# Global Hypersurfaces of Section for Geodesic Flows on Convex Hypersurfaces

Sunghae Cho and Dongho Lee

###### Abstract.

We construct a global hypersurface of section for the geodesic flow of a
convex hypersurface in Euclidean space admits an isometric involution. This
generalizes the Birkhoff annulus to higher dimensions.

##  1\. Introduction

Global surfaces of section, an idea introduced by Poincaré in his work on
celestial mechanics [Poi87] and also explored by Birkhoff [Bir66], feature
prominently in the literature on the 3-dimensional dynamics. They allow us to
reduce the dynamics of vector fields on 3-manifolds to the dynamics of surface
diffeomorphisms. Ghys [Ghy09] called the existence of a global surface of
section as a paradise for dynamicists, since it eliminates technical
difficulties and allows to investigate the pure nature of dynamical systems.

Beyond their versatile utility, the existence of the global surfaces of
section already gives us some information about the dynamics. Unless the given
3-manifold fibers over a circle, a global surface of section must have a
boundary which is a set of periodic orbits. Hence the existence of periodic
orbits is an essential obstruction, which cannot be overcome easily. Kuperberg
[Kup94] introduced an example of a nonvanishing vector field on $S^{3}$
without periodic orbits. Ginzburg [Gin99] has given examples of the
Hamiltonian system without periodic orbits. The horocycle flow on unit
cotangent bundle of higher genus surfaces provides another class of
Hamiltonian flows without periodic orbits.

This obstruction can be removed in the case of Reeb dynamics in dimension
three. The _Weinstein conjecture_ [Wei79] asserts the existence of at least
one periodic Reeb orbit on any compact contact manifold. Hofer [Hof93] proved
the Weinstein conjecture on $S^{3}$, and Taubes [Tau07] proved the Weinstein
conjecture in dimension three in a full generality using embedded contact
homology. The work of Hofer was followed by the joint work with Wysocki and
Zehnder [HWZ98] constructing disk-like global surface of section on a
dynamically convex $S^{3}$. Recent developments along this way can be found in
[HSa11], [HMSa15], and [SaH18]. [CDHR23], [CM22] and [CKMS22] contain results
about the existence of a Birkhoff section for Reeb flows in dimension three, a
generalization of a global surface of section that allows immersed boundary.

The notion of global surface of section can be generalized to higher
dimensions as mentioned by Birkhoff [Bir66]. However, there are only few
results known in higher dimensional dynamics even in the case of Reeb flows,
due to the difficulty caused by the instability of the boundary of global
hypersurface of section. Moreno and van Koert [MvK22] resolved this problem by
imposing a symmetry which guarantees the existence of high dimensional
invariant set, and provided a concrete example of a global hypersurface of
section arising from the restricted three-body problem.

Our main result concerns the existence of global hypersurfaces of section of
the geodesic flows on compact convex hypersurfaces in Euclidean space which
satisfying certain conditions.

######  Theorem 1.1.

Let $M\subset\mathbb{R}^{n+1}$ be a regular closed hypersurface and $N\subset
M$ be a codimension 1 submanifold. Assume that there exists a tubular
neighborhood $\nu(N)$ of $N$ and an isometric involution $i:\nu(N)\to\nu(N)$
whose fixed point locus is $N$. In addition, assume that $M$ has positive
sectional curvature. Then the geodesic flow on $ST^{*}M$ admits a global
hypersurface of section

$P=\left\\{(x,y)\in ST^{*}M:x\in N,\langle y,\nu_{x}\rangle\geq 0\right\\},$

where $\nu$ is a fixed normal vector field of $N$ with respect to $M$.
Moreover, the return map $\Psi:\mathring{P}\to\mathring{P}$ extends smoothly
to the boundary of $P$.

We note that the positivity of sectional curvature is equivalent to the
convexity of the hypersurface. We first prove the existence part in 3.10, and
4.5 establishes that we can extend the return map smoothly to the boundary.
Note that if the dimension of $M$ is 2, our theorem reduces to a special case
of the theorem of Birkhoff [Bir66] which established the existence of a global
surface of section of geodesic flow of a 2-sphere with positive curvature. In
this sense, our result can be regarded as a generalization of the Birkhoff
annulus to higher dimensions.

###  1.1. Acknowledgement

Sunghae Cho and Dongho Lee were supported by National Science Foundation of
Korea Grant NRF2023005562 funded by the Korean Government.

##  2\. Preliminaries

###  2.1. Global Hypersurfaces of Section and Open Book Decompositions

Let $Y$ be a smooth closed manifold, i.e. a compact manifold without boundary,
and $X$ be a non-vanishing vector field on $Y$. A global hypersurface of
section for $X$ is an embedded submanifold $P\subset Y$ of codimension 1 with
(possibly empty) boundary $\partial P=B$ such that

  * $\cdot$

the vector field $X$ is transverse to the interior $\mathring{P}$ of $P$,

  * $\cdot$

the boundary $B$ is $X$-invariant; in other words, $X$ is tangent to $B$,

  * $\cdot$

for any $p\in Y$, there exist $t_{+},t_{-}>0$ such that
$Fl^{X}_{t_{+}}(p),Fl^{X}_{-t_{-}}(p)\in P$.

If $P$ is a global hypersurface of section, we can define the (first) return
time $\tau_{p}$ for each $p\in\mathring{P}$ by
$\tau_{p}=\min\\{t>0:Fl^{X}_{t}(p)\in P\\}$ and the (first) return map by
$\Psi(p)=Fl^{X}_{\tau_{p}}(p)$.

######  Example 2.1.

Let $Y$ be $ST^{*}S^{n}$, the unit cotangent bundle of the $n$-sphere with the
standard metric, and $X$ be the unit geodesic vector field on $Y$. Consider
$Y$ as a subset of
$T^{*}\mathbb{R}^{n+1}\simeq\mathbb{R}^{n+1}\times\mathbb{R}^{n+1}$ and use
the coordinates $(x,y)=(x_{0},\cdots,x_{n},y_{0},\cdots,y_{n})$. Consider the
submanifold

$P=\left\\{(x,y)\in Y:x_{0}=0,y_{0}\geq 0\right\\},$

which is the set of upward directions on the equator. Then one can see that
$P$ is a global hypersurface of section, the boundary of $P$ is the unit
cotangent bundle of the equator of $S^{n}$, and the first return map is the
identity.

Open book decompositions are closely related to the global hypersurfaces of
section. An open book decomposition on a closed manifold $Y$ is a pair
$(B,\pi)$ of a codimension 2 closed submanifold $B$ and a map $\pi$ from
$Y\setminus B$ to $S^{1}\subset\mathbb{C}$ which satisfies the following.

  * $\cdot$

The normal bundle of $B$ is trivial. We call $B$ the binding. Let $\xi:B\times
D^{2}\to\nu(B)$ be a fixed trivialization of the normal bundle, which is
identified with a tubular beighborhood, of $B$.

  * $\cdot$

The map $\pi$ is a fiber bundle such that
$(\pi\circ\xi)(b;r,\theta)=e^{i\theta}$ on $\nu(B)\setminus B$, where
$(r,\theta)$ is a polar coordinate on $D^{2}$. We call the closure of each
fiber $\overline{\pi^{-1}(e^{i\theta})}=P_{\theta}$ the page. Note that
$\partial P_{\theta}=B$ for any $\theta$.

Let $X$ be a vector field on $Y$, and $(B,\pi)$ be an open book on $Y$. If $X$
is transverse to each page $P_{\theta}$ and tangent to $B$, then we say $X$ is
adapted to $(B,\pi)$.

######  Lemma 2.2.

Let $Y$ be a closed manifold, $B\subset Y$ be a codimension 2 closed
submanifold, $X$ be a vector field on $Y$, and $\pi$ be a map from $Y\setminus
B$ to $S^{1}$. Assume that

  1. (1)

$d_{p}\pi(X)>0$ for any $p\in Y$,

  2. (2)

$B$ has a trivial tubular neighborhood, say $B\times D^{2}$, such that
$\pi(b;r,\theta)=e^{i\theta}$,

  3. (3)

$X$ is tangent to $B$.

Then $(B,\pi)$ is an open book decomposition of $Y$, which $X$ is adapted to.

###### Proof.

By (1), $\pi$ is a submersion. We can remove an open tubular neighborhood
$\nu(B)$ of $B$ on which (2) holds, then $\pi|_{Y\setminus\nu(B)}$ is still a
submersion. In particular, (2) guarantees that $\pi|_{\partial\nu(B)}$ is a
submersion. Since $\pi|_{Y\setminus\nu(B)}$ is proper, we can apply the
Ehresmann fibration theorem and conclude that $\pi$ defines a fiber bundle.
With (2), we can see that $(B,\pi)$ is an open book decomposition on $Y$.

Since $d\pi(X)\neq 0$, $X$ cannot be tangent to the level sets of $\pi$. This
means that $X$ is transverse to each page. With (3), we can see that $X$ is
adapted to $(B,\pi)$. ∎

If a vector field $X$ is adapted to $(B,\pi)$, then each page
$P_{\theta}=\pi^{-1}(e^{i\theta})$ can be regarded as a candidate for the
global hypersurface of section. There might exist an orbit of $X$ which does
not return to the page in a finite time. Such an orbit should be asymptotic to
the boundary as $t$ becomes large. A discussion about a case of dimension 3
can be found in [HWZ98]. In 3.5, we will show that the case of unbounded
return time does not appear in our setting.

###  2.2. Symplectic Manifolds and Contact Manifolds

Let $W$ be a manifold without boundary and $\omega$ be a 2-form on $W$. If
$\omega$ is a non-degenerate closed form, we say $\omega$ is a symplectic form
and $(W,\omega)$ is a symplectic manifold. A symplectomorphism is a
diffeomorphism between symplectic manifolds which preserves the symplectic
form. Let $W^{\prime}\subset W$ be a submanifold such that
$\omega|_{W^{\prime}}$ is also a symplectic form on $W^{\prime}$. We call
$W^{\prime}$ a symplectic submanifold. A diffeomorphism between manifolds
induces a symplectomorphism between cotangent bundles via pullback. Also, if
$N$ is a submanifold of $M$, then $T^{*}N$ is a symplectic submanifold of
$T^{*}M$.

Let $(W,\omega)$ be a symplectic manifold with a boundary, and assume that
$\omega$ has a primitive $\lambda$, i.e. $\omega=d\lambda$. A vector field $X$
such that $i_{X}\omega=\lambda$ is called Liouville vector field. If a
Liouville vector field $X$ exists and defines an outward vector field on the
boundary, we call $(W,\lambda)$ a Liouville domain.

Given a smooth function $H:W\to\mathbb{R}$, we can associate a vector field
$X_{H}$ to $H$ via

$\left(i_{X_{H}}\right)\omega(-)=\omega(X_{H},-)=dH(-).$

This is well-defined by non-degeneracy of $\omega$. We say $X_{H}$ is a
Hamiltonian vector field, and in this sense, we call $H$ a Hamiltonian
function or simply Hamiltonian. If a diffeomorphism can be written as a time
1-flow of a Hamiltonian vector field, we say it is a Hamiltonian
diffeomorphism. Note that we can also use time-dependent Hamiltonian
$H:W\times\mathbb{R}\to\mathbb{R}$ and get time-dependent Hamiltonian
diffeomorphism. By definition, a Hamiltonian vector field $X_{H}$ vanishes at
a point $p$ if and only if $d_{p}H=0$. A Hamiltonian diffeomorphism is a
symplectomorphism, as
$\left(Fl^{X_{H}}\right)^{*}\omega=\mathcal{L}_{X_{H}}\omega=di_{X_{H}}\omega=0$
by the Cartan magic formula. An important observation is that the Hamiltonian
vector field of an autonomous Hamiltonian is tangent to the regular level set
of the generating Hamiltonian. In other words, the value of $H$ is preserved
under the flow of $X_{H}$. More detailed explanations with examples about
symplectic geometry and Hamiltonian mechanics can be found in [Arn89],
[CdS01], [Ber01], [HZ11] or [MS17].

Let $H,F:W\to\mathbb{R}$ be Hamiltonians. The Poisson bracket is defined by
$\\{H,F\\}:=\omega(X_{H},X_{F})$. It’s clear that the Poisson bracket is
alternating. If $(W,\omega)=(T^{*}\mathbb{R}^{n},\omega_{\mathrm{std}})$, we
have the following formula, which can be found for example in the chapter 1 of
[MS17]

$\\{H,F\\}=\sum_{j}\frac{\partial H}{\partial y_{j}}\frac{\partial F}{\partial
x_{j}}-\frac{\partial H}{\partial x_{j}}\frac{\partial F}{\partial x_{j}}.$

Let $Y$ be a $(2n+1)$-dimensional manifold, and $\xi$ be a $2n$-dimensional
distribution on $Y$. Then we can locally write $\xi$ as $\ker\alpha$ for some
1-form $\alpha$. Assume that $\xi$ is coorientable, i.e. $TW/\xi$ is
orientable. Then we can find a globally defined 1-form $\alpha$. If
$\alpha\wedge(d\alpha)^{n}$ is a volume form, we say $\alpha$ is a contact
form, $\xi=\ker\alpha$ is a contact structure, and $(W,\xi)$ is a contact
manifold. A contact form $\alpha$ defines a unique vector field $R$ such that
$\alpha(R)=1$, $i_{R}d\alpha=0$. We call $R$ the Reeb vector field. Note that
the Reeb vector field is always non-vanishing. A standard example of a contact
manifold is the boundary of a Liouville domain $(W,\lambda)$ with contact form
$\lambda|_{\partial W}$, and a regular level set of Hamiltonian function
$H:(W,d\lambda)\to\mathbb{R}$ with the same contact form. In the second case,
the Hamiltonian vector field $X_{H}$ restricted to the regular level set of
$H$ is the Reeb vector field. We can consider an open book decomposition
$(B,\pi)$ on a contact manifold $(Y,\ker\alpha)$ which the Reeb vector field
adapted to. The close relationship between contact structures on a manifold
and an open book decomposition is explored in [Gir02].

###  2.3. Geodesic Flow as a Hamiltonian Flow

Let $(M,g)$ be a complete Riemannian manifold. For each $x\in M$ and $v\in
T_{p}M$, we have a unique geodesic $\gamma_{x,v}$ with the initial condition
$\gamma_{x,v}(0)=x$, $\dot{\gamma}_{x,v}(0)=v$. The geodesic flow is a
1-parameter family of diffeomorphisms on $TM$ defined by
$\Phi_{t}(x,v)=(\gamma_{x,v}(t),\dot{\gamma}_{x,v}(t))$. By differentiating
$\Phi_{t}$ by $t$, we get the geodesic vector field on $TM$ which generates
the geodesic flow.

The metric $g$ on $TM$ induces the dual metric $g^{*}$ on $T^{*}M$ by natural
pairing, and we can also define the (co-)geodesic flow and (co-)geodesic
vector field on $T^{*}M$.

######  Proposition 2.3.

The geodesic vector field on $T^{*}M$ is a Hamiltonian vector field with
Hamiltonian

$H(x,y):=\frac{1}{2}\left(||y||_{g^{*}}^{2}-1\right).$

where $T^{*}M$ is equipped with a canonical symplectic form $\omega=\sum
dx_{i}\wedge dy_{i}$

###### Proof.

See the proof of Theorem 2.3.1 in [FvK18] or Theorem 1.5.2 in [Gei08]. ∎

The regular level set $H^{-1}(0)$ is a unit cotangent bundle $ST^{*}M$, which
naturally is a contact manifold, whose Reeb vector field is Hamiltonian vector
field $X_{H}$ restricted to $ST^{*}M$.

For notational convenience, we write a point $(x_{1},\ldots,x_{n})$ in
$\mathbb{R}^{n}$ by $\vec{x}$ so that $x=(x_{0},\vec{x})$ is a point in
$\mathbb{R}^{n+1}$. For a function $f:\mathbb{R}^{n+1}\to\mathbb{R}$, we
denote the partial derivative $\partial_{x_{i}}f$ by $f_{i}$, and the gradient
vector field $(f_{0},\ldots,f_{n})$ by $\nabla f$. We will also write the
Hessian matrix of $f$ which is computed in $\mathbb{R}^{n+1}$ by
$\text{Hess}(f)=(\partial_{x_{i}}\partial_{x_{j}}f)_{ij}=(f_{ij})$. We use
coordinate $(y_{0},\cdots,y_{n})$ for the cotangent fiber
$T^{*}_{x}\mathbb{R}^{n+1}$, and also for the cotangent fiber of a
hypersurface contained in $\mathbb{R}^{n+1}$.

Let $f:\mathbb{R}^{n+1}\to\mathbb{R}$ be a smooth function, and $0$ be a
regular value. The level set $M=f^{-1}(0)$ is an $n$-dimensional Riemannian
manifold, whose metric is inherited from $\mathbb{R}^{n+1}$. We can embed
$T^{*}M$ into $T^{*}\mathbb{R}^{n+1}$ by

$T^{*}M=\left\\{(x,y)\in T^{*}\mathbb{R}^{n+1}:f(x)=0,\,\,y\cdot\nabla
f=0\right\\}.$

Here, $\cdot$ is a standard inner product on $\mathbb{R}^{n+1}$, and we
identified $T^{*}M$ to $TM$ by the metric on $M$. Let
$\tilde{H}=\frac{1}{2}\left(||y||^{2}-1\right)$ on $T^{*}\mathbb{R}^{n+1}$,
and $H=\tilde{H}|_{T^{*}W}$. Define
$\tilde{f},g:T^{*}\mathbb{R}^{n+1}\to\mathbb{R}$ by

$\tilde{f}(x,y)=f(x),\quad g(x,y)=y\cdot\nabla f$

so that $T^{*}M$ is an intersection $\tilde{f}^{-1}(0)\cap g^{-1}(0)$.

######  Proposition 2.4.

Let $W$ be a symplectic manifold, and $\tilde{H}$ be a Hamiltonian on $W$.
Consider smooth functions $f,g:W\to\mathbb{R}$ with $c_{1},c_{2}$ as the
regular values of $f,g$ such that $V=f^{-1}(c_{1})\cap g^{-1}(c_{2})\subset W$
is a symplectic submanifold of codimension 2. Let $H=\tilde{H}|_{V}$. Then the
Hamiltonian vector field $X_{H}$ is given by

$X_{H}=X_{\tilde{H}}-\frac{\\{g,\tilde{H}\\}}{\\{g,f\\}}X_{f}-\frac{\\{f,\tilde{H}\\}}{\\{f,g\\}}X_{g}.$

###### Proof.

We can write $X_{H}$ as $X_{H}=X_{\tilde{H}}+aX_{f}+bX_{g}$ for some functions
$a,b$. Since $X_{H}$ is defined on the level set of $f$ and $g$, we must have
$X_{H}(f)=0=X_{H}(g)$. We also have $X_{f}(f)=0=X_{g}(g)$ from the definition.
It follows that

$\displaystyle 0$
$\displaystyle=X_{\tilde{H}}(f)+bX_{g}(f)=\\{f,\tilde{H}\\}+b\\{f,g\\},$
$\displaystyle 0$
$\displaystyle=X_{\tilde{H}}(g)+aX_{f}(g)=\\{g,\tilde{H}\\}+a\\{g,f\\}.$

Putting $a,b$ into the first equation yields the result. ∎

######  Remark 2.5.

The formula of 2.4 can be directly generalized to the case of any
$2k$-functions, say $V=f_{1}^{-1}(c_{1})\cap\cdots\cap f_{2k}^{-1}(c_{2k})$.
The coefficients of $X_{f_{i}}$’s in that case will be a solution of a linear
equation consisting of Poisson brackets.

From a straightforward computation, we have

$X_{\tilde{H}}=\sum_{j}y_{j}\frac{\partial}{\partial x_{j}},\quad
X_{\tilde{f}}=-\sum_{j}\frac{\partial f}{\partial
x_{j}}\frac{\partial}{\partial y_{j}}.$

Using the formula for the Poisson bracket, we have

$\displaystyle\\{\tilde{f},g\\}$ $\displaystyle=\sum\frac{\partial f}{\partial
x_{j}}\frac{\partial g}{\partial y_{j}}=\sum\left(\frac{\partial f}{\partial
x_{j}}\right)^{2}=||\nabla f||^{2},$ $\displaystyle\\{\tilde{f},\tilde{H}\\}$
$\displaystyle=\sum\frac{\partial f}{\partial
x_{j}}\frac{\partial\tilde{H}}{\partial y_{j}}=\sum\frac{\partial f}{\partial
x_{j}}y_{j}=y\cdot\nabla f=0,$ $\displaystyle\\{g,\tilde{H}\\}$
$\displaystyle=\sum\frac{\partial g}{\partial
x_{j}}\frac{\partial\tilde{H}}{\partial
y_{j}}=\sum_{j}\frac{\partial}{\partial
x_{j}}\left(\sum_{i}y_{i}\frac{\partial f}{\partial
x_{i}}\right)y_{j}=\sum_{i,j}\frac{\partial^{2}f}{\partial x_{i}\partial
x_{j}}y_{i}y_{j}=\mathrm{Hess}(f)(y,y).$

To sum up, we have the following.

######  Theorem 2.6.

Let $f:\mathbb{R}^{n+1}\to\mathbb{R}$ be a smooth function, and $c$ be a
regular value of $f$. Let $M=f^{-1}(c)$. Then, the geodesic vector field on
$T^{*}M$ is given by

$X_{H}=\sum_{j}y_{j}\frac{\partial}{\partial
x_{j}}-\frac{\mathrm{Hess}(f)_{x}(y,y)}{\|\nabla
f(x)\|^{2}}\sum_{j}f_{j}(x)\frac{\partial}{\partial y_{j}}.$

Alternatively, this formula can be derived by orthogonal projection.

###  2.4. Sectional Curvature of Hypersurfaces

An important quantity associated to the Riemannian manifold $(M,g)$ is the
sectional curvature $K_{M}(-,-)$, which is defined for a point $x\in M$ and
two vectors $v,w$ in $T_{p}M$. This can be regarded as the Gauss curvature of
a 2-dimensional subspace ‘spanned’ by $v,w$. For the precise definition and
properties of sectional curvature, we refer to [Mil63], [KN63], [Spi79], or
[dC92]. Let $N\subset M$ be an oriented hypersurface, and $\nu$ be the unit
normal vector field on $N$.

######  Proposition 2.7.

[[dC92], Theorem 6.2.5] Let $S(v)=\nabla_{v}\nu$. Then the following formula
holds.

$K_{N}(v,w)=K_{M}(v,w)+\frac{\langle S(v),v\rangle\langle
S(w),w\rangle-\langle S(v),w\rangle^{2}}{\langle v,v\rangle\langle
w,w\rangle-\langle v,w\rangle^{2}}$

The formula becomes simpler in the case of a hypersurface in the Euclidean
space, whose sectional curvature vanishes.

######  Corollary 2.8.

[[dC92], Theorem 6.2.5] Let $f:\mathbb{R}^{n+1}\to\mathbb{R}$ be a function,
and $M=f^{-1}(c)$ be a regular hypersurface. Let $v,w$ be orthonormal vector
fields on $TM$. Then we have the following.

$K_{M}(v,w)=\langle S(v),v\rangle\langle S(w),w\rangle-\langle
S(v),w\rangle^{2}.$

##  3\. A Global Hypersurface of Section of the Geodesic Flow on a Convex
Hypersurface

###  3.1. Setting

We consider the case $f:\mathbb{R}^{n+1}\to\mathbb{R}$ is a smooth function
and $0$ is a regular value. Let $M$ be the regular level set $f^{-1}(0)$. We
assume the following conditions for $f$.

  1. (A1)

For any point $(x_{0},\vec{x})$ in $M$ such that $|x_{0}|$ is small enough,
$f(x_{0},\vec{x})=f(-x_{0},\vec{x})$.

  2. (A2)

The Hessian of $f$ is positive definite. Explicitly, for any $(x,y)\in
T^{*}M$, $y\neq 0$, $\mathrm{Hess}(f)_{x}(y,y)>0$.

Here is a classical theorem which explains the role of (A1). For the proof,
see [KN69] Chapter VII.8.

######  Lemma 3.1.

Let $(M,g)$ be a Riemannian manifold and $N$ be a closed submanifold. Assume
that there exist a tubular neighborhood $\nu(N)$ of $N$ and an isometric
involution $i:\nu(N)\to\nu(N)$, i.e., $i$ is an isometry, $i\neq\mathrm{Id}$
and $i^{2}=\mathrm{Id}$. In addition, assume that $N$ is the fixed point locus
of $i$. Then $N$ is a totally geodesic submanifold. In other words, the
geodesic vector field of $T^{*}M$ restricted to $T^{*}N$ is tangent to
$T^{*}N$.

Let $N$ be the codimension $1$ submanifold $N=M\cap\\{(x,y):x_{0}=0\\}.$ Then
it is clear that $N$ is a fixed point set of a locally defined isometric
involution $i_{M}(x_{0},\vec{x})=(-x_{0},\vec{x})$, and it follows that $N$ is
a totally geodesic submanifold from 3.1. Let $Y=ST^{*}M$ and $B=ST^{*}N$. Then
$B$ can be written as

$B=\\{(x,y)\in ST^{*}M:x_{0}=0,y_{0}=0\\}.$

We note the triviality of the normal bundle of $B$ in $Y$.

######  Lemma 3.2.

Let $M$ be a Riemannian manifold diffeomorphic to an $n$-sphere, and $N\subset
M$ be a codimension $1$ closed submanifold. Then $ST^{*}N$ has a trivial
normal bundle in $ST^{*}M$.

###### Proof.

We’ll first show that the normal bundle $\nu(N)$ of $N$ is trivial, which is
equivalent to the orientability of $N$. Suppose not, then for any section $s$
of $\nu(N)$ which meets zero section transversely, there exists a loop
$\gamma$ such that $s|_{\gamma}$ has odd number of zeros. It follows that the
intersection form $[N]\cdot[\gamma]$ in $\mathbb{Z}_{2}$-coordinate is not
$0$, which is a contradiction because
$H_{n-1}(S^{n};\mathbb{Z}_{2})=H_{1}(S^{n};\mathbb{Z}_{2})=0$.

Now, let $\nu_{M}(N)\simeq N\times(-\epsilon,\epsilon)\subset M$. For $x\in
N$, we have $T_{x}M=\mathbb{R}\oplus T_{x}N$, implying

$ST_{x}M=\left\\{(t,v)\in\mathbb{R}\oplus T_{x}N:t^{2}+\lVert
v\rVert^{2}=1\right\\},$

where $\lVert\cdot\rVert$ is the metric inherited from $M$. Thus, the normal
fiber at $p=(x,v)\in ST^{*}N$ is

$\nu_{ST^{*}M}(ST^{*}N)_{p}=\nu_{ST_{x}^{*}M}(ST_{x}^{*}N)_{v}\oplus\nu_{M}(N)_{x}\simeq\nu_{S^{n-1}}(S^{n})_{v}\oplus\nu_{M}(N)_{x},$

where $S^{n-1}$ is embedded in $S^{n}$ along the equator. ∎

Condition (A2) implies that $f$ is a convex function, and $M$ bounds a compact
convex domain. It follows that $M$ is diffeomorphic to $S^{n}$, where the
diffeomorphism can be explicitly written as $x\mapsto x/||x||$, assuming that
$0$ is in the interior of the bounding region of $M$. Since $M$ is convex and
$0$ is in the interior of the domain $M$ bounds, this is well-defined.

###  3.2. Existence of a Global Hypersurface of Section

With the setting of the previous section, we define a map $\pi:Y\setminus B\to
S^{1}\subset\mathbb{C}$ by

$\pi(x,y)=\frac{x_{0}+iy_{0}}{|x_{0}+iy_{0}|}.$

The angular form is defined by

$\Theta=i\cdot
d\log\pi=\frac{y_{0}dx_{0}-x_{0}dy_{0}}{x_{0}^{2}+y_{0}^{2}}=\frac{\theta}{x_{0}^{2}+y_{0}^{2}}.$

Put $X_{H}$ which was computed in 2.6 into $\theta$, we have the following.

$\theta(X_{H})=y_{0}^{2}+x_{0}^{2}\frac{\mathrm{Hess}(f)_{x}(y,y)}{\|\nabla
f(x)\|^{2}}\frac{f_{0}(x)}{x_{0}}.$

We define a function $A=A(x,y)$ by

$A(x,y)=\frac{\mathrm{Hess}(f)_{x}(y,y)}{\|\nabla
f(x)\|^{2}}\frac{f_{0}(x)}{x_{0}}$

so that $\theta(X_{H})=A(x,y)x_{0}^{2}+y_{0}^{2}$.

######  Proposition 3.3.

Under the assumptions (A1) and (A2), there exists $\varepsilon>0$, which only
depends on the function $f$, such that $A(x,y)>\varepsilon$ for any $(x,y)\in
Y\setminus B$.

###### Proof.

Since $0$ is a regular value of $f$ and $Y$ is compact, there exists some
$C>0$ such that $0<||\nabla f||^{2}\leq C$ on $Y$. Also, by the compactness of
$Y$, condition (A2) implies that there exists some $\delta>0$ such that
$\mathrm{Hess}(f)_{x}(y,y)>\delta$ for any $(x,y)\in Y$.

By condition (A1), we have $f_{0}|_{x_{0}=0}=0$. Thus, we can perform a Taylor
expansion with respect to $x_{0}$:

$\displaystyle f(x_{0},\vec{x})$
$\displaystyle=f(0,\vec{x})+\frac{1}{2}f_{00}(0,\vec{x})x_{0}^{2}+O(x_{0}^{3}),$
$\displaystyle\frac{f_{0}(x_{0},\vec{x})}{x_{0}}$
$\displaystyle=f_{00}(0,\vec{x})+O(x_{0}).$

Since $\mathrm{Hess}(f)_{x}(y,y)>\delta$, we have
$f_{00}(x)=\mathrm{Hess}(f)_{x}((y_{0},0),(y_{0},0))>\delta$ for any $x$. We
can choose a small $\eta>0$ such that for $|x_{0}|<\eta$,
$f_{0}/x_{0}>\delta/2$. Consequently, we have $A(x,y)>\delta^{2}/2C$ for
$|x_{0}|<\eta$.

For $|x_{0}|\geq\eta$, we only need to bound $f_{0}/x_{0}$. However,
$f_{0}(0,\vec{x})=0$ and $f_{00}>\delta$ implies that $f_{0}(x_{0},\vec{x})>0$
if $x_{0}>0$ and $f_{0}(x_{0},\vec{x})<0$ if $x_{0}<0$. By the compactness of
$Y\cap\\{|x_{0}|\geq\eta\\}$, there exists some $\delta_{1}>0$ such that
$f_{0}/x_{0}>\delta_{1}$. This implies $A(x,y)>\delta\delta_{1}/C$ for
$|x_{0}|\geq\eta$. Taking
$\varepsilon=\min(\delta^{2}/2C,\delta\delta_{1}/C)$, we obtain the desired
lower bound. ∎

######  Theorem 3.4.

Under the assumptions (A1) and (A2), $\pi:Y\setminus B\to S^{1}$ defines an
open book decomposition, which the geodesic vector field is adapted to.

###### Proof.

We will apply 2.2 to this situation. From 3.3, it is evident that there exists
$\varepsilon>0$ such that

$\Theta(X_{H})=\frac{A(x,y)x_{0}^{2}+y_{0}^{2}}{x_{0}^{2}+y_{0}^{2}}>\varepsilon$

for any $(x,y)\in Y\setminus B$. Since $\Theta=i\cdot d\log\pi$, the first
condition on $\pi$ and $X_{H}$ in 2.2 is satisfied.

Now consider the trivial tubular neighborhood of $B$, the existence of which
is guaranteed by 3.2, denoted as

$\displaystyle\nu(B)$ $\displaystyle\simeq B\times D^{2}$ $\displaystyle(x,y)$
$\displaystyle\mapsto(\vec{x},\vec{y};x_{0},y_{0})$

where $x_{0},y_{0}$ are small enough. Note that $\pi(b,r,\theta)=e^{i\theta}$.

Lastly, since $N$ is a totally geodesic submanifold by (A1), the geodesic
vector field $X_{H}$ is tangent to $ST^{*}N=B$. Therefore, we can apply 2.2
and obtain the desired result. ∎

######  Theorem 3.5.

Under the assumptions (A1) and (A2), the geodesic flow on $Y=ST^{*}M$ admits a
global hypersurface of section, which is given by

$P=\\{(x,y)\in Y:x_{0}=0,y_{0}\geq 0\\}.$

###### Proof.

The hypersurface $P$ corresponds to a page $\pi^{-1}(i)$ of the open book
constructed in 3.4. To conclude, we need to demonstrate that the return time
is bounded. It suffices to show that there exists $t\in\mathbb{R}_{>0}$ such
that $\pi(Fl^{X_{H}}_{t}(x,y))=\pi(x,y)$. From 3.3, we observe the existence
of $\varepsilon>0$ such that $\Theta(X_{H})>\varepsilon$. Thus, we have

$\int_{0}^{2\pi/\varepsilon}i\cdot
d\log\pi(X_{H})>\int_{0}^{2\pi/\varepsilon}\varepsilon dt>2\pi.$

By the intermediate value theorem, we can conclude that there exists a
positive number $\tau<2\pi/\varepsilon$ such that $\int_{0}^{\tau}i\cdot
d\log\pi(X_{H})=2\pi$, which means that
$\pi(Fl^{X_{H}}_{\tau}(x,y))=\pi(x,y).$ It means that $\tau$ is a bounded
positive finite return time for $(x,y)$. The boundedness of the negative
return time can be demonstrated similarly. ∎

######  Corollary 3.6.

Under the assumptions (A1) and (A2), the geodesic flow on $M$ admits an
$S^{1}$-family of global hypersurfaces of section.

###### Proof.

We have an open book decomposition with bounded return time, so we can choose
any page of the open book, let’s say $\pi^{-1}(e^{i\theta})=P_{\theta}$, as a
global hypersurface of section. ∎

Note that if we take $n=2$, then $M$ is 2-sphere so $P=P_{\pi/2}$ in 3.5 is
the Birkhoff annulus. In this sense, our construction can be regarded as a
generalization of the Birkhoff annulus.

###  3.3. Relation with Sectional Curvature

We can apply 2.8 on our hypersurface $M$. The unit normal vector $\nu$ is
$\frac{\nabla f}{||\nabla f||}$, and we have

$S\left(\frac{\partial}{\partial x_{i}}\right)=\frac{\partial}{\partial
x_{i}}\frac{\nabla f}{||\nabla f||}=\sum_{j}\left(\frac{f_{ij}}{||\nabla
f||}-\frac{\sum_{k}f_{k}f_{ki}f_{j}}{||\nabla
f||^{3}}\right)\frac{\partial}{\partial x_{j}}.$

It follows that

$S(v,w)=\frac{\sum_{i,j}v_{i}f_{ij}w_{i}}{||\nabla
f||}-\frac{\sum_{i,j,k}v_{i}f_{ik}f_{k}f_{j}w_{j}}{||\nabla
f||^{3}}=\frac{\mathrm{Hess}(f)(v,w)}{||\nabla f||}$

for $v,w\in TM$. The second term vanishes because $w\in TM$ implies that
$w\cdot\nabla f=\sum_{j}w_{j}f_{j}=0$.

Put this into 2.8, and we get the following.

######  Proposition 3.7.

Let $v,w$ be orthogonal unit tangent vectors of a point $x\in M$. Then,

$K_{M}(v,w)=\frac{\mathrm{Hess}(f)_{x}(v,v)\mathrm{Hess}(f)_{x}(w,w)-\mathrm{Hess}(f)_{x}(v,w)^{2}}{||\nabla
f(x)||^{2}}.$

Hence, the sign of $K_{M}$ depends on the sign of the ($2\times 2$)-minors of
$\mathrm{Hess}(f)$. Since the formula holds for any $v$ and $w$, we have the
following.

######  Corollary 3.8.

The sectional curvature $K_{M}(\sigma)_{x}$ is positive for any point $x\in M$
and plane $\sigma\subset T_{x}M$ if and only if $\mathrm{Hess}(f)_{x}$ is
either positive definite or negative definite.

###### Proof.

Since $\mathrm{Hess}(f)_{x}$ is symmetric bilinear form, there exists a basis
$\mathfrak{B}$ of $T_{x}M$ that diagonalizes $\mathrm{Hess}(f)_{x}$, and it
continuously varies with $x$. Let

$[\mathrm{Hess}(f)_{x}]_{\mathfrak{B}}=\mathrm{diag}(\lambda_{1},\cdots,\lambda_{n}).$

Since $K_{M}>0$, we must have $\lambda_{i}\lambda_{j}>0$ for any $i,j$. It
means that $\lambda_{i}$ all have the same sign, which implies that
$\mathrm{Hess}(f)_{x}$ is either positive definite or negative definite. Since
the process depends on $x$ continuously, the sign of $\lambda_{i}$ cannot
change since $\lambda_{i}\neq 0$. The converse follows from the formula in
3.7. ∎

The following corollary can also be found in various literature, for example
[Sac60].

######  Corollary 3.9.

Let $f:\mathbb{R}^{n+1}\to\mathbb{R}$ and $M=f^{-1}(0)\subset\mathbb{R}^{n+1}$
be a regular hypersurface. Assume that $M$ has a positive sectional curvature.
Then $M$ is diffeomorphic to the $n$-sphere.

###### Proof.

By 3.8, $\mathrm{Hess}(f)$ is either positive definite or negative definite.
If $\mathrm{Hess}(f)$ is positive definite, $f^{-1}(-\infty,0]$ is convex and
we can get the result. If $\mathrm{Hess}(f)$ is negative definite, then for
$\bar{f}=-f$, $M=\bar{f}^{-1}(0)$, and $\mathrm{Hess}(\bar{f})$ is positive
definite, and we can use the same argument. ∎

Now we can formulate 3.5 in another form.

######  Theorem 3.10.

Let $M=f^{-1}(0)\subset\mathbb{R}^{n+1}$ be a hypersurface satisfying (A1),
and $M$ has a positive sectional curvature. Then there exists an open book
decomposition $(\pi,B)$ of $ST^{*}M$ to which the geodesic vector field
$X_{H}$ is adapted, as given in 3.4. Moreover, there exists a global
hypersurface of section $P$, as given in 3.5.

###### Proof.

Since $K_{M}>0$, 3.8 implies that $f$ is either positive definite or negative
definite. If $f$ is positive definite, we’re done. If $f$ is negative
definite, define $\tilde{f}=-f$. Then $\tilde{f}$ is positive definite,
$\tilde{f}^{-1}(0)=f^{-1}(0)$, and the condition (A1) still holds. Moreover,
the formula in 2.6 for $f$ and $\tilde{f}$ are the same. Thus we get the
result. ∎

###  3.4. Topology of the Global Hypersurface of Section

Using the same notations from the previous sections, we will explore the
topology of $N$ and $P$.

######  Lemma 3.11.

Under the assumptions (A1) and (A2), $N$ is diffeomorphic to $S^{n-1}$.

###### Proof.

We can express $N$ as

$N=\left\\{(0,\vec{x})\in\mathbb{R}^{n+1}:f(0,\vec{x})=0\right\\}\subset\left\\{(0,\vec{x})\in\mathbb{R}^{n+1}:\vec{x}\in\mathbb{R}^{n}\right\\}\simeq\mathbb{R}^{n}.$

The convexity of $f$ is preserved if we restrict $f$ to the subspace
$\left\\{(0,\vec{x}):\vec{x}\in\mathbb{R}^{n}\right\\}$, so we obtain the
result. ∎

######  Proposition 3.12.

The global hypersurface of section $P$ constructed in 3.5 is diffeomorphic to
$T_{\leq 1}^{*}S^{n-1}$, which is the subset of $T^{*}S^{n-1}$ consisting of
covectors of length $\leq 1$. The boundary $B=\partial P$ is homeomorphic to
$ST^{*}S^{n-1}$.

###### Proof.

Let $p$ be a diffeomorphism from the upper hemisphere $H^{n}\subset S^{n}$ to
the closed disk $D^{n}$. Define a map $\phi:P\to T^{*}_{\leq 1}N$ by
$\phi(x,y)=(x,p(y))$, then it’s clear that $\phi$ is a diffeomorphism. With
3.11, we can conclude the result. ∎

##  4\. Return Map

###  4.1. General Properties

In this subsection, we investigate the general properties of the global
hypersurfaces of sections for Reeb vector fields. Let $\alpha$ be a contact
form on a manifold $Y$ and $R$ be its Reeb vector field. Assume that there
exists an open book decomposition $(B,\pi)$ of $Y$ to which $R$ is adapted.
Let $P=P_{\theta}=\pi^{-1}(e^{i\theta})$ be a page, and assume that the first
return map $\Psi:\mathring{P}\to\mathring{P}$ is well-defined, i.e. the return
time $\tau$ is bounded for each point $p\in P$.

######  Lemma 4.1.

The interior of $P$ is a symplectic manifold with symplectic form $d\alpha$.

###### Proof.

Let $\dim Y=2n+1$. Since $Y$ is contact, $\alpha\wedge d\alpha^{n}$ never
vanishes. Since $\mathring{P}$ is transverse to $R$, we can take a local frame
$(X_{1},\cdots,X_{2n})$ of $\mathring{P}$ such that $(R,X_{1},\cdots,X_{2n})$
is a local frame of $Y$, and

$\alpha\wedge
d\alpha^{n}(R,X_{1},\cdots,X_{2n})=\alpha(R)d\alpha^{n}(X_{1},\cdots,X_{2n})\neq
0.$

It means that $d\alpha$ is a non-degenerate closed 2-form on $\mathring{P}$,
so it’s a symplectic form. ∎

######  Proposition 4.2.

For the return map $\Psi:\mathring{P}\to\mathring{P}$, we have

$\Psi^{*}\alpha-\alpha=d\tau.$

In particular, $\Psi$ is a symplectomorphism.

###### Proof.

This is a generalization of the well-known fact in dimension 3, which can be
found, for example, in [ABHSa17] or [FvK18]. We have
$Fl^{R}_{\tau_{p}}(p)=\Psi(p)$. Differentiating both sides and plugging in a
vector field $X$, we have

$dFl^{R}_{\tau_{p}}(p)X+(d\tau_{p}(X))R=d_{p}\Psi(X).$

Since $Fl^{R}$ preserves $\alpha$, we have that

$\Psi^{*}\alpha(X)=\alpha(d\Psi(X))=(Fl^{R})^{*}\alpha(X)+\alpha(R)d\tau(X)=\alpha(X)+d\tau(X).$

The second statement follows if we take exterior derivative on both sides. ∎

######  Remark 4.3.

If we equip $d\alpha$ to the whole $P$, which is a manifold with boundary,
then $(P,d\alpha)$ is not a Liouville domain. More precisely, $d\alpha$
degenerates at the boundary of $P$. The Reeb vector field $R$ is tangent to
$\partial P$ and $i_{R}d\alpha=0$, indicating that $d\alpha$ degenerates on
the boundary. But even though $d\alpha$ degenerates at the boundary,
$(\partial P,\ker(\alpha|_{\partial P}))$ is a contact manifold.

###  4.2. Extension to the Boundary

In the preceding sections, we constructed a global hypersurface of section $P$
for the geodesic flow on a convex hypersurface $M=f^{-1}(0)$ in
$\mathbb{R}^{n+1}$ and we obtained a return map
$\Psi:\mathring{P}\to\mathring{P}$. As demonstrated in 4.1, $\mathring{P}$ is
a symplectic manifold, with its symplectic form being the restriction of the
standard symplectic form on $T^{*}M$. Furthermore, as indicated in 4.2, $\Psi$
is a symplectomorphism.

Now we investigate the boundary behavior of the return map in terms of the
defining function $f$. Given that $(\partial_{x_{0}},\partial_{y_{0}})$ forms
a normal symplectic frame of a contact submanifold $B=ST^{*}N$, it follows
from Lemma 8.1 of [MvK22] the linearized Hamiltonian flow decomposes into two
blocks; one corresponding to $TB$ and the other to
$\langle\partial_{x_{0}},\partial_{y_{0}}\rangle$. Let
$Z_{N}=\begin{pmatrix}x_{0}(t)\\\ y_{0}(t)\end{pmatrix}$ be the normal part of
a flowline linearized near $B$. Then we have

$\dot{Z}_{N}=J_{N}S_{N}Z_{N}$

where $J_{N}$ is the normal part of the standard complex structure on
$TY=TT^{*}M$ and $S_{N}$ is a matrix called the normal Hessian.

######  Proposition 4.4.

[MvK22] If the normal Hessian is positive definite, the return map extends to
the boundary smoothly, and such an extension is unique.

######  Theorem 4.5.

Under the assumptions (A1) and (A2), or the assumption of 3.10, the return map
$\Psi:\mathring{P}\to\mathring{P}$ can be smoothly extended to the boundary.

###### Proof.

Instead of working with a normal Hessian, we will use linearized flow.
According to Proposition 8.2 of [MvK22], we must have

$\Theta(X_{H})=\frac{Z_{N}^{t}S_{N}Z_{N}}{Z_{N}^{t}Z_{N}}+O(1)$

near $(x_{0},y_{0})=(0,0)$. From the previous section, we also have

$\Theta(X_{H})=\frac{A(x,y)x_{0}^{2}+y_{0}^{2}}{x_{0}^{2}+y_{0}^{2}}=\frac{Z_{N}^{t}\mathrm{diag}\left(\frac{\mathrm{Hess}(f)_{x}(y,y)}{||\nabla
f(x)||^{2}}f_{00}(x),1\right)Z_{N}}{Z_{N}^{t}Z_{N}}+O(x_{0}).$

It follows that

$S_{N}=\mathrm{diag}\left(\frac{\mathrm{Hess}(f)_{x}(y,y)}{||\nabla
f(x)||^{2}}f_{00}(x),1\right).$

From the condition (A2), we observe that $\mathrm{Hess}(f)(y,y)$ and $f_{00}$
are always positive, and it follows that $S_{N}$ is positive definite. From
4.4, we get the result. If we impose the positive sectional curvature
condition, it follows that $\mathrm{Hess}(f)(y,y)$ and $f_{00}$ have the same
sign, which leads to the same result. ∎

######  Example 4.6.

3.4 can be applied to ellipsoids, since they are convex and satisfying the
symmetry condition (A1). In particular, consider an ellipsoid given by

$E=\left\\{x\in\mathbb{R}^{n+1}:\frac{x_{0}^{2}}{a_{0}^{2}}+x_{1}^{2}+\cdots+x_{n}^{2}-1=0\right\\},$

where $a_{0}\in\mathbb{R}_{>0}$. The global hypersurface of section
constructed in 3.4 is the upper-hemisphere bundle on the equator,

$P=\left\\{((0,\vec{x}),(y_{0},\vec{y}))\in
T^{*}\mathbb{R}^{n+1}\,:\,||\vec{x}||=1,\,\langle\vec{x},\vec{y}\rangle=0,\,y_{0}^{2}+||\vec{y}||^{2}=1,\,y_{0}\geq
0\right\\}.$

The return map $\Psi$ on $P$ can be computed in terms of elliptic integrals

$\Psi\left((0,\vec{x}),(y_{0},\vec{y})\right)=\left(\left(0,\vec{x}\cos{G(\|\vec{y}\|)}+\frac{\vec{y}}{\|\vec{y}\|}\sin{G(\|\vec{y}\|)}\right),\left(y_{0},\vec{y}\cos{G(\|\vec{y}\|)}-\|\vec{y}\|\vec{x}\sin{G(\|\vec{y}\|)}\frac{}{}\right)\right),$

where $G$ is given by

$G(t)=-\frac{t(1-a_{0}^{2})}{a_{0}}F\left(2\pi\,\Big{|}\,\frac{-(1-a_{0}^{2})(1-t^{2})}{a_{0}^{2}}\right)+\frac{t}{a_{0}}\Pi\left(1-t^{2};2\pi\,\Big{|}\,\frac{-(1-a_{0}^{2})(1-t^{2})}{a_{0}^{2}}\right).$

where $F$ is the elliptic integral of the first kind and $\Pi$ is the elliptic
integral of the third kind.

This can be regarded as a specific case of a _hypersurface of revolution_. Let
$f:\mathbb{R}^{n+1}\to\mathbb{R}$ satisfy (A1) globally, (A2) and the third
condition; if $||\vec{x}||=||\vec{x}\,^{\prime}||$, then
$f(x_{0},\vec{x})=f(x_{0},\vec{x}\,^{\prime})$. For convenience, we assume
that $f(0,\vec{x})=0$ if and only if $||\vec{x}||=1$. The hypersurface of
revolution is given by a regular level set $M=f^{-1}(0)$. In this case, we can
parametrize the set $M\cap\left\\{(x_{0},x_{1},0,\cdots,0)\right\\}$ by
$(a(\phi),\cos\phi,0,\cdots,0)$, where $a$ is a function on $\phi$. Note that
in the case of ellipsoid, $a(\phi)=a_{0}\sin\phi$. We can compute the return
map with a help of the _Clairaut integral_ [Arn89], which has a same form as
in the case of ellipsoid with

$G(t):=t\int_{0}^{2\pi}\frac{\sqrt{(1-t^{2})\sin^{2}{\sigma}+\\{a^{\prime}(\arcsin{(\sqrt{1-t^{2}}\sin{\sigma})})\\}^{2}}}{1-(1-t^{2})\sin^{2}{\sigma}}d\sigma.$

It’s straightforward to see that the return map $\Psi$ is a Hamiltonian
diffeomorphism generated by $H(x,y)=\left(\int G\right)(||\vec{y}||)$.

Sending $a(\phi)$ to 0, the dynamics converges to a billiard on the unit disk
$D=\left\\{\vec{x}\in\mathbb{R}^{n}:||\vec{x}||\leq 1\right\\}$, which is
defined on a set

$Y_{0}=\left\\{(\vec{x},\vec{y})\in T^{*}\mathbb{R}^{n}:||\vec{x}||\leq
1,\,||\vec{y}||=1,\,\vec{y}\text{ points inward if }||\vec{x}||=1\right\\}.$

The global hypersurface of section converges to
$P_{0}=\left\\{(\vec{x},\vec{y})\in Y_{0}:||\vec{x}||=1\right\\}$. We write
$\vec{y}=\vec{y}_{T}+\vec{y}_{N}$ for $(\vec{x},\vec{y})\in P_{0}$, where
$\vec{y}_{T}$ is a component tangent to $\partial D$, and and $\vec{y}_{N}$ is
a component normal to $\partial D$. Then the return map converges to the
second iterate of the high-dimensional billiard map

$\Psi\left(\vec{x},\vec{y}\right)=\left(\vec{x}\cos{G_{0}(\|\vec{y}_{T}\|)}+\frac{\vec{y}_{T}}{\|\vec{y}_{T}\|}\sin{G_{0}(\|\vec{y}_{T}\|)},\,\vec{y_{T}}\cos{G_{0}(\|\vec{y}_{T}\|)}-\|\vec{y}_{T}\|\vec{x}\sin{G_{0}(\|\vec{y_{T}}\|)}\right),$

where $G_{0}(t)=4\arccos t$, which agrees with the function $G$ with
$a(\phi)=0$.

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11institutetext: Department of Astrophysics, IMAPP, Radboud University
Nijmegen, P. O. Box 9010, 6500 GL Nijmegen, the Netherlands, 11email:
<EMAIL_ADDRESS>22institutetext: Institute of Astronomy, KU Leuven,
Celestijnenlaan 200D, 3001, Leuven, Belgium

# One size does not fit all: Evidence for a range of mixing efficiencies in
stellar evolution calculations

C. Johnston 1122

(Received 14 April 2021; accepted 13 July 2021)

###### Abstract

Context. Internal chemical mixing in intermediate- and high-mass stars
represents an immense uncertainty in stellar evolution models. In addition to
extending the main-sequence lifetime, chemical mixing also appreciably
increases the mass of the stellar core. Several studies have made attempts to
calibrate the efficiency of different convective boundary mixing mechanisms,
with sometimes seemingly conflicting results.

Aims. We aim to demonstrate that stellar models regularly under-predict the
masses of convective stellar cores.

Methods. We gather convective core mass and fractional core hydrogen content
inferences from numerous independent binary and asteroseismic studies, and
compare them to stellar evolution models computed with the MESA stellar
evolution code.

Results. We demonstrate that core mass inferences from the literature are
ubiquitously more massive than predicted by stellar evolution models without
or with little convective boundary mixing.

Conclusions. Independent of the form of internal mixing, stellar models
require an efficient mixing mechanism that produces more massive cores
throughout the main sequence to reproduce high-precision observations. This
has implications for the post-main sequence evolution of all stars which have
a well developed convective core on the main sequence.

###### Key Words.:

Asteroseismology - Stars: interiors - Stars: evolution - Stars: oscillations
(including pulsations) - Stars: binaries: eclipsing

##  1 Introduction

The treatment of convection and of convective boundary mixing has major
implications for the results of stellar evolutionary models (Maeder, 2009;
Kippenhahn et al., 2012). As it is inherently a 3D process, the 1D
descriptions of convection required for modern stellar structure and evolution
calculations, often achieved using the Mixing Length Theory (MLT, Böhm-
Vitense, 1958) or variations thereof, have shortcomings. Although MLT does
well in approximating the deep interiors of a convective region, its
limitations become apparent at the boundaries of convective regions. Namely,
MLT does not capture the scope of processes that occur over different distance
scales at the interface of a turbulent convective zone and a stably stratified
radiative zone. In such transition regions, phenomena including convective
penetration, overshooting, turbulent entrainment, bulk shear mixing, and the
generation of internal gravity waves which induce mixing in the adjacent
radiative region may occur (Renzini, 1987; Arnett et al., 2019). Additionally,
1D simplifications of convection result in a single, hard convective boundary
(based either on the Schwarzschild or Ledoux criterion), rather than the
dynamical, fluctuating boundary that is often observed in 2- and 3D
hydrodynamical simulations (Meakin & Arnett, 2007; Arnett et al., 2015;
Cristini et al., 2017; Higl et al., 2021). As a result of ignoring the non-
local extensions of convection, simplified 1D descriptions of convection
underestimate the transport of chemical species both at the convective
boundary and throughout the adjacent radiative region. More specifically, in
the absence of some form of explicitly included extra chemical mixing, 1D
stellar models with a convective nuclear burning core underestimate the
transport of fresh chemical species from the envelope into the core where they
can participate in burning. This implies that models which do not include
extra mixing underestimate the mass of the convective core at any time along a
star’s evolution when the core is convective (Bressan et al., 1981; Stothers &
Chin, 1981; Bertelli et al., 1985; Maeder & Meynet, 1987).

To account for the missing chemical mixing induced at convective boundaries,
1D stellar evolution models introduce ad-hoc parameterised mixing profiles
that are stitched to convective regions. However, each parameterised mechanism
only serves to address one aspect of the many processes occurring at, or
beyond, a convective boundary (Meakin & Arnett, 2007; Arnett et al., 2015;
Rogers & McElwaine, 2017). Furthermore, all of these parameterised
descriptions have at least one free parameter that scales the ‘efficiency’ of
the chemical mixing induced by that mechanism. Numerous studies have attempted
to observationally calibrate the efficiency of different mixing mechanisms
both at the convective boundary and in the radiative envelope, however, these
studies often report conflicting efficiencies for the same mechanisms. The
focus of this paper is to discuss the complexities of observationally
constraining the efficiency of mixing mechanisms in stellar models, to
introduce convective core mass as a robust point of comparison for the
implementation of chemical mixing in 1D stellar models, and to consider the
consequences of a range of inferred mixing efficiencies in model calculations.

##  2 Chemical mixing in the literature

A complete chemical mixing profile should include the effects of entrainment,
penetration, overshooting, semi-convection, as well as mixing induced by
rotational instabilities and internal gravity waves (Langer, 2012; Maeder,
2009; Hirschi et al., 2014; Arnett et al., 2015; Rogers & McElwaine, 2017;
Salaris & Cassisi, 2017). Unfortunately, discussion of chemical mixing
mechanisms in the literature suffers from a problem of mixed identities.
Numerous studies have attempted to constrain the impact of chemical mixing in
different forms, be it any of the mechanisms previously mentioned. However,
more often than not, these studies focus on constraining the efficiency of a
single mixing mechanism, fixing the efficiency of the other mechanisms or
ignoring them outright. In the process of doing this, one effectively enforces
that the mixing mechanism under investigation in practice becomes a proxy for
all of the chemical transport present. Thus, when comparing results of
different studies, one must account for both the physical mechanisms that were
explicitly included, as well as the absence of effects from any physical
mechanisms that were not included.

Given modern observational quantities, their associated precision, and
underlying model assumptions and degeneracies, we are unable to disentangle
the contributions of individual mixing mechanisms to the point of robustly
calibrating multiple mixing mechanisms in a single target (Valle et al., 2017,
2018; Constantino & Baraffe, 2018; Aerts et al., 2018; Johnston et al.,
2019c). Instead, different observables provide constraints on the overall
mixing history of a star. We will briefly review the types of observations
used to constrain chemical mixing in the literature.

###  2.1 Eclipsing Binaries

If the two components of a double-lined spectroscopic binary system eclipse,
one can model the eclipses and radial velocities to determine absolute masses
and radii of the components. Combined with effective temperature estimates
from spectroscopy, modelling eclipsing binaries provides some of the most
precise determination of fundamental stellar properties possible, with quoted
uncertainties to better than one percent on mass and radius (Torres et al.,
2010; Serenelli et al., 2021). Furthermore, binaries afford the assumption
that both components have the same age and initial chemical composition,
providing additional powerful constraints.

The standard approach in the literature is to fit stellar evolution models (or
isochrones) to the observed mass, radius, and temperature of the binary
components to determine how much internal mixing, if any, is required to
explain observations of intermediate- to high-mass stars with a convective
core on the main sequence. This approach has been used to investigate the
impact of rotational mixing (Brott et al., 2011b; Schneider et al., 2014b;
Ekström et al., 2018), convective penetration (Schroder et al., 1997; Pols et
al., 1997; Ribas et al., 2000; Guinan et al., 2000; Claret & Torres, 2016),
and convective overshooting (Stancliffe et al., 2015; Higl & Weiss, 2017;
Claret & Torres, 2019; Costa et al., 2019; Tkachenko et al., 2020). However,
notable works claim that chemical mixing efficiency cannot be uniquely
determined using this approach, even with such precise mass and radius
estimates (Valle et al., 2018; Constantino & Baraffe, 2018; Johnston et al.,
2019b). Similarly, systems that display apsidal motion have been used to
investigate chemical mixing through its influence on the central condensation
of the star over time (Claret & Giménez, 2010; Claret, 2019).

We note, however, that the fundamental stellar properties used in this
modelling method are not directly sensitive to the efficiency of any
particular mixing mechanism individually. Instead this methodology is
sensitive to the efficiency of any process which alters the radius and
temperature of the star. Through the transport of hydrogen from the envelope
into the core, chemical mixing increases the mass (and luminosity) of the
core, and extends the main-sequence lifetime. These changes to the core
translate to changes in the overall temperature and radius of the star at a
given age, which is what is estimated in binary modelling. Furthermore, in
addition to the transport of chemicals in the interior, very rapid rotation
causes changes to the radius and temperature at the surface of the star. Thus,
modelling the fundamental properties of binaries is sensitive to the total
internal mixing history which transports fresh hydrogen from the radiative
region into the convective core throughout the stellar lifetime, as well as
any modification to the surface properties due to rapid rotation. To this end,
we should be careful to not conflate the inferred overall mixing efficiency
from binary modelling with the efficiency of a single mixing mechanism.

###  2.2 Stellar clusters

Given the assumption that the members of a stellar cluster are formed at
approximately the same time and with a similar initial chemical composition,
the observed morphology of the colour-magnitude diagram can be used to
calibrate internal mixing mechanisms in stellar models. Several studies have
demonstrated that the width of the main sequence is sensitive to the mixing
history of stars, and used this sensitivity to estimate the effect of
overshooting and/or rotational mixing on stellar models (VandenBerg et al.,
2006; Castro et al., 2014; Martinet et al., 2021).

Similar to the width of the main sequence, the morphology of the extended
main-sequence turn off (eMSTO) is used to test the implementation of mixing
mechanisms in stellar models. While the eMSTO phenomenon is commonly
associated with rapid rotation and its consequences (Dupree et al., 2017;
Kamann et al., 2018; Bastian et al., 2018; Georgy et al., 2019), studies have
shown that the eMSTO of young massive clusters can be reproduced by stellar
evolution models with convective boundary mixing calibrated from
asteroseismology (Yang & Tian, 2017; Johnston et al., 2019a). Alternatively,
binary evolution and its byproducts have been invoked as a means of explaining
the eMSTO phenomenon (Schneider et al., 2014a; Gosnell et al., 2015; Beasor et
al., 2019). As in the case of eclipsing binaries, this type of analysis is not
directly sensitive to the efficiency of a given mixing mechanism, but rather
to the total mixing across the evolution of a star.

###  2.3 Asteroseismology

Regions of the stellar interior can be driven to deviate from hydrostatic
equilibrium due to a build up of radiation caused by partial ionisation zones,
the blocking of convective flux, or other mechanisms. Depending on where these
perturbations occur within the stellar interior, the restoring force will be
dominated by either the pressure force or the buoyancy force. If a
perturbation is regularly driven, it produces a standing pressure or buoyancy
wave propagating throughout the star, resulting in surface brightness and
velocity variations. The frequency of the standing wave is determined by the
chemical and thermodynamical conditions of the regions that the waves travel
through. Asteroseismology is then the study of the stellar interior via the
observation of and modelling of stellar oscillations (Aerts et al., 2010;
Aerts, 2021).

The frequencies of g modes (buoyancy waves) in particular are sensitive to the
near-core chemical gradient (Miglio et al., 2008) and rotation rate (Bouabid
et al., 2013). Asteroesismology of stars oscillating in p modes and in g modes
with 1.3 $\leq$ M $\leq$ 24 M⊙ has revealed the need for a wide range of
convective boundary mixing efficiencies to reproduce observed pulsation
frequencies (Briquet et al., 2007; Handler, 2013; Moravveji et al., 2016;
Schmid & Aerts, 2016; Szewczuk & Daszyńska-Daszkiewicz, 2018; Mombarg et al.,
2019; Pedersen et al., 2021). Additionally, asteroseismology of stars
pulsating in p modes (pressure waves) between 1.15-1.5 M⊙ has revealed the
need for a range of convective mixing efficiencies to reproduce observed
frequencies (Deheuvels et al., 2010; Deheuvels & Michel, 2011; Silva Aguirre
et al., 2011; Salmon et al., 2012; Angelou et al., 2020; Viani & Basu, 2020;
Noll et al., 2021). Similar to binaries and clusters, the modelling of
pulsation frequencies (for both p and g modes) is not sensitive to the
efficiency of a single mixing mechanism. Instead asteroseismology is sensitive
to the total influence of chemical mixing that alters the core mass and
radius, as well as the chemical gradient near the core (high order g modes)
and the bulk density of the star (low order p and g modes). However, studies
have demonstrated that g-mode asteroseismology is able to differentiate
between the morphology and temperature gradient of the near core mixing
profile for models of the same mass evolved to the same core hydrogen content
(Pedersen et al., 2018; Michielsen et al., 2019).

##  3 Convective core masses

Despite the remarkable precision provided by eclipse modelling and/or
asteroseismology, modelling methodologies cannot uniquely constrain the
efficiency of an individual mixing mechanism (Valle et al., 2018; Aerts et
al., 2018). Unfortunately, parameter correlations in stellar evolution models
can produce several models with different masses, ages, and amounts of
internal mixing that all reproduce observations reasonably well within the
uncertainties. As such, it is often the case that any amount of internal
mixing can be used to match observations, even when the bulk stellar
properties or oscillation frequencies are known to a high precision (e.g.,
Aerts et al., 2018; Constantino & Baraffe, 2018; Johnston et al., 2019c, b;
Michielsen et al., 2019; Sekaran et al., 2021). Furthermore, both binary and
asteroseismic studies have demonstrated that reasonably similar fits to the
data can be obtained using models with different types of internal mixing
mechanisms (Moravveji et al., 2015; Claret & Torres, 2017; Mombarg et al.,
2019; Pedersen et al., 2021) and for different bulk chemical compositions
(Claret & Torres, 2017, 2018).

Although model degeneracies prevent the direct constraining of mixing
mechanism efficiencies, Johnston et al. (2019b); Tkachenko et al. (2020)
demonstrated that 1% precision on mass and radius estimates from eclipsing
binaries result in inferences on the mass and hydrogen content of the
convective core to a precision of $\sim 10$ % or better. A similar result has
been found through asteroseismic analysis of both p and g mode pulsating stars
with a convective core on the main sequence (Mazumdar et al., 2006; Briquet et
al., 2011; Johnston et al., 2019c; Angelou et al., 2020; Mombarg et al., 2021;
Pedersen et al., 2021).

A few studies have performed comparative model fits between models with an
exponentially decaying diffusive overshooting with a radiative temperature
gradient in the overshooting region and models with step penetrative
convection with an adiabatic expansion of the convective core. These studies
demonstrate that the best matching solutions from either model have core
masses, masses, and ages, which largely agree within uncertainties (Claret &
Torres, 2018; Tkachenko et al., 2020; Mombarg et al., 2021; Pedersen et al.,
2021). However, it should be noted that solutions with different metallicities
tend to have different core mass inferences, masses, and ages as well. This
highlights that core mass inferences can function as a robust proxy for the
overall influence of chemical mixing, even when the efficiency of the
implemented mechanism cannot be constrained. Furthermore, this implies that
core mass inferences can serve as a robust comparison point for solutions that
do not use the same implementation of chemical mixing.

Figure 1: Convective core mass (Mcc) vs initial mass (M) of the star. Solid
and dashed lines correspond to fractional core mass at specified points in
evolution. Colour of region corresponds to fractional core hydrogen content as
indicated by colour-bar. Colour of individual marker corresponds to inferred
fractional core hydrogen content of best model. Squares with orange
uncertainties denote values obtained from binary systems, circles with blue
uncertainties denote values obtained from binary systems with at least one
pulsating component, and triangles with grey uncertainties denote values
obtained from pulsating stars. References for values and their uncertainties
are listed in Table 1.

To this end, we collect a large sample of convective core mass and core
hydrogen content inferences (the mass and remaining hydrogen content within
the Schwarzschild boundary) for core-hydrogen burning stars from the
literature and show them in Fig. 1. The values, their associated
uncertainties, and references are listed in Table 1. The core mass inferences
are plotted against predictions from non-rotating models with $Z=0.014$ and
$Y=0.276$ (computed with mesa r10398; Paxton et al., 2019) with a minimum
amount of internal chemical mixing (included in the form of diffusive
exponential overshooting with $f_{ov}$=0.005). The models span 1.1 ¡ M ¡ 25
M⊙. The colour of the background denotes the remaining fraction of the core-
hydrogen content relative to the amount at the zero-age main sequence. The
colour of the markers correspond to the inferred fractional core-hydrogen
content of that star. If the model predictions were accurate, the colour of
the marker would match the colour of the background where the point is
situated. Instead, the inferred values are systematically shifted upwards with
respect to their predicted location, indicating that the observations are
better reproduced by models with more massive convective cores that are less
progressed along their main-sequence evolution. This sample is composed of
inferences from p and g mode (and hybrid) pulsating stars, eclipsing binaries,
and pulsating stars in binaries, spanning a wide mass range (1.14 ¡ M ¡ 24
M⊙).

The diversity of this sample i) confirms the robust need for more massive
convective cores across a wide stellar mass range, and ii) demonstrates that
models including only one mixing mechanism with a single efficiency cannot
reproduce the range of core masses displayed in Fig. 1. Instead, we find that
the range of observations can be reproduced by considering stellar models with
internal mixing profiles that span a wide range of efficiencies. This result
is consistent with the fact that the observables used in modelling efforts are
not directly sensitive to a particular chemical mixing mechanism, but rather
to the overall influence of chemical mixing on the stellar temperature,
radius, core properties, and near core chemical gradient. Thus, we interpret
the diversity of inferred core masses to result from the combined effect of
multiple mixing mechanisms active in different stars. We adopt a
representative mixing profile consisting of convective boundary mixing in the
form of diffusive exponential overshooting (Freytag et al., 1996) with
$f_{ov}\in[0.005,0.04]H_{p}$, and mixing in the radiative zone induced by
internal gravity waves with base efficiencies in the range of $D_{\rm
IGW}\in[1-100]$ cm2 s-1. This range is derived by first considering the range
of efficiencies for each mechanism reported in the literature, and then
trimming the ranges to include all values for which we can reproduce the
values reported in Table 1. We stress that this representative profile and
efficiency ranges are only a proxy for the total amount of internal chemical
mixing, and should not be considered as absolute metric. We remark instead
that this range of efficiencies can be interpreted as arising from different
mechanisms producing an overall different effect in different stars.

##  4 Implications beyond the main sequence

The modification of the convective core mass during the main sequence
propagates throughout the remainder of the star’s evolution (Stothers & Chin,
1981). Kaiser et al. (2020) demonstrated that variable convective boundary
mixing efficiencies lead to large differences in the resulting helium-core
mass at the terminal-age main sequence (¿30%), the carbon/oxygen core mass at
the end of core helium burning (up to 70%), and in the observed surface
abundances. These differences necessarily have consequences for the stellar
end product, although Kaiser et al. (2020) only focused on supernova
progenitors. Such variations in stellar end products have also been
demonstrated for varying degrees of rapid rotation (Brott et al., 2011a;
Ekström et al., 2012), as well as for the inclusion of convective entrainment
in 1D models (Scott et al., 2021).

Figure 2 shows the predicted values of the helium core mass at the end of the
main sequence for stars between 1.1$\leq$M$\leq$25 M⊙. The grey region
contains those predicted values for models which include the representative
range of chemical mixing efficiencies which reliably reproduces the range of
inferred core masses shown in Fig 1. These predictions are compared to
predictions by non-rotating MIST (red, Choi et al., 2016) and PARSEC (blue,
Bressan et al., 2012) models. The result of this comparison is that we predict
a wide range of possible resultant helium core masses, compared to the MIST or
PARSEC models which adopt a single efficiency for one chemical mixing
mechanism, and that the predicted range of helium core masses increases with
mass within the available range.

The prediction of a range of helium core masses has numerous implications
beyond main-sequence evolution and to other fields of astronomy. Crucially,
this range suggests that there is not a singular one-to-one relation between
the birth-mass of the star and the mass of the end-product, as can be seen in
the range of the helium core masses. Kaiser et al. (2020) demonstrated that
this can have an impact on the results of supernova simulations, but did not
consider masses below 15 M⊙. The mass limit for a star to produce a neutron
star end-product canonically depends on the remnant core mass exceeding the
Chandrasekhar limit of M${}_{\rm crit}\approx 1.44$ M⊙. Typically, a remnant
core with such a mass is thought to be achieved for stars with initial masses
M${}_{\rm init}>\sim 8-9$ M⊙, considering possible accretion onto the core at
later evolutionary stages (Heger et al., 2003; Fryer & Hungerford, 2005;
Camenzind, 2007; Kippenhahn et al., 2012). This limiting mass is denoted by
the dotted black line in Fig. 2. Interestingly we see that this critical
remnant core mass is already reached for stars with initial masses as low as
M${}_{\rm init}>\sim 7$ M⊙ for models with the most internal mixing that is
considered. This lower limit does not consider possible core mass enhancement
through later phases of shell burning or convective boundary mixing during
later core burning phases.

Furthermore, studies have demonstrated that convective boundary mixing is
favoured to explain observations of evolved single stars (Montalbán et al.,
2013; Constantino et al., 2017; Bossini et al., 2017; Arentoft et al., 2017;
den Hartogh et al., 2019), and/or binary interaction products, such as
subdwarfs (Constantino & Baraffe, 2018; Ostrowski et al., 2021). However, the
study of binaries and stellar populations is also dependent upon
considerations of internal mixing in stellar models. Particularly, the
pathways of binary evolution products depend on the radius and central
condensation of an evolving star to determine the probability, duration, and
efficiency of mass transfer.

Figure 2: Predicted helium core mass at the terminal-age main sequence for
models with CBM represented by overshooting and internal gravity wave mixing
(grey) compared against predictions from MIST models (Choi et al., 2016) in
red and PARSEC models (Bressan et al., 2012) in blue. The dotted black line
denotes the fractional core mass required to produce a remnant helium core
with $M=1.44\leavevmode\nobreak\ {\rm M_{\odot}}$.

##  5 Conclusions

It is easy to review the results of all the attempts to calibrate different
internal mixing mechanisms and think that they disagree. However, given the
nature of the sensitivity of modern observables combined with the limitations
of 1D stellar models, we argue that these results do agree. Irrespective of
the type of internal mixing mechanism implemented, models require a mechanism
to modify the core mass, internal chemical gradients, and surface properties
in order to reproduce observations. This is supported by the fact that
different studies have been able to independently reproduce the behaviour of
commonly used observables with different implementations of mixing mechanisms
such as convective penetration, overshooting, entrainment, rotation, and
internal gravity waves (Brott et al., 2011b; Ekström et al., 2012; Staritsin,
2013; Moravveji et al., 2015; Claret & Torres, 2017; Tkachenko et al., 2020;
Pedersen et al., 2021; Scott et al., 2021).

Consulting the diversity of results from the literature, it is clear that a
wide range of internal mixing efficiencies are required to reproduce
observations even when considering a single mixing mechanism. When considering
a single efficiency for a single mixing mechanism, models predictions are
unintentionally biased. Based on the diverse sample of inferred core masses we
have collected, we instead argue that a range of efficiencies needs to be
considered to function as a proxy for the various mixing processes present at
the convective boundary which are commonly ignored in standard 1D models. As
this in turn produces a range of helium core masses at the terminal-age main
sequence, this range of efficiencies should be accounted for in evolutionary
calculations for both single and binary stars, nucleosynthetic yield
predictions, and population synthesis efforts beyond the main sequence.

###### Acknowledgements.

CJ would like to thank the referee for their comments which improved the
manuscript. CJ thanks M.G. Pedersen, J.S.G. Mombarg, G. Angelou, L. Viani, and
C. Neiner for sharing their results to be included in this manuscript, as well
as D.M. Bowman, A. Tkachenko, and C. Aerts for their useful discussion on
topics concerning asteroseismology and chemical mixing profiles, and M. Abdul-
Masih and A. Escorza for their comments on the manuscript. This work has
received funding from NOVA, the European Research Council under the European
Union’s Horizon 2020 research and innovation programme (N∘670519:MAMSIE), and
from the Research Foundation Flanders under grant agreement G0A2917N
(BlackGEM). The computational resources and services used in this work were
provided by the VSC (Flemish Supercomputer Center), funded by the Research
Foundation - Flanders (FWO) and the Flemish Government - department EWI to PI
Johnston.

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##  Appendix A Sample

Table 1: Sample of estimated stellar masses and inferred convective core
masses and fractional core-hydrogen content.

Star |  Mass [M⊙] |  Mcc [M⊙] |  Xc/Xini |  | Star |  Mass [M⊙] |  Mcc [M⊙] |  Xc/Xini |
---|---|---|---|---|---|---|---|---|---
KIC 12009504 | $1.19\pm 0.07$ | $0.08\pm 0.02$ | 0.21 | (a) | KIC 7760680 | $3.37\begin{subarray}{c}+0.01\\\ -0.06\end{subarray}$ | $0.62\begin{subarray}{c}+0.02\\\ -0.00\end{subarray}$ | 0.53 | (c)
KIC 6225718 | $1.26\pm 0.03$ | $0.08\pm 0.04$ | 0.60 | (a) | KIC 8057661 | $9.52\begin{subarray}{c}+0.02\\\ -0.03\end{subarray}$ | $1.76\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | 0.18 | (c)
KIC 10454113 | $1.27\pm 0.02$ | $0.08\pm 0.03$ | 0.59 | (a) | KIC 8255796 | $5.73\begin{subarray}{c}+0.10\\\ -0.03\end{subarray}$ | $0.49\begin{subarray}{c}+0.10\\\ -0.00\end{subarray}$ | 0.02 | (c)
KIC 5184732 | $1.20\pm 0.01$ | $0.06\pm 0.001$ | 0.26 | (a) | KIC 8381949 | $6.27\begin{subarray}{c}+0.79\\\ -0.00\end{subarray}$ | $1.53\begin{subarray}{c}+0.07\\\ -0.10\end{subarray}$ | 0.57 | (c)
KIC 12009504 | $1.20\pm 0.01$ | $0.07\pm 0.02$ | 0.21 | (a) | KIC 8459899 | $3.36\begin{subarray}{c}+0.00\\\ -0.03\end{subarray}$ | $0.84\begin{subarray}{c}+0.03\\\ -0.00\end{subarray}$ | 0.34 | (c)
KIC 7206837 | $1.44\pm 0.04$ | $0.17\pm 0.03$ | 0.41 | (a) | KIC 8714886 | $6.49\begin{subarray}{c}+0.00\\\ -0.08\end{subarray}$ | $0.92\begin{subarray}{c}+0.01\\\ -0.00\end{subarray}$ | 0.28 | (c)
KIC 12258514 | $1.24\pm 0.02$ | $0.09\pm 0.01$ | 0.07 | (a) | KIC 8766405 | $3.49\begin{subarray}{c}+0.00\\\ -0.23\end{subarray}$ | $0.55\begin{subarray}{c}+0.03\\\ -0.00\end{subarray}$ | 0.22 | (c)
KIC 7510397 | $1.36\pm 0.04$ | $0.11\pm 0.03$ | 0.09 | (a) | KIC 9020774 | $3.41\begin{subarray}{c}+0.21\\\ -0.28\end{subarray}$ | $0.70\begin{subarray}{c}+0.07\\\ -0.17\end{subarray}$ | 0.78 | (c)
KIC 8228742 | $1.33\pm 0.05$ | $0.11\pm 0.01$ | 0.06 | (a) | KIC 9715425 | $4.72\begin{subarray}{c}+0.45\\\ -0.20\end{subarray}$ | $0.91\begin{subarray}{c}+0.01\\\ -0.06\end{subarray}$ | 0.48 | (c)
KIC 2710594 | $1.50\begin{subarray}{c}+0.13\\\ -0.09\end{subarray}$ | $0.16\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | 0.67 | (b) | KIC 10526294 | $3.64\begin{subarray}{c}+0.10\\\ -0.21\end{subarray}$ | $0.41\begin{subarray}{c}+0.01\\\ -0.06\end{subarray}$ | 0.23 | (c)
KIC 3448365 | $1.49\begin{subarray}{c}+0.01\\\ -0.07\end{subarray}$ | $0.12\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | 0.63 | (b) | KIC 10536147 | $7.50\begin{subarray}{c}+0.07\\\ -0.41\end{subarray}$ | $2.17\begin{subarray}{c}+0.06\\\ -0.14\end{subarray}$ | 0.98 | (c)
KIC 4846809 | $1.49\begin{subarray}{c}+0.05\\\ -0.02\end{subarray}$ | $0.15\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | 0.58 | (b) | KIC 11360704 | $4.47\begin{subarray}{c}+1.54\\\ -0.00\end{subarray}$ | $1.20\begin{subarray}{c}+0.00\\\ -0.66\end{subarray}$ | 0.47 | (c)
KIC 5114382 | $1.61\begin{subarray}{c}+0.02\\\ -0.14\end{subarray}$ | $0.16\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | 0.36 | (b) | KIC 11971405 | $3.67\begin{subarray}{c}+0.21\\\ -0.72\end{subarray}$ | $0.62\begin{subarray}{c}+0.11\\\ -0.03\end{subarray}$ | 0.46 | (c)
KIC 5522154 | $1.72\begin{subarray}{c}+0.28\\\ -0.22\end{subarray}$ | $0.19\begin{subarray}{c}+0.03\\\ -0.02\end{subarray}$ | 0.97 | (b) | KIC 12258330 | $3.53\begin{subarray}{c}+0.04\\\ -0.13\end{subarray}$ | $0.69\begin{subarray}{c}+0.03\\\ -0.00\end{subarray}$ | 0.68 | (c)
KIC 5708550 | $1.99\begin{subarray}{c}+0.01\\\ -0.61\end{subarray}$ | $0.20\begin{subarray}{c}+0.01\\\ -0.06\end{subarray}$ | 0.98 | (b) | KIC 4930889 A | $4.89\begin{subarray}{c}+1.49\\\ -1.09\end{subarray}$ | 0.54 | 0.07 | (d)
KIC 5788623 | $1.47\begin{subarray}{c}+0.01\\\ -0.05\end{subarray}$ | $0.12\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | 0.70 | (b) | KIC 4930889 B | $3.47\begin{subarray}{c}+1.40\\\ -0.57\end{subarray}$ | 0.60 | 0.68 | (d)
KIC 6468146 | $1.59\begin{subarray}{c}+0.01\\\ -0.04\end{subarray}$ | $0.17\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | 0.61 | (b) | KIC 6352430 A | $3.23\begin{subarray}{c}+1.98\\\ -0.56\end{subarray}$ | 0.56 | 0.59 | (d)
KIC 6468987 | $1.57\begin{subarray}{c}+0.13\\\ -0.16\end{subarray}$ | $0.20\begin{subarray}{c}+0.02\\\ -0.02\end{subarray}$ | 0.74 | (b) | KIC 6352430 B | $1.34\begin{subarray}{c}+0.68\\\ -1.14\end{subarray}$ | 0.08 | 0.77 | (d)
KIC 6678174 | $1.77\begin{subarray}{c}+0.23\\\ -0.22\end{subarray}$ | $0.18\begin{subarray}{c}+0.02\\\ -0.02\end{subarray}$ | 0.20 | (b) | KIC 10080943 A | $1.99\begin{subarray}{c}+0.37\\\ -0.49\end{subarray}$ | 0.15 | 0.08 | (d)
KIC 6935014 | $1.45\begin{subarray}{c}+0.03\\\ -0.03\end{subarray}$ | $0.16\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | 0.86 | (b) | KIC 10080943 B | $1.85\begin{subarray}{c}+0.41\\\ -0.45\end{subarray}$ | 0.17 | 0.30 | (d)
KIC 6953103 | $1.88\begin{subarray}{c}+0.12\\\ -0.19\end{subarray}$ | $0.25\begin{subarray}{c}+0.02\\\ -0.03\end{subarray}$ | 0.48 | (b) | V578 Mon A | $14.55\pm 0.09$ | $5.12\pm 0.44$ | 0.82 | (e)
KIC 7023122 | $1.51\begin{subarray}{c}+0.00\\\ -0.04\end{subarray}$ | $0.16\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | 0.96 | (b) | V578 Mon B | $10.30\pm 0.06$ | $3.21\pm 0.37$ | 0.89 | (e)
KIC 7365537 | $1.70\begin{subarray}{c}+0.30\\\ -0.26\end{subarray}$ | $0.18\begin{subarray}{c}+0.03\\\ -0.03\end{subarray}$ | 0.97 | (b) | V453 Cyg A | $14.95\pm 0.35$ | $4.81\pm 0.35$ | 0.41 | (e)
KIC 7380501 | $1.72\begin{subarray}{c}+0.00\\\ -0.35\end{subarray}$ | $0.14\begin{subarray}{c}+0.01\\\ -0.03\end{subarray}$ | 0.19 | (b) | V453 Cyg B | $11.91\pm 0.57$ | $3.66\pm 0.53$ | 0.66 | (e)
KIC 7434470 | $1.48\begin{subarray}{c}+0.01\\\ -0.16\end{subarray}$ | $0.17\begin{subarray}{c}+0.01\\\ -0.02\end{subarray}$ | 0.64 | (b) | V478 Cyg A | $16.74\pm 0.73$ | $6.04\pm 0.85$ | 0.58 | (e)
KIC 7583663 | $1.47\begin{subarray}{c}+0.27\\\ -0.16\end{subarray}$ | $0.16\begin{subarray}{c}+0.03\\\ -0.02\end{subarray}$ | 0.64 | (b) | V478 Cyg B | $16.39\pm 0.71$ | $5.97\pm 0.84$ | 0.57 | (e)
KIC 7939065 | $1.48\begin{subarray}{c}+0.02\\\ -0.06\end{subarray}$ | $0.14\begin{subarray}{c}+0.00\\\ -0.01\end{subarray}$ | 0.66 | (b) | AH Cep A | $16.26\pm 0.18$ | $6.20\pm 0.62$ | 0.74 | (e)
KIC 8364249 | $1.52\begin{subarray}{c}+0.23\\\ -0.05\end{subarray}$ | $0.17\begin{subarray}{c}+0.03\\\ -0.01\end{subarray}$ | 0.66 | (b) | AH Cep B | $14.86\pm 0.70$ | $5.22\pm 0.77$ | 0.70 | (e)
KIC 8375138 | $1.47\begin{subarray}{c}+0.05\\\ -0.07\end{subarray}$ | $0.15\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | 0.69 | (b) | V346 Cen A | $13.76\pm 0.40$ | $4.14\pm 0.35$ | 0.36 | (e)
KIC 8645874 | $1.52\begin{subarray}{c}+0.02\\\ -0.01\end{subarray}$ | $0.14\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | 0.88 | (b) | V346 Cen B | $8.40\pm 0.10$ | $2.04\pm 0.35$ | 0.73 | (e)
KIC 8836473 | $1.48\begin{subarray}{c}+0.11\\\ -0.01\end{subarray}$ | $0.14\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | 0.34 | (b) | V573 Car A | $16.62\pm 0.45$ | $6.65\pm 0.45$ | 0.87 | (e)
KIC 9480469 | $1.49\begin{subarray}{c}+0.05\\\ -0.11\end{subarray}$ | $0.14\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | 0.82 | (b) | V573 Car B | $12.49\pm 0.17$ | $4.37\pm 0.32$ | 0.95 | (e)
KIC 9595743 | $1.42\begin{subarray}{c}+0.03\\\ -0.04\end{subarray}$ | $0.09\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | 0.94 | (b) | V1032 Sco A | $18.97\pm 0.77$ | $7.44\pm 0.89$ | 0.60 | (e)
KIC 9751996 | $1.74\begin{subarray}{c}+0.01\\\ -0.02\end{subarray}$ | $0.17\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | 0.29 | (b) | V1032 Sco B | $10.47\pm 0.24$ | $3.29\pm 0.24$ | 0.87 | (e)
KIC 10467146 | $1.77\begin{subarray}{c}+0.02\\\ -0.27\end{subarray}$ | $0.16\begin{subarray}{c}+0.01\\\ -0.02\end{subarray}$ | 0.22 | (b) | V380 Cyg A | $15.00\pm 0.50$ | $3.85\pm 0.15$ | 0.00 | (e)
KIC 11080103 | $1.53\begin{subarray}{c}+0.02\\\ -0.09\end{subarray}$ | $0.16\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | 0.76 | (b) | V380 Cyg B | $8.32\pm 0.75$ | $2.26\pm 0.59$ | 0.78 | (e)
KIC 11099031 | $1.57\begin{subarray}{c}+0.03\\\ -0.21\end{subarray}$ | $0.17\begin{subarray}{c}+0.01\\\ -0.02\end{subarray}$ | 0.41 | (b) | CW Cep A | $13.01\pm 0.07$ | $4.38\pm 0.62$ | 0.77 | (e)
KIC 11294808 | $1.45\begin{subarray}{c}+0.20\\\ -0.04\end{subarray}$ | $0.13\begin{subarray}{c}+0.02\\\ -0.00\end{subarray}$ | 0.27 | (b) | CW Cep B | $11.95\pm 0.09$ | $3.86\pm 0.48$ | 0.79 | (e)
KIC 11456474 | $1.49\begin{subarray}{c}+0.14\\\ -0.13\end{subarray}$ | $0.13\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | 0.25 | (b) | U Oph A | $5.10\pm 0.05$ | $1.01\pm 0.11$ | 0.65 | (e)
KIC 11721304 | $1.43\begin{subarray}{c}+0.04\\\ -0.03\end{subarray}$ | $0.15\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | 0.89 | (b) | U Oph B | $4.60\pm 0.05$ | $0.93\pm 0.08$ | 0.74 | (e)
KIC 11754232 | $1.53\begin{subarray}{c}+0.02\\\ -0.04\end{subarray}$ | $0.15\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | 0.90 | (b) | V621 Per A | $11.81\pm 0.30$ | $2.80\pm 0.21$ | 0.84 | (e)
KIC 11907454 | $1.47\begin{subarray}{c}+0.01\\\ -0.10\end{subarray}$ | $0.13\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | 0.83 | (b) | HD165246 B | $3.8\begin{subarray}{c}+0.4\\\ -0.5\end{subarray}$ | $0.42\begin{subarray}{c}+0.4\\\ -0.2\end{subarray}$ | 1.00 | (f)
KIC 11826272 | $1.52\begin{subarray}{c}+0.04\\\ -0.07\end{subarray}$ | $0.16\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | 0.51 | (b) | HD165246 A | $23.7\begin{subarray}{c}+1.1\\\ -1.4\end{subarray}$ | $10.2\begin{subarray}{c}+1.3\\\ -1.6\end{subarray}$ | 0.76 | (f)
KIC 11917550 | $1.51\begin{subarray}{c}+0.00\\\ -0.10\end{subarray}$ | $0.16\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | 0.67 | (b) | 2052 Oph | $8.9\pm 0.7$ | $1.70\pm 0.2$ | 0.41 | (g)
KIC 11920505 | $1.45\begin{subarray}{c}+0.03\\\ -0.01\end{subarray}$ | $0.11\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | 0.88 | (b) | HD 43317 | $5.8\begin{subarray}{c}+0.1\\\ -0.2\end{subarray}$ | $1.33\pm 0.05$ | 0.76 | (h)
KIC 12066947 | $1.49\begin{subarray}{c}+0.02\\\ -0.01\end{subarray}$ | $0.14\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | 0.65 | (b) | HIC 12258514 | $1.20\pm 0.05$ | $0.05\pm 0.01$ |  | (i)
KIC 3240411 | $5.34\begin{subarray}{c}+0.31\\\ -0.07\end{subarray}$ | $1.56\begin{subarray}{c}+0.00\\\ -0.02\end{subarray}$ | 0.68 | (c) | KIC 8228742 | $1.25\pm 0.08$ | $0.06\pm 0.02$ |  | (i)
KIC 3459297 | $3.70\begin{subarray}{c}+0.00\\\ -0.18\end{subarray}$ | $0.24\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | 0.05 | (c) | KIC 7510397 | $1.22\pm 0.06$ | $0.07\pm 0.03$ |  | (i)
KIC 3865742 | $5.66\begin{subarray}{c}+0.00\\\ -0.08\end{subarray}$ | $1.05\begin{subarray}{c}+0.03\\\ -0.00\end{subarray}$ | 0.22 | (c) | KIC 10162436 | $1.30\pm 0.06$ | $0.09\pm 0.02$ |  | (i)
KIC 4936089 | $3.66\begin{subarray}{c}+0.01\\\ -0.00\end{subarray}$ | $0.43\begin{subarray}{c}+0.00\\\ -0.00\end{subarray}$ | 0.35 | (c) | KIC 5773345 | $1.49\pm 0.09$ | $0.15\pm 0.01$ |  | (i)
KIC 4939281 | $5.38\begin{subarray}{c}+1.18\\\ -0.34\end{subarray}$ | $1.31\begin{subarray}{c}+0.01\\\ -0.27\end{subarray}$ | 0.48 | (c) | KIC 3456181 | $1.39\pm 0.07$ | $0.11\pm 0.03$ |  | (i)
KIC 5309849 | $5.67\begin{subarray}{c}+0.01\\\ -0.00\end{subarray}$ | $0.75\begin{subarray}{c}+0.00\\\ -0.00\end{subarray}$ | 0.15 | (c) | KIC 8179536 | $1.22\pm 0.09$ | $0.03\pm 0.03$ |  | (i)
KIC 5941844 | $3.75\begin{subarray}{c}+0.03\\\ -0.02\end{subarray}$ | $0.90\begin{subarray}{c}+0.04\\\ -0.27\end{subarray}$ | 0.94 | (c) |  $\beta$ Cma | $13.5\pm 0.5$ | $3.5\pm 0.3$ | 0.18 | (j)
KIC 6462033 | $7.45\begin{subarray}{c}+0.13\\\ -0.00\end{subarray}$ | $1.08\begin{subarray}{c}+0.05\\\ -0.00\end{subarray}$ | 0.17 | (c) |  $\beta$ Cru | $14.5\pm 0.5$ | $4.13\pm 0.5$ | 0.27 | (k)
KIC 6780397 | $6.29\begin{subarray}{c}+0.01\\\ -0.00\end{subarray}$ | $0.55\begin{subarray}{c}+0.00\\\ -0.00\end{subarray}$ | 0.05 | (c) | KIC 9850387 | $1.66\pm 0.01$ | $0.16\pm 0.01$ | 0.24 | (l)
KIC 7630417 | $6.92\begin{subarray}{c}+0.01\\\ -0.00\end{subarray}$ | $0.91\begin{subarray}{c}+0.00\\\ -0.00\end{subarray}$ | 0.15 | (c) |  |  |  |  |

(a) Angelou et al. 2020 ; (b) Mombarg et al. 2021 ; (c) Pedersen et al. 2021 ;
(d) Johnston et al. 2019c; (e) Tkachenko et al. 2020; (f) Johnston et al.
2021; (g) Briquet et al. 2011; (h) Buysschaert et al. 2018; (i) Viani & Basu
2020; (j) Mazumdar et al. 2006; (k) Cotton et al. in review; (l) Sekaran et
al. 2021