ implies $c(N)=c({\mathbb{CP}}^{n})$. See
Remark 3.4.

Next, the contact and K-contact structures on the total space of a principal
circle bundle over a symplectic base manifold are more involved. Let
$(N,\omega)$ be a compact symplectic manifold of dimension $2n$ with
$[\omega]$ being a primitive integral cohomology class, and let $\pi\colon
M\to N$ be a principal circle bundle with first Chern class $[\omega]$. By
Boothby and Wang ([1]), there is a connection 1-form $\alpha$ on $M$ such that
$d\alpha=\pi^{*}(\omega)$, $(M,\alpha)$ is a compact contact manifold, and the
Reeb flow of $\alpha$ generates a free circle action which is exactly the
principal circle action of the circle bundle. In this case, we call the
principal circle bundle $\pi\colon M\to N$ a Boothby-Wang fibration. For any
Riemannian metric $g_{N}$ on $N$ compatible with $\omega$,
$g=\pi^{*}(g_{N})+\alpha\otimes\alpha$ defines an adapted metric on $M$ (see
Section 3), so $(M,\alpha,g)$ becomes a K-contact manifold; it is of rank one
since each Reeb orbit of $\alpha$ is closed. Note that we can have many
different choices of $g_{N}$, hence of $g$. Assume now that $N$ admits a
Hamiltonian torus $T$ action, we will see that we can lift the $T$-action on
$N$ to a $T$-action on $M$ preserving the connection one form $\alpha$ and the
metric $g=\pi^{*}(g_{N})+\alpha\otimes\alpha$ if $g_{N}$ is $T$-invariant on
$N$. Let $S^{1}$ be the fiber transformation group of the principal circle
bundle $\pi\colon M\to N$. Then the $T\times S^{1}$-action preserves $\alpha$
and $g$ on $M$, so $T\times S^{1}$ is in the isometry group of the K-contact
manifold $(M,\alpha,g)$. We will show that other than the regular K-contact
structures above, for each $2\leq k\leq\dim(T\times S^{1})$, $M$ admits a
K-contact structure of rank $k$ such that its closed Reeb orbits are in one to
one correspondence with the fixed points of the Hamiltonian $T$-action on $N$.

######  Theorem 3.

Let $(N,\omega)$ be a compact symplectic manifold of dimension $2n$ with
$[\omega]$ being a primitive integral class, and let $\pi\colon M\to N$ be a
principal circle bundle with first Chern class $[\omega]$. Assume $N$ admits a
Hamiltonian torus $T^{l}$-action with $1\leq l\leq n$, then for any $2\leq
k\leq l+1$, there exists a K-contact structure on $M$ of rank $k$ whose closed
Reeb orbits are in one to one correspondence with the fixed points of the
$T^{l}$-action on $N$. If $(N,\omega)$ is a Kähler manifold, then the above
K-contact structures on $M$ are Sasakian structures.

In the Boothby-Wang fibration $\pi\colon M\to N$, for the regular contact form
$\alpha$ with $d\alpha=\pi^{*}(\omega)$, the quotient space by the Reeb flow
is just the manifold $N$. For each K-contact form in the conclusion of Theorem
3, the closure of the Reeb flow is $T^{k}$, the quotient space by the Reeb
flow is not a Hausdorff space, it is locally the quotient of a Cartesian space
by an affine action of a countable group; for a general description of such a
quotient space, see [19, Theorem 1.3].

In Theorem 3, in particular, if the Hamiltonian $T$-action on $N$ has exactly
$n+1$ isolated fixed points, then the $2n+1$-dimensional compact manifold $M$
admits a K-contact structure with exactly $n+1$ closed Reeb orbits. Similar to
the symplectic case, $n+1$ is the minimal number of closed Reeb orbits that a
compact K-contact manifold of dimension $2n+1$ can have (see [26] or [4]).

There is the claim by Rukimbira that compact K-contact manifolds of dimension
$2n+1$ with exactly $n+1$ closed Reeb orbits are finitely covered by
$S^{2n+1}$, are homeomorphic to $S^{2n+1}$ if they are simply connected ([27,
28]). In [4], Goertsches, Nozawa and Töben give a $7$-dimensional counter
example of the claim, the Stiefel manifold
$V_{2}({\mathbb{R}}^{5})=SO(5)/SO(3)$, which is simply connected, and is a
principal circle bundle over the Grassmannian manifold of oriented 2-planes in
${\mathbb{R}}^{5}$. We will show that this example belongs to a general family
of counter examples, the Stiefel manifold
$V_{2}({\mathbb{R}}^{n+2})=SO(n+2)/SO(n)$, which is a principal circle bundle
over the Grassmannian manifold of oriented 2-planes in ${\mathbb{R}}^{n+2}$
for any $n\geq 3$ odd, see Example 7.1. We use Theorems 2 and 3 to give a
collection of examples of compact simply connected K-contact manifolds of
dimension $2n+1$ with exactly $n+1$ closed Reeb orbits which are not
homeomorphic to $S^{2n+1}$. This is Theorem 4 below.

######  Theorem 4.

There are many simply connected compact K-contact manifolds of dimension
$2n+1$ with exactly $n+1$ isolated closed Reeb orbits which are not
homeomorphic to $S^{2n+1}$, while having the real cohomlogy ring of
$S^{2n+1}$. See Section 7 for the examples constructed.

In [16], for compact contact toric manifolds of Reeb type, a subclass of
K-contact manifolds of maximal rank, in the regular contact case, the author
gives sufficient conditions for the manifolds to be homeomorphic to the
spheres. In [16], we use Boothby-Wang’s claim that the quotient space of a
regular contact manifold is a symplectic manifold. In this paper, we use
Boothby-Wang’s converse claim that a principal circle bundle over a symplectic
manifold with a symplectic form representing an integral cohomology class is a
contact manifold. The current results are for more general K-contact manifolds
than the contact toric manifolds of Reeb type in [16].

Now we summarize the structure of the paper. In Section 2, we prove Theorem 1.
In Section 3, we prove Theorem 2. In Section 4, we recall the Boothby-Wang
fibration, and state the contact and K-contact structures on the total space.
In Section 5, we lift the Hamiltonian torus action on the symplectic base
manifold to the total space of the Boothby-Wang fibration. In Section 6, we
perturb the K-contact structures on the total space of the Boothby-Wang
fibration and prove Theorem 3. In Section 7, we use Theorems 2 and 3 to
construct the collection of examples of simply connected compact K-contact
manifolds of dimension $2n+1$ with exactly $n+1$ closed Reeb orbits which are
not homeomorphic to $S^{2n+1}$, but are real cohomology spheres, proving
Theorem 4. In Section 8, through the Boothby-Wang fibration, we give a family
of examples of simply connected K-contact manifolds of dimension $2n+1$ which
are not real cohomology spheres, and admit K-contact structures with more than
$n+1$ number of closed Reeb orbits, using our Theorem 3.

##  2\. proof of Theorem 1

In this section, we give a proof of Theorem 1. We first prove a lemma on the
fundamental group of the total space of a principal circle bundle. In this
section, we only use Lemma 2.1 (1), we will later use Lemma 2.1 (2).

######  Lemma 2.1.

Let $N$ be a compact manifold with a degree 2 primitive integral cohomology
class $x$, and let $\pi\colon M\to N$ be a principal circle bundle over $N$
with Euler class $x$. Then the following claims hold.

  1. (1)

If $\pi_{1}(N)=1$ and $H^{2}(N;{\mathbb{Z}})={\mathbb{Z}}$, then
$\pi_{1}(M)=1$.

  2. (2)

If $N$ is symplectic with a primitive integal symplectic class $[\omega]$,
$H^{2}(N;{\mathbb{Z}})={\mathbb{Z}}$, and $N$ admits a Hamiltonian circle
action with isolated fixed points, then $\pi_{1}(M)=1$.

###### Proof.

(1) Consider the homotopy exact sequence for the circle bundle $\pi\colon M\to
N$:

(2.2)
$\cdots\to\pi_{2}(N)\to\pi_{1}(S^{1})\to\pi_{1}(M)\to\pi_{1}(N)\to\pi_{0}(S^{1})\to\cdots.$

Since $\pi_{1}(N)=1$, we have $\pi_{2}(N)=H_{2}(N;{\mathbb{Z}})$, and
$H^{2}(N;{\mathbb{Z}})=H_{2}(N;{\mathbb{Z}})$ (by the universal coefficient
theorem). By considering the universal $S^{1}$-bundle $ES^{1}\to BS^{1}$, a
classifying map $N\to BS^{1}$, and a diagram chasing, we see that the map

$\pi_{2}(N)=H_{2}(N;{\mathbb{Z}})=H^{2}(N;{\mathbb{Z}})={\mathbb{Z}}\to\pi_{1}(S^{1})=H_{1}(S^{1};{\mathbb{Z}})=H^{1}(S^{1};{\mathbb{Z}})={\mathbb{Z}}$

in (2.2) is multiplication by the Euler class $x\in
H^{2}(N;{\mathbb{Z}})={\mathbb{Z}}$ of the $S^{1}$-bundle $\pi\colon M\to N$.
Since $x$ is primitive integral, (up to a sign) the above map is
multiplication by $1$, so the map $\pi_{2}(N)\to\pi_{1}(S^{1})$ is surjective,
by the exact sequence (2.2), we get

$\pi_{1}(M)=1.$

(2) If $N$ is a compact symplectic manifold which admits a Hamiltonian circle
action with isolated fixed points, then the minimum of the moment map is a
single point, by [12], $\pi_{1}(N)=\pi_{1}(\mbox{minimum})=1$, so the claim
follows from (1). ∎

######  Remark 2.3.

In the proof of Lemma 2.1, we need the map $\pi_{2}(N)\to\pi_{1}(S^{1})$ to be
surjective to claim $\pi_{1}(M)=1$. The condition
$H^{2}(N;{\mathbb{Z}})={\mathbb{Z}}$ is just an easy one to claim this. One
could have other conditions for the claim.

###### Proof of Theorem 1.

(1) Suppose we have the isomorphism
$H^{*}(N;{\mathbb{R}})={\mathbb{R}}[x]/x^{n+1}$ as a ring. Then
$H^{\mbox{odd}}(N;{\mathbb{R}})=0$. So the Gysin long exact sequence for the
circle bundle $\pi\colon M\to N$ in ${\mathbb{R}}$-coefficients splits into a
short exact sequence

$0\to
H^{2i+1}(M;{\mathbb{R}})\overset{\pi_{*}}{\longrightarrow}H^{2i}(N;{\mathbb{R}})\overset{f}{\longrightarrow}H^{2i+2}(N;{\mathbb{R}})\overset{\pi^{*}}{\longrightarrow}H^{2i+2}(M;{\mathbb{R}})\to
0,$

where $\pi_{*}$ is integration along the fiber, $f$ is multiplication by the
Euler class $x$ of the bundle $\pi\colon M\to N$, and $\pi^{*}$ is pull back.
The ring isomorphism $H^{*}(N;{\mathbb{R}})={\mathbb{R}}[x]/x^{n+1}$ implies
that $f$ is an isomorphism for each $0\leq 2i\leq 2n-2$, and is $0$ when
$2i=2n$, so $H^{*}(M;{\mathbb{R}})\cong H^{*}(S^{2n+1};{\mathbb{R}})$ as
rings.

Conversely, if $H^{*}(M;{\mathbb{R}})\cong H^{*}(S^{2n+1};{\mathbb{R}})$ as
rings, then $H^{k}(M;{\mathbb{R}})=0$ for $1\leq k\leq 2n$ and
$H^{2n+1}(M;{\mathbb{R}})={\mathbb{R}}$. The long Gysin exact sequence

$\cdots\to H^{*+1}(M;{\mathbb{R}})\to H^{*}(N;{\mathbb{R}})\to
H^{*+2}(N;{\mathbb{R}})\to H^{*+2}(M;{\mathbb{R}})\to\cdots$

implies that $f\colon H^{*}(N;{\mathbb{R}})\to H^{*+2}(N;{\mathbb{R}})$ is an
isomorphism for each $0\leq*\leq 2n-1$, and is $0$ when $*=2n$, hence
$H^{2i}(N;{\mathbb{R}})={\mathbb{R}}$ for all $0\leq 2i\leq 2n$, and
$H^{\mbox{odd}}(N;{\mathbb{R}})=0$, so we have the ring isomorphism
$H^{*}(N;{\mathbb{R}})={\mathbb{R}}[x]/x^{n+1}\cong
H^{*}({\mathbb{CP}}^{n};{\mathbb{R}})$.

(2) Consider the long Gysin exact sequence for the circle bundle $\pi\colon
M\to N$ in ${\mathbb{Z}}$-coefficients:

$\cdots\to H^{*+1}(M;{\mathbb{Z}})\to H^{*}(N;{\mathbb{Z}})\to
H^{*+2}(N;{\mathbb{Z}})\to H^{*+2}(M;{\mathbb{Z}})\to\cdots$

and follow the arguments in (1). In the process of the arguments, we need to
note that $x\in H^{2}(N;{\mathbb{Z}})$ implies that $x^{i}$ is an integral
class for each $2\leq i\leq n$, and

$H^{2i}(N;{\mathbb{Z}})\overset{f}{\longrightarrow}H^{2i+2}(N;{\mathbb{Z}})$

which is multiplication by $x$, is an isomorphism if and only if $x^{i}$ and
$x^{i+1}$ are generators for the two corresponding cohomology groups, where
$0\leq i\leq n-1$.

(3) If $M$ is homeomorphic to $S^{2n+1}$, then $\pi_{1}(M)=1$ and
$H^{*}(M;{\mathbb{Z}})\cong H^{*}(S^{2n+1};{\mathbb{Z}})$ as rings; by (2.2),
$\pi_{1}(N)=1$, and by (2), $H^{*}(N;{\mathbb{Z}})={\mathbb{Z}}[x]/x^{n+1}$ as
a ring. Conversely, if $\pi_{1}(N)=1$ and
$H^{*}(N;{\mathbb{Z}})={\mathbb{Z}}[x]/x^{n+1}$ as a ring, then by Lemma 2.1
(1), $\pi_{1}(M)=1$, and by (2), $H^{*}(M;{\mathbb{Z}})\cong
H^{*}(S^{2n+1};{\mathbb{Z}})$ as rings. By the generalized Poincaré conjecture
([9, 29]), $M$ is homeomorphic to $S^{2n+1}$. ∎

##  3\. proof of Theorem 2

In this section, we first prove Theorem 2, then we make a few very helpful
remarks for understanding the theorem.

###### Proof of Theorem 2.

(1) When $N$ admits a Hamiltonian $S^{1}$-action with exactly $n+1$ isolated
fixed points, using the fact that the moment map is a Morse function on $N$,
we obtain the integral cohomology groups of $N$ as follows (see [13]):

(3.1)
$H^{2i}(N;{\mathbb{Z}})={\mathbb{Z}},\,\,\,\mbox{and}\,\,\,H^{2i+1}(N;{\mathbb{Z}})=0\,\,\,\mbox{for
all $0\leq i\leq n$}.$

In particular, $H^{2}(N;{\mathbb{Z}})={\mathbb{Z}}$. By Lemma 2.1 (2),
$\pi_{1}(N)=1$. Then the claim follows from Theorem 1 (3).

(2) When $N$ admits a Hamiltonian $S^{1}$-action with fixed point set
consisting of $n+1$ isolated points, if $c_{1}(N)=(n+1)x$, then by [13,
Theorem 1], the largest weight of the $S^{1}$-action at all the fixed points
on $N$ is equal to the length of the moment map image interval (which is an
integer since $[\omega]$ is an integral cohomology class). In this case, by
[13, Theorem 2], $H^{*}(N;{\mathbb{Z}})={\mathbb{Z}}[x]/x^{n+1}$ holds as a
ring, and we also have the total Chern class isomorphism $c(N)\cong
c({\mathbb{CP}}^{n})=(1+x)^{n+1}$. By (1), $M$ is homeomorphic to $S^{2n+1}$.
∎

######  Remark 3.2.

In Theorem 2, $x=[\omega]$ is primitive integral implies that $x^{i}$ is an
integral class for each $0\leq i\leq n$. There exists a natural number $a$
such that $\frac{1}{a}x^{n}\in H^{2n}(N;{\mathbb{Z}})$ is primitive integral.
Then (3.1) implies that the integral cohomology ring $H^{*}(N;{\mathbb{Z}})$
is generated by

$1,\,\,\,x,\,\,\,\frac{1}{a_{2}}x^{2},\,\,\,\cdots,\,\,\,\frac{1}{a_{n-2}}x^{n-2},\,\,\,\frac{1}{a}x^{n-1},\,\,\,\frac{1}{a}x^{n},$

where $a_{2}$, $\cdots$, $a_{n-2}$ are natural numbers, and by Poincaré
duality, $a_{i}a_{n-i}=a$ for all $2\leq i\leq n-2$. If we do not have the
following ring isomorphism, i.e.,

$H^{*}(N;{\mathbb{Z}})\neq{\mathbb{Z}}[x]/x^{n+1},$

then let $a_{k}$ be the first natural number which is not equal to $1$, then
by the Gysin exact sequence in ${\mathbb{Z}}$-coefficients:

$\cdots\to H^{*}(M;{\mathbb{Z}})\to H^{*-1}(N;{\mathbb{Z}})\to
H^{*+1}(N;{\mathbb{Z}})\to H^{*+1}(M;{\mathbb{Z}})\to\cdots,$

we get

$H^{2k}(M;{\mathbb{Z}})={\mathbb{Z}}_{a_{k}}$

is torsion. In this case $M$ cannot be homeomorphic to $S^{2n+1}$.

######  Remark 3.3.

In Theorem 2, if the fixed point set of the Hamiltonian circle action on $N$
consists of isolated points but more than $n+1$ isolated points, then $N$ has
more even Betti numbers than ${\mathbb{CP}}^{n}$ implied by the fact that the
moment map is a perfect Morse function; in this case, $M$ is not even a real
cohomology sphere by Theorem 1 (1). See Example 8.1.

######  Remark 3.4.

In Theorem 2, if the fixed points of the Hamiltonian circle action on $N$ are
not necessarily isolated, we may still have the claims (1) and (2). For
example, when the fixed point set of the circle action consists of two
connected components $X$ and $Y$ with $\dim(X)+\dim(Y)+2=\dim(N)$, [18] shows
that this condition is equivalent to the condition that the even Betti numbers
of $N$ are minimal, i.e., $b_{2i}(N)=1$ for each $0\leq 2i\leq 2n$. In this
case, by [18], $N$ either has the integral cohomology ring of
${\mathbb{CP}}^{n}$ or it has the integral cohomology ring of the Grassmannian
of oriented 2-planes in ${\mathbb{R}}^{n+2}$, denoted
$\widetilde{G}_{2}({\mathbb{R}}^{n+2})$ with $n\geq 3$ odd. By [17], the
manifold $N$ is simply connected. In this case, we still have the claims (1)
and (2) of Theorem 2, moreover, in (2), $M$ is homeomorphic to $S^{2n+1}$
implies $c_{1}(N)=(n+1)[\omega]$ by (1) and the results in [18].

######  Remark 3.5.

In Theorem 2, if there is a Hamiltonian torus action on $N$, since $N$ is
compact, there is a finite number of stabilizer groups for the torus action,
so we can find a subcircle action with exactly the same fixed point set as the
torus action. Hence for the claims in Theorem 2, assuming a torus action is
stronger than assuming a circle action on $N$, or assuming a circle action on
$N$ is more general.

##  4\. K-contact structures on the total spaces of principal circle bundles
over symplectic manifolds

Suppose $(M,\alpha)$ is a compact regular contact manifold. Then by Boothby
and Wang [1], $M$ is a principal circle bundle over the Reeb orbit space
$N=M/S^{1}$, $\alpha$ is a connection form in this bundle, and the curvature
form of $\alpha$ projects to a symplectic form $\omega$ on $N=M/S^{1}$ which
represents an integral cocycle, i.e., $d\alpha=\pi^{*}(\omega)$, where
$\pi\colon M\to N$ is the projection, and $[\omega]\in H^{2}(N;{\mathbb{Z}})$.

Conversely, given a compact symplectic manifold $(N,\omega)$, with $[\omega]$
an integral cohomology class, Boothby and Wang also have a converse claim as
follows, this is the part we will use in this paper.

######  Theorem 5.

(Boothby-Wang)[1] Let $(N,\omega)$ be a compact symplectic manifold with
primitive integral symplectic class $[\omega]$. Let $\pi\colon M\to N$ be a
principal circle bundle over $N$ with first Chern class $[\omega]$, and
$\alpha$ be a connection form on $M$ such that $d\alpha=\pi^{*}(\omega)$. Then
$(M,\alpha)$ is a contact manifold with regular contact form $\alpha$ whose
Reeb flow generates exactly the transformation group of the fibers of the
principal circle bundle.

In the Boothby-Wang fibration in Theorem 5 with connection $\alpha$, suppose
$g_{N}$ is any Riemannian metric on $N$ compatible with $\omega$, i.e., there
is an almost complex structure $J_{N}$ on $N$ such that
$g(X,Y)=\omega(X,J_{N}Y)$ for any $X,Y\in TN$, we define an almost complex
structure $J_{M}$ on $\ker(\alpha)\subset TM$ by

$J_{M}X=(J_{N}(\pi_{*}X))_{\mbox{hor}},\quad\forall\,\,X\in\ker(\alpha),$

where $(J_{N}(\pi_{*}X))_{\mbox{hor}}$ is the horizontal lift of the vector
field $J_{N}(\pi_{*}X)$. We have $\pi_{*}(J_{M}X)=J_{N}(\pi_{*}X)$. Then
$g=\pi^{*}(g_{N})\oplus\alpha\otimes\alpha$ is a Riemannian metric on $M$
adapted to $\alpha$, and is invariant under the Reeb flow of $\alpha$, so
$(M,\alpha,g)$ is a K-contact manifold. Note that we may have many different
choices for $g_{N}$ and hence for $g$. Conversely, let $g$ be an adapted
Riemannian metric on $M$ to $\alpha$, and is invariant under the flow of the
Reeb vector field $R$ of $\alpha$. Since $L_{R}g=0$, we have $L_{R}J_{M}=0$,
where $J_{M}$ is a compatible almost complex structure on $\ker(\alpha)$ (see
[32, Lemma 3.2]), so both $g$ and $J_{M}$ project respectively to a metric
$g_{N}$ and an almost complex structure $J_{N}$ on $N$ such that $g_{N}$,
$J_{N}$ and $\omega$ are compatible. So $g$ is of the form
$g=\pi^{*}(g_{N})\oplus\alpha\otimes\alpha$. Moreover, by Hatakeyama [5] and
Morimoto [21], $(M,\alpha,g)$ is a Sasakian structure if and only if
$(N,\omega,J_{N})$ is a Kähler structure. We summarize these as follows.

######  Theorem 6.

[1, 5, 21] In the Boothby-Wang fibration $\pi\colon M\to N$ in Theorem 5 with
connection $\alpha$ such that $d\alpha=\pi^{*}(\omega)$, any K-contact
Riemannian metric on $M$ adapted to $\alpha$ is of the form
$g=\pi^{*}(g_{N})\oplus\alpha\otimes\alpha$, where $g_{N}$ is any Riemannian
metric on $N$ compatible with $\omega$. So $(M,\alpha,g)$ is a K-contact
manifold. Moreover, $(M,\alpha,g)$ is a Sasakian manifold if and only if
$(N,\omega,g_{N})$ is a Kähler manifold.

##  5\. lift a Hamiltonian torus action on the symplectic base to the total
space of the fibration $\pi\colon M\to N$

In this section, assuming that there is a Hamiltonian torus $T$-action on the
compact base symplectic manifold $(N,\omega)$, where $[\omega]$ is primitive
integral, we show how to lift the $T$-action to the total space $M$ of the
principal circle bundle $\pi\colon M\to N$, so there are more symmetries on
$M$ than the principal circle action.

Recall that if $(M,\alpha)$ is a contact manifold with a compact Lie group
$G$-action, then there exists a contact moment map $\psi\colon
M\to\mathfrak{g}^{*}$ defined by

(5.1)
$\psi^{X}=\langle\psi,X\rangle=\alpha(X_{M}),\quad\forall\,\,X\in\mathfrak{g}=\mbox{Lie}(G),$

where $X_{M}$ is the vector field generated by the $X$-action on $M$.

Let $(N,\omega)$ be a compact symplectic manifold with primitive integral
symplectic class $[\omega]$. Let $\pi\colon M\to N$ be a principal circle
bundle with connection form $\alpha$ such that $d\alpha=\pi^{*}(\omega)$, so
$(M,\alpha)$ becomes a compact contact manifold. For the symplectic manifold
$(N,\omega)$, we assume that there is a Hamiltonian Lie group $G$-action with
moment map $\phi\colon N\to\mathfrak{g}^{*}$ satisfying

$d\phi^{X}=d\langle\phi,X\rangle=-i_{X_{N}}\omega,\quad\forall\,\,X\in\mathfrak{g},$

where $X_{N}$ is the vector field generated by the $X$-action on $N$. It is
known that the Lie algebra $\mathfrak{g}=\,$Lie$(G)$-action on $N$ lifts to
the total space $M$ by the formula

(5.2)
$X_{M}=\widetilde{X_{N}}+\phi^{X}\frac{\partial}{\partial\theta},\quad\forall\,\,X\in\mathfrak{g},$

where $\widetilde{X_{N}}$ is the horizontal lift of $X_{N}$ to $M$, and
$\frac{\partial}{\partial\theta}$ is the vector field generated by the
principal circle action on $M$, it is also the Reeb vector field of the
contact 1-form $\alpha$. For this lift, one may refer to Chapter 6 of the book
by Ginzburg, Guillemin and Karshon [3] or the works [7, 8, 11] by Kostant and
Kirillov. We can check that $X_{M}$ commutes with
$\frac{\partial}{\partial\theta}$ for all $X\in\mathfrak{g}$, and (5.2) gives
a Lie algebra action of $\mathfrak{g}$ on $M$, i.e.,

$[X_{M},Y_{M}]=\widetilde{[X,Y]_{N}}+\phi^{[X,Y]}\frac{\partial}{\partial\theta},\quad\forall\,\,X,Y\in\mathfrak{g}.$

Note that since $\alpha$ is the connection form, also the contact form on $M$,
by (5.1), the lifted Lie algebra action on $M$ produces a contact moment map

$\psi^{X}=\alpha(X_{M})=\phi^{X},\quad\forall\,\,X\in\mathfrak{g}.$

This can be expected, since the contact moment map defined by the contact form
in (5.1) is the same as the moment map for the presymplectic form
$\pi^{*}(\omega)$ on the presymplectic manifold $(M,\pi^{*}(\omega))$.

The lift (5.2) uses the connection form to define the horizontal lift
$\widetilde{X_{N}}$. Two different connections $\alpha$ and $\alpha^{\prime}$
such that $d\alpha=\pi^{*}(\omega)$ and $d\alpha^{\prime}=\pi^{*}(\omega)$
differ by the pull back by $\pi^{*}$ of an invariant closed 1-form on $N$. In
certain situations, the lift does not depend on the choice of the connection,
it is uniquely determined by the symplectic form $\omega$ and the moment map
$\phi$ on $N$, for example, when a maximal torus of $G$ has a fixed point on
$N$, or when $N$ is compact and $H^{1}(N,{\mathbb{R}})=0$. The reader may
check this, or refer to page 101 in [3]. We do not use this point in this
paper, we just need to choose a connection $\alpha$ such that
$d\alpha=\pi^{*}(\omega)$.

In this paper, our concern is the closures of the contact flows, hence we are
only concerned about torus actions on $N$. We now assume that there is a
Hamiltonian torus $T$ action on the compact base symplectic manifold $N$. It
is known that $\dim(T)\leq\frac{1}{2}\dim(N)$. Since $N$ is compact, the fixed
point set of the Hamiltonian $T$-action on $N$ is not empty. The moment map
$\phi$ of the $T$-action on $N$ is defined up to translation in
$\mathfrak{t}^{*}$ by a constant, so we may assume that at a fixed point of
$T$, $\phi$ achieves a value in the integral lattice of $\mathfrak{t}^{*}$.
Since $[\omega]$ is an integral class, then all the fixed points of $T$
achieve moment map values in the integral lattice of $\mathfrak{t}^{*}$. By
Example 6.10 in [3], the Lie algebra $\mathfrak{t}$-action on $M$ determines a
Lie group $T$-action on $M$. Hence we lifted the Hamiltonian $T$-action on the
base manifold $N$ to the total space $M$ of the principal circle bundle
$\pi\colon M\to N$. Now we have a $T\times S^{1}$-action on $M$, where the
$S^{1}$-action is the principal circle action on $M$ rotating the fibers of
the bundle $\pi\colon M\to N$.

Let $\psi\colon M\to\mathfrak{t}^{*}\times{\mathbb{R}}^{*}$ be the contact
moment map of the $T\times S^{1}$-action. View $\psi$ as a linear function on
$\mathfrak{t}\times{\mathbb{R}}$, then the image of $\psi$ lies in a
hyperplane in $\mathfrak{t}^{*}\times{\mathbb{R}}^{*}$ given by

$\psi^{\theta}=\langle\psi,\frac{\partial}{\partial\theta}\rangle=\alpha\Big{(}\frac{\partial}{\partial\theta}\Big{)}=1,$

where we are using $\frac{\partial}{\partial\theta}$ both for the Reeb vector
field of $\alpha$, the vector field generated by the principal $S^{1}$-action,
and for the Lie algebra element in ${\mathbb{R}}=\,$Lie$(S^{1})$. Up to a
translation by a constant, this moment map image can be identified with the
moment map image of the $T$-action on $N$, hence it is a convex polytope.

Recall that by Theorem 6, we have K-contact structures $(\alpha,g)$ on $M$. We
can check that for the lifted vector field $X_{M}$ in (5.2), we have

$L_{X_{M}}\alpha=0,\quad\forall\,\,X\in\mathfrak{t},$

so $\alpha$ is $T$-invariant, hence is $T\times S^{1}$-invariant. We can
choose the Riemannian metric $g_{N}$ on $N$ to be $T$-invariant. So the
Riemannian metric $g=\pi^{*}(g_{N})+\alpha\otimes\alpha$ on $M$ adapted to
$\alpha$ is $T\times S^{1}$-invariant. Hence the abelian Lie group $T\times
S^{1}$ is in the isometry group of the K-contact manifold $(M,\alpha,g)$.

We summarize the main points of this section as follows.

######  Theorem 7.

Let $(N,\omega)$ be a compact symplectic manifold of dimension $2n$ with
primitive integral symplectic class $[\omega]$, and $\pi\colon M\to N$ be a
principal circle bundle with connection form $\alpha$ such that
$d\alpha=\pi^{*}(\omega)$. If there is a Hamiltonian torus $T$-action on $N$
with $1\leq\dim(T)\leq n$, then the $T$-action can be lifted to a $T$-action
on $M$, and the K-contact structure $(\alpha,g)$ on $M$ in Theorem 6 can be
made to be $T\times S^{1}$-invariant, where $S^{1}$ is the principal
$S^{1}$-action on $M$.

##  6\. changing the K-contact structures on the total space of the Boothby-
Wang fibration $\pi\colon M\to N$ — proof of Theorem 3

Let $(N,\omega)$ be a compact symplectic manifold with primitive integral
symplectic class $[\omega]$. By Theorem 6, the total space of the principal
circle bundle $\pi\colon M\to N$ with connection $\alpha$ such that
$d\alpha=\pi^{*}(\omega)$ has regular K-contact structures $(M,\alpha,g)$. By
Theorem 7, if there is a Hamiltonian torus $T$-action on $N$, then the
$T$-action can be lifted to a $T$-action on $M$ such that the structure
$(\alpha,g)$ are $T\times S^{1}$-invariant. In this section, we show that we
can perturb the contact form $\alpha$ to a new contact form $\alpha^{\prime}$,
and there exists a K-contact structure $(\alpha^{\prime},g^{\prime})$ on $M$
such that the contact flow of $\alpha^{\prime}$ has closure $T\times S^{1}$ in
the isometry group of $(M,g^{\prime})$. Moreover, we show that the closed Reeb
orbits of $\alpha^{\prime}$ are in one to one correspondence with the fixed
points of the $T$-action on $N$. If $(N,\omega)$ is a Kähler manifold, then
the K-contact structure $(\alpha^{\prime},g^{\prime})$ on $M$ is a Sasakian
structure. Using this, we will give the proof of Theorem 3.

First we have the following observation given by Yamazaki.

######  Proposition 6.1.

[31] Let $(M,\alpha)$ be a compact $2n+1$-dimensional contact manifold, and
$\varphi_{t}$ be the contact flow of $\alpha$. If there exists a torus $T$
with $1\leq\dim(T)\leq n+1$, a smooth effective $T$-action on $M$, and a
homomorphism $h\colon{\mathbb{R}}\to T$ with dense image such that
$\varphi_{t}=T\circ(h(t))$, then there exists an adapted Riemannian metric $g$
to $\alpha$ such that $(M,\alpha,g)$ is a K-contact manifold.

The idea of the proof of Proposition 6.1 is summarized below. Since the Reeb
flow preserves $\alpha$, $\varphi_{t}=T\circ(h(t))$, and $h(t)$ is dense in
$T$, the $T$-action preserves $\alpha$. Then the $T$-action preserves
$d\alpha$, and in the subbundle $\ker(\alpha)$, we can choose $T$-invariant
Riemannian metric $g_{0}$ and $T$-invariant almost complex structure $J$ such
that $(g_{0},J,d\alpha)$ are compatible. Then we have a $T$-invariant
Riemannian metric $g=g_{0}+\alpha\otimes\alpha$ on $M$. In particular, the
Reeb vector field is Killing for $g$. So $(M,\alpha,g)$ is a K-contact
manifold.

######  Proposition 6.2.

Let $(N,\omega)$ be a compact symplectic manifold of dimension $2n$ with a
primitive integral symplectic class $[\omega]$, and $\pi\colon M\to N$ be a
principal circle bundle with connection form $\alpha$ such that
$d\alpha=\pi^{*}(\omega)$. Assume $(N,\omega)$ is equipped with a Hamiltonian
torus $T$ action with $1\leq\dim(T)\leq n$. Then the following claims hold.

  1. (1)

There exists a K-contact structure $(\alpha^{\prime},g^{\prime})$ on $M$ such
that the closure of the Reeb flow of $\alpha^{\prime}$ in the isometry group
of $(M,g^{\prime})$ is $T\times S^{1}$, where $S^{1}$ is the principal
$S^{1}$-transformation group on $M$. The closed Reeb orbits of
$\alpha^{\prime}$ are in one to one correspondence with the fixed points of
the $T$-action on $N$.

  2. (2)

If $(N,\omega)$ is a Kähler manifold, then the K-contact structure in $(1)$ is
Sasakian.

###### Proof.

(1) By Theorem 7, we have a $T\times S^{1}$-action on $M$ preserving $\alpha$,
and the $1\times S^{1}$-action generates the Reeb vector field
$\frac{\partial}{\partial\theta}$ of $\alpha$. The contact moment map
$\psi\colon M\to\mathfrak{t}^{*}\times{\mathbb{R}}^{*}$ of the $T\times
S^{1}$-action on $M$ has image, which is a convex polytope, lying in the
hyperplane in $\mathfrak{t}^{*}\times{\mathbb{R}}^{*}$ away from the origin
given by

(6.3)
$\psi^{\theta}=\langle\psi,\frac{\partial}{\partial\theta}\rangle=\alpha\Big{(}\frac{\partial}{\partial\theta}\Big{)}=1.$

Here we are using $\frac{\partial}{\partial\theta}$ both for the vector field
on $M$ and for the Lie algebra element in ${\mathbb{R}}=\,$Lie$(S^{1})$. Since
(6.3) holds and the image of $\psi$ is compact, we can find a generic element
$\xi\in\mathfrak{t}\times{\mathbb{R}}=\,$Lie$(T\times S^{1})$, an element
whose one parameter subgroup $\exp(t\xi)$ is dense in $T\times S^{1}$, such
that

$\psi^{\xi}=\langle\psi,\xi\rangle=\alpha(\xi_{M})>0,$

where $\xi_{M}$ is the vector field on $M$ generated by the
$\exp(t\xi)$-action. Let

$\alpha^{\prime}=\frac{\alpha}{\alpha(\xi_{M})}.$

Then $\alpha^{\prime}$ is a new contact form on $M$ whose Reeb vector field is
$\xi_{M}$. The contact manifold $(M,\alpha^{\prime})$ and the torus $T\times
S^{1}$ satisfy the condition of Proposition 6.1, where $\dim(T\times
S^{1})\leq n+1$. By Proposition 6.1, there exists a K-contact structure
$(\alpha^{\prime},g^{\prime})$ on $M$ such that the closure of the Reeb flow
of $\alpha^{\prime}$ in the isometry group of $(M,g^{\prime})$ is $T\times
S^{1}$.

Now we prove the second claim. Let $\phi\colon N\to\mathfrak{t}^{*}$ be the
moment map of the $T$-action on $N$. Let
$\xi=(\xi_{1},\xi_{2})\in\mathfrak{t}\times{\mathbb{R}}$ (we can take
$\xi_{2}=\frac{\partial}{\partial\theta}$ if $\xi_{1}$ is suitably chosen). Up
to a translation of the moment map $\phi$ by a constant, we may assume that
$\phi^{\xi_{1}}\neq 0$. First assume $n\in N$ is a fixed point for the
$T$-action on $N$. Since $\xi_{1}\in\mathfrak{t}$ is generic, $n$ is a fixed
point of $\exp(t\xi_{1})$, so ${\xi_{1}}_{N}(n)=0$. Then for any
$m\in\pi^{-1}(n)$, the horizontal lift $\widetilde{{\xi_{1}}_{N}}(m)=0$. By
(5.2),

${\xi_{1}}_{M}(m)=0+\phi^{\xi_{1}}(n)\frac{\partial}{\partial\theta}(m).$

So ${\xi_{1}}_{M}$ and ${\xi_{2}}_{M}$ are linearly dependant, both in the
direction of $\frac{\partial}{\partial\theta}$, so the $\xi_{M}$-action
through the point $m$ produces a closed Reeb orbit for the 1-form
$\alpha^{\prime}$. Conversely, if $n\in N$ is not a fixed point for the
$T$-action on $N$, then ${\xi_{1}}_{N}(n)\neq 0$ since
$\xi_{1}\in\mathfrak{t}$ is generic. Then for any $m\in\pi^{-1}(n)$,

${\xi_{1}}_{M}(m)=\widetilde{{\xi_{1}}_{N}}(m)+\phi^{\xi_{1}}(n)\frac{\partial}{\partial\theta}(m),\,\,\,\mbox{where}\,\,\,\widetilde{{\xi_{1}}_{N}}(m)\neq
0.$

So ${\xi_{1}}_{M}$ and ${\xi_{2}}_{M}$ are linearly independant; so the
closure of the Reeb orbit through $m$ generated by $\xi_{M}$ is at least
2-dimensional since $\xi$ is generic, hence the Reeb orbit through $m$ is not
closed.

(2) If $(N,\omega)$ is a Kähler manifold, by Theorem 6, for the regular
contact form, the connection form $\alpha$ on $M$, the K-contact structure
$(\alpha,g)$ on $M$ is a Sasakian structure. By Theorem 7, this Sasakian
structure is $T\times S^{1}$-invariant. Hence the perturbed Reeb vector field
$\xi_{M}$ of $\alpha^{\prime}$ satisfies

$L_{\xi_{M}}g=0,\,\,\,\Big{[}\xi_{M},\frac{\partial}{\partial\theta}\Big{]}=0,\,\,\,\alpha(\xi_{M})>0,$

where $\frac{\partial}{\partial\theta}$ represents the Reeb vector field of
$\alpha$. By [32, Theorem 7.2], the new K-contact structure
$(\alpha^{\prime},g^{\prime})$ on $M$ in (1) is also Sasakian. ∎

######  Lemma 6.4.

Let $(N,\omega)$ be a compact symplectic manifold admitting a Hamiltonian
torus $T^{l}$-action. Then for each $1\leq j\leq l$, there exists a
Hamiltonian subtorus $T^{j}\subset T^{l}$ action such that the fixed point set
of the $T^{j}$-action is exactly the fixed point set of the $T^{l}$-action.

###### Proof.

Since $N$ is compact, there exists a Hamiltonian $S^{1}\subset T^{l}$ action
such that the fixed point set of the $S^{1}$-action is exactly the fixed point
set of the $T^{l}$-action. Take any subgroup of $T^{l}/S^{1}$ and identify it
with a subgroup $T^{\prime}\subset T^{l}$, then the $S^{1}\times
T^{\prime}$-action is Hamiltonian and has fixed point set exactly the same as
the fixed point set of the $S^{1}$-action, which is also the fixed point set
of the $T^{l}$-action. ∎

We can now prove Theorem 3.

###### Proof of Theorem 3.

By Lemma 6.4, for each $1\leq j\leq l$, there exists a Hamiltonian subtorus
$T^{j}\subset T^{l}$ action such that the fixed point set of the
$T^{j}$-action is exactly the fixed point set of the $T^{l}$-action. For each
fixed $T^{j}$, apply Proposition 6.2 for the Hamiltonian $T^{j}$-action on
$N$, we obtain the claims. ∎

For a better understanding of Proposition 6.2, we give a standard example the
sphere $S^{2n+1}$ below. Some of the information appearing below is given in
[25, 31].

######  Example 6.5.

Consider the unit sphere $S^{2n+1}$

$\\{(z_{1},z_{2},\cdots,z_{n+1})\in{\mathbb{C}}^{n+1}\,|\,|z_{1}|^{2}+|z_{2}|^{2}+\cdots+|z_{n+1}|^{2}\\}$

with the standard contact form

$\alpha=\sum(x_{i}dy_{i}-y_{i}dx_{i})$

induced from the standard symplectic form $\omega=2\sum dx_{i}\wedge dy_{i}$
on ${\mathbb{C}}^{n+1}$. The Reeb vector field of $\alpha$ is

$R=\sum\Big{(}x_{i}\frac{\partial}{\partial
y_{i}}-y_{i}\frac{\partial}{\partial x_{i}}\Big{)}.$

This vector field is induced by the ${\mathbb{R}}$-action, the Reeb flow of
$\alpha$, given by

(6.6)
$t\cdot(z_{1},z_{2},\cdots,z_{n+1})=(e^{it}z_{1},e^{it}z_{2},\cdots,e^{it}z_{n+1}),\quad\mbox{where}\,\,t\in{\mathbb{R}}.$

This flow generates a free $S^{1}$-action, hence the contact form $\alpha$ is
regular. We have an $S^{1}$-prinpical bundle $S^{2n+1}\to
S^{2n+1}/S^{1}\approx{\mathbb{CP}}^{n}$, the hopf fibration. A K-contact
metric adapted to $\alpha$ is

$g=g_{0}+\alpha\otimes\alpha,$

where

$g_{0}=\sum\big{(}dz_{i}\otimes
d\bar{z}_{i}\big{)}|_{\ker(\alpha)}=\sum\big{(}dr_{i}\otimes
dr_{i}+r_{i}^{2}d\theta_{i}\otimes d\theta_{i}\big{)}|_{\ker(\alpha)}.$

Now we take $\lambda_{1},\lambda_{2},\cdots,\lambda_{n+1}$ to be rationally
linearly independant real numbers, and take the vector field $R_{\lambda}$
given by

$R_{\lambda}=\sum\lambda_{i}\Big{(}x_{i}\frac{\partial}{\partial
y_{i}}-y_{i}\frac{\partial}{\partial x_{i}}\Big{)}.$

Then we take a perturbed contact one-form of $\alpha$:

(6.7)
$\alpha^{\prime}=\frac{\alpha}{\alpha(R_{\lambda})}=\sum\frac{(x_{i}dy_{i}-y_{i}dx_{i})}{\sum(\lambda_{i}(x_{i}^{2}+y_{i}^{2}))}.$

The Reeb vector field of $\alpha^{\prime}$ is $R_{\lambda}$, and $R_{\lambda}$
is generated by the ${\mathbb{R}}$-action on $S^{2n+1}$ given by

(6.8)
$t\cdot(z_{1},z_{2},\cdots,z_{n+1})=(e^{i\lambda_{1}t}z_{1},e^{i\lambda_{2}t}z_{2},\cdots,e^{i\lambda_{n+1}t}z_{n+1}),\quad\mbox{where}\,\,t\in{\mathbb{R}}.$

Note that there is a natural $T^{n+1}$-action on $S^{2n+1}$ induced from the
$T^{n+1}$-action on ${\mathbb{C}}^{n+1}$. The ${\mathbb{R}}$-action (6.8) acts
as a subgroup of $T^{n+1}$. The action in (6.8) has $n+1$ number of closed
orbits, given by $(0,\cdots,e^{i\lambda_{i}t}z_{i},\cdots,0)$, where
$i=1,\cdots,n+1$. They correspond to the $n+1$ number of fixed points
$[(0,\cdots,e^{i\lambda_{i}t}z_{i},\cdots,0)]$ for the
$T^{n+1}/S^{1}=T^{n}$-action on $S^{2n+1}/S^{1}\approx{\mathbb{CP}}^{n}$,
where the $S^{1}$-action on $S^{2n+1}$ is generated by the Reeb vector field
$R$, or the action given by (6.6). Other Reeb orbits of $\alpha^{\prime}$ are
not closed. Note that the element
$\xi=(\lambda_{1},\lambda_{2},\cdots,\lambda_{n+1})\in{\mathbb{R}}^{n+1}=\,$Lie$(T^{n+1})$
is generic, i.e., the subgroup $\exp(t\xi)$ is dense in $T^{n+1}$, since the
$\lambda_{i}$s’ are rationally linearly independant.

By Proposition 6.1, there exists a K-contact metric adapted to
$\alpha^{\prime}$, it can be chosen to be

$g^{\prime}=g^{\prime}_{0}+\alpha^{\prime}\otimes\alpha^{\prime},\,\,\,\mbox{where}\,\,\,g^{\prime}_{0}=\frac{g_{0}}{\alpha(R_{\lambda})}|_{\ker(\alpha^{\prime})}.$

The K-contact structures $(\alpha,g)$ and $(\alpha^{\prime},g^{\prime})$ are
Sasakian structures.

######  Remark 6.9.

In [31], a perturbed contact form
$\alpha^{\prime}=\sum\frac{1}{\lambda_{i}}(x_{i}dy_{i}-y_{i}dx_{i})$ on
$S^{2n+1}$ is used.

######  Remark 6.10.

Similarly as in Example 6.5, if
$\xi=(\lambda_{1},\lambda_{2},\cdots,\lambda_{n+1})\in{\mathbb{R}}^{n+1}=\,$Lie$(T^{n+1})$
such that $\exp(t\xi)$ is dense in a $k$-dimensional torus $T^{k}$ of
$T^{n+1}$ with $k<n+1$, then by Proposition 6.1, we have a K-contact structure
on $S^{2n+1}$ of rank $k$. The corresponding contact form then has non-
isolated closed Reeb orbits. This is a different way of constructing K-contact
structures of smaller rank from the construction in Theorem 3, where the
smaller dimensional torus has the same set of closed Reeb orbits as the bigger
original torus.

##  7\. examples of simply connected compact K-contact manifolds with minimal
closed Reeb orbits which are not homeomorphic to the spheres — proof of
Theorem 4

In this section, we give a collection of examples of simply connected compact
K-contact manifolds with minimal closed Reeb orbits but are not homeomorphic
to the spheres. These examples are all real cohomology spheres.

First, recall that by Theorem 2, if $(N,\omega)$ is a compact symplectic
manifold of dimension $2n$ equipped with a Hamiltonian $S^{1}$-action with
$n+1$ isolated fixed points, $[\omega]=x$ is primitive integral, and
$H^{*}(N;{\mathbb{Z}})={\mathbb{Z}}[x]/x^{n+1}$, then the total space $M$ of
the principal circle bundle over $N$ with first Chern class $x=[\omega]$ is
homeomorphic to $S^{2n+1}$. By Theorem 3, the manifold $M$ has a K-contact
structure of rank $2$ which has exactly $n+1$ closed Reeb orbits. In
particular, if $N={\mathbb{CP}}^{n}$, then $M$ is $S^{2n+1}$. The sphere
$S^{2n+1}$ has a standard regular K-contact (Sasakian) structure of rank 1.
Since ${\mathbb{CP}}^{n}$ admits a Hamiltonian $T^{n}$-action with $n+1$
isolated fixed points, $S^{2n+1}$ admits a K-contact (Sasakian) structure of
any rank $2\leq k\leq n+1$ which has $n+1$ number of closed Reeb orbits by
Theorem 3.

Next, using Theorems 2 and 3, we will get a collection of K-contact manifolds
of dimension $2n+1$ with $n+1$ number of closed Reeb orbits which are not
homeomorphic to $S^{2n+1}$. In Example 7.1, we have a family of examples for
$n\geq 3$ odd. In Examples 7.1 and 7.2, we might have different spaces of $N$
other than the known manifolds mentioned in the examples.

######  Example 7.1.

Let $(N,\omega)$ be a compact symplectic manifold of dimension $2n$ admitting
a Hamiltonian $S^{1}$-action with exactly $n+1$ isolated fixed points, where
$n\geq 3$ is odd, and $x=[\omega]$ is primitive integral. Assume the integral
cohomology ring $H^{*}(N;{\mathbb{Z}})$ is generated by

$1,\,x,\,\cdots,\,x^{\frac{n-1}{2}},\,\frac{1}{2}x^{\frac{n+1}{2}},\,\cdots,\,\frac{1}{2}x^{n-1},\,\frac{1}{2}x^{n}.$

For example, $N=\widetilde{G}_{2}({\mathbb{R}}^{n+2})$, the Grassmannian of
oriented two planes in ${\mathbb{R}}^{n+2}$ for $n\geq 3$ odd, is such a
manifold, see [13]; since $\widetilde{G}_{2}({\mathbb{R}}^{n+2})$ is a
coadjoint orbit of $SO(n+2)$, it is a Kähler manifold. Let $\pi\colon M\to N$
be the principal $S^{1}$-bundle over $N$ with first Chern class $x$. By
Theorem 2 (1), $M$ is not homeomorphic to $S^{2n+1}$. By Theorem 1 (1), $M$
has the real cohomology ring of $S^{2n+1}$. By Lemma 2.1 (2),

$\pi_{1}(M)=1.$

To get the precise integral cohomology groups (ring) of $M$, we can use the
Gysin exact sequence for the circle bundle $\pi\colon M\to N$:

$\cdots\to H^{*}(M;{\mathbb{Z}})\to H^{*-1}(N;{\mathbb{Z}})\to
H^{*+1}(N;{\mathbb{Z}})\to H^{*+1}(M;{\mathbb{Z}})\to\cdots,$

and get

$H^{*}(M;{\mathbb{Z}})=\left\\{\begin{array}[]{ll}{\mathbb{Z}}&\mbox{if
$*=0,2n+1$},\\\ {\mathbb{Z}}_{2}&\mbox{if $*=n+1$},\\\ 0&\mbox{other
$*$s'}.\end{array}\right.$

By Theorem 3, the simply connected manifold $M$ admits a K-contact structure
of rank $2$ with exactly $n+1$ closed Reeb orbits. In particular, if
$N=\widetilde{G}_{2}({\mathbb{R}}^{n+2})$, then
$M=V_{2}({\mathbb{R}}^{n+2})=SO(n+2)/SO(n)=O(n+2)/O(n)$, the Stiefel manifold
of orthonormal 2-frames in ${\mathbb{R}}^{n+2}$. The maximal torus of
$SO(n+2)$ for $n\geq 3$ odd is $T^{\frac{n+1}{2}}$. As a coadjoint orbit of
$SO(n+2)$, $\widetilde{G}_{2}({\mathbb{R}}^{n+2})$ admits a Hamiltonian
$T^{\frac{n+1}{2}}$-action with $n+1$ isolated fixed points (see [22]), by
Theorem 3, we can have K-contact structures of different ranks on
$V_{2}({\mathbb{R}}^{n+2})$ which have exactly $n+1$ closed Reeb orbits. Since
$\widetilde{G}_{2}({\mathbb{R}}^{n+2})$ is Kähler, these K-contact structures
on $V_{2}({\mathbb{R}}^{n+2})$ are Sasakian.

######  Example 7.2.

Let $(N,\omega)$ be a compact symplectic manifold of dimension $10$ admitting
a Hamiltonian $S^{1}$-action with exactly 6 isolated fixed points, where
$x=[\omega]$ is primitive integral. Assume the integral cohomology ring
$H^{*}(N;{\mathbb{Z}})$ is generated by

$1,\,x,\,\frac{1}{3}x^{2},\,\frac{1}{6}x^{3},\,\frac{1}{18}x^{4},\,\frac{1}{18}x^{5}.$

For example, a 10-dimensional coadjoint orbit of the Lie group $G_{2}$,
denoted $\mathcal{O}$, is such a manifold, see [14]. As a coadjoint orbit of
$G_{2}$, $\mathcal{O}$ is a Kähler manifold. Let $\pi\colon M\to N$ be the
principal circle bundle over $N$ with first Chern class $x$. By Theorem 2 (1),
$M$ is not homeomorphic to $S^{11}$. By Theorem 1 (1), $M$ has the real
cohomology ring of $S^{11}$. By Lemma 2.1 (2),

$\pi_{1}(M)=1.$

By the Gysin exact sequence for the circle bundle $\pi\colon M\to N$, we can
get the precise integral cohomology groups (ring) of $M$:

$H^{*}(M;{\mathbb{Z}})=\left\\{\begin{array}[]{ll}{\mathbb{Z}}&\mbox{if
$*=0,11$},\\\ {\mathbb{Z}}_{3}&\mbox{if $*=4$},\\\ {\mathbb{Z}}_{2}&\mbox{if
$*=6$},\\\ {\mathbb{Z}}_{3}&\mbox{if $*=8$},\\\ 0&\mbox{other
$*$s'}.\end{array}\right.$

By Theorem 3, the simply connected $11$-dimensional manifold $M$ admits a
K-contact structure of rank 2 with exactly $6$ closed Reeb orbits. In
particular, if $N=\mathcal{O}$, it can be written as a quotient by $U(2)$,
i.e., $\mathcal{O}=G_{2}/U(2)$. The Lie group $G_{2}$ is simply connected; by
[10, Theorems 2 and 11], the principal circle bundles over
$\mathcal{O}=G_{2}/U(2)$, as a group, is generated by $G_{2}/SU(2)$, where
$SU(2)$ is the semi-simple part of $U(2)$. Hence the principal circle bundle
over $\mathcal{O}=G_{2}/U(2)$ with first Chern class $x$, the generator of
$H^{2}(\mathcal{O},{\mathbb{Z}})$, is $M=G_{2}/SU(2)$. By Theorem 3, the rank
2 K-contact structure on $M=G_{2}/SU(2)$ is Sasakian. Moreover, the maximal
torus $T^{2}$ of $G_{2}$ acts on $\mathcal{O}=G_{2}/U(2)$ in a Hamiltonian way
also with $6$ isolated fixed points, see [14]. Theorem 3 gives a K-contact
(Sasakian) structure of rank 3 on $M=G_{2}/SU(2)$ with exactly $6$ isolated
closed Reeb orbits.

######  Example 7.3.

Let $(N,\omega)$ be a compact symplectic manifold of dimension $6$ admitting a
Hamiltonian $S^{1}$-action with exactly $4$ isolated fixed points, where
$x=[\omega]$ is primitive integral. Assume the integral cohomology ring
$H^{*}(N;{\mathbb{Z}})$ is generated by

$1,\,\,x,\,\,\frac{1}{5}x^{2},\,\,\frac{1}{5}x^{3}.$

For example, $V_{5}$, a fano manifold, is such a manifold. The fact that a
compact $6$-dimensional Hamiltonian $S^{1}$-manifold with exactly 4 isolated
fixed points can have the above integral cohomology ring structure is
discovered by Tolman in [30]. In [20], McDuff gives $V_{5}$ as such an
example, and shows that $V_{5}$ is the only such manifold up to
$S^{1}$-equivariant symplectomorphism; moreover, $V_{5}$ is an
$S^{1}$-invariant Kähler manifold. Let $\pi\colon M\to N$ be the principal
circle bundle over $N$ with first Chern class $x$. By Theorem 2 (1), $M$ is
not homeomorphic to $S^{7}$. By Theorem 1 (1), $M$ has the real cohomology
ring of $S^{7}$. By Lemma 2.1 (2),

$\pi_{1}(M)=1.$

The Gysin exact sequence for the circle bundle $\pi\colon M\to N$ gives the
precise integral cohomology groups (ring) of $M$:

$H^{*}(M;{\mathbb{Z}})=\left\\{\begin{array}[]{ll}{\mathbb{Z}}&\mbox{if
$*=0,7$},\\\ {\mathbb{Z}}_{5}&\mbox{if $*=4$},\\\ 0&\mbox{other
$*$s'}.\end{array}\right.$

By Theorem 3, the simply connected $7$-dimensional manifold $M$ admits a
K-contact structure of rank 2 with exactly $4$ closed Reeb orbits. Due to the
uniqueness of $N=V_{5}$, the K-contact structure on $M$ is a Sasakian
structure.

######  Example 7.4.

Let $(N,\omega)$ be a compact symplectic manifold of dimension $6$ admitting a
Hamiltonian $S^{1}$-action with eactly $4$ isolated fixed points, where
$x=[\omega]$ is primitive integral. Assume the integral cohomology ring
$H^{*}(N;{\mathbb{Z}})$ is generated by

$1,\,\,x,\,\,\frac{1}{22}x^{2},\,\,\frac{1}{22}x^{3}.$

For example, $V_{22}$, a fano manifold, is such a manifold. Similar to Example
7.3, $V_{22}$ is the unique such manifold up to $S^{1}$-equivariant
symplectomorphism, and it is an $S^{1}$-invariant Kähler manifold, see [30,
20]. Let $\pi\colon M\to N$ be the principal circle bundle over $N$ with first
Chern class $x$. By Theorem 2 (1), $M$ is not homeomorphic to $S^{7}$. By
Theorem 1 (1), $M$ has the real cohomology ring of $S^{7}$. By Lemma 2.1 (2),

$\pi_{1}(M)=1.$

The Gysin exact sequence for the circle bundle $\pi\colon M\to N$ gives the
integral cohomology groups (ring) of $M$:

$H^{*}(M;{\mathbb{Z}})=\left\\{\begin{array}[]{ll}{\mathbb{Z}}&\mbox{if
$*=0,7$},\\\ {\mathbb{Z}}_{22}&\mbox{if $*=4$},\\\ 0&\mbox{other
$*$s'}.\end{array}\right.$

By Theorem 3, the simply connected $7$-dimensional manifold $M$ admits a
K-contact structure (of rank $2$) which has exactly $4$ closed Reeb orbits.
Due to the uniqueness of $N=V_{22}$, the K-contact structure on $M$ is a
Sasakian structure.

For the examples in Example 7.3 and 7.4, I do not know if the 7-dimensional
contact manifolds, the total spaces of the circle bundles over $V_{5}$ and
$V_{22}$ are known, and if they have names.

##  8\. examples of simply connected K-contact manifolds which are not real
cohomology spheres

In this section, through the Boothby-Wang fibration, we give a family of
examples of simply connected K-contact manifolds of dimension $2n+1$ which are
not real cohomology spheres, and admit K-contact structures with more than
$n+1$ number of isolated closed Reeb orbits, by our Theorem 3.

######  Example 8.1.

Let $(N,\omega)$ be a compact symplectic manifold of dimension $2n$ admitting
a Hamiltonian $S^{1}$-action with exactly $n+2$ isolated fixed points, where
$x=[\omega]$ is primitive integral. By [15], $n$ must be even. For example,
$\widetilde{G}_{2}({\mathbb{R}}^{n+2})$, the Grassmannian of oriented two
planes in ${\mathbb{R}}^{n+2}$ with $n>2$ even, is such a manifold, and
$\widetilde{G}_{2}({\mathbb{R}}^{n+2})$ is a Kähler manifold as a coadjoint
orbit of $SO(n+2)$, see [15]. Let $\pi\colon M\to N$ be a principal circle
bundle over $N$ with first Chern class $x$. By Theorem 6, $(M,\alpha)$ is a
K-contact manifold, where $\alpha$ is a connection one form on $M$ such that
$\pi^{*}(\omega)=d\alpha$. By [15], the manifold $N$ has odd Betti numbers all
zero, and even Betti numbers as follows:

$b_{2i}=1\,\,\,\mbox{for each $0\leq 2i\leq 2n$ and $2i\neq n$,
and}\,\,\,b_{n}=2.$

By Theorem 1 (1), $M$ is not a real cohomology sphere. We can use the Gysin
exact sequence in ${\mathbb{R}}$-coefficients

$\cdots\to H^{*}(M;{\mathbb{R}})\to H^{*-1}(N;{\mathbb{R}})\to
H^{*+1}(N;{\mathbb{R}})\to H^{*+1}(M;{\mathbb{R}})\to\cdots$

to get the non-zero Betti numbers of $M$:

$b_{0}=b_{n}=b_{n+1}=b_{2n+1}=1.$

(We can also use the Gysin exact sequence in ${\mathbb{Z}}$-coefficients to
obtain the integral cohomology of $M$.) We still have
$H^{2}(N;{\mathbb{Z}})={\mathbb{Z}}$ ([15]), by Lemma 2.1 (2),

$\pi_{1}(M)=1.$

By Theorem 3, the simply connected manifold $M$ admits a K-contact structure
of rank 2 with $n+2$ closed Reeb orbits. In particular, if
$N=\widetilde{G}_{2}({\mathbb{R}}^{n+2})$ with $n>2$ even, then
$M=V_{2}({\mathbb{R}}^{n+2})=SO(n+2)/SO(n)=O(n+2)/O(n)$, the Stiefel manifold
of orthonormal 2-frames in ${\mathbb{R}}^{n+2}$. The maximal torus of
$SO(n+2)$ with $n>2$ even is $T^{\frac{n}{2}+1}$. The Kähler manifold
$\widetilde{G}_{2}({\mathbb{R}}^{n+2})$ admits a Hamiltonian
$T^{\frac{n}{2}+1}$-action with $n+2$ isolated fixed points ([22]). By Theorem
3, we have K-contact (Sasakian) structures on $V_{2}({\mathbb{R}}^{n+2})$ of
different ranks with $n+2$ closed Reeb orbits.

## References

  * [1] W. M. Boothby and H. C. Wang, On contact manifolds, Annals of Math., 68, No. 3, 1958, 721-734.
  * [2] C. P. Boyer and K. Galicki, A note on toric contact geometry, J. Geom. Phys. 35 (2000), 288-298.
  * [3] V. Ginzburg, V. Guillemin and Y. Karshon, Moment maps, cobordisms, and Hamiltonian group actions, American Mathematical Society Mathematical Surveys and Monographs 98, 2002.
  * [4] O. Goertsches, H. Nozawa, and D. Töben, Equivariant cohomology of K-contact manifolds, Math. Ann. (2012) 354,1555-1582.
  * [5] Y. Hatakeyama, Some notes on differentiable manifolds with almost contact structures, Tôhoku Math. J. 15, 1963, 176-181.
  * [6] D. Jang, Almost complex torus manifolds - a problem of Petrie type, Proceedings of Ameri. Math. Soc., bf 152, no. 7, 2024, 3153-3164.
  * [7] A. A. Kirillov, Unitary representations of nilpotent Lie groups, Russian Math. Surveys, 17, (1962), 53-104.
  * [8] A. A. Kirillov, Merits and demerits of the orbit method, Bull. Amer. Math. Soc. 36 (1999), 433-488.
  * [9] B. Kleiner and J. Lott, Notes on Perelman’s papers, Geometry and Topology 12 (2008), 2587-2858.
  * [10] S. Kobayashi, Principal fibre bundles with the $1$-dimensional toroidal group, Töhoku Math. J. 8 (1956), 29-45.
  * [11] B. Kostant, Quantization and unitary representations, in Modern Analysis and Applications, Springer Lecture Notes in Mathematics 170, p. 87-107, New York, Springer-Verlag, 1970.
  * [12] H. Li, $\pi_{1}$ of Hamiltonian $S^{1}$-manifolds, Proceedings of Ameri. Math. Soc., vol. 131, no. 11, 3579-3582, 2003.
  * [13] H. Li, Hamiltonian circle actions with minimal isolated fixed points, Math. Z., 2023.
  * [14] H. Li, Hamiltonian $S^{1}$-manifolds which are like a coadjoint orbit of $G_{2}$, arXiv:2403.01825.
  * [15] H. Li, Hamiltonian circle actions with almost minimal isolated fixed points, Journal of Geometry and Physics, 163 (2021), 104141.
  * [16] H. Li, Spherical contact toric manifolds, Proceedings of Ameri. Math. Soc., vol. 151, no. 1, 2023, 349-353.
  * [17] H. Li, M. Olbermann and D. Stanley, One connectivity and finiteness of Hamiltonian $S^{1}$-manifolds with minimal fixed sets, J. of Lond. Math. Soc. (2) 92 (2015), no. 2, 284-310.
  * [18] H. Li and S. Tolman, Hamiltonian circle actions with minimal fixed sets, International Journal of Mathematics 23, no. 8 (2012), 1250071.
  * [19] Y. Lin and D. Miyamoto, Riemannian foliations and quasifolds, arXiv: 2309.15166v1.
  * [20] D. McDuff, Some $6$-dimensional Hamiltonian $S^{1}$ manifolds, J. of Topology, 2 (2009), no. 3, 589-623.
  * [21] A. Morimoto, On normal almost contact structures with a regularity, Tôhoku Math. J. 16, (1964), 90-104.
  * [22] D. Morton, GKM manifolds with low Betti numbers, Ph.D thesis, University of Illinois at Urbana-Champaign, 2011.
  * [23] S. B. Myers and N. E. Steenrod, The group of isometries of a Riemannian manifold, Ann. Math. 40(2), 400-416 (1939).
  * [24] T. Petrie, Torus actions on homotopy complex projective spaces, Invent. Math. 20, (1973), 139-146.
  * [25] P. Rukimbira, The dimension of leaf closures of K-contact flows, Ann. Global Anal. Geom. 12, 103-108 (1994).
  * [26] P. Rukimbira, Topology and closed characteristics of K-contact manifolds, Bull. Belg. Math. Soc. Simon Stevin 2, 349-356 (1995).
  * [27] P. Rukimbira, On K-contact manifolds with minimal number of closed characteristics, Proceedings of the Ameri. Math. Soc., vol. 127, no. 11, 3345-3351, 1999.
  * [28] P. Rukimbira, Spherical rigidity via contact dynamics, Bull. Belg. Math. Soc. 7 (2000), 563-569.
  * [29] S. Smale, Generalized Poincaré’s conjecture in dimensions greater than four, Ann. of Math. (2) 74 (1961), 391-406.
  * [30] S. Tolman, On a symplectic generalization of Petrie’s conjecture, Trans. Amer. Math. Soc., 362 (2010), 3963-3996.
  * [31] T. Yamazaki, A construction of K-contact manifolds by a fiber join, Tôhoku Math. J. 51 (1999), 433-446.
  * [32] K. Yano and M. Kon, Structures on Manifolds, Series in Pure Mathematics, World Scientific 3 (1984).