# Quantum gates with weak van der Waals interactions of neutral Rydberg atoms

Xiao-Feng Shi  School of Physics and Optoelectronic Engineering, Xidian
University, Xi’an 710071, China  Yan Lu  School of Physics and Optoelectronic
Engineering, Xidian University, Xi’an 710071, China

###### Abstract

Neutral atoms are promising for large-scale quantum computing, but accurate
neutral-atom entanglement depends on large Rydberg interactions which strongly
limit the interatomic distances. Via a phase accumulation in detuned Rabi
cycles enabled by a Rydberg interaction of similar magnitude to the Rydberg
Rabi frequency, we study a controlled-phase gate with an arbitrary phase and
extend it to the controlled-NOT gate. The gates need only three steps for
coupling one Rydberg state, depend on easily accessible van der Waals
interaction that naturally arises between distant atoms, and have no rotation
error in the weak interaction regime. Importantly, they can work with very
weak interactions so that well-separated qubits can be entangled. The gates
are sensitive to the irremovable fluctuation of Rydberg interactions, but can
still have a fidelity over 98% with realistic position fluctuation of qubits
separated over 20 $\mu$m.

##  I introduction

Neutral atoms can be rapidly entangled when they are excited to high-lying
Rydberg states, which renders the possibility to use neutral atoms for large-
scale quantum computing Saffman  _et al._ (2010); Saffman (2016); Weiss and
Saffman (2017); Adams  _et al._ (2020); Wu  _et al._ (2021); Morgado and
Whitlock (2021). There have been several experiments demonstrating
entanglement between individual neutral atoms by Rydberg interactions Wilk
_et al._ (2010); Isenhower  _et al._ (2010); Zhang  _et al._ (2010); Maller
_et al._ (2015); Jau  _et al._ (2016); Zeng  _et al._ (2017); Levine  _et al._
(2018); Picken  _et al._ (2019); Levine  _et al._ (2019); Graham  _et al._
(2019); Jo  _et al._ (2020); Madjarov  _et al._ (2020) primarily via the
blockade mechanism Jaksch  _et al._ (2000) where two nearby Rydberg atoms
should possess a strong interaction $V$. The value of $V$ drops quickly when
the qubit spacing increases, so it is necessary to place the qubits close
enough for the blockade condition to hold. For example, the interatomic
distance was in the range $[3.6,~{}5.7]~{}\mu$m in recent experiments of high-
fidelity neutral-atom entanglement Levine  _et al._ (2018, 2019); Graham  _et
al._ (2019); Madjarov  _et al._ (2020). On the other hand, the yet to be
large-scale quantum processor is supposed to host a large number of qubits in
which entanglement operations between distant qubits are required in a general
computational task. To tackle this issue, Ref. Weimer  _et al._ (2012)
suggested a method to entangle two qubits separated by a chain of ancillary
qubits by adiabatically following the many-body ground state of the qubit
chain coupled via the Rydberg blockade mechanism, and Ref. Cesa and Martin
(2017) proposed to entangle distant qubits by coupling two targeted qubits
with a group of ancillary atoms via Rydberg interactions. These methods depend
on strong interactions of nearby Rydberg atoms. For example, Ref. Weimer  _et
al._ (2012) analyzed a model by assuming a large enough $V$ enabled by a small
average interatomic spacing $1~{}\mu$m, so that the entanglement of two logic
qubits separated by $20~{}\mu$m would require a coherent control over around
19 ancillary qubits.

In this work, we analyze an entangling gate between two well-separated neutral
atoms by partially exciting them to Rydberg states. In contrast to the gate
protocols dependent on the blockade mechanism, we find that a tiny $V$ can
rapidly generate entanglement between ground hyperfine levels via detuned
ground-Rydberg Rabi oscillations, and a controlled phase gate with an
arbitrary phase $\theta$ can be created accurately. The intrinsic fidelity of
the gate is perfect when $V$ is frozen and when the Rydberg-state decay is
ignored, but for realistic setups where there will be fluctuations of qubit
positions Graham  _et al._ (2019) and Rydberg-state decay, the gate can still
have a fidelity $0.99$ with qubit spacing over $20~{}\mu$m. Compared to the
Rydberg gate by a dynamical phase shift $Vt/\hbar$ from the Rydberg
interactions Jaksch  _et al._ (2000), our gate does not require the Rydberg
Rabi frequency $\Omega$ to be much larger than $V/\hbar$ and, thus, a high-
fidelity implementation is less technically demanding, where $h(\hbar)$ is the
(reduced) Planck constant. Compared to the Rydberg gates in Shi (2017) that
work with similar condition $V/\hbar\sim\Omega$ but need five pulses to couple
multiple Rydberg states, the gate in this work needs fewer pulses to couple
only one Rydberg state. In contrast to previous Rydberg gates based on Förster
resonances Shi (2017); Beterov  _et al._ (2016); Petrosyan  _et al._ (2017);
Beterov  _et al._ (2018a, b); Khazali and Molmer (2020); Beterov  _et al._
(2020); Stojanovic (2021), our gate depends on van der Waals interactions that
naturally appear in well separated atoms without resorting to external fields
for tuning a Förster resonance. Since entangling operations between well-
separated qubits are necessary for large-scale quantum computing, our method
can simplify the quantum circuit in quantum computing because otherwise many
standard Rydberg gates are needed to entangle two distant qubits.

The remainder of this article is organized as follows. In Sec. II, we study
the controlled-phase gate with an arbitrary phase, and then extend it to a
CNOT gate. In Sec. III, we study the achievable gate fidelity with realistic
parameters and fluctuation of qubit positions. Section IV gives a comparison
between the gate protocols here and previous Rydberg gates with distant
qubits. A brief summary is given in Sec. V.

Figure 1: A controlled-phase gate with an arbitrary phase $\theta$ mediated by
a weak Rydberg interaction $V$ that satisfies the condition $\theta=-2\pi
V/\sqrt{\hbar^{2}\Omega_{\text{t}}^{2}+V^{2}}$ (up to a trivial addend of an
integer times $2\pi$). The first and third pulses excite and deexcite the
Rydberg excitation in the control qubit. At the middle of the second pulse a
$\pi$ phase change is inserted in the Rabi frequency via modifying the phases
in the Rydberg lasers. The phase $\theta$ is tunable via adjustment of the
ratio between $\Omega_{\text{t}}$ and $V$.

##  II Two-qubit controlled gates with weak van der Waals interactions

###  II.1 A controlled-phase gate with an arbitrary phase $\theta$

We first present the protocol for a controlled-phase gate and then extend it
to a CNOT gate in Sec. II.2. In particular, we consider a quantum entanglement
operation

$\displaystyle\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$
$\displaystyle\longmapsto$
$\displaystyle\\{|00\rangle,|01\rangle,|10\rangle,e^{i\theta}|11\rangle\\},$
(1)

by exciting two qubits to Rydberg states via external laser fields, where
$\theta$ is tunable via adjustment of the frequency and magnitude of the laser
field, and $|0\rangle$ and $|1\rangle$ are the two states of a qubit defined
by the two hyperfine-Zeeman ground substates of a heavy alkali-metal atom such
as 87Rb and 133Cs. External laser fields are sent to the qubits for exciting
$|1\rangle$ to a Rydberg state $|r\rangle$. Equation (1) is realized
effectively by three pulses, where “effectively” means that the total number
of pulses is four, but since the middle two pulses only differ by a phase
which can be rapidly inserted in the laser fields, the total number of pulses
is effectively three as shown in Fig. 1. Pulses 1 and 3 are for exciting and
deexciting the control qubits with Rabi frequencies of magnitude
$|\Omega_{\text{c}}|$, and pulse 2 is for exciting the target qubit with Rabi
frequencies of magnitude $|\Omega_{\text{t}}|$. Upon the application of pulse
1 with a $\pi$ rotation $|1\rangle\rightarrow|r\rangle$ for the control qubit
enabled by the Hamiltonian $\Omega_{\text{c}}|r\rangle\langle 1|/2+$H.c. in a
rotating frame, the input eigenstates evolve as

$\displaystyle|01\rangle$ $\displaystyle\longmapsto$
$\displaystyle|01\rangle,$ $\displaystyle|10\rangle$
$\displaystyle\longmapsto$ $\displaystyle-i|r0\rangle,$
$\displaystyle|11\rangle$ $\displaystyle\longmapsto$
$\displaystyle-i|r1\rangle,$

where the other input state $|00\rangle$ does not evolve since it is not
excited because the energy separation between $|0\rangle$ and $|1\rangle$ is
$E_{\text{hyper}}=h\times 6.8$ GHz for 87Rb which is orders of magnitude
larger than ($\hbar$ times) the Rydberg Rabi frequency $2\pi\times 0.8$ MHz
that we will consider. Then, the probability
$2(\hbar\Omega_{\text{t}}/E_{\text{hyper}})^{2}$ Saffman and Walker (2005) of
Rydberg excitation of $|0\rangle$ is on the order of $10^{-8}$, which can be
ignored.

The Hamiltonian for pulse 2 sent upon the target qubit is
$\pm\Omega_{\text{t}}|r\rangle\langle 1|/2+$H.c., where the sign $+(-)$
applies for the first (second) half of pulse 2. During the first half of pulse
2, laser fields are sent to the target qubit for the transition
$|1\rangle\rightarrow|r\rangle$ with a Rabi frequency $\Omega_{\text{t}}$ for
a pulse duration $t_{0}=2\pi/\sqrt{\Omega_{\text{t}}^{2}+V^{2}/\hbar^{2}}$,
where $V$ is the Rydberg interaction of the state $|rr\rangle$. As previous
studied in Refs. Shi (2017, 2019, 2018); Levine  _et al._ (2019), the
interaction $V$ will detune the transition between $|r1\rangle$ and
$|rr\rangle$ so that the state rotation has a generalized Rabi frequency
$\overline{\Omega_{\text{t}}}\equiv\sqrt{\Omega_{\text{t}}^{2}+V^{2}/\hbar^{2}}$.
In contrast to the resonant Rabi rotation between two states where one full
Rabi cycle results in a phase change $\pi$ to the state, the detuned Rabi
cycle will lead to a phase change
$\theta/2=-\pi[1+V/(\hbar\overline{\Omega_{\text{t}}})]$ Shi (2017), so that
the state evolution in the first half of pulse 2 is

$\displaystyle|01\rangle$ $\displaystyle\longmapsto$ $\displaystyle
e^{-it_{0}\hat{H}(\Omega_{\text{t}})/\hbar}|01\rangle,$
$\displaystyle-i|r0\rangle$ $\displaystyle\longmapsto$
$\displaystyle-i|r0\rangle,$ $\displaystyle-i|r1\rangle$
$\displaystyle\longmapsto$ $\displaystyle-ie^{i\theta/2}|r1\rangle,$

where

$\displaystyle\hat{H}(\Omega_{\text{t}})$ $\displaystyle=$
$\displaystyle\hbar[\Omega_{\text{t}}|0r\rangle\langle 01|]/2+\text{H.c.}$

The second half of pulse 2 is similar to its first half, but with a Rabi
frequency $-\Omega_{\text{t}}$. With a radiation duration $t_{0}$, the input
state $|01\rangle$ returns to itself since its total time-evolution operator
in the second pulse
$e^{-it_{0}\hat{H}(-\Omega_{\text{t}})/\hbar}e^{-it_{0}\hat{H}(\Omega_{\text{t}})/\hbar}$
reduces to an identity because
$\hat{H}(-\Omega_{\text{t}})+\hat{H}(\Omega_{\text{t}})=0$. On the other hand,
remarkably, the change of the sign of the Rydberg Rabi frequency does not
change the picture of the detuned Rabi oscillation, so that another $\theta/2$
phase change appears for the input state $|11\rangle$. As a consequence, the
state evolution is

$\displaystyle e^{-it_{0}\hat{H}(\Omega_{\text{t}})/\hbar}|01\rangle$
$\displaystyle\longmapsto$ $\displaystyle|01\rangle,$
$\displaystyle-i|r0\rangle$ $\displaystyle\longmapsto$
$\displaystyle-i|r0\rangle,$ $\displaystyle-ie^{i\theta/2}|r1\rangle$
$\displaystyle\longmapsto$ $\displaystyle-ie^{i\theta}|r1\rangle,$ (2)

in the second half of pulse 2.

Pulse 3 is similar to pulse 1 but with a $\pi$ phase change to the Rabi
frequency which becomes $-\Omega_{\text{t}}$. The $\pi$ rotation restores the
state of the control qubit back to the ground state, leading to

$\displaystyle|01\rangle$ $\displaystyle\longmapsto$
$\displaystyle|01\rangle,$ $\displaystyle-i|r0\rangle$
$\displaystyle\longmapsto$ $\displaystyle|10\rangle,$ $\displaystyle-
ie^{i\theta}|r1\rangle$ $\displaystyle\longmapsto$ $\displaystyle
e^{i\theta}|11\rangle.$

With the understanding that the input state $|00\rangle$ does not evolve in
the rotating frame, the state transform in Eq. (1) is realized as shown in
Fig. 1.

Figure 2: A CNOT gate with a pulse sequence similar to that in Fig. 1 when
$\theta=\pi$. (a,b) During pulse 2, both qubit states $|0\rangle$ and
$|1\rangle$ are excited to the Rydberg state $|r\rangle$ via two different
low-lying $p$-orbital states. (c) When the Rydberg Rabi frequencies for
$|0\rangle\rightarrow|r\rangle$ and $|1\rangle\rightarrow|r\rangle$ are both
$\Omega_{\text{t}}/\sqrt{2}$, it is like that the state
$|\overline{1}\rangle\equiv(|0\rangle+|1\rangle)/\sqrt{2}$ is excited with a
Rabi frequency $\Omega_{\text{t}}$ during pulse 2, while the other state
$|\overline{0}\rangle\equiv(|0\rangle-|1\rangle)/\sqrt{2}$ stays intact. (d)
By a similar pulse sequence as in Fig. 1 with the modification of pulse 2
shown in (c) with the condition $\hbar\Omega_{\text{t}}/V=\sqrt{3}$, a state
mapping diag$\\{1,1,-1,1\\}$ is realized in the basis
$\\{|0\overline{0}\rangle,|0\overline{1}\rangle,|1\overline{0}\rangle,|1\overline{1}\rangle\\}$,
which is the standard CNOT gate after expanding $|\overline{0(1)}\rangle$.

###  II.2 A CNOT gate

The controlled phase gate studied in Sec. II.1 becomes a CZ gate when
$\theta=\pi$, which has the same entangling power Williams (2011) as a CNOT
gate. On the other hand, the pulse sequence in Sec. II.1 can be slightly
modified so as to directly create a CNOT gate,

$\displaystyle\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$
$\displaystyle\longmapsto$
$\displaystyle\\{|00\rangle,|01\rangle,|11\rangle,|10\rangle\\}.$ (3)

The CNOT gate sequence shown in Fig. 2 is similar to that described in Fig. 1
with two modifications. First, pulse 1 and pulse 3 use the same set of laser
fields, i.e., no need to have a relative sign difference in the Rabi
frequencies as in Fig. 1. Second, in pulse 2, we replace the Rydberg-
excitation Hamiltonian $\pm\hbar\Omega_{\text{t}}|r\rangle\langle 1|/2+$H.c.
in Fig. 1 by

$\displaystyle\pm\hbar\Omega_{\text{t}}|r\rangle(\langle 0|+\langle
1|)/(2\sqrt{2})+\text{H.c.},$ (4)

i.e., (i) the first half of pulse 2 requires two sets of laser fields, one for
the Rydberg excitation
$|0\rangle\xrightarrow[]{\Omega_{\text{t}}/\sqrt{2}}|r\rangle$, and the other
for the Rydberg excitation
$|1\rangle\xrightarrow[]{\Omega_{\text{t}}/\sqrt{2}}|r\rangle$, and (ii) the
second half of pulse 2 requires two sets of laser fields, one for the Rydberg
excitation $|0\rangle\xrightarrow[]{-\Omega_{\text{t}}/\sqrt{2}}|r\rangle$,
and the other for the Rydberg excitation
$|1\rangle\xrightarrow[]{-\Omega_{\text{t}}/\sqrt{2}}|r\rangle$. This requires
simultaneous excitation of both qubit states to a Rydberg state. For two-
photon Rydberg excitations, different intermediate states should be used with
an example shown in Figs. 2(a) and 2(b). To understand the time evolution of
the wavefunctions for different input states in pulse 2, we follow Ref. Shi
(2020a) and consider
$\\{|0\overline{0}\rangle,|0\overline{1}\rangle,|1\overline{0}\rangle,|1\overline{1}\rangle\\}$,
where $|\overline{1}(\overline{0})\rangle=(|0\rangle\pm|1\rangle)/\sqrt{2}$ as
shown in Fig. 2(c). In this basis, Eq. (4) becomes

$\displaystyle\hat{\bar{H}}(\pm\Omega_{\text{t}})=\pm\hbar\Omega_{\text{t}}|r\rangle\langle\overline{1}|/2+\text{H.c.}$

With the condition $\hbar\Omega_{\text{t}}/V=\sqrt{3}$ (corresponding to
$\theta=\pi$ in the picture of Sec. II), the first half and the second half of
pulse 2, each with duration $t_{0}=\sqrt{3}\pi/\Omega_{\text{t}}$, lead to

$\displaystyle|0\overline{1}\rangle$ $\displaystyle\longmapsto$ $\displaystyle
e^{-it_{0}\hat{{\bar{H}}}(\Omega_{\text{t}})/\hbar}|0\overline{1}\rangle\longmapsto|0\overline{1}\rangle,$
$\displaystyle-i|r\overline{0}\rangle$ $\displaystyle\longmapsto$
$\displaystyle-i|r\overline{0}\rangle\longmapsto-i|r\overline{0}\rangle,$
$\displaystyle-i|r\overline{1}\rangle$ $\displaystyle\longmapsto$
$\displaystyle|r\overline{1}\rangle\longmapsto i|r\overline{1}\rangle,$

for which a detailed proof is given above Eq. (2). Figure 2(c) shows that the
Rydberg deexcitation via pulse 3 leads to a CNOT gate of Eq. (3) with the fact
that

$\displaystyle|\overline{1}\rangle\langle\overline{1}|-|\overline{0}\rangle\langle\overline{0}|$
$\displaystyle=$ $\displaystyle|0\rangle\langle 1|+|1\rangle\langle 0|.$

##  III Gate fidelity

The gates can be fast. The total gate duration is
$t_{\text{g}}=2\pi(1/\Omega_{\text{c}}+2/\overline{\Omega_{\text{t}}})$
according to the pulse sequence in Fig. 1 or Fig. 2. For a CZ or CNOT gate
where $\theta=\pi$, we should have the condition

$\displaystyle\Omega_{\text{t}}=\sqrt{3}V/\hbar$ (5)

so that the total gate duration is
$t_{\text{g}}=2\pi(1/\Omega_{\text{c}}+\sqrt{3}/\Omega_{\text{t}})$. Rydberg
Rabi frequencies around $2\pi\times 4.6$ MHz were used for entanglement of
individual neutral atoms in Ref. Graham  _et al._ (2019), so the CZ or CNOT
gates can have a short duration $t_{\text{g}}=0.59~{}\mu$s if similar Rabi
frequencies are employed.

A high fidelity (about $99\%$) can be realized with our gates. Because the
gate sequences for the CZ and the CNOT gates are similar, we consider the CZ
gate as an example. In order to avoid slow gate speed, a Rydberg interaction
around or over $h\times 0.5$ MHz is desirable. Then, higher Rydberg states are
preferred so that the Rydberg interaction can still be around $h\times 0.5$
MHz even if the qubits are separated by tens of $\mu$m. Among the previous
demonstrations of neutral atom quantum gates, the highest $d$-orbital Rydberg
state of 87Rb atoms ever used had a principal quantum number $n=97$ Isenhower
_et al._ (2010); Zhang  _et al._ (2010). The excitation of $s$-orbital Rydberg
states with $n=102$ was demonstrated in an ensemble of 87Rb atoms where
coherent many-body phenomena were observed Dudin and Kuzmich (2012). To avoid
the anisotropic interaction of $d$-orbital Rydberg states, we consider
$s$-orbital Rydberg states whose interaction is highly isotropic in real space
Walker and Saffman (2008). For these reasons, we analyze the gate performance
with $|r\rangle\equiv|97S_{1/2},m_{J}=1/2\rangle$ and Rydberg Rabi frequencies

$\displaystyle\Omega_{\text{c}}=\Omega_{\text{t}}=2\pi\times
0.8~{}\text{MHz},$ (6)

so that the duration will be $3.4~{}\mu$s for a CZ gate. We note that a
similar Rydberg Rabi frequency $2\pi\times 0.81$ MHz was used in Zhang  _et
al._ (2010) for entanglement of individual 87Rb atoms via the excitation of
the $97d$ states. The Rydberg interaction of $|rr\rangle$ for well separated
qubits is characterized by $C_{6}/\mathcal{L}^{6}$ Walker and Saffman (2008);
Saffman  _et al._ (2010), where $C_{6}=h\times 39.5$ THz$\mu$m6 Shi  _et al._
(2014) is the van der Waals coefficient and $\mathcal{L}$ is the qubit
spacing. To analyze the performance of a CZ gate where $\theta=\pi$ in Eq.
(1), the condition in Eq. (5) requires that the distance between the centers
of the traps for the two qubits should be $20.99~{}\mu$m with Eq. (6) [which
leads to $V\approx h\times 0.46$ MHz according to Eq. (5)].

There are intrinsic errors from the Rydberg-state decay and the position
fluctuations of the atomic qubits when the traps are turned off during the
gate sequence (there is also an error due to Doppler dephasing but it is not
an intrinsic issue since it is removable by the methods in, e.g., Refs
Ryabtsev  _et al._ (2011); Shi (2020b)). The former can be estimated as
$E_{\text{decay}}=T_{\text{Ryd}}/\tau$, where $\tau$ is the lifetime of the
Rydberg state and $T_{\text{Ryd}}$ is the total duration for the input state
to stay in the Rydberg state averaged over the four input eigenstates,

$\displaystyle T_{\text{Ryd}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{\alpha\in\\{0,1\\}}\sum_{|\psi(0)\rangle\in\\{|01\rangle,|10\rangle,|11\rangle\\}}\int\Big{[}|\langle\alpha
r|\psi(t)\rangle|^{2}$ $\displaystyle+|\langle
r\alpha|\psi(t)\rangle|^{2}+2|\langle rr|\psi(t)\rangle|^{2}\Big{]}dt,$

where $|\psi(t)\rangle$ is the wavefunction calculated by unitary time
evolution from the initial state $|\psi(0)\rangle$. Numerical simulation shows
$T_{\text{Ryd}}\approx 1.52\times 2\pi/\Omega_{\text{c}}$ in general and
$T_{\text{Ryd}}\approx 1.91~{}\mu$s with Eq. (6), and $\tau$ is $0.311$ ms (or
1.10 ms) at room temperature (or at $4.2$ K) Beterov  _et al._ (2009), leading
to $E_{\text{decay}}=6.14\times 10^{-3}$ (or $1.74\times 10^{-3})$.

Figure 3: (a) Schematic of the position fluctuation of the two atomic qubits
in real space. The trap centers are at
$(x_{\text{c}},y_{\text{c}},z_{\text{c}})$ and
$(L+x_{\text{t}},y_{\text{t}},z_{\text{t}})$, while the locations of the
qubits can deviate from the centers of the traps. The longitudinal and
transverse r.m.s. position fluctuations of the atoms in the traps are
$\sigma_{z}$ and $\sigma_{\perp}$ along the $\mathbf{z}$ and $\mathbf{x}$ (or
$\mathbf{y}$) directions. The gate sequence proceeds after the traps are
switched off so that the atoms fly freely. (b) Rydberg excitation of the qubit
state $|1\rangle\equiv^{87}$Rb$|5S_{1/2},F=2,m_{F}=2\rangle$ via a largely
detuned intermediate state $|e\rangle$. The red and blue shadows in (a)
indicate the lower and upper laser fields in (b) used for exciting the Rydberg
states from $|1\rangle$.

The fluctuation of the qubit positions contributes the major error because our
gate works perfectly only when the interaction is equal to the desired value.
There will be random position fluctuation of the qubits before the traps are
switched off, and the atoms fly freely during the gate sequence. This leads to
fluctuation of the qubit spacing which results in fluctuation of $V$. A
numerical investigation of this matter requires information for realistic
r.m.s. qubit-position fluctuations in reference to the trap center. Though
most publications on entanglement experiments did not show such details, Ref.
Graham  _et al._ (2019) showed that the longitudinal and transverse r.m.s.
position fluctuations of the atoms in their traps were
$\sigma_{z0}=1.47~{}\mu$m and $\sigma_{\perp 0}=0.27~{}\mu$m, respectively. In
order to incorporate the effect of the free flight of the atoms during the
gate sequence, we would like to modify the values of r.m.s. fluctuations. The
change of the atomic locations during the gate sequence will change the
interatomic distance. For an atomic temperature around $T_{a}=10~{}\mu$K
(e.g., qubits with $T_{a}=5,~{}10,~{}15~{}\mu$K were studied in the
entanglement experiments of Refs. Picken  _et al._ (2019), Levine  _et al._
(2018), and Graham  _et al._ (2019), respectively), the r.m.s. distance for a
qubit to fly during the gate sequence is about
$\ell=v_{\text{rms}}t_{\text{g}}$, where $v_{\text{rms}}=\sqrt{k_{B}T_{a}/m}$
is the r.m.s. speed of the atom along one of the three directions in the 3D
space, $T_{a}$ is the effective temperature of the atom, and $k_{B}$ and $m$
are the Boltzmann constant and the atomic mass, respectively. For a free
flight of qubits, the average change of distance will be
$v_{\text{rms}}t_{\text{g}}/2$ for a duration $t_{\text{g}}$. So, we analyze
the fluctuation of the qubit positions by increasing the r.m.s. fluctuations
to

$\displaystyle\sigma_{z}$ $\displaystyle=$ $\displaystyle\sigma_{z0}+\ell/2,$
$\displaystyle\sigma_{\perp}$ $\displaystyle=$ $\displaystyle\sigma_{\perp
0}+\ell/2,$

which leads to $(\sigma_{z},\sigma_{\perp})=(1.52,~{}0.32)~{}\mu$m in the
condition of Eq. (6).

The schematic of position fluctuation is shown in Fig. 3(a). We suppose that
the quantization axis is along $\mathbf{x}$, and $\pi$ polarized Rydberg laser
lights travel along $\mathbf{z}$. The Rydberg excitation scheme is shown in
Fig. 3(b). The centers of the traps for the control and target qubits are
$(0,0,0)$ and $(L,0,0)$, respectively. Because of the finiteness of the trap
depths, the positions of the control and target qubits are
$(x_{\text{c}},y_{\text{c}},z_{\text{c}})$ and
$(L+x_{\text{t}},y_{\text{t}},z_{\text{t}})$, respectively, where the
distribution function

$\displaystyle\mathscr{G}(\zeta)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi}\sigma_{\perp}}e^{-\zeta^{2}/(2\sigma_{\perp}^{2})}$

characterizes
$\zeta\in\\{x_{\text{c}},y_{\text{c}},x_{\text{t}},y_{\text{t}}\\}$, and

$\displaystyle\mathscr{G}(\zeta)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi}\sigma_{z}}e^{-\zeta^{2}/(2\sigma_{z}^{2})}$

is the distribution function for $\zeta=z_{\text{c}}$ or $z_{\text{t}}$. In
the numerical simulation, the value of $V$ is calculated by using the actual
distance of the qubits
$\mathcal{L}=$$\sqrt{(x_{\text{c}}-x_{\text{t}}-L)^{2}+(y_{\text{c}}-t_{\text{t}})^{2}+(z_{\text{c}}-z_{\text{t}})^{2}}$.
The average fidelity is sampled with

$\displaystyle\overline{\mathcal{F}}$ $\displaystyle=$
$\displaystyle\Pi_{\zeta}\int\mathscr{G}(\zeta)d\zeta\mathcal{F},$

where the fidelity $\mathcal{F}$ is defined by Pedersen  _et al._ (2007)

$\displaystyle\mathcal{F}$ $\displaystyle=$
$\displaystyle\left[|\text{Tr}(U^{\dagger}\mathscr{U})|^{2}+\text{Tr}(U^{\dagger}\mathscr{U}\mathscr{U}^{\dagger}U)\right]/20.$

Here, $\mathscr{U}$ is the actual gate matrix evaluated by using the unitary
dynamics with the Rydberg-state decay ignored, and $U$=diag$\\{1,1,1,-1\\}$ is
the ideal gate matrix in the basis
$\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$. The integral for each
$\zeta$ is approximated by taking discrete values; for example, for
$x_{\text{c}}$, the sampling is approximated by
$\sum\mathscr{G}(x_{\text{c}})\mathcal{F}/\sum\mathscr{G}(x_{\text{c}})$ with
$x_{\text{c}}\in\\{-1.5,-1.5+\delta,-1.5+2\delta,~{}\cdots,~{}1.5-2\delta,1.5-\delta,1.5\\}\sigma_{\perp}$,
and the total gate fidelity is sampled by similar approximations for
$\\{y_{\text{c}},z_{\text{c}},x_{\text{t}},y_{\text{t}},z_{\text{t}}\\}$. The
numerical simulation is resource-demanding, but we can obtain an estimate
about the convergence of the integration via slowly varying the steps. For
$\delta$ equal to $0.25,0.2,0.15,0.12$, and $0.1$, the sampled average
fidelities are $0.9910,0.9912,0.9914,0.9920$, and $0.9920$ respectively, from
which we estimate $\overline{\mathcal{F}}=0.992$. With Rydberg-state decay
included, the intrinsic fidelity would be
$\overline{\mathcal{F}}-E_{\text{decay}}=0.986$ (or $0.990$) at $300$ K (or
4.2 K).

##  IV Comparison with other methods

The gates in this work have several advantages compared to other gates Jaksch
_et al._ (2000); Shi (2017, 2019); Beterov  _et al._ (2016) that can also work
with weak interactions in distant Rydberg atoms.

The first advantage is that our gate does not have rotation errors in the weak
interaction regime. It was proposed in Ref. Jaksch  _et al._ (2000) that phase
gates can be created via the dynamical phase shift of the van der Waals
interaction between two weakly interacting Rydberg atoms as experimentally
demonstrated in Ref. Jo  _et al._ (2020). The method of Jaksch  _et al._
(2000) involves the excitation $|11\rangle\rightarrow|rr\rangle$ which has an
intrinsic error $2V^{2}/(\hbar^{2}\Omega^{2})$ due to the failure for exciting
the state $|rr\rangle$ from $|11\rangle$ as analyzed in Ref. Saffman and
Walker (2005). In contrast, the intrinsic accuracy of the gate here is limited
only by the Rydberg-state decay in the weak interaction regime.

The second advantage is that our gate depends on van der Waals interaction
that naturally appears between two distant Rydberg atoms. As has been
experimentally tested Jo  _et al._ (2020), such a strategy allows an easy
access to an experimental demonstration. In comparison, by using Förster
resonance one can also design entangling gates between well-separated qubits
Beterov  _et al._ (2016, 2020, 2018a, 2018b) but fine tuning via external
fields is required Safinya  _et al._ (1981); Anderson  _et al._ (1998);
Mourachko  _et al._ (1998); Westermann S.  _et al._ (2006); Nipper  _et al._
(2012); Richards and Jones (2016); Kondo  _et al._ (2016); Ryabtsev  _et al._
(2010); Tretyakov  _et al._ (2014); Faoro  _et al._ (2015); Tretyakov  _et
al._ (2017); Cheinet  _et al._ (2020). However, it will be useful to
investigate, if possible, an extension of our method to the regime of Förster
resonance. If it is indeed possible, high-fidelity entanglement should be
reachable with qubits separated even longer than the estimate in this work.
This is because the Förster resonant interaction falls off as
$1/\mathcal{L}^{3}$, while the van der Waals interaction falls off as
$1/\mathcal{L}^{6}$ when $\mathcal{L}$ increases.

The third advantage is that our method can more easily realize a CZ gate
compared to previous high-fidelity gates based on weak van der Waals
interactions Shi (2017, 2019). Compared to Ref. Shi (2017) which uses five
pulses for coupling two different Rydberg states, the gate here needs only
three steps to couple one Rydberg state. Compared to Shi (2019) which proposes
a quantum entangling gate that should be repeated several times to form a CZ
gate when assisted by single-qubit gates, our method can realize a controlled-
phase gate (such as CZ) with any desired phase by only three laser pulses as
shown in Fig. 1.

##  V conclusions

We study a controlled-phase gate based on weak van der Waals interactions
between neutral Rydberg atoms. The gate is realized via phase accumulations in
detuned Rabi oscillations enabled by weak Rydberg interactions. The gate is
easily realizable and can have a high fidelity because it only needs three
steps to couple one type of Rydberg state, has no intrinsic rotation errors in
the weak interaction regime, and does not depend on fine tuning of Förster
resonant processes. We use practical parameters to show that it is possible to
create a CZ gate in two qubits separated by $21~{}\mu$m with a fidelity about
$0.990~{}($or $0.986)$ at 4.2 K (or at room temperature). We also show that a
similar pulse sequence can lead to a CNOT gate which is realizable with
similarly small van der Waals interactions. The theory brings hope to simplify
the quantum circuit for large-scale quantum computing in which entanglement
between distant qubits is required.

## ACKNOWLEDGMENTS

This work is supported by the National Natural Science Foundation of China
under Grants No. 12074300 and No. 11805146, the Natural Science Basic Research
plan in Shaanxi Province of China under Grant No. 2020JM-189, and the
Fundamental Research Funds for the Central Universities.

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# On bisectors in quaternionic hyperbolic space

Igor A.R. Almeida, Jaime L.O. Chamorro, Nikolay Gusevskii Corresponding
author.

###### Abstract

In this paper, we study a problem related to geometry of bisectors in
quaternionic hyperbolic geometry. We develop some of the basic theory of
bisectors in quaternionic hyperbolic space ${\rm H^{n}_{\mathbb{Q}}}$. In
particular, we show that quaternionic bisectors enjoy various decompositions
by totally geodesic submanifolds of ${\rm H^{n}_{\mathbb{Q}}}$. In contrast to
complex hyperbolic geometry, where bisectors admit only two types of
decomposition (described by Mostow and Goldman), we show that in the
quaternionic case geometry of bisectors is more rich. We describe an infinite
family of different decompositions of bisectors in ${\rm H^{n}_{\mathbb{Q}}}$
by totally geodesic submanifolds of ${\rm H^{n}_{\mathbb{Q}}}$ isometric to
complex hyperbolic space ${\rm H^{n}_{\mathbb{C}}}$.

MSC: 32H20; 20H10; 22E40; 57S30; 32G07; 32C16

Keywords: Quaternionic hyperbolic space; bisectors.

## Introduction

It is well known that the rank one symmetric spaces of non-compact type are
real, complex and quaternionic hyperbolic spaces, or a Cayley hyperbolic
plane. In order to construct discrete groups of isometries in these
geometries, one needs an appropriate notion of polyhedra, which becomes
nontrivial in the absence totally geodesic hypersurfaces from which to form a
polyhedron’s faces in spaces with non-constant sectional curvature. This
happens in complex and quaternionic hyperbolic geometries, and in geometry of
the Cayley hyperbolic plane. Since the faces of Dirichlet fundamental
polyhedra are in bisectors, it is natural to use bisectors as the building
blocks for polyhedra in these geometries. Therefore, it is necessary to
understand the geometric structure of such hypersurfaces. In complex
hyperbolic geometry, it was done by Mostow [9] and Goldman [7]. They showed
that bisectors in complex hyperbolic space ${\rm H^{n}_{\mathbb{C}}}$ admit a
decomposition into complex hyperplanes and into totally real totally geodesic
submanifolds (meridian decomposition). Goldman also proved that these
decompositions are unique. In this paper, we will show that some basic results
from complex hyperbolic geometry carry over to the quaternionic case. But, it
will be shown that the geometry of bisectors in quaternionic hyperbolic space
is more rich. First, we prove an analogue of the Mostow decomposition of
bisectors in ${\rm H^{n}_{\mathbb{Q}}}$. Then we show that any bisector in
quaternionic hyperbolic space ${\rm H^{n}_{\mathbb{Q}}}$ is a union of totally
geodesic submanifolds of ${\rm H^{n}_{\mathbb{Q}}}$ isometric to ${\rm
H^{n}_{\mathbb{C}}}$ intersecting in a common point. In some sense, this is an
analogue of the Goldman meridian decomposition. We call such decompositions
the fan decompositions. We will show that such decompositions are not unique.
The existence of fan decompositions implies, in particular, that any bisector
in quaternionic hyperbolic space is starlike with respect to any point in its
real spine.

The paper is organized as follows. In Section 1, we summarize some basic
results about geometry of quaternionic hyperbolic space. In Section 2, we
study geometry of bisectors in quaternionic hyperbolic space.

##  1 Preliminaries

In this section, we recall some basic results related to quarternions and
geometry of projective and hyperbolic spaces.

###  1.1 Quaternions

First, we recall some basic facts about the quaternions we need. The
quaternions $\mathbb{Q}$ are the $\mathbb{R}$-algebra generated by the symbols
$i,j,k$ with the relations

$l^{2}=j^{2}=k^{2}=-1,\ \ \ ij=-ji=k,\ \ \ jk=-kj=1,\ \ \ ki=-ik=j.$

So, $\mathbb{Q}$ is a skew field and a 4-dimensional division algebra over the
reals.

Let $a\in\mathbb{Q}$. We write $a=a_{0}+a_{1}i+a_{2}j+a_{3}k$,
$a_{i}\in\mathbb{R}$, then by definition

$\bar{a}=a_{0}-a_{1}i-a_{2}j-a_{3}k,\ \ \ {\rm Re}\ a=a_{0},\ \ \ {\rm Im}\
a=a_{1}i+a_{2}j+a_{3}k.$

Note that, in contrast with the complex numbers, ${\rm Im}\ a$ is not a real
number (if $a_{i}\neq 0$ for some $i=1,2,3$), and that conjugation obeys the
rule

$\overline{ab}=\bar{b}\bar{a}.$

Also, we define $|a|=\sqrt{a\bar{a}}$. We have that if $a\neq 0$ then

$a^{-1}=\bar{a}/|a|^{2}.$

In what follows, we will identify the reals numbers with ${\mathbb{R}}1$ and,
usually use the subfield $\mathbb{C}\subset\mathbb{Q}$ generated by $1$ and
$i$.

Two quaternions $a$ and $b$ are called similar if there exists $\lambda\neq 0$
such that

$a=\lambda b\lambda^{-1}.$

By replacing $\lambda$ by $\tau=\lambda/|\lambda|$, we may always assume
$\lambda$ to be unitary.

The following proposition was proved in [4].

###### Proposition 1.1

Two quaternions $a$ and $b$ are similar if and only if ${\rm Re}\ a={\rm Re}\
b$ and $|a|=|b|$. Moreover, every similarity class contains a complex number,
unique up to conjugation.

###### Corollary 1.1

Any quaternion $a$ is similar to a unique complex number $b=b_{0}+b_{1}i$ such
that $b_{1}\geq 0.$

Also, this proposition implies that every quaternion is similar to its
conjugate.

Example: $jzj^{-1}=\bar{z}$ for all $z\in\mathbb{C}$.

We say that $a\in\mathbb{Q}$ is imaginary if ${\rm Re}(a)=0$. Let us suppose
that $a$ is imaginary and $|a|=1$. Then $a^{2}=-1$. This implies that the real
span of $1$ and $a$ is a subfield of $\mathbb{Q}$ isomorphic to the field of
complex numbers. We denote this subfield by $\mathbb{C}(a)$. It is easy to
prove that any subfield of $\mathbb{Q}$ containing real numbers and isomorphic
to the field of complex numbers is of the form $\mathbb{C}(a)$ for some $a$
imaginary with $|a|=1$.

The following was proved in [5].

###### Proposition 1.2

Let $a$ be as above. Then the centralizer of $a$ is $\mathbb{C}(a)$.

More generally, for any $\lambda\in\mathbb{Q}$, let $\mathbb{C}(\lambda)$ also
denote the real span of $1$ and $\lambda$.

###### Proposition 1.3

Let $\lambda\in\mathbb{Q}\setminus\mathbb{R}$. Then the centralizer of
$\lambda$ is $\mathbb{C}(\lambda)$.

###  1.2 Hyperbolic spaces

In this section we discuss two models for the hyperbolic spaces, its isometry
group and totally geodesics submanifolds. The references for this section is
[5] and [10].

####  1.2.1 Projective Model

We denote by $\mathbb{F}$ one of the real division algebras $\mathbb{R}$,
$\mathbb{C}$, or $\mathbb{Q}$. Let us write $\mathbb{F}^{n+1}$ for a right
$\mathbb{F}$ \- vector space of dimension $n+1$. The $\mathbb{F}$\- projective
space $\mathbb{P}\mathbb{F}^{n}$ is the manifold of right $\mathbb{F}$\- lines
in $\mathbb{F}^{n+1}$. Let $\pi$ denote a natural projection from
$\mathbb{F}^{n+1}\setminus\\{0\\}$ to the projective space
$\mathbb{P}\mathbb{F}^{n}$.

Let $\mathbb{F}^{n,1}$ denote a $(n+1)$-dimensional $\mathbb{F}$-vector space
equipped with a Hermitian form $\Psi=\langle-,-\rangle$ of signature $(n,1)$.
Then there exists a (right) basis in $\mathbb{F}^{n+1}$ such that the
Hermitian product is given by $\langle v,w\rangle=v^{*}J_{n+1}w,$ where
$v^{*}$ is the Hermitian transpose of $v$ and $J_{n+1}=(a_{ij})$ is the
$(n+1)\times(n+1)$-matrix with $a_{ij}=0$ for all $i\neq j$, $a_{ii}=1$ for
all $i=1,\ldots,n$, and $a_{ii}=-1$ when $i=n+1$.

That is,

$\langle
v,w\rangle=\bar{v}_{1}w_{1}+\ldots+\bar{v}_{n}w_{n}-\bar{v}_{n+1}w_{n+1},$

where $v_{i}$ and $w_{i}$ are coordinates of $v$ and $w$ in this basis. We
call such a basis in $\mathbb{F}^{n,1}$ an orthogonal basis defined by a
Hermitian form $\Psi=\langle-,-\rangle$.

Let $V_{-},V_{0},V_{+}$ be the subsets of $\mathbb{F}^{n,1}\setminus\\{0\\}$
consisting of vectors where $\langle v,v\rangle$ is negative, zero, or
positive respectively. Vectors in $V_{0}$ are called null or isotropic,
vectors in $V_{-}$ are called negative, an vectors in $V_{+}$ are called
positive. Their projections to $\mathbb{P}\mathbb{F}^{n}$ are called
isotropic, negative, and positive points respectively.

The projective model of hyperbolic space ${\rm H^{n}_{\mathbb{F}}}$ is the set
of negative points in $\mathbb{P}\mathbb{F}^{n}$, that is, ${\rm
H^{n}_{\mathbb{F}}}=\pi(V_{-}).$

We will consider ${\rm H^{n}_{\mathbb{F}}}$ equipped with the Bergman metric
[5]:

$d(p,q)=\cosh^{-1}\\{|\Psi(v,w)|[\Psi(v,v)\Psi(w,w)]^{-1/2}\\},$

where $p,q\in{\rm H^{n}_{\mathbb{F}}}$, and $\pi(v)=p,\pi(w)=q$.

The boundary $\partial{\rm H^{n}_{\mathbb{F}}}=\pi(V_{0})$ of ${\rm
H^{n}_{\mathbb{F}}}$ is the sphere formed by all isotropic points.

Let ${\rm U}(n,1;\mathbb{F})$ be the unitary group corresponding to this
Hermitian form $\Phi$. If $g\in{\rm U}(n,1;\mathbb{F})$, then $g(V_{-})=V_{-}$
and $g(v\lambda)=(g(v))\lambda$, for all $\lambda\in\mathbb{F}$. Therefore
${\rm U}(n,1;\mathbb{F})$ acts in $\mathbb{P}\mathbb{F}^{n}$, leaving ${\rm
H^{n}_{\mathbb{F}}}$ invariant.

The group ${\rm U}(n,1;\mathbb{F})$ does not act effectively in ${\rm
H^{n}_{\mathbb{F}}}$. The kernel of this action is the center ${\rm
Z}(n,1;\mathbb{F})$. Thus the projective group ${\rm PU}(n,1;\mathbb{F})={\rm
U}(n,1;\mathbb{F})/{\rm Z}(n,1;\mathbb{F})$ acts effectively. In spite of this
fact we shall find it convenient to deal with ${\rm U}(n,1;\mathbb{F})$ rather
than ${\rm PU}(n,1;\mathbb{F})$. The center $\mathbb{Z}(n,1,\mathbb{F})$ in
${\rm U}(n,1;\mathbb{F})$ is $\\{\pm{\rm E}\\}$ if $\mathbb{F}=\mathbb{R}$ or
$\mathbb{Q}$, and is the circle group $\\{\lambda{\rm E}:|\lambda|=1\\}$ if
$\mathbb{F}=\mathbb{C}$. Here ${\rm E}$ is the identity transformation of
$\mathbb{F}^{n,1}$.

It is well-known, see for instance [5], that ${\rm PU}(n,1;\mathbb{F})$ acts
transitively in ${\rm H^{n}_{\mathbb{F}}}$ and doubly transitively on
$\partial{\rm H^{n}_{\mathbb{F}}}$.

We remark that

  * •

if $\mathbb{F}=\mathbb{R}$ then ${\rm H^{n}_{\mathbb{F}}}$ is a real
hyperbolic space ${\rm H^{n}_{\mathbb{R}}}$,

  * •

if $\mathbb{F}=\mathbb{C}$ then ${\rm H^{n}_{\mathbb{F}}}$ is a complex
hyperbolic space ${\rm H^{n}_{\mathbb{C}}}$,

  * •

if $\mathbb{F}=\mathbb{Q}$ then ${\rm H^{n}_{\mathbb{F}}}$ is a quaternionic
hyperbolic space ${\rm H^{n}_{\mathbb{Q}}}$.

It is easy to show [5] that ${\rm H^{1}_{\mathbb{Q}}}$ is isometric to ${\rm
H^{4}_{\mathbb{R}}}$.

####  1.2.2 The ball model

In this section, we consider the space $\mathbb{F}^{n,1}$ equipped by an
orthogonal basis

$e=\\{e_{1},\ldots,e_{n},e_{n+1}\\}.$

For any $v\in\mathbb{F}^{n,1}$, we write $v=(z_{1},\ldots,z_{n},z_{n+1})$,
where $z_{i}$, $i=1,\ldots,n+1$ are coordinates of $v$ in this basis.

If $v=(z_{1},\ldots,z_{n},z_{n+1})\in V_{-}$, the condition $\langle
v,v\rangle<0$ implies that $z_{n+1}\neq 0.$ Therefore, we may define a set of
coordinates $w=(w_{1},\ldots,w_{n})$ in ${\rm H^{n}_{\mathbb{F}}}$ by
$w_{i}(\pi(z))=z_{i}z_{n+1}^{-1}.$ In this way ${\rm H^{n}_{\mathbb{F}}}$
becomes identified with the ball

$B=B(F)=\\{w=(w_{1},\ldots,w_{n})\in\mathbb{F}^{n}:\Sigma_{i=1}^{n}|w_{i}|^{2}<1\\}.$

With this identification the map $\pi:V_{-}\rightarrow{\rm
H^{n}_{\mathbb{F}}}$ has the coordinate representation $\pi(z)=w$, where
$w_{i}=z_{i}z_{n+1}^{-1}$.

####  1.2.3 Totally geodesic submanifolds

We will need the following result, see [5], which describes all totally
geodesic submanifolds of ${\rm H^{n}_{\mathbb{F}}}$.

Let $F$ be a subfield of $\mathbb{F}$. An $F$-unitary subspace of
$\mathbb{F}^{n,1}$ is an $F$-subspace of $\mathbb{F}^{n+1}$ in which the
Hermitian form $\Phi$ is $F$-valued. An $F$-hyperbolic subspace of
$\mathbb{F}^{n,1}$ is an $F$-unitary subspace in which the Hermitian form
$\Phi$ is non-degenerate and indefinite.

###### Proposition 1.4

Let $M$ be a totally geodesic submanifold of ${\rm H^{n}_{\mathbb{F}}}$. Then
either

(a) $M$ is the intersection of the projectivization of an $F$-hyperbolic
subspace of $\mathbb{F}^{n,1}$ with ${\rm H^{n}_{\mathbb{F}}}$ for some
subfield $F$ of $\mathbb{F}$, or

(b) $\mathbb{F}=\mathbb{Q}$, and $M$ is a 3-dimensional totally geodesic
submanifold of a totally geodesic quaternionic line ${\rm H^{1}_{\mathbb{Q}}}$
in ${\rm H^{n}_{\mathbb{Q}}}$.

From the last proposition follows that

  * •

in the real hyperbolic space ${\rm H^{n}_{\mathbb{R}}}$ any totally geodesic
submanifold is isometric to ${\rm H^{k}_{\mathbb{R}}},\ k=1,\ldots,n,$

  * •

in the complex hyperbolic space ${\rm H^{n}_{\mathbb{C}}}$ any totally
geodesic submanifod is isometric to ${\rm H^{k}_{\mathbb{C}}},\ k=1,\ldots,n,$
or to ${\rm H^{k}_{\mathbb{R}}},\ k=1,\ldots,n,$

  * •

in the quaternionic hyperbolic space ${\rm H^{n}_{\mathbb{Q}}}$ any totally
geodesic submanifold is isometric to ${\rm H^{k}_{\mathbb{Q}}},\
k=1,\ldots,n,$ or to ${\rm H^{k}_{\mathbb{C}}},\ k=1,\ldots,n,$ or to ${\rm
H^{k}_{\mathbb{R}}},\ k=1,\ldots,n,$ or to a 3-dimensional totally geodesic
submanifold of a totally geodesic quaternionic line ${\rm
H^{1}_{\mathbb{Q}}}$.

In what follows we will use the following terminology:

  * •

A totally geodesic submanifold of ${\rm H^{n}_{\mathbb{Q}}}$ isometric to
${\rm H^{1}_{\mathbb{Q}}}$ is called a quaternionic geodesic.

  * •

A totally geodesic submanifold of ${\rm H^{n}_{\mathbb{Q}}}$ isometric to
${\rm H^{1}_{\mathbb{C}}}$ is called a complex geodesic.

  * •

A totally geodesic submanifold of ${\rm H^{n}_{\mathbb{Q}}}$ isometric to
${\rm H^{2}_{\mathbb{R}}}$ is called a real plane.

A basic property of ${\rm H^{n}_{\mathbb{Q}}}$ is that two distinct points in
${\rm H^{n}_{\mathbb{Q}}}\cup\partial{\rm H^{n}_{\mathbb{F}}}$ span a unique
quaternionic geodesic. We also remark that any $2$-dimensional totally
geodesic submanifold of a totally geodesic quaternionic line ${\rm
H^{1}_{\mathbb{Q}}}$ is isometric to ${\rm H^{1}_{\mathbb{C}}}$.

###### Proposition 1.5

Let $V$ be a subspace of $\mathbb{F}^{n,1}$. Then each linear isometry of $V$
into $\mathbb{F}^{n,1}$ can be extended to an element of ${\rm
U}(n,1;\mathbb{F})$.

This is a particular case of the Witt theorem, see [10].

###### Corollary 1.2

Let $S\subset{\rm H^{n}_{\mathbb{F}}}$ be a totally geodesic submanifold. Then
each linear isometry of $S$ into ${\rm H^{n}_{\mathbb{F}}}$ can be extended to
an element of the isometry group of ${\rm H^{n}_{\mathbb{F}}}$.

An interesting class of totally geodesic submanifolds of the quaternionic
hyperbolic space ${\rm H^{n}_{\mathbb{Q}}}$ are submanifods which we call
totally geodesic submanifolds of complex type, or simply, submanifolds of
complex type. Their construction is the following. Let
$\mathbb{C}^{n+1}(a)\subset\ \mathbb{Q}^{n+1}$ be the subset of vectors in
$\mathbb{Q}^{n+1}$ with coordinates in $\mathbb{C}(a)$, where $a$ is a
imaginary quaternion, $|a|=1$. Then $\mathbb{C}^{n+1}(a)$ is a vector space
over the field $\mathbb{C}(a)$. The projectivization of $\mathbb{C}^{n+1}(a)$,
denoted by ${\rm M}^{n}(\mathbb{C}(a))$, is a projective submanifold of
$\mathbb{P}\mathbb{Q}^{n}$ of real dimension $2n$. We call this submanifold
${\rm M}^{n}(\mathbb{C}(a))$ a projective submanifold of complex type of
maximal dimension. It is clear that the space $\mathbb{C}^{n+1}(a)$ is
indefinite. The intersection ${\rm M}^{n}(\mathbb{C}(a))$ with ${\rm
H^{n}_{\mathbb{Q}}}$ is a totally geodesic submanifold of ${\rm
H^{n}_{\mathbb{Q}}}$, called a totally geodesic submanifold of complex type of
maximal dimension. It was proven in [5] that all these submanifolds are
isometric, and, moreover, they are globally equivalent with respect to the
isometry group of ${\rm H^{n}_{\mathbb{Q}}}$, that is, for any two such
submanifolds ${\rm M}$ and ${\rm N}$ there exists an element $g\in{\rm
PU}(n,1;\mathbb{Q})$ such that ${\rm M=g(N)}$. In particular, all of them are
globally equivalent with respect to ${\rm PU}(n,1;\mathbb{Q})$ to the
canonical totally geodesic complex submanifold ${\rm H^{n}_{\mathbb{C}}}$
defined by $\mathbb{C}^{n+1}\subset\mathbb{Q}^{n+1}$. This corresponds to the
canonical subfield of complex numbers
$\mathbb{C}=\mathbb{C}(i)\subset\mathbb{Q}$ in the above.

If $V^{k+1}\subseteq\mathbb{C}^{n+1}(a)$ is a subspace of complex dimension
$k+1$, then its projectivization $W$ is called a projective submanifold of
complex type of complex dimension $k$. When
$V^{k+1}\subseteq\mathbb{C}^{n+1}$, then its projectivization $W$ is called a
canonical projective submanifold of complex type of complex dimension $k$ In
this case, we will denote $W$ as $\mathbb{P}\mathbb{C}^{k}$.

If $V^{k+1}\subseteq\mathbb{C}^{n+1}(a)$ is indefinite, then the intersection
of its projectivization with ${\rm H^{n}_{\mathbb{Q}}}$ is a totally geodesic
submanifold of ${\rm H^{n}_{\mathbb{Q}}}$. We call this submanifold of ${\rm
H^{n}_{\mathbb{Q}}}$ a totally geodesic submanifold of complex type of complex
dimension $k$. When $V^{k+1}\subseteq\mathbb{C}^{n+1}$, we call this totally
geodesic submanifold a canonical totally geodesic submanifold of complex type
of complex dimension $k$, or a canonical complex hyperbolic submanifold of
dimension $k$ of ${\rm H^{n}_{\mathbb{Q}}}$. In this case, we will denote this
submanifold as ${\rm H^{k}_{\mathbb{C}}}$.

####  1.2.4 Stabilizators of totally geodesic submanifolds

Let $M$ be a totally geodesic submanifold in ${\rm H^{n}_{\mathbb{F}}}$. Let
$I(M)$ denote the subgroup of ${\rm PU}(n,1;\mathbb{F})$ which leaves $M$
invariant.

If $F$ is a subfield of $\mathbb{F}$, we let
${\mathcal{N}}^{+}(F,\mathbb{F})=\\{\lambda\in\mathbb{F}^{+};\lambda
F\lambda^{-1}=F\\}$, where $\mathbb{F}^{+}$ denotes the subgroup in
$\mathbb{F}$ of elements with norm one.

The following propositions can be found in [5], pp.74.

###### Proposition 1.6

Let $M$ be a totally geodesic submanifold, such that $M$ is the intersection
of the projectivization of an $F$-hyperbolic subspace of $\mathbb{F}^{n,1}$
with ${\rm H^{n}_{\mathbb{F}}}$ for some subfield $F$ of $\mathbb{F}$. Then
the elements $g\in I(M)$ are of the form:

$g=\left(\begin{array}[]{rr}A\lambda&0\\\ 0&B\end{array}\right),$

where $A\in U(m,1;F)$, $\lambda\in{\mathcal{N}}^{+}(F,\mathbb{F})$ and $B\in
U(n-m;\mathbb{F})$.

###### Proposition 1.7

Let $M$ be a 3-dimensional totally geodesic submanifold of a totally geodesic
quaternionic line ${\rm H^{1}_{\mathbb{Q}}}$ in ${\rm H^{n}_{\mathbb{Q}}}$.
Then the elements $g\in I(M)$ are of the form:

$g=\left(\begin{array}[]{rr}A&0\\\ 0&B\end{array}\right),\text{ where
}A=\left(\begin{array}[]{rr}a&b\\\ -\varepsilon b&\varepsilon
a\end{array}\right)\in U(1,1;\mathbb{Q}),$

$\varepsilon=\pm 1$ and $B\in U(n-1;\mathbb{Q})$.

####  1.2.5 A little more about the isometry group of the quaternionic
hyperbolic space

Let us consider the complex hyperbolic space ${\rm H^{n}_{\mathbb{C}}}$. It
has a natural complex structure related to its isometry group, and its
isometry group is generated by the holomorphic isometry group, which is the
projective group ${\rm PU}(n,1;\mathbb{C})$ and the anti-holomorphic isometry
$\sigma$ induced by complex conjugation in $\mathbb{C}^{n+1}$. This anti-
holomorphic isometry corresponds to the unique non-trivial automorphism of the
field of complex numbers.

Quaternionic hyperbolic space ${\rm H^{n}_{\mathbb{Q}}}$ has no complex
structure related to its isometry group. Its isometry group sometimes is
denoted by ${\rm PSp}(n,1)$ because in the orthogonal basis of
$\mathbb{Q}^{n,1}$ the group ${\rm U}(n,1;\mathbb{Q})$ can be considered as a
group of symplectic matrices preserving the Hermitian form
$\Psi=\langle-,-\rangle$ of signature $(n,1)$, that is, ${\rm Sp}(n,1)$

We recall that if $f:\mathbb{Q}\rightarrow\mathbb{Q}$ is an automorphism of
$\mathbb{Q}$, then $f$ is an inner automorphism of $\mathbb{Q}$, that is,
$f(q)=aqa^{-1}$ for some $a\in\mathbb{Q}$.

It follows from the fundamental theorem of projective geometry (see Theorem
2.26 in [2]) that each projective map
$L:\mathbb{P}\mathbb{Q}^{n}\rightarrow\mathbb{P}\mathbb{Q}^{n}$ is induced by
a semilinear (linear) map
$\tilde{L}:\mathbb{Q}^{n+1}\rightarrow\mathbb{Q}^{n+1}$.

It is easy to see that if a projective map
$L:\mathbb{P}\mathbb{Q}^{n}\rightarrow\mathbb{P}\mathbb{Q}^{n}$ is induced by
a semilinear map $\tilde{L}:\mathbb{Q}^{n+1}\rightarrow\mathbb{Q}^{n+1}$,
$\tilde{L}(v)=ava^{-1}$, $v\in\mathbb{Q}^{n+1}$, $a\in\mathbb{Q}$, then it is
induced also by a linear map $v\mapsto av$. Therefore, the projective group of
$\mathbb{P}\mathbb{Q}^{n}$ is the projectivization of the linear group of
$\mathbb{Q}^{n+1}$.

It follows from the above that if $L:{\rm H^{n}_{\mathbb{Q}}}\rightarrow{\rm
H^{n}_{\mathbb{Q}}}$ is an isometry, then $L$ is induced by a linear isometry
$\tilde{L}$ in ${\rm Sp}(n,1)$, which is the projectivization of the linear
group ${\rm U}(n,1;\mathbb{Q})={\rm Sp}(n,1)$.

Next we consider a curious map, which is an isometry of the quaternionic
hyperbolic space, that has no analogue in geometries over commutative fields.
Let $\tilde{L_{a}}:v\mapsto av$, $v\in\mathbb{Q}^{n+1}$, $a\in\mathbb{Q}$, $a$
is not real. The projectivization of this linear map defines a non-trivial map
$L_{a}:\mathbb{P}\mathbb{Q}^{n}\rightarrow\mathbb{P}\mathbb{Q}^{n}$. We remark
that in projective spaces over commutative fields, like $\mathbb{R}$ and
$\mathbb{C}$, this map $L_{a}$ is identity. It easy to see that
$\tilde{L_{a}}\in{\rm U}(n,1;\mathbb{Q})$ if and only if $|a|=1$.

###### Proposition 1.8

The fixed point set of $L_{a}$ is a totally geodesic submanifold of complex
type of maximal dimension in ${\rm H^{n}_{\mathbb{Q}}}.$ This submanifold is
globally equivalent to the canonical complex hyperbolic submanifold ${\rm
H^{n}_{\mathbb{C}}}$ of ${\rm H^{n}_{\mathbb{Q}}}$.

Proof: The proof follows from Proposition 1.3.

##  2 Bisectors in quaternionic hyperbolic space

In ${\rm H^{n}_{\mathbb{F}}}$, as it follows from Proposition 1.4, totally
geodesic (real) hypersurfaces exist only when $\mathbb{F}=\mathbb{R}$.
Therefore, when $\mathbb{F}\neq\mathbb{R}$, a reasonable substitute are the
bisectors which are closed as possible to being totally geodesic. A
comprehensive study of bisectors in complex hyperbolic space the reader can
find in the Goldman book [7].

Giraud [6] and Mostow [9] described the structure of bisectors in complex
hyperbolic space ${\rm H^{n}_{\mathbb{C}}}$ in terms of a foliation by totally
geodesic complex submanifolds of maximal dimension. Goldman [7] proved that
bisectors enjoy another decomposition into totally real, totally geodesic
submanifolds, which is called the meridian decomposition. It is important that
these decompositions are unique.

In this section, we will describe various decompositions of bisectors in
quaternionic hyperbolic space. We show that geometry of bisectors in
quaternionic hyperbolic space is more rich than in complex hyperbolic
geometry. First, we prove an analogue of the Mostow decomposition of bisectors
in ${\rm H^{n}_{\mathbb{Q}}}$. Finally, we will show that any bisector in
quaternionic hyperbolic space ${\rm H^{n}_{\mathbb{Q}}}$ is the union of
totally geodesic submanifolds of ${\rm H^{n}_{\mathbb{Q}}}$ isometric to ${\rm
H^{n}_{\mathbb{C}}}$ intersecting in a common point. In some sense, this is an
analogue of the Goldman meridian decomposition. We call such decompositions
the fan decompositions. The existence of fan decompositions implies, in
particular, that any bisector in quaternionic hyperbolic space is starlike
with respect to any point in its real spine.

The proof of these results are based on the classification of reflective
submanifolds in quaternionic hyperbolic space [8].

###  2.1 Bisectors in Hyperbolic spaces

Let $p_{1},p_{2}$ be two distinct points in ${\rm H^{n}_{\mathbb{F}}}$. The
bisector equidistant from $p_{1}$ and $p_{2}$, or the bisector of
$\\{p_{1},p_{2}\\}$ is defined as

$B(p_{1},p_{2})=\\{p\in{\rm H^{n}_{\mathbb{F}}}:d(p_{1},p)=d(p_{2},p)\\}.$

An equidistant hypersurface or bisector in ${\rm H^{n}_{\mathbb{F}}}$ is a
subset $B=B(p_{1},p_{2})$ for some pair of points $p_{1},p_{2}$. In this case,
we will say that $B$ is equidistant from $p_{1}$ (or equidistant from
$p_{2}$).

Next we consider the following two cases: (1) $\mathbb{F}=\mathbb{C}$, and (2)
$\mathbb{F}=\mathbb{Q}$.

####  2.1.1 Bisectors in complex hyperbolic space

In this section, we recall some basic results on bisectors in complex
hyperbolic space.

Let $\\{p_{1},p_{2}\\}$ be as above. Then there exists the unique complex
geodesic $\Sigma\subset{\rm H^{n}_{\mathbb{C}}}$ spanned by $p_{1}$ and
$p_{2}$. Following Mostow [9], we call $\Sigma$ the complex spine with respect
to the pair $\\{p_{1},p_{2}\\}$. The spine of $B$ (with respect to
$\\{p_{1},p_{2}\\}$) equals

$\sigma\\{p_{1},p_{2}\\}=B(p_{1},p_{2})\cap\Sigma=\\{p\in\Sigma:d(p_{1},p)=d(p_{2},p)\\}$

that is, the orthogonal bisector of the geodesic segment joining $p_{1}$ and
$p_{2}$ in $\Sigma$.

The following result is due to Giraud [6] and Mostow [9], see also [7].

###### Theorem 2.1

Let $B,\Sigma$ and $\sigma$ be as above. Let $\Pi_{\Sigma}:{\rm
H^{n}_{\mathbb{C}}}\rightarrow\Sigma$ be orthogonal projection onto $\Sigma$.
Then

$B=\Pi_{\Sigma}^{-1}(\sigma)=\bigcup_{s\in\sigma}\Pi_{\Sigma}^{-1}(s)$

.

The complex hyperplanes $\Pi_{\Sigma}^{-1}(s)$, for $s\in\sigma$, are called
the slices of $B$ (with respect to $\\{p_{1},p_{2}\\}$), and the decomposition
of a bisector into slices is called the slice decomposition, or the Mostow
decomposition.

Since orthogonal projection $\Pi_{\Sigma}:{\rm
H^{n}_{\mathbb{C}}}\rightarrow\Sigma$ is a real analytic fibration, we obtain
that a bisector is a real analytic hypersurface in ${\rm H^{n}_{\mathbb{C}}}$
diffeomorphic to $\mathbb{R}^{2n-1}$.

It was shown by Goldman [7] that the slices, the spine and complex spine of a
bisector $B$ depend intrinsically on the hypersurface $B$, and not on the
defining pair $\\{p_{1},p_{2}\\}$.

###### Theorem 2.2

Suppose that $\\{p_{1},p_{2}\\}$ and $\\{p^{\prime}_{1},p^{\prime}_{2}\\}$ are
two pairs of distinct points in complex hyperbolic space ${\rm
H^{n}_{\mathbb{C}}}$ such that $B=B(p_{1},p_{2})$ and
$B^{\prime}=B(p^{\prime}_{1},p^{\prime}_{2})$ are equal. Then the slices
(respectively spine, complex spine) of $B$ with respect to $\\{p_{1},p_{2}\\}$
equal the slices (respectively spine, complex spine) of $B^{\prime}$ with
respect to $\\{p^{\prime}_{1},p^{\prime}_{2}\\}$.

Goldman’s proof of this theorem is based on the following: the complex
hyperbolic space is a complex analytic manifold; a bisector is Levi-flat and
its maximal holomorphic submanifolds are its slices.

The endpoints of the spine of a bisector $B$ are called the vertices of $B$.
Since a geodesic $\sigma\in{\rm H^{n}_{\mathbb{C}}}$ is completely determined
by the unordered pair $\partial\sigma\subset\partial{\rm H^{n}_{\mathbb{C}}}$
consisting of its endpoints, bisectors are completely parametrized by
unordered pairs of distinct points in $\partial{\rm H^{n}_{\mathbb{C}}}$.
Therefore, there is a duality between bisectors and geodesics. We associate to
every bisector a geodesic, its spine $\sigma$. Conversely, if
$\sigma\subset{\rm H^{n}_{\mathbb{C}}}$ is a geodesic, then there exists a
unique bisector $B=B_{\sigma}$ with the spine $\sigma$. Given a geodesic
$\sigma$, there exists a unique complex geodesic $\Sigma\supset\sigma$. Let
$R_{\sigma}:\Sigma\rightarrow\Sigma$ be the unique reflection whose fixed-
point set is $\sigma$. Take an arbitrary point
$p_{1}\in\Sigma\smallsetminus\sigma$ and let $p_{2}=R_{\sigma}(p_{1})$. Then

$B=B(p_{1},p_{2})=\Pi_{\Sigma}^{-1}(\sigma)$

is a bisector having $\sigma$ as its spine. Therefore, we have

###### Theorem 2.3

There is a natural bijective correspondence between bisectors in ${\rm
H^{n}_{\mathbb{C}}}$ and geodesics in ${\rm H^{n}_{\mathbb{C}}}$.

Next, we recall so called the meridianal decomposition of bisectors in complex
hyperbolic space invented by Goldman [7].

The following result shows that bisectors decompose into totally real geodesic
submanifolds as well as into complex totally geodesic submanifolds of complex
dimension $n-1$.

###### Theorem 2.4

Let $\sigma\in{\rm H^{n}_{\mathbb{C}}}$ be a real geodesic. For each $k\geq
2$, the bisector $B$ having spine $\sigma$ is the union of all
$\mathbb{R}^{k}$-planes containing $\sigma$.

Following Goldman, we refer to the $\mathbb{R}^{n}$-planes containing the
spine as the meridians of the bisector $B$.

We remark that in contrast to the Mostow decomposition, which is a non-
singular foliation, the meridianal decomposition is a singular foliation
having the geodesic $\sigma$ as its singular set.

Goldman’s proof of Theorem 2.4 is based on a sufficiently complicate
computation. As a corollary of the existence of the meridianal decomposition,
we have the following.

###### Theorem 2.5

Bisectors in complex hyperbolic space are starlike with respect to any point
in its real spine.

###  2.2 Bisectors in quaternionic hyperbolic space

In this section, we will study bisectors in $n$\- dimensional quaternionic
hyperbolic space ${\rm H^{n}_{\mathbb{Q}}}$.

As we have seen in the previous section, the orthogonal projections play an
important role in the study of bisectors. The first problem we consider when
the orthogonal projections onto totally geodesic submanifolds have totally
geodesic fibers.

####  2.2.1 Orthogonal projections and reflective submanifolds

It is elementary that in real hyperbolic space ${\rm H^{n}_{\mathbb{R}}}$, the
orthogonal projections onto totally geodesic submanifolds have totally
geodesic fibers. It is not true, in general, in hyperbolic spaces with non-
constant sectional curvature.

For instance, if $M$ is a totally geodesic submanifold of real dimension one,
that is, a real geodesic, in complex hyperbolic space ${\rm
H^{2}_{\mathbb{C}}}$, then the fibers of the orthogonal projection onto $M$
are not totally geodesic, because in this case the fibers have real dimension
$3$, but we know that any totally geodesic submanifold in ${\rm
H^{2}_{\mathbb{C}}}$ has real dimension $1$ or $2$.

Another example: let $M$ be a totally real totally geodesic plane in ${\rm
H^{3}_{\mathbb{C}}}$ , that is, $M$ is isometric to ${\rm
H^{2}_{\mathbb{R}}}$. Then, using again the classification of totally geodesic
submanifolds in ${\rm H^{3}_{\mathbb{C}}}$, it is easy to prove that the
fibers of the orthogonal projection onto $M$ are not totally geodesic.

One more example: let $M$ be a totally geodesic submanifold in quaternionic
hyperbolic space ${\rm H^{2}_{\mathbb{Q}}}$ of complex type of real dimension
$2$, that is, $M$ is isometric to ${\rm H^{1}_{\mathbb{C}}}$. Then it can be
shown that the fibers of the orthogonal projection onto $M$ are not totally
geodesic.

We will show that this problem is related to so-called reflective submanifolds
of hyperbolic spaces.

First, we recall the definition of reflective submanifolds.

Let $N$ be a Riemannian manifold and $M$ a submanifold of $N$. Then $M$ is
called reflective if the geodesic reflection of $N$ in $M$ is a globally well-
defined isometry of $M$.

Since any reflective submanifold is a connected component of the fixed point
set of an isometry, it is totally geodesic.

Also, we recall the definition of symmetric submanifolds.

A submanifold $M$ of a Riemannian manifold $N$ is called symmetric if for each
point $p$ in $M$ there exists an isometry $I_{p}$ of $N$ such that

$I_{p}(p)=p,\ \ \ I_{p}(M)=M,\ \ \ (I_{p})_{*}X=-X,\ \ \ (I_{p})_{*}Y=Y$

for all $X\in t,Y\in\nu_{p}M.$

Here we denote by $T_{p}M$ the tangent space of $M$ at $p$, by $\nu_{p}M$ the
normal space of $M$ at $p$, and by $(I_{p})_{*}$ the differential of $I_{p}$.

For symmetric spaces there is the following useful criterion.

###### Proposition 2.1

A totally geodesic submanifold of a simply connected Riemannian symmetric
space is symmetric if and only if it is reflective.

As a reflective submanifold is symmetric, at each point there exists a totally
geodesic submanifold normal to it. In symmetric spaces this normal submanifold
is also reflective. Since all hyperbolic spaces are symmetric this result
holds for any hyperbolic space.

Let $M$ be a reflective submanifold in a symmetric simply connected Riemannian
space $N$, and $p\in M$. Let $M_{p}^{\bot}$ denote its orthogonal complement
at $p$, that is, the connected, complete, totally geodesic submanifold of $N$
with $TM_{p}^{\bot}=\nu_{p}M$.

The following theorems describe all reflective submanifolds in complex and
quaternionic hyperbolic spaces, see [8].

###### Theorem 2.6

Let $M$ be a reflective submanifold in complex hyperbolic space ${\rm
H^{n}_{\mathbb{C}}}$. Then $M$ is either isometric to ${\rm
H^{k}_{\mathbb{C}}}$, $k=1,\ldots,n-1$, or to ${\rm H^{n}_{\mathbb{R}}}$.

###### Theorem 2.7

Let $M$ be a reflective submanifold in quaternionic hyperbolic space ${\rm
H^{n}_{\mathbb{Q}}}$. Then $M$ is either isometric to ${\rm
H^{k}_{\mathbb{Q}}}$, $k=1,\ldots,n-1$, or to ${\rm H^{n}_{\mathbb{C}}}$.

###### Corollary 2.1

Let $M$ be a reflective submanifold in complex hyperbolic space ${\rm
H^{n}_{\mathbb{C}}}$, and $p\in M$. Then $M_{p}^{\bot}$ is reflective, and it
is isometric to ${\rm H^{n-k}_{\mathbb{C}}}$ if $M$ is isometric to ${\rm
H^{k}_{\mathbb{C}}},$ and $M_{p}^{\bot}$ is isometric to ${\rm
H^{n}_{\mathbb{R}}}$ if $M$ is isometric to ${\rm H^{n}_{\mathbb{R}}}$.

###### Corollary 2.2

Let $M$ be a reflective submanifold in quaternionic hyperbolic space ${\rm
H^{n}_{\mathbb{Q}}}$. Then $M_{p}^{\bot}$ is reflective, and it is isometric
to ${\rm H^{n-k}_{\mathbb{Q}}}$ if $M$ is isometric to ${\rm
H^{k}_{\mathbb{Q}}}$, and $M_{p}^{\bot}$ is isometric to ${\rm
H^{n}_{\mathbb{C}}}$ if $M$ is isometric to ${\rm H^{n}_{\mathbb{C}}}$.

All this implies the following results.

###### Theorem 2.8

Let $M$ be a totally geodesic submanifold in complex hyperbolic space ${\rm
H^{n}_{\mathbb{C}}}$. Then the fibers of the orthogonal projection onto $M$
are totally geodesic if and only if $M$ is either isometric to ${\rm
H^{k}_{\mathbb{C}}}$, $k=1,\ldots,n-1$, or to ${\rm H^{n}_{\mathbb{R}}}$.

###### Theorem 2.9

Let $M$ be a totally geodesic submanifold in quaternionic hyperbolic space
${\rm H^{n}_{\mathbb{Q}}}$. Then the fibers of the orthogonal projection onto
$M$ are totally geodesic if and only if $M$ is either isometric to ${\rm
H^{k}_{\mathbb{Q}}}$, $k=1,\ldots,n-1$, or to ${\rm H^{n}_{\mathbb{C}}}$.

####  2.2.2 Mostow decomposition of bisectors in quaternionic hyperbolic space

In this section, we describe the Mostow decomposition of bisectors in
quaternionic hyperbolic space ${\rm H^{n}_{\mathbb{Q}}}$.

Let $p_{1},p_{2}$ be two distinct points in ${\rm H^{n}_{\mathbb{Q}}}$, and
let $B(p_{1},p_{2})$ be the bisector of $\\{p_{1},p_{2}\\}$, that is,

$B(p_{1},p_{2})=\\{p\in{\rm H^{n}_{\mathbb{Q}}}:d(p_{1},p)=d(p_{2},p)\\}.$

Then it follows from projective geometry that there exists the unique
quaternionic geodesic $\Sigma\subset{\rm H^{n}_{\mathbb{Q}}}$ spanned by
$p_{1}$ and $p_{2}$. We call $\Sigma$ the quaternionic spine with respect to
the pair $\\{p_{1},p_{2}\\}$. The real spine of $B$ (with respect to
$\\{p_{1},p_{2}\\}$) equals

$\sigma\\{p_{1},p_{2}\\}=B(p_{1},p_{2})\cap\Sigma=\\{p\in\Sigma:d(p_{1},p)=d(p_{2},p)\\}$

that is, the orthogonal bisector of the geodesic segment joining $p_{1}$ and
$p_{2}$ in $\Sigma$. Since $\Sigma$ is isometric to ${\rm
H^{4}_{\mathbb{R}}}$, it follows that $\sigma$ is isometric to ${\rm
H^{3}_{\mathbb{R}}}$.

Let $\Pi_{\Sigma}:{\rm H^{n}_{\mathbb{Q}}}\rightarrow\Sigma$ be orthogonal
projection onto $\Sigma$. It follows from Theorem 2.9 that the fibers of $\Pi$
are totally geodesic submanifolds of ${\rm H^{n}_{\mathbb{Q}}}$ isometric to
${\rm H^{n-1}_{\mathbb{Q}}}$.

First, we prove the following proposition.

###### Proposition 2.2

For any $p\in{\rm H^{n}_{\mathbb{Q}}}\setminus\Sigma$, and $q\in\Sigma$, the
geodesics from $\Pi_{\Sigma}(p)$ to $p$ and $q$ are orthogonal and span a
totally real totally geodesic $2$-plane.

Proof: To prove this, we will use the ball model for ${\rm
H^{n}_{\mathbb{Q}}}$, see Section 1.2.2. In what follows, we identify ${\rm
H^{n}_{\mathbb{Q}}}$ with the unit ball $\mathbb{D}$ in $\mathbb{Q}^{n}$,

$\mathbb{D}=\\{(z_{1},z_{2},\ldots,z_{n})\in\mathbb{Q}^{n}:|z_{1}|^{2}+|z_{2}|^{2}+\ldots+|z_{n}|^{2}<1\\}.$

We write $(z_{1},z_{2},\ldots,z_{n})=(z_{1},z^{\prime})$, where
$z^{\prime}=(z_{2},\ldots,z_{n})\in\mathbb{Q}^{n-1}.$ If
$z_{2}=z_{3}=\ldots=z_{n}=0$, we denote $z^{\prime}$ by $0^{\prime}$.

Since the isometry group of ${\rm H^{n}_{\mathbb{Q}}}$ acts transitively on
the set of quaternionic geodesics in ${\rm H^{n}_{\mathbb{Q}}}$, we can assume
without loss of generality that
$\Sigma=\\{(z,0^{\prime})\in\mathbb{D}:z\in\mathbb{Q}\\}$. Therefore, in this
case, if $p=(z,z^{\prime})\in\mathbb{D}$, we have that
$\Pi_{\Sigma}(p)=(z,0^{\prime})$.

Let $p=(z,z^{\prime})\in\mathbb{D}\setminus\Sigma$,
$p^{\prime}=\Pi_{\Sigma}(p)=(z,0^{\prime})$, and
$q=(w,0^{\prime})\in\Sigma,w\in\mathbb{Q}$. Let $P=(z,z^{\prime},1)^{t}$,
$P^{\prime}=(z,0^{\prime},1)^{t}$, $Q=(w,0^{\prime},1)^{t}$ be vectors in
$\mathbb{Q}^{n,1}$ representing the points $p,p^{\prime},q,$ respectively.

Now we compute the Hermitian triple product $\langle P,P^{\prime},Q\rangle$.
We have

$\langle P,P^{\prime},Q\rangle=\langle P,P^{\prime}\rangle\langle
P^{\prime},Q\rangle\langle Q,P\rangle=\\\
(\bar{z}z-1)(\bar{z}w-1)(\bar{w}z-1)=\\\ (|z|^{2}-1)|\bar{z}w-1|^{2}.$

So, the Hermitian triple product $\langle P,P^{\prime},Q\rangle$ is real.
Therefore, the points $p,p^{\prime},q,$ lie in a totally real geodesic plane
in ${\rm H^{n}_{\mathbb{Q}}}$.

Also we need so-called the Pythagorean theorem in hyperbolic plane ${\rm
H^{2}_{\mathbb{R}}}$, see [3].

###### Theorem 2.10

Let $a,b,c$ be vertices of a right triangle with right angle at $a$. Then

$\cosh d(b,c)=\cosh d(a,b)\cosh d(a,c).$

Now we are ready to prove the Mostow decomposition theorem in quaternionic
hyperbolic geometry.

###### Theorem 2.11

Let $B,\Sigma$ and $\sigma$ be as above. Let $\Pi_{\Sigma}:{\rm
H^{n}_{\mathbb{Q}}}\rightarrow\Sigma$ be orthogonal projection onto $\Sigma$.
Then

$B=\Pi_{\Sigma}^{-1}(\sigma)=\bigcup_{s\in\sigma}\Pi_{\Sigma}^{-1}(s)$

.

Proof: Let $p\in\Pi_{\Sigma}^{-1}(\sigma)$. First, we have that
$d(\Pi_{\Sigma}(p),p_{1})=d(\Pi_{\Sigma},p_{2})$. By applying Proposition 2.2,
we have that the points $p,\ \Pi_{\Sigma}(p),\ p_{1}$ lie in a totally real
geodesic plane in ${\rm H^{n}_{\mathbb{Q}}}$. The same is true for the points
$p,\ \Pi_{\Sigma}(p),\ p_{2}.$ Then it follows from the Pythagorean theorem
applied to the right triangles with vertices $p,\ \Pi_{\Sigma}(p),\ p_{1}$ and
$p,\ \Pi_{\Sigma}(p),\ p_{2}$ that $d(p_{1},p)=d(p_{2},p)$, that is $p\in B.$

We call the quaternionic hyperplanes $\Pi_{\Sigma}^{-1}(p)$, for $p\in\sigma$,
the slices of the bisector $B$ (with respect to $\\{p_{1},p_{2}\\}$). It is
clear that any two distinct slices of $B$ are ultraparallel.

Since orthogonal projection $\Pi_{\Sigma}:{\rm
H^{n}_{\mathbb{Q}}}\rightarrow\Sigma$ is a real analytic fibration, we have:

###### Corollary 2.3

A quaternionic bisector is a real analytic hypersurface in ${\rm
H^{n}_{\mathbb{Q}}}$ diffeomorphic to $\mathbb{R}^{4n-1}$.

Now we show that the quaternionic spine $\Sigma$ and the real spine $\sigma$
of $B$ depend intrinsically on $B$, and not on the pair $\\{p_{1},p_{2}\\}$
used to define $B$. As was remarked above, Goldman’s proof of the complex
hyperbolic analogue of this fact is based on the following: the complex
hyperbolic space is a complex analytic manifold; a bisector is Levi-flat and
its maximal holomorphic submanifolds are its slices. Since quaternionic
hyperbolic space has no natural complex structure, this proof does not work in
the quaternioic case. We prove this result using some elementary facts in
quaternionic projective geometry.

###### Theorem 2.12

Let us suppose that $\\{p_{1},p_{2}\\}$ and
$\\{p^{\prime}_{1},p^{\prime}_{2}\\}$ be two pairs of distinct points in ${\rm
H^{n}_{\mathbb{Q}}}$ such that the bisectors $B=B(p_{1},p_{2})$ and
$B^{\prime}=B^{\prime}(p^{\prime}_{1},p^{\prime}_{2})$ are equal. Then the
slices (respectively, quaternioic spine, real spine) of $B$ with respect to
$\\{p_{1},p_{2}\\}$ equal the slices (respectively, quaternioic spine, real
spine) of $B^{\prime}$ with respect to $\\{p^{\prime}_{1},p^{\prime}_{2}\\}$.

Proof: First we show that the slices of $B$ coincide with the slices of
$B^{\prime}$.

We need the following facts from projective geometry: any two different
projective submanifolds in quaternionic projective space
$\mathbb{P}\mathbb{Q}^{n}$, $n>1$, of quaternionic dimension $n-1$ are
transversal along their intersection; every intersection of projective
submanifolds in $\mathbb{P}\mathbb{Q}^{n}$ is a projective submanifold in
$\mathbb{P}\mathbb{Q}^{n}$. Also, we recall that any totally geodesic
submanifold of ${\rm H^{n}_{\mathbb{Q}}}$ isometric to ${\rm
H^{k}_{\mathbb{Q}}}$, $k=1,\ldots,n-1$, is the intersection of a projective
submanifold in $\mathbb{P}\mathbb{Q}^{n}$ of dimension $k$ with ${\rm
H^{n}_{\mathbb{Q}}}$.

Let $s^{\prime}$ be a slice of $B^{\prime}$. Since $B=B^{\prime}$, then
$s^{\prime}$ is either a slice of $B$, or $s^{\prime}$ intersects a slice $s$
of $B$ transversally. Suppose that $s\cap s^{\prime}\neq\emptyset$ and $s\neq
s^{\prime}$. Since $s\subset B$ and $s^{\prime}\subset B$, and $B$ is a smooth
submanifold of ${\rm H^{n}_{\mathbb{Q}}}$ of real dimension $4n-1$, it follows
from transversality of $s$ and $s^{\prime}$ along their intersection that
$\dim_{\mathbb{R}}s+\dim_{\mathbb{R}}s^{\prime}-(4n-1)=\dim_{\mathbb{R}}(s\cap
s^{\prime})$. This implies that $\dim_{\mathbb{R}}(s\cap s^{\prime})=4n-7$.
This is impossible, since $s\cap s^{\prime}$ is the intersection of a
projective submanifold in $\mathbb{P}\mathbb{Q}^{n}$ with ${\rm
H^{n}_{\mathbb{Q}}}$. Therefore, we get that $s=s^{\prime}$. So, the slices of
$B^{\prime}$ are slices of $B$. Since each slice is orthogonal to $\Sigma$,
any pair of distinct slices of bisector are ultraparallel and their unique
common orthogonal quaternionic geodesic is equal to $\Sigma$.

Therefore, $B$ completely determines the quaternionic spine $\Sigma$. Since
$\sigma=B\cap\Sigma$, the bisector $B$ uniquely determines its real spine.

####  2.2.3 Automorphism of bisectors

If $N$ is a totally geodesic submanifold of ${\rm H^{n}_{\mathbb{Q}}}$
isometric to ${\rm H^{3}_{\mathbb{R}}}$, then there exists the unique
quaternionic geodesic $M$ which contains $N$. Then the pair $\\{M,N\\}$
defines a bisector $B$ whose quaternionic spine equals to $M$ and real spine
equals to $N$. Indeed, take a point $p$ in $M\setminus N$. Let $\gamma$ be the
unique real geodesic in $M$ which contains $p$ and which is orthogonal to $N$.
Let $p^{\prime}\in\gamma$ be the point symmetric to $p$. It is clear that $M$
is the quaternionic spine of the bisector $B=B(p,p^{\prime})$, and $N$ is the
real spine of $B$.

Since the group of isometries $PU(n,1,\mathbb{Q})$ of ${\rm
H^{n}_{\mathbb{Q}}}$ acts transitively on such pairs $\\{M,N\\}$, we get that
$PU(n,1,\mathbb{Q})$ acts transitively on bisectors. Since the quaternionic
geodesic $M$ containing $N$ is unique, we have that a bisector in ${\rm
H^{n}_{\mathbb{Q}}}$ is defined uniquely by its real spine. Furthermore, the
stabilizer of a bisector in $PU(n,1,\mathbb{Q})$ equals the stabilizer of its
real spine. This group is described in Proposition 1.7.

####  2.2.4 Orthogonality of totally geodesic submanifolds in ${\rm
H^{n}_{\mathbb{Q}}}$ isometric to ${\rm H^{n}_{\mathbb{C}}}$

Suppose that $M$ and $N$ are totally geodesic submanifolds in ${\rm
H^{n}_{\mathbb{Q}}}$ isometric to ${\rm H^{n}_{\mathbb{C}}}$.

Let $I_{M}$ and $I_{N}$ denote the geodesic reflections in $M$ and $N$
respectively. Since $M$ and $N$ are reflective, $I_{M}$ and $I_{N}$ are
isometries of ${\rm H^{n}_{\mathbb{Q}}}$. The fixed point set of $I_{M}$
equals to $M$, and the fixed point set of $I_{N}$ equals to $N$.

The proof of the following theorem is standard.

###### Theorem 2.13

The following conditions are equivalent:

  1. 1.

$I_{M}$ and $I_{N}$ commute,

  2. 2.

$I_{M}(N)=N$,

  3. 3.

$I_{N}(M)=M$,

  4. 4.

$M$ and $N$ intersect orthogonally in ${\rm H^{n}_{\mathbb{Q}}}$.

We recall that ${\rm H^{n}_{\mathbb{C}}}$ has a totally geodesic submanifold
isometric to ${\rm H^{n}_{\mathbb{R}}}$.

###### Theorem 2.14

Let $M$ be a totally geodesic submanifold of ${\rm H^{n}_{\mathbb{Q}}}$
isometric to ${\rm H^{n}_{\mathbb{C}}}$. Then for any totally geodesic
submanifold $S\subset M$ isometric to ${\rm H^{n}_{\mathbb{R}}}$ there exists
a totally geodesic submanifold $N$ isometric to ${\rm H^{n}_{\mathbb{C}}}$
such that $S=M\cap N$ and $N$ is orthogonal to $M$ along $S$.

Proof: To prove this, we will use the ball model for ${\rm
H^{n}_{\mathbb{Q}}}$, see Section 1.2.2. In what follows, we identify ${\rm
H^{n}_{\mathbb{Q}}}$ with the unit ball $\mathbb{D}$ in $\mathbb{Q}^{n}$,

$\mathbb{D}=\\{(q_{1},q_{2},\ldots,q_{n})\in\mathbb{Q}^{n}:|q_{1}|^{2}+|q_{2}|^{2}+\ldots+|q_{n}|^{2}<1\\}.$

Let us consider the following subsets of $\mathbb{D}$:

$\mathbb{D}_{\mathbb{C}}=\\{(z_{1},z_{2},\ldots,z_{n})\in\mathbb{C}^{n}:|z_{1}|^{2}+|z_{2}|^{2}+\ldots+|z_{n}|^{2}<1\\},$

$\mathbb{D}_{\mathbb{R}}=\\{(x_{1},x_{2},\ldots,x_{n})\in\mathbb{R}^{n}:|x_{1}|^{2}+|x_{2}|^{2}+\ldots+|x_{n}|^{2}<1\\},$

where $\mathbb{C}=\\{z=x+iy:x,y\in\mathbb{R}\\}$ and $\mathbb{R}$ is the
standard subfield of real numbers of $\mathbb{Q}$.

Then $\mathbb{D}_{\mathbb{C}}$ is a totally geodesic submanifold of
$\mathbb{D}$ isometric to ${\rm H^{n}_{\mathbb{C}}}$, and
$\mathbb{D}_{\mathbb{R}}$ is a totally geodesic submanifold of
$\mathbb{D}_{\mathbb{C}}$ isometric to ${\rm H^{n}_{\mathbb{R}}}$.

Since the group of isometries of ${\rm H^{n}_{\mathbb{Q}}}$ acts transitively
on the set of pairs of the form $(M,S)$ in ${\rm H^{n}_{\mathbb{Q}}}$, we can
assume without loss of generality that $M=\mathbb{D}_{\mathbb{C}}$ and
$S=\mathbb{D}_{\mathbb{R}}$.

Let $a\in\mathbb{Q}$ be purely imaginary, $|a|=1$. Then $a^{2}=-1$. Let
$\mathbb{C}(a)$ be the subfield of $\mathbb{Q}$ spanned by $1$ and $a$. Denote
by $\mathbb{D}_{a}$ the following set in $\mathbb{D}$ :

$\mathbb{D}_{a}=\\{(a_{1},a_{2},\ldots,a_{n})\in\mathbb{C}(a)^{n}:|a_{1}|^{2}+|a_{2}|^{2}+\ldots+|a_{n}|^{2}<1\\}.$

We know, see Section 1.2.3, that $\mathbb{D}_{a}$ is a totally geodesic
submanifold of $\mathbb{D}$ isometric to ${\rm H^{n}_{\mathbb{C}}}$. We remark
that $\mathbb{D}_{a}=\mathbb{D}_{\mathbb{C}}$ if $a=i$. It is clear that
$\mathbb{D}_{\mathbb{C}}\cap\mathbb{D}_{a}=\mathbb{D}_{\mathbb{R}}$, provided
that $a\neq i$.

Let $V_{\mathbb{C}}$, $V_{a}$, $V_{\mathbb{R}}$ denote the sets of vectors in
$\mathbb{Q}^{n,1}$ with coordinates in $\mathbb{C}$, $\mathbb{C}(a)$, and
$\mathbb{R}$ respectively. Then we have that $\mathbb{D}_{\mathbb{C}}$,
$\mathbb{D}_{a}$, $\mathbb{D}_{\mathbb{R}}$ are projectivizations of
$V_{\mathbb{C}}$, $V_{a}$, and $V_{\mathbb{R}}$ respectively.

Consider the following semi-linear map
$L_{a}:\mathbb{Q}^{n,1}\longrightarrow\mathbb{Q}^{n,1}$ given by
$L_{a}(v)=ava^{-1}$, $v\in\mathbb{Q}^{n,1}$. Then it follows that $L_{a}(v)=v$
for all $v\in V_{a}$, see Section 1.2.5. In particular, $L_{i}(v)=v$ for all
$v\in V_{i}$, that is, $L_{i}(v)=v$ for all $v\in V_{\mathbb{C}}.$ We know
that projectivization of this semi-linear map $L$ is equal to projectivization
of the linear map $L^{\prime}(v)=av.$ We have that $L^{\prime}\circ
L^{\prime}=-I$, where $I$ is the identity map of $\mathbb{Q}^{n,1}$. This
implies that projectivization of $L^{\prime}_{a}$ is an isometric involution
of $\mathbb{D}$ whose fixed point set equals to $\mathbb{D}_{a}$.

Since $ik=-ki$, we have that $L^{\prime}_{i}\circ
L^{\prime}_{k}=-L^{\prime}_{k}\circ L^{\prime}_{i}$. It follows that
projectivizations of $L^{\prime}_{i}$ and $L^{\prime}_{k}$ are commuting
involutions whose fixed points sets are
$\mathbb{D}_{i}=\mathbb{D}_{\mathbb{C}}$ and $\mathbb{D}_{k}$.

Let again $M=\mathbb{D}_{\mathbb{C}}$ and $S=\mathbb{D}_{\mathbb{R}}$. We
define $N$ to be $\mathbb{D}_{k}$. Then $M\cap N=S.$ Also, let $I_{M}$ be
projectivization of $L^{\prime}_{i}$, and let $I_{N}$ be projectivization of
$L^{\prime}_{k}$. Therefore, we get that $I_{M}$ and $I_{N}$ are geodesic
reflections in $M$ and $N$ respectively such that $I_{M}(N)=N$, $I_{N}(M)=M$,
$I_{M}$ and $I_{N}$ commute.

We remark that $I_{M}$ acts on $N$ as a geodesic reflection in $S\subset N$,
and $I_{N}$ acts on $M$ as a geodesic reflection in $S\subset M.$

####  2.2.5 Fan decomposition of bisectors in quaternionic hyperbolic space

In this section, we show that any bisector $B$ in quaternionic hyperbolic
space ${\rm H^{n}_{\mathbb{Q}}}$ is the union of totally geodesic submanifolds
in ${\rm H^{n}_{\mathbb{Q}}}$ isometric to ${\rm H^{n}_{\mathbb{C}}}$, all
passing through a point $o$ in ${\rm H^{n}_{\mathbb{Q}}}$, that is,

$B=\bigcup_{\alpha}{N_{\alpha}}(o),$

where $N_{\alpha}(o)$ is a totally geodesic submanifolds in ${\rm
H^{n}_{\mathbb{Q}}}$ isometric to ${\rm H^{n}_{\mathbb{C}}}$ passing through a
point $o$ in ${\rm H^{n}_{\mathbb{Q}}}$.

This decomposition in somewhat is similar to the Goldman meridional
decomposition of bisectors in complex hyperbolic space. We call a such
decomposition a fan decomposition of a bisector in ${\rm H^{n}_{\mathbb{Q}}}$.

We call the point $o\in{\rm H^{n}_{\mathbb{Q}}}$ the center of a fan
decomposition of $B$ and the submanifolds $N_{\alpha}(o)$ the complex blades
of a fan decomposition of $B$.

A fan decomposition of a bisector $B$ is defined by its center $o$ and its
complex blades.

First, we define the center and the complex blades of a fan decomposition of a
bisector.

Let $B$ be a bisector in ${\rm H^{n}_{\mathbb{Q}}}$. Let $\Sigma$ and $\sigma$
be its quaternionic and real spine respectively.

The center of a fan decomposition of $B$ is any point $o\in\sigma$.

The complex blades of a fan decomposition of $B$ passing through a point $o$
are constructed as follows.

Let $\gamma$ be the unique real geodesic in $\Sigma$ passing through the point
$o$ orthogonal to $\sigma$. Let $p_{1}$, $p_{2}$ be distinct points in
$\gamma$ symmetric with respect to $o$. Then it is clear that
$B=B(p_{1},p_{2})$. Let $M$ be a totally geodesic submanifold of ${\rm
H^{n}_{\mathbb{Q}}}$ isometric to ${\rm H^{n}_{\mathbb{C}}}$ containing
$\gamma$. Since $M$ is totally geodesic, we have that $B_{M}=M\cap B$ is a
bisector in $M$. We denote by $\Sigma_{M}$ the complex spine of $B_{M}$ and by
$\sigma_{M}$ the real spine of $B_{M}$. Then $\Sigma_{M}$ is a totally
geodesic submanifold of $\Sigma$ isometric to ${\rm H^{1}_{\mathbb{C}}}$
containing $\gamma$, and $\sigma_{M}$ is a real geodesic in $\Sigma_{M}$
passing through the point $o$ orthogonal to $\gamma$. By applying Goldman’s
meridianal decomposition, we have that there exists a totally geodesic
submanifold $S\subset B_{M}$ isometric to ${\rm H^{n}_{\mathbb{R}}}$
containing $\sigma_{M}$, $S$ is a meridian of $B_{M}$. Let us choose any such
$S$ and fix it. It follows from Theorem 2.14 that there exists a totally
geodesic submanifold $N$ isometric to ${\rm H^{n}_{\mathbb{C}}}$ such that
$S=M\cap N$ and $N$ is orthogonal to $M$ along $S$. We define such a
submanifold $N$ to be a complex blade of a fan decomposition of $B$.

It is seen that a complex blade of a fan decomposition of a bisector $B$
centered at $o$ is defined by $o$, by a totally geodesic submanifold $M$ of
${\rm H^{n}_{\mathbb{Q}}}$ isometric to ${\rm H^{n}_{\mathbb{C}}}$ containing
$\gamma$, and by a meridian of the bisector $B_{M}$.

The following proposition was proved in [1]

###### Proposition 2.3

Let $p=(p_{1},p_{2},p_{3})$ be an ordered triple of distinct points,
$p_{i}\in\mathbb{P}\mathbb{Q}^{n}$. Then there exists a projective submanifold
$W\subset\mathbb{P}\mathbb{Q}^{n}$ of complex type of complex dimension $2$
passing through the points $p_{i}$, that is, $p_{i}\in W$, $i=1,2,3.$
Moreover, this submanifold $W$ can be chosen, up to the action of ${\rm
PU}(n,1;\mathbb{Q})$, to be the canonical complex submanifold
$\mathbb{P}\mathbb{C}^{2}\subset\mathbb{P}\mathbb{Q}^{n}$.

###### Theorem 2.15

Let $B$ and $o\in\sigma$ as above. Then $B$ is the union of its complex blades
containing the point $o$.

Proof: Let $\gamma$ be the unique real geodesic in $\Sigma$ passing through
the point $o$ orthogonal to $\sigma$. Let $p_{1}$, $p_{2}$ be distinct points
in $\gamma$ symmetric with respect to $o$. Then it is clear that
$B=B(p_{1},p_{2})$

Let $M$ be a totally geodesic submanifold of ${\rm H^{n}_{\mathbb{Q}}}$
isometric to ${\rm H^{n}_{\mathbb{C}}}$ containing $\gamma$. Then $B_{M}=M\cap
B$ is a bisector in $M$. We denote by $\Sigma_{M}$ the complex spine of
$B_{M}$ and by $\sigma_{M}$ the real spine of $B_{M}$. Then $\Sigma_{M}$ is a
totally geodesic submanifold of $\Sigma$ isometric to ${\rm
H^{1}_{\mathbb{C}}}$ containing $\gamma$, and $\sigma_{M}$ is a real geodesic
in $\Sigma_{M}$ passing through the point $o$ orthogonal to $\gamma$. By
applying Goldman’s meridianal decomposition, we have that there exists a
totally geodesic submanifold $S\subset B_{M}$ isometric to ${\rm
H^{n}_{\mathbb{R}}}$ containing $\sigma_{M}$, $S$ is a meridian of $B_{M}$.
Let us choose any such $S$ and fix it. Let $N$ be a complex blade defined by
$M$ and $S$, that is, $S=M\cap N$ and $N$ is orthogonal to $M$ along $S$.

Let $I_{N}$ denote geodesic reflection in $N$. Then it follows Theorem 2.14
that $M$ is invariant with respect to $I_{N}$, and, moreover,
$I_{N}(p_{1})=p_{2}$, $I_{N}(p_{2})=p_{1}$.

We first show that $N\subset B$. It suffices to show that for any $p\in N$ we
have that $d(p,p_{1})=d(p,p_{2}).$ But this is clear because

$d(p,p_{1})=d(I_{N}(p),(I_{N}(p_{1}))=d(p,p_{2}).$

Conversely, we will show that each $p\in B$ lies in a complex blade containing
the point $o$.

Let $p\in B$. Then it follows from Proposition 2.3 that there exists a totally
geodesic submanifold $K$ of ${\rm H^{n}_{\mathbb{Q}}}$ isometric to ${\rm
H^{n}_{\mathbb{C}}}$ passing through the points $p$, $p_{1}$, $p_{2}$. Then
$B_{K}=K\cap B$ is a bisector in $K$. Since $p\in B$, it follows that $p\in
B_{K}$. By applying Goldman’s meridianal decomposition, we have that there
exists a meridian $S_{K}$ of $B_{K}$ containing $p$. By applying Theorem 2.14,
we get there exists a totally geodesic submanifold $L$ of ${\rm
H^{n}_{\mathbb{Q}}}$ isometric to ${\rm H^{n}_{\mathbb{C}}}$ orthogonal to $K$
along $S_{K}$, that is, $L$ is a complex blade containing $p$. It follows from
the above that $L\subset B$. Since, by construction, $p\in L$, the result
follows.

It is seen that two distinct complex blades intersect in the point $o$.

As a corollary of this theorem we have the following:

###### Theorem 2.16

Bisectors in quaternionic hyperbolic space ${\rm H^{n}_{\mathbb{Q}}}$ are
starlike with respect to any point in its real spine.

Proof: Let $B$ be a bisector in ${\rm H^{n}_{\mathbb{Q}}}$ and $o$ be a point
in $\sigma$. Let $p$ be an arbitrary point in $B$. By applying Theorem 2.15,
we have that there exists a totally geodesic submanifold $M\subset B$
isometric to ${\rm H^{n}_{\mathbb{Q}}}$ containing $o$ and $p$. This implies
that the geodesic passing through $o$ and $p$ lies in $B$.

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