# Conformal Geodesics Cannot Spiral

Peter Cameron, Maciej Dunajski & Paul Tod

###### Abstract

We show that conformal geodesics on a Riemannian manifold cannot spiral: there
does not exist a conformal geodesic which becomes trapped in every
neighbourhood of a point.

In memory of Bernd Schmidt (1941–2023)

##  1 Introduction

A pseudo-Riemannian manifold, $(M,g)$ defines a distinguished set of curves
which are metric geodesics of $g$. These curves are solutions to a set of 2nd
order ODEs and are uniquely specified locally by an initial point and an
initial tangent direction. However, with the exception of null geodesics in
the Lorentzian case, there is in general no relation between the metric
geodesics of two conformally related metrics. By analogy with metric
geodesics, conformal geodesics can be thought of as a distinguished set of
curves defined on a conformal manifold $(M,[g])$. These curves are solutions
to a system of conformally invariant 3rd order ODEs and are uniquely specified
locally by an initial position, unit tangent direction and perpendicular
acceleration [12, 1, 2, 11]. Although metric geodesics have been studied
extensively, relatively little is known about conformal geodesics.

Conformal geodesics were systematically introduced into general relativity by
Friedrich and Schmidt [6]. Their motivation was to construct good co-ordinate
systems for the local study of conformal boundaries, as introduced in
Penrose’s process of conformal compactification. Normal or Gaussian co-
ordinates based on timelike geodesics cannot be expected to behave well near
such conformal boundaries - recall for example how, in compactified Minkowski
space, timelike geodesics do not pass through null infinity but instead all
focus at timelike infinity. It was shown in [6] (following an earlier
suggestion in [10]) that one could instead define conformal normal or
conformal Gaussian co-ordinates constructed from conformal geodesics. These
would provide good co-ordinates near the conformal boundaries of
asymptotically Minkowskian, asymptotically de Sitter or anti-de Sitter
spacetimes.

An unresolved issue stemming from this earlier work, which was also known as
an unsolved problem in Riemannian-signature conformal geometry, was the
question of spiralling for conformal geodesics. A curve can be said to spiral
at a point $p_{*}$ if it enters and remains in every neighbourhood of $p_{*}$
but does not pass through $p_{*}$ itself. It is a classical result, based on
the existence of geodesically-convex neighbourhoods [5], that metric geodesics
cannot spiral, although until now this has not been shown for conformal
geodesics. It is a question of interest both abstractly and because spiralling
raises the possibility of a new kind of co-ordinate singularity to guard
against.

In [11], spiralling was ruled out in some special cases using 1st integrals,
and integrability (see also [3] for integrable, non-spiralling examples on
four-manifolds). For example, it was shown that conformal geodesics defined on
a Riemannian Einstein manifold cannot spiral. The conformal geodesic equations
were also integrated for some example geometries and it was observed that the
solutions did not spiral.

In this paper we will prove a general no-spiralling theorem for conformal
geodesics (Theorem 2.3). We begin by considering the analogous result for
metric geodesics. The standard proof that metric geodesics cannot spiral
relies on the existence of a geodesically convex neighbourhood at each point
[5, Proposition 4.2]. Such a proof will not work for conformal geodesics
since, despite their variational origins [4], these are not local length
minimisers. With this in mind, we begin by constructing a proof of the no-
spiralling theorem for metric geodesics which does not rely on length
minimisation arguments. This will then form the basis of our proof in the
conformal geodesic case.

### Acknowledgements

MD has been partially supported by STFC grants ST/P000681/1, and ST/T000694/1.
PC acknowledges the support of the Royal Society University Research
Fellowship URF$\backslash$R1$\backslash$211216 and the Todd/Goddard Fund of St
John’s College, Cambridge. We are also grateful to the anonymous referee for
their insightful comments.

##  2 The Conformal Geodesic Equations

Conformal geodesics are curves defined on a conformal manifold $(M,[g])$,
which we assume to be Riemannian. Given any choice of metric $g\in[g]$, these
curves can be defined by the following set of conformally invariant third
order differential equations [11]:

$\begin{split}\nabla_{u}a&=-\left(|a|^{2}+L(u,u)\right)u+L^{\\#}u\end{split}$
(2.1)

where $u$ is the unit tangent vector to the curve (i.e $g(u,u)=1$),
$a:=\nabla_{u}u$ is the perpendicular acceleration (i.e $g(u,a)=0$) and
$|a|:=\sqrt{g(a,a)}$. The Schouten tensor $L:T_{p}M\times
T_{p}M\mapsto\mathbbm{R}$ is defined by

$L=\frac{1}{d-2}\left(Ric-\frac{R}{2(d-1)}g\right)$ (2.2)

in terms of the Ricci tensor $Ric$, the Ricci scalar $R$ and the dimension of
the manifold $d$. We denote by $L^{\\#}:T_{p}M\mapsto T_{p}M$ the
corresponding endomorphism defined by $g(L^{\\#}u,v)=L(u,v)$ for all vector
fields $u,v$.

Changing the metric to $\hat{g}=\Omega^{2}g$, where
$\Omega:M\rightarrow\mathbbm{R}$ results in changes to the Schouten tensor,
the Levi–Civita connection, unit tangent vector and the acceleration

$\displaystyle\hat{L}$ $\displaystyle=$ $\displaystyle
L-\nabla\Upsilon+\Upsilon\otimes\Upsilon-\frac{1}{2}|\Upsilon|^{2}g,$
$\displaystyle\hat{\nabla}_{v}w$ $\displaystyle=$
$\displaystyle\nabla_{v}w+\Upsilon(w)v+\Upsilon(v)w-g(v,w)\Upsilon^{\sharp},$
$\displaystyle\hat{u}$ $\displaystyle=$ $\displaystyle\Omega^{-1}u,$
$\displaystyle\hat{a}$ $\displaystyle=$
$\displaystyle\Omega^{-2}(a-\Upsilon^{\sharp}+\Upsilon(u)u),$

where $\Upsilon\equiv\Omega^{-1}d\Omega$, and $\Upsilon^{\sharp}$ is a vector
field defined by $\Upsilon(w)=g({\Upsilon^{\sharp}},{w})$. It is now a matter
of explicit calculation to verify that the conformal geodesic equations (2.1)
are conformally invariant.

In the rest of the paper we shall fix a metric $g$ in the conformal class, and
consider (2.1) for this choice of metric. We will denote the set of unit
vectors in $T_{p}M$ by $S(T_{p}M)$ and $S\left(TM\right)$ will denote the
corresponding unit tangent bundle over $M$.

Picard’s theorem applied to equation (2.1) shows that, locally, a conformal
geodesic is uniquely specified by an initial point $p\in M$, an initial unit
tangent, $u_{0}$, at $p$ and an initial acceleration, $a_{0}$, at $p$ which is
perpendicular to $u_{0}$ [11, Theorem 1.1]. We will denote the conformal
geodesic with these initial conditions and arc length parameter $t$ by
$\gamma_{p,u_{0},a_{0}}(t)$.

######  Definition 2.1.

A curve, $\gamma$, with arc length parameter $t$, spirals towards a point
$p_{*}\in M$ if for any neighbourhood $N$ containing $p_{*}$, there exists
some $T$ such that $\gamma(t)\in N$ for all $t>T$.

We demand that the conformal geodesic has a well defined unit tangent vector
and thus exclude the trivial case where it consists only of a single point.

Throughout this paper we will refer to balls in $(M,g)$ and in various tangent
spaces. We make the following definition

$\begin{split}B(q,r):=\\{q^{\prime}\in M:d(q,q^{\prime})<r\\},\text{
}\bar{B}(q,r):=\\{p^{\prime}\in M:d(q,q^{\prime})\leq r\\}\end{split}$ (2.3)

where $d:M\times M\rightarrow\mathbbm{R}_{\geq 0}$ denotes the Riemannian
distance function induced by the metric $g$.

We will also refer to balls in the tangent space $T_{u_{0}}(T_{p}M)$, where
$u_{0}\in S(T_{p}M)$. We identify $T_{u_{0}}(T_{p}M)$ with $T_{p}M$ and these
balls are defined using the metric $g$, which in turn defines a notion of
length on the vector space $T_{p}M$.

###  2.1 Proof for Metric Geodesics

######  Theorem 2.2.

Let $(M,g)$ be a Riemannian manifold. Then geodesics on $(M,g)$ cannot spiral.

Proof (without using length minimisation arguments)111It was pointed out to us
by the referee that since this proof does not rely on length minimisation
arguments, Theorem 2.2 also applies to geodesics of any affine connection.:

  * •

Step 1: Define the exponential map and geodesic ball at $p\in M$ and establish
that all metric geodesics through $p$ reach the boundary of this geodesic
ball.

The exponential map, $\text{exp}_{p}$ at a point $p\in M$ [9, Definition 1.10]
is a diffeomorphism from some neighbourhood of $U\subset T_{p}M$ containing 0
to a neighbourhood $V\subset M$ containing $p$. A geodesic ball is any open
ball centred on $p$ and contained in $V$. The exponential map at $p$ allows us
to construct a co-ordinate system on any geodesic ball. Geodesic segments
foliate the geodesic ball with the point $p$ removed. For the purposes of this
proof, the important property of the geodesic ball is that all metric
geodesics through $p$ must reach its boundary.

  * •

Step 2: Show that for any compact $K\subset M$ there exists $\epsilon>0$ such
that for all $p\in K$, $B(p,\epsilon)$ is a geodesic ball.

We define a function on $M$ which maps each point $p\in M$ to the injectivity
radius at $p$ (defined to be the supremum of the set $\\{r\in\mathbbm{R}_{\geq
0}:B(p,r)\text{ is a geodesic ball}\\}$), capped at 1 to avoid any
complications involving infinite sizes. Since the exponential map defined at
any $p\in M$ is a local diffeomorphism, this function is strictly positive. By
the continuity of the injectivity radius [8, Proposition 2.1.10], it is also
continuous. The result then follows from the fact that a continuous function
on a compact set is bounded and attains its bounds.

  * •

Step 3: Use the fact that metric geodesics intersect the boundary of the
geodesic ball to deduce that they cannot spiral.

Suppose the metric geodesic $\gamma$ spirals towards some $p_{*}\in M$. Then
for any $R<d(p_{*},\gamma(0))$ there is a unique $T>0$ such that
$d(p_{*},\gamma(T))=R$ and $d(p_{*},\gamma(t))<R$ for all $t>T$. Denote the
point $\gamma(T)$ by $p$. This is the point at which $\gamma$ enters
$B(p_{*},R)$ for the final time.

Let $K\subset M$ containing $p_{*}$ be compact. Choose $R=\epsilon/2$ (where
$\epsilon$ is as in Step 2) and find the corresponding $p$ (reducing
$\epsilon$ if necessary so that $R<d(p_{*},\gamma(0))$. By reducing $\epsilon$
further if necessary, we can assume that $p\in K$. Then, using the results of
Step 2, $B(p,\epsilon)$ is a geodesic ball. From Step 1 we know that $\gamma$
must leave this geodesic ball at some later parameter value. In doing so, it
must once again reach a distance of at least $R=\epsilon/2$ from $p_{*}$. This
is a contradiction since we assumed that $d(p_{*},\gamma(t))<R$ for all $t>T$.
This argument is illustrated in Figure 1. ∎

Figure 1: We find $T\in\mathbbm{R}$ and $\epsilon>0$ such that there is a
geodesic ball of radius $\epsilon$ centred at $p:=\gamma(T)$, where
$d(p_{*},\gamma(T))=\epsilon/2$ and $d(p_{*},\gamma(t))<\epsilon/2$ for all
$t>T$. However, we have also shown that $\gamma$ must leave this geodesic ball
and hence reach a distance of $\epsilon/2$ from $p_{*}$ at some $t>T$. This is
our contradiction. We show geodesic balls as circles for ease of illustration.

###  2.2 Outline of Proof for Conformal Geodesics

The exponential map at $p\in M$ was useful because it allowed us to identify a
subset of $M$ which was foliated by segments of metric geodesics through $p$.
Inspired by this, we will define an exponential map adapted to conformal
manifolds which allows us to see how conformal geodesics foliate some subset
of $M$.

Despite some additional complexities in the conformal geodesic case, we are
able to prove a no-spiralling theorem by following three steps analogous to
those used in the proof of Theorem 2.2. These three steps correspond to the
three main sections of this paper.

######  Theorem 2.3.

Let $(M,[g])$ be a conformal manifold such that metrics in $[g]$ are smooth.
Then conformal geodesics on $(M,[g])$ cannot spiral.

Outline of Proof (see Section 5 for the complete proof using results derived
in Sections 3 and 4):

  * •

Step 1: Define an exponential map adapted to conformal geodesics and identify
a domain and range on which it is a homeomorphism.

In Section 3, we define a conformal geodesic analogue of the exponential map
at some $(p,u_{0})\in S(TM)$. Unlike for metric geodesics, there is no subset
of the domain of this exponential map on which it is a homeomorphism onto some
ball centred at $p$. Instead, in Section 3.3 we find that it is a
homeomorphism onto some set which we call a heart at $p$ (see Definition 3.5).
We say that this heart has direction $u_{0}$. Crucially, in Section 3.4 we are
able to show that conformal geodesics re-intersect the boundary of this heart.
This is analogous to the fact that metric geodesics through some point re-
intersect the boundary of a geodesic ball centred at that point.

Result of Step 1 (Theorem 3.7): For any $(p,u_{0})\in S(TM)$, any conformal
geodesic with unit tangent $u_{0}$ at $p$ re-intersects the boundary of a
heart at $p$ with direction $u_{0}$.

  * •

Step 2: Define a notion of size for a heart and show that given any compact
$K\subset M$, there exists $\epsilon>0$ such that there is a heart at $p$ with
direction $u_{0}$ which has size at least $\epsilon$, for any $p\in K$ and any
$u_{0}\in S(T_{p}M)$.

In the metric geodesic case we considered geodesic balls in $M$ since these
had an obvious size associated to them. In the conformal geodesic setting it
is simpler to consider certain half balls in $T_{u_{0}}(T_{p}M)$ which are
mapped to hearts in $M$. The size of a heart is defined by considering the
radii of balls in $M$ which have $u_{0}$ tangent to their boundary and are
contained inside this heart. Note that this size has units of length,
regardless of the dimension of the conformal manifold. We show that as we vary
$p$ and $u_{0}$ (as well as the radius of the domain ball222This is important
since we need to rule out making the heart have zero size by simply choosing
the domain to be trivial. To avoid this, we choose the domain to be the ball
in $T_{u_{0}}(T_{p}M)$ with $u_{0}$ perpendicular to its boundary and radius
equal to  $\frac{1}{2}\times\text{sup}\\{r:\text{$\exp_{p,u_{0}}$ is a
homeomorphism on the ball of radius $r$ with $u_{0}$ perpendicular to its
boundary}\\}.$ ) in $T_{u_{0}}(T_{p}M)$ we are able to find a lower bound for
this size which is strictly positive and varies continuously with $p$ and
$u_{0}$. The result follows from the fact that a continuous function on a
compact set is bounded and attains its bound.

Result of Step 2 (Theorem 4.2): Let $(M,[g])$ be a conformal manifold such
that metrics in $[g]$ are smooth and let $W$ be a compact subset of $S(TM)$.
Then there exists some $\epsilon>0$ such that at any $(p,u_{0})\in W$ there is
a heart $H_{p,u_{0}}$ at $p$ with direction $u_{0}$ which has size at least
$\epsilon$.

  * •

Step 3: Using the fact that a conformal geodesic with unit tangent $u_{0}$ at
$p$ must re-intersect the boundary of $H_{p,u_{0}}$, deduce that conformal
geodesics cannot spiral.

Suppose $\gamma$ is a conformal geodesic which spirals towards some $p_{*}\in
M$. We consider a metric geodesic ball centred on $p_{*}$ (i.e. a set
$B(p_{*},r)$ for some $r$) which is entered by $\gamma$ for the final time at
some point $p$ where it has unit tangent $u_{0}$ (i.e. $\gamma$ enters the
ball at $p$ and never re-intersects its boundary). Using steps 1 and 2, we
show that it is possible to choose this ball (and the corresponding $p$ and
$u_{0}$) such that it is contained inside some heart $H_{p,u_{0}}$ with
position $p$ and direction $u_{0}$ (see (3.21)). By Theorems 3.7 and 4.2,
$\gamma$ must re-intersect the boundary of this heart, so we deduce that it
must also re-intersect the boundary of the ball. This is our contradiction. ∎

Result of Step 3 (Theorem 2.3): Let $(M,[g])$ be a conformal manifold such
that metrics in $[g]$ are smooth. Then conformal geodesics on $(M,[g])$ cannot
spiral.

##  3 The Exponential Map for Conformal Geodesics

In this section we will complete Step 1 of the proof of Theorem 2.3 as
outlined in Section 2.2.

Step 1: Define an exponential map adapted to conformal geodesics and identify
a domain and range on which it is a homeomorphism.

For any $(p,u_{0})\in S\left(TM\right)$, we define the following analogue of
the exponential map appropriate to conformal geodesics (see [9, Definition
1.10] for a definition of the exponential map for metric geodesics):

######  Definition 3.1 (The Exponential Map).

$\begin{split}\exp_{p,u_{0}}:T_{u_{0}}\left(S\left(T_{p}M\right)\right)\supset
N_{0}&\rightarrow M\\\
A&\mapsto\begin{cases}\gamma_{p,u_{0},A_{\perp}/|A|^{2}}\left(2\pi|A|\right)&\text{if
}A\neq 0\\\ p&\text{if }A=0\end{cases}\end{split}$ (3.4)

where $A_{\perp}:=A-g(A,u_{0})u_{0}$ denotes the component of $A$ which is
perpendicular to $u_{0}\in S(T_{p}M)$ and $N_{0}$ is a neighbourhood of 0 in
$T_{u_{0}}\left(S\left(T_{p}M\right)\right)$ on which this map is well defined
(see [11, Theorem 1.1]). This map is a continuous function of $A$, as we will
see in the next section.

###  3.1 The Directional Derivative of The Exponential Map

The exponential map for metric geodesics is usually analysed by considering
its directional derivatives. We will do the same thing for the conformal
geodesic exponential map in Definition 3.1. Calculating these directional
derivatives at $A=0$ will allow us to understand this map in neighbourhoods of
$p$.

The directional derivative at 0 is

$\begin{split}D\left(\exp_{p,u_{0}}\right)_{0}:T_{0}(N_{0})&\rightarrow
T_{p}M\\\ A&\mapsto\frac{d}{d\lambda}\exp_{p,u_{0}}\left(\lambda
A\right)|_{\lambda=0}\\\
&=\frac{d}{d\lambda}\gamma_{p,u_{0},A_{\perp}/\lambda|A|^{2}}\left(2\pi\lambda|A|\right)|_{\lambda=0}.\\\
\end{split}$ (3.5)

To calculate this, we use a power series expansion:

$\begin{split}\gamma_{p,u_{0},a_{0}}(t)&=p+\sum_{n=0}^{\infty}\frac{1}{n!}t^{n+1}u^{(n)}(0).\end{split}$
(3.6)

This expression (and subsequent similar expressions) should be understood in
terms of co-ordinates.

Using repeated applications of the conformal geodesic equation (2.1) and
working in geodesic normal co-ordinates centred at $p$, we can express the
power series (3.6) in terms of $u_{0}$ and $a_{0}$ (as well as the Schouten
tensor and its derivatives evaluated at $p$ and derivatives of the Christoffel
symbols evaluated at $p$). We see that, for a conformal geodesic with initial
data $\left(p,u_{0},A_{\perp}/\left(\lambda|A|^{2}\right)\right)$, we have

$\begin{split}u^{\prime}(0)&=a_{0}=A_{\perp}/\left(\lambda|A|^{2}\right)=O(\lambda^{-1})\\\
u^{\prime\prime}(0)&=\nabla_{u}a|_{t=0}+O(1)=-\frac{1}{\lambda^{2}}\frac{|A_{\perp}|^{2}}{|A|^{4}}u_{0}+O(1)\end{split}$
(3.7)

where $u_{0}$ denotes the initial unit tangent velocity and by $O(1)$ we mean
terms which are bounded in the limit $\lambda\rightarrow 0$. Continuing
recursively we have

$u^{(n)}(0)=\nabla_{u}^{(n)}u|_{t=0}+O(\lambda^{-(n-2)})=O(\lambda^{-n})$
(3.8)

as $\lambda\rightarrow 0$. Here $\nabla_{u}^{(n)}$ denotes the directional
derivative $\nabla_{u}$ applied $n$ times. In particular all terms involving
the Schouten tensor $L$ and its derivatives, as well as terms involving
derivatives of the Christoffel symbols, are $O(\lambda^{-(n-2)})$ (i.e. they
are sub-leading in the limit $\lambda\rightarrow 0$). To calculate the
directional derivative (3.5) we must evaluate the power series (3.6) at
$t=2\pi\lambda|A|=O(\lambda)$. We see that the terms in this power series
which are first order in $\lambda$ are exactly those that arise in Euclidean
space, where $L\equiv 0$. For the Euclidean metric on $\mathbbm{R}^{n}$,
conformal geodesics with initial data $(p,u_{0},a_{0})$ are circles with
equations of the form

$x(t)=p+u_{0}|a_{0}|^{-1}\sin(|a_{0}|t)+a_{0}|a_{0}|^{-2}\left(1-\cos(|a_{0}|t)\right).$
(3.9)

The directional derivative of the exponential map at 0 is therefore

$D(\exp_{p,u_{0}})_{0}:A\mapsto
u_{0}\frac{|A|^{2}}{|A_{\perp}|}\sin\left(2\pi\frac{|A_{\perp}|}{|A|}\right)+A_{\perp}\frac{|A|^{2}}{|A_{\perp}|^{2}}\left(1-\cos\left(2\pi\frac{|A_{\perp}|}{|A|}\right)\right).$
(3.10)

Note that this expression is finite in the limit $A_{\perp}\rightarrow 0$.

It is clear from the definition of the exponential map that if $A$ and
$A^{\prime}$ are related by a reflection in a plane perpendicular to $u_{0}$
(i.e. $A_{\perp}=A^{\prime}_{\perp}$ and $|A|=|A^{\prime}|$), then
$\exp_{p,u_{0}}(A)=\exp_{p,u_{0}}(A^{\prime})$. Since our aim is to find a
region on which the exponential map is a homeomorphism, we restrict our
attention to the half space $g(A,u_{0})\geq 0$. Based on equation (3.10), we
define the following quantities related to $A_{\perp},A\in T_{p}M$

$\begin{split}\sin\theta&:=\frac{|A_{\perp}|}{|A|}\text{,
}\theta\in[0,\pi/2]\\\
\hat{a}_{0}&:=\frac{A_{\perp}}{|A_{\perp}|}.\end{split}$ (3.11)

We have assumed that $A_{\perp}\neq 0$ and $A\neq 0$ since this will simplify
expressions later on, although these cases can be obtained from the results
below by taking the limits $A_{\perp}\rightarrow 0$ and $A\rightarrow 0$
respectively.

We see that a vector $A$ in the half space $g(u_{0},A)\geq 0$ is uniquely
defined by $\theta$, $\hat{a}_{0}$ and $|A|$. The vector $\hat{a}_{0}$ defines
a “quarter space” in $T_{p}M$ by $g(A,\hat{a}_{0})\geq 0$ and $g(A,u_{0})\geq
0$. The variable $\theta$ is the angle between $A$ and the vector333For fixed
$(u_{0},\hat{a}_{0})$ we will draw sketches in the $u_{0}-\hat{a}_{0}$ half
space defined by $g(A,u_{0})\geq 0$. This actually corresponds to considering
$\theta\in[0,\pi/2]$ for both $\hat{a}_{0}$ and -$\hat{a}_{0}$. $u_{0}$.

In terms of $|A|$, $\theta$ and $\hat{a}_{0}$, we can re-write the directional
derivative (3.10) as

$\begin{split}D(\exp_{p,u_{0}})_{0}:A\mapsto
u_{0}\frac{|A|}{\sin\theta}\sin\left(2\pi\sin\theta\right)+\hat{a}_{0}\frac{|A|}{\sin\theta}\left(1-\cos\left(2\pi\sin\theta\right)\right).\end{split}$
(3.12)

We therefore have the following leading order expression for the exponential
map

$\exp_{p,u_{0}}(|A|,\theta,\hat{a}_{0})=p+u_{0}\frac{|A|}{\sin\theta}\sin\left(2\pi\sin\theta\right)+\hat{a}_{0}\frac{|A|}{\sin\theta}\left(1-\cos\left(2\pi\sin\theta\right)\right)+O(|A|^{3}).$
(3.13)

If we hold $\hat{a}_{0}$ and $\sin\theta/|A|$ fixed while varying $|A|$, we
see from Definition 3.1 that the image of this curve under the exponential map
describes a conformal geodesic parameterised by $|A|$. Suppose we extend
$u_{0}$, $\hat{a}_{0}$ to obtain an orthonormal basis of $T_{p}M$. Using the
metric geodesic exponential map, this basis corresponds to a set of geodesic
normal co-ordinates for a neighbourhood of $p\in M$. This co-ordinate
expression for the conformal geodesic is the same as the co-ordinate
expression for a circle in $\mathbbm{R}^{n}$ to first order in $|A|$, with the
$O(|A|^{2})$ terms vanishing. Recall that conformal geodesics in Euclidean
space are exactly circles (all higher order terms in $|A|$ vanish). By
analogy, if we do a power series expansion of metric geodesics in these same
co-ordinates we also find that the 2nd order term in the expansion parameter
vanishes. Recall that metric geodesics in Euclidean space are exactly straight
lines (i.e. there exist co-ordinates in which all non-linear terms in the
expansion parameter vanish).

###  3.2 The Image of a Ray Under the Exponential Map

In this section we will study the image of a ray in the half space
$g(u_{0},A)\geq 0$ originating from $A=0$. Such a ray is defined by fixing
$\theta$ and $\hat{a}_{0}$. As in Section 3.1, $\hat{a}_{0}$ defines a quarter
space in $T_{p}M$ via the conditions $g(u_{0},A)\geq 0$, $g(\hat{a}_{0},A)\geq
0$ and $\theta$ denotes the angle between $A$ and $u_{0}$.

######  Lemma 3.2.

Let $(p,u_{0})\in S(TM)$. Under the exponential map at $(p,u_{0})$, a ray from
$A=0$ defined by $\hat{a}_{0}$ and $\theta=\theta_{0}$ is mapped to a curve in
$M$ whose tangent at $p$ lies in the half space of
span$\\{u_{0},\hat{a}_{0}\\}$ defined by $g(u_{0},\hat{a}_{0})\geq 0$ and
makes an angle of $\theta=\pi\sin\theta_{0}$ with $u_{0}$ (see Figure 2).

Proof: The directional derivative at 0 in the direction tangent to a
particular ray is given by equation (3.12) and is non-zero provided $A\neq 0$
and $\theta\neq\pi/2$ (i.e. $A_{\perp}\neq A$). This directional derivative
gives the tangent direction at $p$ of the curve which is the image of this ray
under the exponential map. We see that this image curve lies in the same
quarter space as $A$ and that the angle, $\theta$, between the directional
derivative and the vector $u_{0}$ satisfies

$\begin{split}\tan\theta&=\frac{1-\cos(2\pi\sin\theta_{0})}{\sin(2\pi\sin\theta_{0})}\\\
&=\tan(\pi\sin\theta_{0})\\\ \implies\theta&=\pi\sin\theta_{0}.\end{split}$
(3.14)

∎

Figure 2: The directional derivative in some direction in the
$u_{0}-\hat{a}_{0}$ plane which makes an angle of $\theta$ with the positive
$u_{0}$ axis is a vector in the same plane making an angle of $\pi\sin\theta$
with the positive $u_{0}$ axis.

###  3.3 The Exponential Map is a Homeomorphism onto a Heart

Lemma 3.2 tells us that the exponential map takes distinct rays through the
origin in $T_{p}M$ to curves with distinct tangent directions at $p$. We will
show that for rays confined to a wedge in $T_{p}M$, we can find a lower bound
on the arc length parameter (equivalently a lower bound on $|A|$ in Definition
3.4) at which their image curves can re-intersect. This tells us that the
exponential map is a homeomorphism on this wedge. We will then consider a
union of such wedges. We find that as we increase the opening angle of the
wedge towards $\pi$, the lower bound on the arc length parameter we obtain
decreases to $0$. We conclude that the exponential map is a homeomorphism on
some region in the half space $g(A,u_{0})\geq 0$ with cross-sections like the
one shown on the right of Figure 6.

######  Definition 3.3.

For any $u_{0}\in S(T_{p}M)$, let $C(u_{0},r)$ denote the open ball in
$T_{u_{0}}(T_{p}M)$ of radius $r$ (defined using the metric $g$, where we
identify $T_{u_{0}}(T_{p}M)$ with $T_{p}M$) which has $u_{0}$ inward pointing
and perpendicular to its boundary at $p$.

Figure 3: The exponential map is a homeomorphism on a ball in $T_{p}M$ with
$u_{0}$ perpendicular and inward pointing on its boundary.

######  Theorem 3.4.

Let $(p,u_{0})\in S\left(TM\right)$. Then the exponential map at $(p,u_{0})$
is a homeomorphism on an open ball $C(u_{0},r)$ for some $r>0$.

In Section 3.1 we calculated an expression for the directional derivative of
the exponential map at $A=0$. We now calculate the derivative at $A\neq 0$. To
do this we calculate the terms in $\exp_{p,u_{0}}(A+\delta
A)-\exp_{p,u_{0}}(A)$ which are first order in the variation $\delta A$.

Recall that for any $u_{0}\in S(T_{p}M)$ and $A\in T_{u_{0}}(T_{p}M)$ we can
identify $T_{u_{0}}(T_{p}M)$ with $T_{p}M$ and write $A$ as a linear
combination of the perpendicular unit vectors $u_{0}$ and $\hat{a}_{0}$ (but
note that if $A$ is parallel to $u_{0}$ then no such $\hat{a}_{0}$ is defined
or required). It follows that any $\delta A\in T_{u_{0}}T_{p}M$ can be written
as $\delta A=\alpha u_{0}+\beta\hat{a}_{0}+\gamma\hat{a}^{\prime}_{0}$, where
$\hat{a}^{\prime}_{0}$ is a unit vector perpendicular to both $u_{0}$ and
$\hat{a}_{0}$ and we take $\gamma\geq 0$ without loss of generality. Once
again note that if $A$ is parallel to $u_{0}$ then $\hat{a}_{0}$ is not
required. Equation (3.13), suggests it will be more convenient to express
$\delta A$ in terms of variations in $|A|$ and the quantities $\sin\theta$ and
$\hat{a}_{0}$ introduced in (3.11). Similar to before we assume that
$A_{\perp}\neq 0$ (i.e. $\sin\theta\neq 0$) since this will simplify the
expressions, however this case can once again be obtained by taking the limit
$A_{\perp}\rightarrow 0$ in the final result and setting $\beta$ to zero. We
have

$\begin{split}\delta|A|&=\alpha\cos\theta+\beta\sin\theta+O(2),\\\
\delta\sin\theta&=\frac{\beta\cos^{2}\theta}{|A|}-\frac{\alpha\sin\theta\cos\theta}{|A|}+O(2),\\\
\delta\hat{a}_{0}&=\frac{\gamma}{|A|\sin\theta}\hat{a}^{\prime}_{0}+O(2).\end{split}$
(3.15)

where by $O(2)$ we mean terms which are second order in $\alpha$, $\beta$ and
$\gamma$ (i.e. second order in $\delta A$).

By the chain rule, we have

$\begin{split}D(\exp_{p,u_{0}})_{A}(\delta
A)&=\frac{\partial\exp_{p,u_{0}}(A)}{\partial|A|}\delta|A|+\frac{\partial\exp_{p,u_{0}}(A)}{\partial\sin\theta}\delta\sin\theta+\frac{\partial\exp_{p,u_{0}}(A)}{\partial\hat{a}_{0}^{\alpha}}\delta\hat{a}_{0}^{\alpha}.\end{split}$
(3.16)

Since we are interested in calculating $D(\exp_{p,u_{0}})_{A}(\delta A)$ we
will now drop higher order terms in $\delta A$ (i.e. the $O(2)$ terms in
(3.15)). Evaluating each of the three terms in (3.16) separately, we have

$\begin{split}\frac{\partial\exp_{p,u_{0}}(A)}{\partial|A|}\delta|A|=&\left[\alpha\cos\theta+\beta\sin\theta\right]\\\
&\times\left[u_{0}\frac{\sin\left(2\pi\sin\theta\right)}{\sin\theta}+\hat{a}_{0}\frac{\left(1-\cos\left(2\pi\sin\theta\right)\right)}{\sin\theta}+O(|A|^{2})\right]\\\
\frac{\partial\exp_{p,u_{0}}(A)}{\partial\sin\theta}\delta\sin\theta=&\left[\beta\cos^{2}\theta-\alpha\sin\theta\cos\theta\right]\\\
&\times\left[u_{0}\left(\frac{2\pi\cos(2\pi\sin\theta)}{\sin\theta}-\frac{\sin(2\pi\sin\theta)}{\sin^{2}\theta}\right)\right.\\\
&\left.+\hat{a}_{0}\left(\frac{2\pi\sin(2\pi\sin\theta)}{\sin\theta}-\frac{1-\cos(2\pi\sin\theta)}{\sin^{2}\theta}\right)+O(|A|^{2})\right]\\\
\frac{\partial\exp_{p,u_{0}}(A)}{\partial\hat{a}^{\alpha}_{0}}\delta\hat{a}^{\alpha}_{0}=&\gamma\hat{a}_{0}^{\prime}\left[\frac{1-\cos(2\pi\sin\theta)}{\sin^{2}\theta}+O(|A|^{2})\right].\end{split}$
(3.17)

We are interested in when $D(\exp_{p,u_{0}})_{A}(\delta A)$ can be zero. It is
straightforward to check that if $\theta\in\left[0,\theta_{0}\right)$ for some
$\theta_{0}\in\left[0,\frac{\pi}{2}\right)$, each of the three terms above are
linearly independent at $|A|=0$ (assuming they are non-zero), and hence also
at $|A|<c_{1}$, for some constant $c_{1}$ which depends on $\theta_{0}$. It
follows that $D(\exp_{p,u_{0}})_{A}(\delta A)$ can only vanish if each of the
three terms above vanish individually. We calculate the norm of each quantity
and define functions $f_{i}(\theta,|A|)$ ($i=1,2,3$) as follows.

$\begin{split}\left|\frac{\partial\exp_{p,u_{0}}(A)}{\partial|A|}\delta|A|\right|=&2\left|\alpha\cos\theta+\beta\sin\theta\right|\times\underbrace{\left[\frac{\sin(\pi\sin\theta)}{\sin\theta}+O(|A|^{2})\right]}_{f_{1}(\theta,|A|)}\\\
\left|\frac{\partial\exp_{p,u_{0}}(A)}{\partial\sin\theta}\delta\sin\theta\right|=&2\left|\beta\cos^{2}\theta-\alpha\sin\theta\cos\theta\right|\\\
&\times\underbrace{\left[\frac{\sin^{2}(\pi\sin\theta)}{\sin^{4}\theta}+\frac{\pi^{2}}{\sin^{2}\theta}-\frac{\pi\sin(2\pi\sin\theta)}{\sin^{3}\theta}+O(|A|^{2})\right]^{1/2}}_{f_{2}(\theta,|A|)}\\\
\left|\frac{\partial\exp_{p,u_{0}}(A)}{\partial\hat{a}^{\alpha}_{0}}\delta\hat{a}^{\alpha}_{0}\right|=&2\gamma\times\underbrace{\left[\frac{\sin^{2}(\pi\sin\theta)}{\sin^{2}\theta}+O(|A|^{2})\right]}_{f_{3}(\theta,|A|)}\end{split}$
(3.18)

Plots of $f_{1}(\theta,0)$ and $f_{2}(\theta,0)$ are shown in Figures 5 and 5
respectively (note that $f_{3}(\theta,0)=f_{1}(\theta,0)^{2}$). We see that
$f_{i}(\theta,0)>0$ ($i=1,2,3$) for $\theta\in[0,\frac{\pi}{2})$, however
$f_{1}(\frac{\pi}{2},0)=f_{3}(\frac{\pi}{2},0)=0$. Now suppose we restrict to
$\theta\in[0,\theta_{0}]$ for some
$\theta_{0}\in\left[0,\frac{\pi}{2}\right)$. Then there exists some constant
$c_{2}>0$ (depending on $\theta_{0}$) such that $f_{i}(\theta,|A|)>0$
($i=1,2,3$) for $\theta\in[0,\theta_{0}]$ and $|A|<c_{2}$. This means that,
for $\theta\in[0,\theta_{0}]$ and $|A|<c_{2}$, the three quantities in (3.17)
all vanish if and only if $\alpha=\beta=\gamma=0$, or equivalently if and only
if $\delta A=0$. It follows that $D(\exp_{p,u_{0}})_{A}$ has non-zero
determinant at any $A$ with $\theta\in[0,\theta_{0}]$ and
$|A|<c:=\text{min}(c_{1},c_{2})$. We refer to such a set as a wedge of radius
$c$ and opening angle $2\theta_{0}$.

Figure 4: Plot of $f_{1}(\theta,0)$. We see that $f_{1}(\theta,0)>0$ for
$\theta\in[0,\frac{\pi}{2})$.

Figure 5: Plot of $f_{2}(\theta,0)$. We see that $f_{2}(\theta,0)>0$ for
$\theta\in[0,\frac{\pi}{2}]$.

As we let $\theta_{0}\rightarrow\frac{\pi}{2}$, we see that $c\rightarrow 0$,
so the radius of the wedge on which we can guarantee that
$D(\exp_{p,u_{0}})_{A}$ has non-zero determinant tends to zero. We conclude
that $D(\exp_{p,u_{0}})_{A}$ has non-zero determinant on a series of wedges
whose radii tend to zero as their opening angle tends to $\pi$. Taking the
union over all such wedges, we find that $D(\exp_{p,u_{0}})_{A}$ has non-zero
determinant on some set with boundary perpendicular to $u_{0}$ at $p$ (see
Figure 6). Such a set contains an open ball in $T_{u_{0}}(T_{p}M)$ which has
$u_{0}$ inward pointing and perpendicular to its boundary - i.e. it contains
$C(u_{0},r)$ for some $r>0$.

Figure 6: The derivative of the exponential map has non-zero determinant on a
series of wedges whose radii decrease to 0 as their opening angle increases
towards $\pi$. Taking the union of such sets gives a set with cross sections
like the one shown on the right of this figure. In particular we conclude that
the derivative of the exponential map has non-zero determinant on $C(u_{0},r)$
for some $r>0$.

Next we show that the exponential map is an injection on this set
$C(u_{0},r)$. Suppose $A\text{, }A^{\prime}\in C(u_{0},r)$. Since $C(u_{0},r)$
is an open ball in $T_{u_{0}}(T_{p}M)$, it is convex and hence
$f(t):=\exp_{p,u_{0}}(A+t(A^{\prime}-A))\in C(u_{0},r)$ for any $t\in[0,1]$.
This function is differentiable on $(0,1)$ so, by the mean value theorem,
there exists some $t_{*}\in(0,1)$ with

$\begin{split}f^{\prime}(t_{*})&=f(1)-f(0)\\\ \implies
D(\exp_{p,u_{0}})_{A+t_{*}(A^{\prime}-A)}(A^{\prime}-A)&=\exp_{p,u_{0}}(A^{\prime})-\exp_{p,u_{0}}(A).\end{split}$
(3.19)

Since $D(\exp_{p,u_{0}})_{A+t_{*}(A^{\prime}-A)}$ has non-zero determinant, it
follows that if $\exp_{p,u_{0}}(A^{\prime})=\exp_{p,u_{0}}(A)$ then we must
have $A=A^{\prime}$.

We conclude that the exponential map is injective on $C(u_{0},r)$ and hence is
a homeomorphism from this set to its image. ∎

This theorem allows us to formally define the notion of a heart at
$(p,u_{0})\in S(TM)$ which was referred to in Section 2.2.

######  Definition 3.5.

A heart at $(p,u_{0})\in S(TM)$ is defined to be a set
$\exp_{p,u_{0}}(C(u_{0},r))$ where $r>0$ is such that $\exp_{p,u_{0}}$ is a
homeomorphism on $C(u_{0},r)$.

The reason for this name is due to the shape of cross-sections of this set. As
the radius of a wedge shrinks to 0, so too must the size of the smallest ball
which contains its image under the exponential map. Furthermore, by Lemma 3.2
we see that as the opening angle of this wedge increases to $\pi$, the opening
angle of the image set at $p$ must increase to $2\pi$ (consider two rays
defined by $\pm\hat{a}_{0}$ and let $\theta_{0}\rightarrow\frac{\pi}{2}$). In
particular, the images of these two rays are both tangent to $-u_{0}$ at $p$.
Consequently, a heart is a set with “heart shaped” cross-sections which
feature a cusp at $p$ (shown on the right of Figure 3)444As pointed out by the
referee, in flat Euclidean space the shape of these cross-sections resembles
that of a cardioid but is in fact slightly different. The image of a round
sphere in $T_{p}M$ of radius $R$ with $u_{0}$ inward pointing and
perpendicular at its boundary has cross-sections with boundary given in plane
polar co-ordinates $(r,\phi)$ by
$r=\frac{4R\sqrt{\pi^{2}-\phi^{2}}}{\phi}\sin\phi$ for $\phi\in(-\pi,\pi]$..

Note that if we choose $|A_{\perp}|=\epsilon^{3}<1$ and $|A|=\epsilon^{2}$
then the vector $A$ will be mapped to a point on a conformal geodesic which
has perpendicular acceleration of size $1/\epsilon$ at $p$. By choosing
$\epsilon$ arbitrarily small, this can be made arbitrarily large while
simultaneously ensuring that $A\in C(u_{0},r)$. As a result, given any $r>0$,
the image set $\exp_{p,u_{0}}(C(u_{0},r))$ will always contain at least some
portion of every conformal geodesic with unit tangent vector $u_{0}$ at $p$,
even those with arbitrarily large perpendicular acceleration at this point.
This is crucial for the lemma below and for our proof of the no-spiralling
theorem (Theorem 2.3) in Section 5.

Example (the Euclidean metric on $\mathbbm{R}^{n}$): In the case where $g$ is
the Euclidean metric on $\mathbbm{R}^{n}$, the $O(|A|^{3})$ terms in equation
(3.13) are identically 0 and conformal geodesics are circles. The images of
distinct rays from 0 in the half space $g(u_{0},A)>0$ are also rays. These do
not re-intersect, so the exponential map is a homeomorphism (in fact a
diffeomorphism) from this half space onto $\mathbbm{R}^{n}$ with the ray from
0 in the direction of $-u_{0}$ removed. This is essentially the familiar
result that we can foliate $\mathbbm{R}^{n}$ by circles tangent to $u_{0}$ at
$p$ (shown for $n=2$ in Figure 7). Since we have insisted that the conformal
geodesic has initial direction $u_{0}$ (rather than $-u_{0}$) we cannot reach
points on the $-u_{0}$ axis. These points correspond to the circle through
infinity with initial direction $u_{0}$ at $p$.

Figure 7: $\mathbbm{R}^{2}$ can be foliated by circles which are tangent to
$u_{0}$ at $p$.

If the Schouten tensor is not identically zero then the higher order terms in
equation (3.13) become important as $|A|$ increases. These terms may cause the
images of the rays to intersect. This would reduce the size of the domain on
which the exponential map is a homeomorphism.

###  3.4 Conformal geodesics re-intersect the heart boundary

Given any $(p,u_{0})\in S\left(TM\right)$, we would like to consider the
largest open ball $C(u_{0},r)$ (see Definition 3.3) on which $\exp_{p,u_{0}}$
is a homeomorphism. It is of course possible this map is a homeomorphism on
$C(u_{0},r)$ for all $r>0$, as is the case in Euclidean space (see Section
3.3). Our proof of the no-spiralling theorem in Section 5 will be simpler if
we work with a finite sized domain with boundary. With this in mind, we make
the following definition

######  Definition 3.6.

Given any $(p,u_{0})\in S\left(TM\right)$, define

$r_{(p,u_{0})}:=\frac{1}{2}\times\min\\{1,\sup\\{r:\exp_{p,u_{0}}\text{ is a
homeomorphism on }C(u_{0},r)\\}\\}.$ (3.20)

The factor of $1/2$ is included since it will be convenient in the proof of
Theorem 4.2 that the exponential map is a homeomorphism onto the heart and its
boundary. We will be particularly interested in the heart which is the image
of the set $C(u_{0},r_{(p,u_{0})})$ under the exponential map at $(p,u_{0})$.
We denote this heart by

$H_{p,u_{0}}=\exp_{p,u_{0}}(C(u_{0},r_{(p,u_{0})})).$ (3.21)

For the proof of the no-spiralling theorem (Theorem 2.3), we will require the
following property of the heart $H_{p,u_{0}}$.

######  Theorem 3.7.

For any $(p,u_{0})\in S(TM)$, any conformal geodesic with unit tangent $u_{0}$
at $p$ re-intersects the boundary of the heart $H_{p,u_{0}}$.

Proof: As discussed in Section 3.1, a conformal geodesic is the image under
the exponential map of a curve in $T_{p}M$ defined by holding $\hat{a}_{0}$
and $\sin\theta/|A|$ constant while varying $|A|$. Therefore, a conformal
geodesic intersects the boundary of $H_{p,u_{0}}$ when this curve reaches the
boundary of $C(u_{0},r_{(p,u_{0})})\subset T_{p}M$. ∎

In fact, the curve in $T_{p}M$ defined by holding $\hat{a}_{0}$ and
$\sin\theta/|A|$ constant reaches the boundary of $C(u_{0},r_{(p,u_{0})})$ at
some $|A|\leq r_{(p,u_{0})}$. Hence from the definition of the exponential map
(Definition 3.1) we see that the arc length parameter distance travelled along
the conformal geodesic before it reaches the boundary of $H_{p,u_{0}}$ is at
most $2\pi r_{(p,u_{0})}$.

##  4 Size of the Heart

The aim of this section is to follow Step 2 of the proof of Theorem 2.3 as
outlined in Section 2.2.

Step 2: Define a notion of size for a heart and show that given any compact
$K\subset M$, there exists $\epsilon>0$ such that there is a heart at $p$ with
direction $u_{0}$ which has size at least $\epsilon$, for any $p\in K$ and any
$u_{0}\in S(T_{p}M)$.

Consider the heart $H_{p,u_{0}}$ and let $k(p,u_{0},r)$ denote the set of all
closed balls in $M$ of radius $r$ with $p$ on their boundary and $u_{0}$
pointing into or tangent to the ball. If $r$ is sufficiently small, then all
balls in $k(p,u_{0},r)$ will be contained inside the closure of $H_{p,u_{0}}$,
which we denote $\bar{H}_{p,u_{0}}$ (see Figure 8).

######  Definition 4.1.

At any point $(p,u_{0})\in S(TM)$, we define the size of the heart
$H_{p,u_{0}}$ by

$\begin{split}R:S\left(TM\right)&\rightarrow\mathbbm{R}_{>0}\\\
(p,u_{0})&\mapsto\sup\\{r:\bar{B}\subset\bar{H}_{p,u_{0}}\forall\bar{B}\in
k(p,u_{0},r)\\}.\end{split}$ (4.22)

Figure 8: This figure shows a sample of balls in the set $k(p,u_{0},r)$. We
have chosen $r<R(p,u_{0})$ which means that all of these balls are contained
in the heart $H_{p,u_{0}}$.

######  Theorem 4.2.

Let $(M,[g])$ be a conformal manifold such that metrics in $[g]$ are smooth
and let $W$ be a compact subset of $S(TM)$. Then there exists some
$\epsilon>0$ such that $R(p,u_{0})\geq\epsilon$ for any $(p,u_{0})\in W$.

To prove this theorem, we begin by considering the following ODE with an
initial condition

$y^{\prime}(t)=f(t,y)\text{ and }y(0)=y_{0}.$ (4.23)

######  Theorem 4.3 ([7, Section V.3 Corollary 3.3 and Section V.4 Corollary
4.1]).

Suppose $f(t,y)$ is of class $C^{m}$, $m\geq 1$, on an open $(t,y)$-set. Then
(4.23) has unique solution $y=\eta(t,y_{0})$ which is of class $C^{m}$ on its
domain of existence.

######  Corollary 4.4.

Suppose the metric $g$ is $C^{4}$. Then the heart boundary is a $C^{2}$
surface away from the cusp.

Proof: We will parameterise conformal geodesics $x^{\alpha}(t)$ by their arc
length, $t$. The conformal geodesic equation can then be written as three
separate 1st order ODEs on some co-ordinate patch

$\begin{split}\frac{dx^{\beta}}{dt}&=u^{\beta}\\\
\frac{du^{\beta}}{dt}&=a^{\beta}-\Gamma^{\beta}_{\gamma\delta}u^{\gamma}u^{\delta}\\\
\frac{da^{\beta}}{dt}&=-\Gamma^{\beta}_{\gamma\delta}u^{\gamma}a^{\delta}-\left(g_{\gamma\delta}a^{\gamma}a^{\delta}+L_{\gamma\delta}u^{\gamma}u^{\delta}\right)u^{\beta}+L^{\beta}_{\gamma}u^{\gamma}.\end{split}$
(4.24)

Since the metric is $C^{4}$, the Christoffel symbols
$\Gamma^{\beta}_{\gamma\delta}$, the Schouten tensor $L$ and the corresponding
endomorphism $L^{\\#}$ are all $C^{2}$. Theorem 4.3 implies that the solution
$x^{\alpha}=\eta^{\alpha}(t,x_{0},u_{0},a_{0})$ is also $C^{2}$. This tells us
that, away from $A=0$, the exponential map considered as a function of
$(p,u_{0},A)$ is $C^{2}$. We conclude that the heart boundary is $C^{2}$ away
from the cusp. ∎

Next we define the following function.

######  Definition 4.5.

$\begin{split}R_{1}(p,u_{0}):=\sup\\{r:\text{any open ball in $M$ of radius
$r$ with $u_{0}$ tangent to }&\\\ \text{ its boundary is contained in
$H_{p,u_{0}}$}\\}&\end{split}$ (4.25)

######  Lemma 4.6.

For any compact $W\subset S(TM)$, there exists $\epsilon>0$ such that
$R_{1}(p,u_{0})\geq\epsilon$ for any $(p,u_{0})\in W$.

Proof: Let $W\subset S(TM)$ be compact and let $(p,u_{0})\in W$. This
specifies a value $r_{(p,u_{0})}$ such that the exponential map at $(p,u_{0})$
is a homeomorphism from the open ball $C(u_{0},r_{(p,u_{0})})$ to the heart
$H_{p,u_{0}}$. Next we choose a unit vector $\hat{a}_{0}$ which is
perpendicular to $u_{0}$. By restricting to elements of $T_{p}M$ in
$\text{span}\\{u_{0},\hat{a}_{0}\\}$, we obtain a curve through 0 on the
boundary of the ball $C(u_{0},r_{(p,u_{0})})\subset T_{u_{0}}(T_{p}M)$ which
we call a circle of radius $r_{(p,u_{0})}$. This circle is mapped by the
exponential map to a curve $\lambda_{p,u_{0},\hat{a}_{0}}$ on the heart
boundary. Finally, we choose some value $x\in(-\pi/2,\pi/2)$. This corresponds
uniquely to a point on this circle (by considering the angle between $A$ and
$u_{0}$). The exponential map takes this point to a point on the curve
$\lambda_{p,u_{0},\hat{a}_{0}}$. Note that by choosing $x\in(-\pi/2,\pi/2)$ we
do not include the problem point at 0 which is mapped to the cusp.

By Corollary 4.4, $\lambda_{p,u_{0},\hat{a}_{0}}$ is a $C^{2}$ curve away from
the cusp. Hence for any $x\in(-\pi/2,\pi/2)$ we can define the following
continuous function

$F(p,u_{0},\hat{a}_{0},x)=\left|\frac{d^{2}}{dx^{2}}\lambda_{p,u_{0},\hat{a}_{0}}(x)\right|.$
(4.26)

There is a subtlety at the point $p$, where the boundary of the heart contains
a cusp and directional derivatives have discontinuities. However, if we
approach $p$ along the curve $\lambda_{p,u_{0},\hat{a}_{0}}$ in two different
directions then the function $F$ tends to a finite limit (note that these two
limits may not be equal). If we join the two ends of
$\lambda_{p,u_{0},\hat{a}_{0}}$ at $p$, then these two limits correspond to
the two values we get for the magnitude of the curvature of
$\lambda_{p,u_{0},\hat{a}_{0}}$ at $p$ if we use one-sided derivatives in
either direction.

This means that, for fixed $(p,u_{0},\hat{a}_{0})$, the function
$F(p,u_{0},\hat{a}_{0},x)$ is bounded. Moreover, this bound is itself a
continuous function of $(p,u_{0},\hat{a}_{0})$. We conclude that if we
consider $(p,u_{0})\in W\subset S(TM)$, $x\in(-\pi/2,\pi/2)$ and any choice of
$\hat{a}_{0}$, then $F(p,u_{0},\hat{a}_{0},x)$ is bounded.

In short, given any $(p,u_{0})\in W$ and any circle of radius $r_{(p,u_{0})}$
in $T_{u_{0}}(T_{p}M)$ with $u_{0}$ perpendicular to its boundary, the
magnitude of the sectional curvature at a point on the image of this circle,
taken in the direction of the circle’s image, is bounded by $1/\epsilon>0$,
for some $\epsilon>0$, where we understand that at $p$ itself we consider two
different sectional curvatures corresponding to the different directions along
the image of the circle.

As a result, for any $(p,u_{0})\in W$, the heart $H_{p,u_{0}}$ must contain
every open ball in $M$ of radius $\epsilon$ which has $u_{0}$ tangent to its
boundary at $p$, i.e. $R_{1}(p,u_{0})\geq\epsilon$ (see Figure 9). ∎

Figure 9: If every sectional curvature at every point on the heart boundary is
bounded above by $\epsilon>0$, then any ball of radius $1/\epsilon$ with
$u_{0}$ tangent to its boundary at $p$ is contained inside the heart.

Next we consider balls in $M$ with $u_{0}$ pointing inwards on the boundary.
We make the following definitions.

######  Definition 4.7.

$B_{1/2}(r,p,u_{0}):=B(r,p)\cap\\{q:g(\exp^{-1}_{p}(q),u_{0})\geq 0.\\}$
(4.27)

Figure 10: $B_{1/2}(r,p,u_{0})$ is the half ball in $M$ of radius $r$ centred
on $p$ consisting of points $q$ such that $g(\exp^{-1}_{p}(q),u_{0})\geq 0$.

Here $\exp_{p}$ denotes the metric geodesic map at $p$ (we can restrict
ourselves to sufficiently small $r$ so that this map is invertible on $B(p,r)$
for all $(p,u_{0})\in W$). The set we have defined consists of points in the
half ball with boundary perpendicular to $u_{0}$ at $p$ which have non-
negative $u_{0}$ co-ordinate, where these co-ordinates are defined using the
metric geodesic map applied to an orthonormal basis of $T_{p}M$ (see Figure
10).

######  Definition 4.8.

$R_{2}(p,u_{0}):=\sup\\{r:B_{1/2}(r,p,u_{0})\subset H_{p,u_{0}}.\\}$ (4.28)

Figure 11: If $R_{1}(p,u_{0})>r$ and $R_{2}(p,u_{0})>2r$ then $R(p,u_{0})>r$.

######  Lemma 4.9.

For any compact $W\subset S(TM)$, there exists $\epsilon>0$ such that
$R_{2}(p,u_{0})>\epsilon$ for any $(p,u_{0})\in W$.

Proof: Let $W\subset S(TM)$ be compact. From Corollary 4.4,
$\exp_{p,u_{0}}(A)$ regarded as a function of $(p,u_{0},A)$ is continuous away
from the cusp. We also saw in Section 3.5 that the boundary of $H_{p,u_{0}}$
is continuous (including at $p$). We conclude that the boundary of
$H_{p,u_{0}}$ varies continuously with $(p,u_{0})$ and hence $R_{2}(p,u_{0})$
is also continuous. The result follows since $R_{2}(p,u_{0})$ is strictly
positive and a continuous function on a compact set is bounded and attains its
bounds. ∎

Combining these two lemmas, we are able to prove Theorem 4.2.

Proof of Theorem 4.2: By Lemmas 4.6 and 4.9, for any compact $W\subset S(TM)$,
there exists $\epsilon>0$ such that $R_{1}(p,u_{0})>\epsilon$ and
$R_{2}(p,u_{0})>2\epsilon$.

The result follows by observing that $R(p,u_{0})>R_{1}(p,u_{0})$ and
$R(p,u_{0})>R_{2}(p,u_{0})$ (Figure 11). ∎

##  5 No-Spiralling Theorem

The aim of this section is to follow Step 3 of the proof of Theorem 2.3 as
outlined in Section 2.2.

Step 3: Using the fact that a conformal geodesic with unit tangent $u_{0}$ at
$p$ must re-intersect the boundary of $H_{p,u_{0}}$, deduce that conformal
geodesics cannot spiral.

Theorem 2.3 Let $(M,[g])$ be a conformal manifold such that metrics in $[g]$
are smooth. Then conformal geodesics on $(M,[g])$ cannot spiral.

Proof: Suppose there is a conformal geodesic, $\gamma$, which spirals towards
some point $p_{*}\in M$. We will show that there exists $r>0$ such that
$B(p_{*},r)\subset H_{p,u_{0}}$, where $p$ and $u_{0}$ are the point and unit
tangent at which $\gamma$ enters $\bar{B}(p_{*},r)$ for the final time. This
would then contradict Theorem 3.7 since if $\gamma$ does not leave
$B(p_{*},r)$ then it cannot re-intersect the boundary of $H_{p,u_{0}}$. This
is illustrated in Figure 12.

Figure 12: If $\gamma$ enters $B(p_{*},R_{W})\subset H_{p,u_{0}}$, then in
order to reach the boundary of $H_{p,u_{0}}$ it must first leave
$B(p_{*},R_{W})$.

Define a compact subset of $S(TM)$ by $W:=\\{(p,u_{0})\subset
S\left(TM\right):p\in\bar{B}(p_{*},1)\\}$. Invoking Theorem 4.2, we can then
define:

$R_{W}:=\inf_{(p,u_{0})\in W}\\{R(p,u_{0})\\}>0.$ (5.29)

So for any $(p,u_{0})\in S\left(TM\right)$ with $d(p,p_{*})\leq 1$, any closed
ball in $M$ of radius at most $R_{W}$ and $u_{0}$ inward pointing or tangent
to its boundary at $p$ is contained inside $\bar{H}_{p,u_{0}}$.

Let $p$ denote the point at which $\gamma$ enters $\bar{B}(p_{*},R_{W})$ for
the final time and let $u_{0}$ denote its unit tangent at this point (so
$u_{0}$ is either inward pointing or tangent to the boundary of
$\bar{B}(p_{*},R_{W})$ at $p$). By construction we then have
$\bar{B}(p_{*},R_{W})\subset H_{p,u_{0}}$. But since $\gamma$ remains trapped
in $B(p_{*},R_{W})$, it cannot re-intersect the boundary of $H_{p,u_{0}}$.
This contradicts Theorem 3.7. ∎

##  6 Summary

We have shown that conformal geodesics defined on a Riemannian conformal
manifold cannot spiral. This proof was inspired by a proof of the no-
spiralling theorem for metric geodesics which did not rely on length
minimisation arguments, instead focusing on properties of the exponential map.
However, in the conformal geodesic version there were some additional
complications. The most striking of these was that the exponential map we
defined for conformal geodesics was not a homeomorphism onto any ball centred
at $p\in M$. Instead we found that given any $(p,u_{0})\in S\left(TM\right)$,
conformal geodesics with unit velocity $u_{0}$ at $p$ foliate some set with
cross-sections which resemble a heart.

In this paper we have only considered Riemannian conformal manifolds. In the
Lorentzian case, our proof fails from the very beginning. In particular, the
exponential map (Definition (3.1)) is not well defined since it is now
possible to have $|A|=0$ for $A\neq 0$.

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# Bosonic fractional Chern insulating state at integer fillings in multi-band
system

Wei-Wei Luo1 Ai-Lei He2 Yi-Fei Wang3 Yuan Zhou1,4<EMAIL_ADDRESS>Chang-De
Gong3,1 1National Laboratory of Solid State Microstructures and Department of
Physics, Nanjing University, Nanjing 210093, China
2Institute for Advanced Study, Tsinghua University, Beijing 100084, China
3Center for Statistical and Theoretical Condensed Matter Physics, and
Department of Physics, Zhejiang Normal University, Jinhua 321004, China
4Department of Mathematics and Physics, Xinjiang Teacher’s College, Urumqi
830043, China

###### Abstract

The integer quantum Hall state occurs when the Landau levels are fully
occupied by the fermions, while the fractional quantum Hall state usually
emerges when the Landau level is partially filled by the strongly correlated
fermions or bosons. Here, we report two fractional Chern insulating states of
the hard-core bosons in a multi-band lattice model hosting topological flat
bands with high Chern number. The previously proposed $\nu=1/3$ fractional
Chern insulating state inherited from the high Chern number $C=2$ of the
lowest topological flat band is revisited by the infinite density matrix
renormalization group algorithm. In particular, we numerically identify a
bosonic $1/2$-Laughlin-like fractional Chern insulating state at the integer
fillings. We show two lower topological flat bands jointly generate an
effective $C=1$ Chern band with half-filling. Furthermore, we find a strictly
particle-hole-like symmetry between the $\nu$ and $3-\nu$ filling in our
model. These findings extend our understanding of quantum Hall states and
offer a new route to realize the novel fractional states in the system with
multi-bands and high-Chern numbers.

Introduction.—The emergence of topological state and topological order opens a
new window to classify phases of matter beyond the Landau’s symmetry-breaking
paradigm [1]. The precise quantization of Hall conductivity
$\sigma_{xy}=C\frac{e^{2}}{h}$ with $C$ a topological invariant Chern number
appears when the Landau levels generated by the external magnetic field are
fully filled. In contrast, the fractional quantum Hall state is observed when
the strongly interacting electrons partially fill a Landau level, whose
topological order can be characterized by the ground state degeneracy on a
compact geometry and the gapless edge excitations described by the chiral
Luttinger liquid [1, 2]. Laughlin proposed a trial wave function that captures
the essence of the fractional quantum Hall state, in which the quasiparticle
excitations host the fractional charges and fractional braiding statistics
[3]. Halperin further generalized the Laughlin state into the two-component
system (noted as Halperin ($mmn$) state) to account for the spin or pseudospin
degrees of freedom [4]. The lattice version of integer quantum Hall state in
the absence of the Landau level, also known as the Chern insulating state, was
first proposed on a honeycomb lattice model with the staggered flux threading
[5], and experimentally realized in the solid state materials [6] and cold
atom systems [7]. The Chern insulators, compared with their continuum
counterpart, can survive in a less strict setup and host more exotic
topological phases [8], and thus provide a promising application platform.
Since the $C=1$ Chern band can be adiabatically connected to the Landau level
[9, 10, 11], the fractional quantum Hall effect was possible to be realized in
the lattice models. It was previously proposed for the ultracold atoms
confined in optical lattice [12, 13, 14], in which the effective magnetic
field was simulated by introducing oscillating quadrupole potential or lattice
rotation. The proposal of topological flat band (TFB) [15, 16, 17] in the
absence of the external magnetic field opens new window to realize the
fractional Chern insulator. Subsequently, a series of fractional Chern
insulators were established in various lattice models hosting the $C=1$ TFB
[18, 19, 20].

Unlike the unit Chern number of the Landau level in the continuum limit, the
lattice model can host high Chern numbers, such as in the Hofstadter model
[21, 22, 23]. Similar bands with high Chern number can also be generated in
the TFB lattice model. Initially, $C=2$ TFB were constructed in a dice lattice
model [24] and three-band triangular lattice model [25], and the systematic
approaches to generate TFBs with arbitrary Chern number were further proposed
in the multilayer systems [26] and the multi-orbital structures [27]. Although
a $C>1$ Chern band can be mapped to a $C$-component Landau level using the
hybrid Wannier states [28], these components are mutually entangled with each
other, in contrast with the case in the usual Halperin state. A series of
fermionic fractional Chern insulators at $\nu=1/(2C+1)$ filling were
established numerically in the TFB models with high Chern number $C$ [25, 29,
30, 31], which can be understood as the $SU(C)$ color-entangled version of
Halperin states [32, 33]. On the other hand, the topological states in bosonic
system are much subtle. The bosonic fractional Chern insulators at
$\nu=1/(C+1)$ filling were also proposed in the TFB models [25, 29, 34]. These
incompressible states can also be understood by the concept of flux attachment
in lattice system [35, 22], yielding a series of fractional Chern insultors at
filling factors $\nu=r/(r|C|+1)$ for bosons, and $\nu=r/(2r|C|+1)$ for
fermions with $r$ an integer [23]. The bosonic integer quantum Hall (BIQH)
state, a topological phase protected by $U(1)$ symmetry but in the absence of
intrinsic topological order, was first perceived in the interacting two-
component boson gases at the integer filling $\nu=1+1$ [36, 37, 38, 39, 40].
Its existence was also predicted by the composite fermion theory at integer
filling $\nu=1$ of $|C|=2$ band ($r=-1$) [23], and numerically confirmed in
different lattice models [41, 42, 43, 44, 45].

So far, the topological nature of the Chern insulators follow the similar way
established in the continuum limit with the Landau levels, i.e., the integer
and fractional quantum Hall effect emerge at the integer and fractional
filling. It is interesting to explore the novel fractional Chern insulators
with no analogue in the continuum limit [22, 46]. In this paper, combining the
exact diagonalization (ED) and infinite density matrix renormalization group
(iDMRG) algorithm, we study the exotic fractional Chern insulators of the
hard-core bosons in a specific multi-band model. The adopted triangular
lattice model hosts two lower TFBs with respective Chern number $C=2$, and
$-1$. We numerically confirm the previously proposed $\nu=1/3$ bosonic
fractional Chern insulating (FCI) state. Its topological order is well
characterized by the $\mathbf{K}=\big{(}\begin{smallmatrix}2&1\\\ 1&2\\\
\end{smallmatrix}\big{)}$-matrix, i.e., a color-entangled $\frac{1}{3}$-FCI
state with the two components originated from the high Chern number $C=2$ of
the lowest TFB. More importantly, an unexpected bosonic FCI state, rather than
the BIQH state, is observed at the integer fillings $\nu=1$ and $\nu=2$. The
twofold ground-state degeneracy and the exact fractional-$1/2$ charge pumping,
as well as the level counting of the low-lying entanglement spectrum, evidence
the emergence of $\nu=1/2$ Laughlin-like FCI state. We identify that the two
lower TFBs produce an effective $C=1$ Chern band with half-filling. In
addition, we find a strict particle-hole-like symmetry between the $\nu$ and
$3-\nu$ filling inherited from the intrinsic time-reversal symmetry of the
specific model.

Model and method.— We consider loading the hard-core bosons with $U(1)$ charge
conservation into a specific three-band triangular-lattice model first
proposed by one of our authors [25]:

$\displaystyle H$ $\displaystyle=$ $\displaystyle\pm
t\sum_{\langle\ij\rangle}\left[b^{\dagger}_{i}b_{j}\exp\left(i\phi_{ij}\right)+\textrm{H.c.}\right]$
(1) $\displaystyle\pm t^{\prime}\sum_{\langle\langle
ij\rangle\rangle}\left[b^{\dagger}_{i}b_{j}\exp\left(i\phi_{ij}\right)+\textrm{H.c.}\right],$

where $b^{\dagger}_{i}$ creates a hard-core boson at site $i$,
$\langle\dots\rangle$ and $\langle\langle\dots\rangle\rangle$ denote the
nearest-neighbor (NN) and the next-nearest-neighbor (NNN) pairs of sites with
the specified phase factor denoted by the arrows (Fig. 1), respectively. Each
unit cell (gray hexagon) contains three inequivalent sites, and thus three
single-particle bands are created. The flatness ratio and topological index of
each band are parameter-dependent (more details see Appendix I). Here, we fix
$t=1$, $t^{\prime}=1/4$ and $\phi=\pi/3$ [25] unless specified, yielding
$C=2$, $-1$, $-1$ in the each band (from bottom to top) and two lower flat
bands with the respective flatness ratio about $15$, $14$, respectively. Those
TFBs mimic the Landau levels in the continuum but with diverse Chern numbers.
We, therefore, construct a multi-band model with high Chern numbers.

Figure 1: (Color online). Schematic structure of the adopted triangular
lattice model. The unit cell is highlighted by a gray hexagon, and the hopping
process between the NN and NNN neighbors with specified gauge are given on the
right hand.

Numerically, we employ the unbiased ED and iDMRG methods to study the
topological order of the potentially topological states. In ED scheme, a
finite system of $N_{\text{orb}}=N_{x}\times N_{y}$ unit cells (the total
number of site $N_{s}=3\times N_{x}\times N_{y}$) with periodic boundary
condition in both directions is considered. The filling factor is defined as
$\nu=N_{b}/N_{\text{orb}}$ with $N_{b}$ the number of hard-core bosons loaded.
Due to the translational symmetry, the many-body eigenstates can be labeled by
the total momentum quantum number $(k_{x},k_{y})$ in unit of
$(2\pi/N_{x},2\pi/N_{y})$. To access larger system sizes and extend numerical
evidences beyond the ED method, we also study the interacting system on a
infinite cylinder (with finite width $L_{y}$) by iDMRG alogrithm [47, 48].
This method allows us to directly simulate the charge pumping process [49, 50,
51, 52] and conveniently explores the underlying topological order by the
entanglement properties in the many-body ground state [53, 54].

FCI state at $\nu=1/3$.—The $\nu=1/3$ bosonic fractional Chern insulator had
been previously studied by the ED method on a torus geometry [25, 55]. The
low-lying energy spectrum in Fig. 2 (a) unambiguously shows that three quasi-
degenerate ground states, being robust against the boundary conditions, evolve
into each other while are well separated from the higher excited states. By
imposing the twisted boundary phases $\theta_{x}$ and $\theta_{y}$ in both
directions, the Chern number of a many-body ground-state is given by
$C=\frac{1}{2\pi}\int_{0}^{2\pi}\int_{0}^{2\pi}d\theta_{x}d\theta_{y}F(\theta_{x},\theta_{y})$,
where
$F(\theta_{x},\theta_{y})=\Im\left(\left\langle\partial_{\theta_{x}}\psi\middle|\partial_{\theta_{y}}\psi\right\rangle-\left\langle\partial_{\theta_{y}}\psi\middle|\partial_{\theta_{x}}\psi\right\rangle\right)$
is the gauge invariant Berry curvature with $\psi$ the ground-state wave
functions [56]. The Berry curvature of the ground-state manifold [Fig. 2 (b)]
contributes a total Chern number $C=2$. The topological order of $\nu=1/3$ FCI
state can be well characterized by the $\mathbf{K}$ matrix in terms of Chern-
Simons field theory [57, 58]. Here, the ground state degeneracy
$d=\det\mathbf{K}=3$ and the total Chern number
$C=\sum_{i,j}(\mathbf{K}^{-1})_{ij}=2$, yielding
$\mathbf{K}=\big{(}\begin{smallmatrix}2&1\\\ 1&2\\\ \end{smallmatrix}\big{)}$.
Alternatively, this $\nu=1/3$ state is also predicted by composite fermion
theory [23] at bosonic filling $\nu=r/(r|C|+1)$ with $r=1$, where one flux
quantum is attached to each boson to form the weakly interacting composite
fermion. Incompressible state is expected to form when they fully fill integer
number of Chern bands.

To complement the ED results on a torus, we study $\nu=1/3$ case on a cylinder
geometry using the iDMRG algorithm. The exact $2/3$ charge pumping after a
flux quanta threading [Fig. 2(c)] agrees with the Chern number calculation
using ED method [25], where each ground state averages contributes to a Chern
number $C=2/3$. Furthermore, two parallel propagating edge modes with the same
level counting $(1,2,5,\cdots)$ are observed in the momentum resolved
entanglement spectrum of the ground state [Fig. 2(d)], in agreement with two
positive eigenvalues in the $\mathbf{K}$-matrix description and the low-lying
structure of edge excitations in $1/3$ FQH systems [53, 59]. Therefore, the
present state is a two-component fractional quantum Hall state, in which the
two-component nature originates from the $C=2$ Chern number of the lowest TFB.
The exact equivalence between the two components render the bosoic FCI state a
color-entangled $\frac{1}{3}$-FCI state [32, 33, 55], which differs from the
general two-component $\nu=1/3+1/3$ case discussed previously in the Bose gas
with two spin states [60, 38], though they share the similar topological
order.

Figure 2: (Color online). $\nu=1/3$ FCI state. (a) Spectral flow as a function
of twist boundary phase $\theta_{y}$; (b) Total Berry curvature of the three
quasi-degenerate ground states at $10\times 10$ mesh points, which shares a
total Chern number $C=2$; (c) $2/3$ charge pumping during the adiabatic flux
quanta insertion; and (d) Momentum-resolved entanglement spectrum revealing
two branches of chiral edge modes, each with degeneracy pattern: 1, 2, 5. The
lattice size $N_{s}=3\times 3\times 5=45$ is used in ED calculation. The
cylinder width $L_{y}=6$ is adopted in iDMRG calculation.

FCI state at $\nu=1$.—It is commonly accepted that the integer quantum Hall
effect emerges when electrons fully fill the Landau levels. The BIQH,
characterized by $\mathbf{K}=\big{(}\begin{smallmatrix}0&1\\\ 1&0\\\
\end{smallmatrix}\big{)}$, were predicted in the lowest Landau level filled
with two-component interacting bosons at total filling $\nu=1+1$ [36, 37].
This state was also proposed in the Hofstadter model when the $C=2$ Chern band
are fully occupied by the hard-core bosons [43]. It is thus interesting to
explore the potential state in the current lattice model at integer filling.

Figure 3: (Color online). $\nu=1$ FCI state. (a) Low-lying energy spectrum for
different lattice sizes; (b) Spectral flow as a function of twist boundary
phase $\theta_{y}$; (c) Total Berry curvature of the two ground states, which
share a total Chern number $|C|=1$; (d) $1/2$ charge pumping during adiabatic
flux quanta insertion; (e) Momentum-resolved entanglement spectrum revealing
one branch of chiral edge mode, with degeneracy pattern: 1, 1, 2, 3, 5, 7. The
lattice size $N_{s}=3\times 3\times 3=27$ is used in ED calculation. The
cylinder width $L_{y}=6$ is adopted in iDMRG calculation.

We study the low-lying energy spectrum of different lattice sizes up to $36$
sites with the ED scheme, which is almost the current computational limit
(Hilbert space dimension reaches $10^{8}$ for a single momentum sector). The
twofold quasi-degenerate ground states, separated from the excited states by a
finite gap, emerge at the fixed momentum sector $(k_{x},k_{y})=(0,0)$,
irrespective of the lattice size [Fig. 3 (a)]. The robustness of the ground
states manifold is also confirmed by the twisted boundary conditions
$\theta_{x/y}$ [Fig. 3(b)], where the two states never mix with higher excited
states. This agrees with the previous statistical rule in $C=1$ TFB system
[19, 25, 61, 20], where two ground states are predicted to emerge at
$(k_{x},k_{y})$ and $(k_{x}+N_{b},k_{y}+N_{b})\mod(N_{x},N_{y})$ sector,
respectively. In contrast, only a single ground state is observed in the $C=2$
Hofstadter bands at $\nu=1$ [43], in which a BIQH state is predicted. The FCI
at integer filling $\nu=1$ is further supported by the iDMRG results. The
charge pumping after a flux quanta threading is exactly $1/2$ [Fig. 3(d)].
This agrees with the existence of two degenerate ground states carrying a
total Chern number $C=1$. Moreover, only single branch of the edge mode is
observed in the momentum resolved entanglement spectrum [Fig. 3(e)], in sharp
contrast with the usual situation of two branch of edge modes in a $C=2$ band.
The corresponding counting sequence $(1,1,2,3,5,7,\cdots)$ is consistent with
that in a $1/2$ Laughlin state. Therefore, we find a $1/2$ Laughlin-like FCI
state at the integer filling $\nu=1$, rather than the expected BIQH state.

The confirmed evidence of $1/2$ Laughlin-like fractional Chern insulator comes
from the many-body Chern number of the ground-state wave function by
integrating the Berry curvature in the first Brillouin zone. The discrete
Berry curvatures [56] are relatively smooth [Fig. 3(c)]. Their summations
yield a precisely quantized Chern number $C_{tot}=1$, or averagely a $C=1/2$
fractional Chern number for each ground state. This is in stark contrast to
the FCI states reported at the bosonic $\nu=1/3$ filling and the fermionic
$\nu=1/5$ filling in this model before [25] and the BIQH state reported in the
bosonic gases [36, 37] and Hofstadter band [43], where the ground state
manifold contributes to a total Chern number $C_{tot}=2$. Considering the fact
of two lower TFBs with $C=2$ and $C=-1$ in our model, it is natural to believe
that the hard-core bosons occupy the two lower TFBs simultaneously, which
effectively generate a $C=1$ topological band but with $\nu=1/2$ filling. It
was previously reported the giving width to the Chern band with high Chern
number can help to stabilize a fractional Chern insulator [30]. Here, the two
TFBs, instead of high Chern number, play the similar role, which may be the
reason why the $1/2$ FCI state remains robust at the integer filling though
the effective flatness ratio of the generated $|C|=1$ Chern band is much
reduced down to about $4$ (details see Appendix I). The less strict condition
for the fractional Chern insulator may provide new platform to realizing the
fractional quantum hall state in multi-band systems.

Robustness of FCI state at $\nu=1$.—Similar band topology but in the absence
of the flat bands were previously constructed in a specific Hofstadter lattice
model [31], where the BIQH state is predicted at integer filling with the
hard-core bosons [43]. It is therefore interesting to study whether the FCI
state at integer filling turns into the BIQH state, or other competing order,
when the flatness of TFBs is lowered. The flatness of TFBs is closely related
to the NNN hopping integral $t^{\prime}$ (Appendix I). We show the low-energy
many-body spectrum $E_{n}-E_{1}$ evolving with $t^{\prime}$ in Fig. 4 (a) on a
torus with fixed $N_{s}=3\times 3\times 3=27$ sites. The twofold ground state
quasi-degeneracy remains evident among $0.12\leq t^{\prime}\leq 0.4$,
indicating the robustness of FCI state at $\nu=1$ filling in this region. When
$t^{\prime}$ is reduced down to below $0.12$, only a single ground state
separated by excited states is observed. In contrast to BIQH, this is a
trivial state with $C=0$. Another phase transition is observed at
$t^{\prime}=0.4$, where the underlying band topology changes (Appendix I). We
do not find any signature of the BIQH state within the current setup.

On the other hand, the FCI state is often challenged by other conventional
ordered states in the strongly correlated systems [18], e.g., the charge-
density wave state may be favorable when the long-range Coulomb interaction
are taken into account [19, 54]. We further discuss the role of NN Coulomb
repulsion $V$ [62] on the FCI state. In the large $V$ limit, the loading
bosons tend to occupy the NNN neighbors, i.e, the same sublattice, to lower
energy contributed by the Coulomb repulsion. Fig. 4 (b) shows a phase
transition occurring at $V_{c}\simeq 1.1$. The FCI state at the integer
filling $\nu=1$ is robust below $V_{c}$ since the two twofold quasi-degeneracy
remains. Above the critical $V_{c}$, a topologically trivial charge-density-
wave state characterized by the threefold quasi-degenerate ground state
emerges, in agreement with the constructed three sublattices in our model.

Figure 4: (Color online). Robustness of $\nu=1$ FCI state. Evolution of low-
lying energy spectrum versus NNN hopping strength $t^{\prime}$ (a) and NN
interaction $V$ (b). System size is $N_{s}=3\times 3\times 3=27$. Colors label
distinct momentum sectors, in particular states at $(0,0)$ momentum sector are
colored blue. In (a), the $\nu=1$ FCI state is dominated between
$0.12<t^{\prime}<0.4$, where the two ground states always share a Chern number
$|C|=1$. Beyond this region, two topologically trivial states emerge. In (b) a
phase transition from the $1/2$ Laughlin-like FCI state to the topologically
trivial charge-density-wave state occurs at $V=1.1$ at integer fillings
$\nu=1$.

Particle-hole symmetry.—Since no band projection is performed here, we are
able to study the system at higher fillings with $\nu>1$. Giving the
unexpected FCI state at $\nu=1$ filling, in which an effective
$\frac{1}{2}$-filled Chern band with $C=1$ is produced. Is it possible that
the BIQH state emerges when the two lower TFBs are fully occupied? Our
numerical results on $\nu=2$ filling show that the present bosonic system
remains the $1/2$ Laughlin-like FCI state. We, therefore, establish a
fractional Chern insulator at the integer fillings in our constructed bosonic
lattice model. It should be emphasized that the energy spectrum at $\nu=2$
filling is exactly the same as that at $\nu=1$ filling. The charge pumping,
and entanglement spectrum, except for a minus sign, and the reversal
propagating direction, respectively, are also the same (Appendix II). Similar
behavior can be further observed at the filling of $\nu=\frac{1}{3}$ and
$\nu=\frac{8}{3}$. It seems that there exist an particle-hole-like symmetry
between $\nu$ and $3-\nu$ filling in our adopted model. The particle-hole
symmetry between the $\nu$ and $1-\nu$ filling of fermions and between $\nu$
and $2-\nu$ filling of two-component bosons in the quantum Hall state was
previously reported when particles partially occupy the lowest Landau level
[40, 63], where the particle-hole symmetry is preserved due to the perfectness
of the Landau level. Here, no such symmetry explicitly exists as revealed by
the band structure. Considering a particle-hole transformation
$\mathcal{P}_{\text{ph}}$ i.e., $b^{{\dagger}}_{i}\leftrightarrow b_{i}$,
yielding
$\mathcal{P}_{\text{ph}}H(\phi)\mathcal{P}^{-1}_{\text{ph}}=H(-\phi)$. We show
that the only difference between the $\phi$ and $-\phi$ is the reversed sign
of the Chern number for each bands. Therefore, an additional particle-hole-
like symmetry between $\nu$ and $3-\nu$ filling exists in our constructed
bosonic lattice model. Such symmetry originates from the intrinsic commute
relation of bosons and the special time-reversal symmetry of the model
Hamiltonian (more details see Appendix II). We emphasize that no particle-hole
symmetry is presented in the corresponding fermionic system.

Summary and discussion.—In summary, we study the fractional Chern insulating
states of the hard-core bosons on a specific three-band triangular lattice
model possessing two lower TFBs with $C=2$ and $C=-1$. Combining the ED and
iDMRG algorithm, we characterize the underlying topological orders by ground
state degeneracy, low energy spectrum, Chern number, together with charge
pumping and entanglement spectrum. At $\nu=1/3$ filling, a color-entangled
$\frac{1}{3}$ FCI state is observed, in which two components come from the
Chern number $C=2$ of the lowest TFB. Particularly, instead of a BIQH state,
we find a robust $1/2$ Laughlin-like fractional Chern insulating state at the
integer fillings. The loading hard-core bosons simultaneously occupy the two
lower TFBs, producing an effective $C=1$ Chern band with $\nu=1/2$-filling. In
addition, a strict particle-hole-like symmetry between the $\nu$ and $3-\nu$
fillings, originated from the intrinsic commute relation of bosons and the
time-reversal symmetry of the specific model, is revealed in present lattice
model. Our results demonstrate the rich family of the novel topological states
in the lattice systems with high Chern number and multi-bands, and add the new
insight into the FCIs.

Further understanding the topological order of the present FCI state at
integer fillings, especially constructing a model fractional state and
performing the overlap with the ground state wave function obtained by the ED
method, is highly desirable. Exploring more exotic FCI states and potential
relation to the generalized Pauli principle in the multi-band system with
nontrivial topology should be interesting for future study. Such multi-band
physics is neglected in most previous studies due to the simplified band
projection. Whether the multi-band induced fractional Chern insulators can
also be realized in the fermionic system is also interesting. Very recently, a
direct transition between $\frac{1}{3}$ Laughlin state and Chern insulating
state driven by the interaction is reported in the Harper-Hofstadter model
loaded by fermions [64], where the lowest Landau level comprises multiple sub-
bands. This may be the clue of multi-band and high Chern number induced
fractional state at integer filling in fermionic system.

The single-occupancy constraint of hard-core bosons may prevent the emergence
of BIQH state in present lattice model. Whether the BIQH state will survive in
this TFB model loaded by soft-core bosons, especially for intermediate
interaction regime, is also an interesting problem that worth delicate
investigation. A first study of three-body hard-core bosons with intermediate
Hubbard interaction strength has not given any clue of BIQH state. Since the
computational effort to study the full phase diagram in current TFB model with
soft-core bosons are much larger than that of hard-core bosons, and more
exotic non-Abelian FCI phases that occupy multiple Chern bands may also
emerge, we will leave the numerically intensive problem in a detailed future
work.

Acknowledgments.— We thank S.-L. Yu, W. Zhu, and T.-S. Zeng for helpful
discussions. This work is supported by the National Nature Science Foundation
of China Grant Nos. 12074175 and 11874325. YZ and CDG also acknowledge the
Ministry of Science and Technology of China under Grant No. 2016YFA0300401.
ALH acknowledges the China Postdoctoral Science Foundation (Nos. 2020M680499).

## Appendix For “Bosonic fractional Chern insulating state at integer fillings
in multi-band system”

The adopted triangular lattice model loaded with the hard-core bosons has been
schematically shown in the main text. The model Hamiltonian can be written as
[25]

$\displaystyle H$ $\displaystyle=$ $\displaystyle\pm
t\sum_{\langle\ij\rangle}\left[b^{\dagger}_{i}b_{j}\exp\left(i\phi_{ij}\right)+\textrm{H.c.}\right]$
(S1) $\displaystyle\pm t^{\prime}\sum_{\langle\langle
ij\rangle\rangle}\left[b^{\dagger}_{i}b_{j}\exp\left(i\phi_{ij}\right)+\textrm{H.c.}\right].$

Each unit cell comprises three inequivalent sites, which naturally produce
three single particle bands. Only the nearest-neighbor (NN) and the next-NN
hopping process of the hard-core bosons are considered. It should be reminded
that no on-site term exist in our model, which is crucial for the for the
particle-hole-like symmetry existing in the present bosonic model as discussed
below. In addition, we restrict our model in the canonical ensemble, which
guarantee the particle number conservation. Even no explicit interactions are
considered, the model system remains strongly correlated due to the nature of
hard-core boson.

##  I Single-particle energy spectrum

The band topology of the present lattice model are parameters dependent. In
Fig. S1, we show the band structure on cylinder evolving with the next-NN
hopping integral $t^{\prime}$ at fixed $\phi=\pi/3$. A topological phase
transition occurs at $t^{\prime}_{c}=1/3$. Below $t^{\prime}_{c}$, the Chern
number for the respective Chern band is $(2,-1,-1)$ (from bottom to top),
while it turns to be $(-4,5,-1)$ for $t^{\prime}>t^{\prime}_{c}$. For large
enough $t^{\prime}>1$, a topological phase with Chern number $(5,-4,-1)$ may
also exist. We emphasize that there exist a special time-reversal-like
symmetry in the present model. The band structure remains unchanged except for
the reversed Chern number for each band when $\phi\rightarrow-\phi$ but with
the other parameters fixed. The chiral edge states shown in Fig. S1 reverse
their propagating direction (not shown).

Figure S1: (Color online). Single-particle spectrum of the three-band
triangular lattice model for different $t^{\prime}$ at fixed $\phi=\pi/3$. The
Chern numbers, from low to high energy bands, for $t^{\prime}<1/3$ topological
phase is $(2,-1,-1)$. (c) Higher-Chern-number bands exists for
$t^{\prime}>1/3$, which is $C=(-4,5,-1)$ at $t^{\prime}=1/2$. Figure S2:
(Color online). $t^{\prime}$-dependence of the flatness ratio of the
respective topological flat band, together with that of the generated $C=1$
Chern band. The flatness ratio of the middle topological flat band is
renormalized by a factor of $10$. $\phi$ is fixed at $\pi/3$.

On the other hand, the flatness ratio of the respective topological band also
varies with parameters as shown in Fig. S2. We define the flatness ratio of
the respective lower topological band by $\Delta_{i}/W_{i}$, where
$\Delta_{i}$, and $W_{i}$ denotes the band gap above, and the bandwidth of,
the selected $i$-th band ($i=1,2$ from bottom to top), respectively. With
fixed $\phi=\pi/3$, the flatness ratio of the lowest topological flat band
(TFB) reaches its maximum at $t^{\prime}=\frac{1}{4}$. It decreases
monotonically with enhanced or weakened $t^{\prime}$. The flatness ratio of
the middle TBF could be higher than $200$ at $t^{\prime}=\frac{1}{5}$. As
mentioned in main text, the two lower TFBs jointly generate an effective $C=1$
Chern bands at integer fillings, which is the reason of the emergence of
$\frac{1}{2}$ Laughlin-like FCI state at integer filling. We also show the
flatness ratio of the generated effective $C=1$ Chern band comprising two
lower TFBs, defined as the $R_{T}=\Delta_{2}/(W_{1}+W_{2}+\Delta_{1})$. The
flatness ratio of this Chern band in the region where the $\frac{1}{2}$
Laughlin-like FCI state at integer fillings is observed is relative high, but
much reduce in comparison with that of the respective TFB. In this sense, the
multi-band physics provides the easy access to the fractional Chern insulating
state.

Figure S3: (Color online). $\nu=1$ (Top) and $\nu=2$ (bottom) FCI states. (a)
$1/2$ charge pumping after one flux insertion. (b) Momentum-resolved
entanglement spectrum reveals one branch of chiral edge mode with degeneracy
pattern: 1, 1, 2, 3, 5, 7, … The two FCI states are time-reversal counterparts
of each other, where the charge pumping differs by a sign and edge mode
differs by the propogating direction.

##  II Particle-hole-like symmetry

Obviously, the band structure in the above section shows no particle-hole
symmetry for fermions. However, the situation in bosonic system is
significantly different from that in the fermionic system due to the intrinsic
statistical law. In a hard-core bosonic system, we apply the particle-hole
transformation $\mathcal{P}$, i.e., $b^{{\dagger}}\leftrightarrow b$, on
Hamiltonian (S1), yielding

$\displaystyle H$ $\displaystyle=$ $\displaystyle\pm
t\sum_{\langle\ij\rangle}\left[b_{j}b^{\dagger}_{i}\exp\left(i\phi_{ij}\right)+\textrm{H.c.}\right]$
(S2) $\displaystyle\pm t^{\prime}\sum_{\langle\langle
ij\rangle\rangle}\left[b_{j}b^{\dagger}_{i}\exp\left(i\phi_{ij}\right)+\textrm{H.c.}\right].$

Due to the commute relation between $\left[b_{i},b_{j}^{{\dagger}}\right]=0$,
we have $\mathcal{P}H(\phi)\mathcal{P}^{-1}=H(-\phi)$. We have shown that the
band structure remains unchanged except for the reversed Chern number for
$\phi\rightarrow-\phi$. Therefore, the only change for topological state is
the reversed propagating direction, or _particle_ to _hole_. Such particle-
hole symmetry may be broken when the mass terms, i.e.,
$b^{{\dagger}}_{i}b_{i}$, are taken into account, by which
$\left[b_{i},b_{i}^{{\dagger}}\right]=1$. Therefore, the particle-hole-like
symmetry found in the present bosonic lattice model originates from the
intrinsic commute relation of the bosons and the unique time-reversal symmetry
of the specific model. This symmetry is naturally absent in the fermionic
system due to the anti-commute relation of fermionic operators.

The particle-hole-like symmetry may be further understood by mapping the hard-
core bosons onto the spin-$\frac{1}{2}$ quantum model. Using the Matsuda-
Matsubara transformation, $b^{{\dagger}}_{i}\rightarrow S_{i}^{+}$,
$b_{i}\rightarrow S_{i}^{-}$ and $n_{i}\rightarrow S_{i}^{z}+\frac{1}{2}$, the
above Hamiltonian can be expressed as

$\displaystyle H$ $\displaystyle=$ $\displaystyle\pm
t\sum_{\langle\ij\rangle}\left[S^{+}_{i}S^{-}_{j}\exp\left(i\phi_{ij}\right)+\textrm{H.c.}\right]$
(S3) $\displaystyle\pm t^{\prime}\sum_{\langle\langle
ij\rangle\rangle}\left[S^{+}_{i}S^{-}_{j}\exp\left(i\phi_{ij}\right)+\textrm{H.c.}\right].$

The particle-hole transformation in the bosonic model is same as the spin flip
process in spin model. In the absence of symmetry breaking term $S_{i}^{z}$,
the above Hamiltonian after spin flip also results in $H(-\phi)$.

In Fig. S3, we plot the charge pumping and entanglement spectrum to show the
particle-hole-like symmetry between the $\nu=1$ and $\nu=2$, in which a $1/2$
Laughlin-like fractional Chern insulating state is realized at integer
fillings. The two cases are almost exactly same, except for the charge (or
hole) pumping process and reversal propagating direction. Similar behavior can
be also found at the integer filling $\nu=1/3$ and $\nu=8/3$, where a color-
entangled $\frac{1}{3}$ FCI state is observed.

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