[a]M. Boukidi

# Probing a 96 GeV Higgs Boson in the Di-Photon Channel at the LHC

R. Benbrik  S. Moretti  S. Semlali

###### Abstract

Recently the CMS collaboration reported a $\sim 3\sigma$ local excess in the
di-photon spectrum at 96 GeV. The same mass range concurs with a $\sim
2\sigma$ local excess in the $b\bar{b}$ invariant mass spectrum in four-jet
events collected at LEP. In this contribution we show that at 1$\sigma$ level
the 2HDM type-III can perfectly fit both excesses simultaneously, while
satisfying all experimental and theoretical constraints.

##  1 Introduction

The discovery of a scalar particle at the Large Hadron Collider (LHC) [1, 2],
has been widely hailed as the opening of a new era in particle physics. Such a
Higgs boson is the first fundamental particle to be found in Nature and the
last remaining undiscovered particle required for the experimental
confirmation of the SM. Even though the measurements of the production and
decay rates of the 125 GeV Higgs boson are currently in good agreement with
the expectations within the SM, the Higgs sector chosen by Nature may not
necessarily be the minimal one of such a construct. There might be additional
real or complex singlets, doublets triplets, and so on, or any mixture of
these. An extended Higgs sector may also be incorporated into a proper
theoretical framework, for example, a specific realisation of the 2-Higgs
Doublet Model (2HDM) can be made part of the Minimal Supersymmetric Standard
Model (MSSM).

Light neutral Higgs bosons searches have been performed at LHC in the di-
photon final state $pp\to\phi\to\gamma\gamma$. In this regard CMS has
identified two excesses [3] with 2$\sigma$ local significance at
$m_{\gamma\gamma}=97.6$ GeV at Run-1 (2012) with 19.7 fb-1 of luminosity and
with 3$\sigma$ local significance at $m_{\gamma\gamma}=95.3$ GeV at Run-2
(2016) with 35.9 fb-1 of luminosity. Around the aforementioned mass range, an
anomaly has remained from LEP data, in the $e^{+}e^{-}\to Z(H\to b\bar{b})$
channel, wherein a 2.3 $\sigma$ local excess was observed in the $b\bar{b}$
invariant mass. [4].

In this work, we aim to investigate such reported excesses within a 2HDM Type-
III with a specific Yukawa texture allowing for some amount of Lepton Flavour
Violation (LFV). This is done to comply with LEP, Tevatron, LHC and low energy
measurements sensitive to not only the SM-like Higgs boson $H$ but also to a
potentially lighter $h$ state playing the role of the object behind the
aforementioned excesses at 96 GeV. In fact, a generic 2HDM contains five
physical Higgs bosons: 2 $\mathcal{CP}$-even $h$ and $H$ ($m_{h}<m_{H}$), one
$\mathcal{CP}$-odd $A$ and a pair of charged Higgs $H^{\pm}$. The 2HDM can be
classified into Type-I, Type-II, lepton-specific, and flipped depending on how
the Higgs doublets couple to the fermions. The 2HDM Type-III corresponds to
the case where each of the two Higgs doublets couples to all fermions
simultaneously. Consequently, tree-level Flavour Changing Neutral Currents
(FCNCs) in the sectors of charged quarks and leptons are being induced. In our
approach, rather than postulating a $Z_{2}$ symmetry (exact or softly-broken)
to control the latter, we assume a generic Yukawa texture that we will
constrain by exploiting theoretical conditions of self-consistency as well as
experimental measurements of masses and couplings.

The contribution is organized as follows: In the next section we shall
introduce briefly the basic notation of 2HDM, We then move on to explain the
features of the two experimental excesses (LEP and LHC). Then we will map one
onto the others by presenting numerical results. We will finally conclude.

##  2 General 2HDM

The most general scalar potential of the 2HDM can be written as [5]:

$\displaystyle\mathcal{V}$
$\displaystyle=m_{11}^{2}\Phi_{1}^{\dagger}\Phi_{1}+m_{22}^{2}\Phi_{2}^{\dagger}\Phi_{2}-\left[m_{12}^{2}\Phi_{1}^{\dagger}\Phi_{2}+\rm{H.c.}\right]+\lambda_{1}(\Phi_{1}^{\dagger}\Phi_{1})^{2}+\lambda_{2}(\Phi_{2}^{\dagger}\Phi_{2})^{2}+\lambda_{3}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{2}^{\dagger}\Phi_{2})~{}$
(1)
$\displaystyle+\lambda_{4}(\Phi_{1}^{\dagger}\Phi_{2})(\Phi_{2}^{\dagger}\Phi_{1})+\frac{1}{2}\left[\lambda_{5}(\Phi_{1}^{\dagger}\Phi_{2})^{2}+\rm{H.c.}\right]+\left\\{\left[\lambda_{6}(\Phi_{1}^{\dagger}\Phi_{1})+\lambda_{7}(\Phi_{2}^{\dagger}\Phi_{2})\right](\Phi_{1}^{\dagger}\Phi_{2})+\rm{H.c.}\right\\}$

Adopting $\mathcal{CP}$-conserving option, and very minimal version of Higgs
couplings, the above potential can be parametrized by seven free parameters,
those are: Higgs masses, $m_{h}$, $m_{H}$, $m_{H^{\pm}}$, $m_{A}$, the ratio
of the vacuum expectation values of the two Higgs doublets fields
$\tan\beta=v_{2}/v_{1}$, the mixing angle of the $\mathcal{CP}$-even Higgs
states $\alpha$, and $m^{2}_{12}$. In the Yukawa sector, the general scalar to
fermions couplings are expressed by:

$\displaystyle-{\cal L}_{Y}$ $\displaystyle=$
$\displaystyle\bar{Q}_{L}Y^{u}_{1}U_{R}\tilde{\Phi}_{1}+\bar{Q}_{L}Y^{u}_{2}U_{R}\tilde{\Phi}_{2}+\bar{Q}_{L}Y^{d}_{1}D_{R}\Phi_{1}+\bar{Q}_{L}Y^{d}_{2}D_{R}\Phi_{2}$
(2) $\displaystyle+$
$\displaystyle\bar{L}Y^{\ell}_{1}\ell_{R}\Phi_{1}+\bar{L}Y^{\ell}_{2}\ell_{R}\Phi_{2}+H.c.$

where $Q_{L}=(u_{L},d_{L})$ and $L=(\ell_{L},\nu_{L})$ are the doublets of
$SU(2)_{L}$, and $Y^{f,\ell}_{1,2}$ denote the $3\times 3$ Yukawa matrices. To
get naturally small FCNCs while inducing flavor violating Higgs signals, we
adopt the description presented in [6] by assuming a flavour symmetry that
suggest a specific texture of the Yukawa matrices, where the non-diagonal
Yukawa couplings, $\tilde{Y}_{ij}$, are given in terms of fermions masses and
dimensionless real parameter,
$\tilde{Y}_{ij}\propto\sqrt{m_{i}m_{j}}/v~{}\chi_{ij}$.
After electroweak symmetry breaking (EWSB) the Yukawa Lagrangian can be
expressed in terms of the mass eigenstates of the Higgs bosons, as follows:

$\displaystyle-{\cal L}^{III}_{Y}$
$\displaystyle=\sum_{f=u,d,\ell}\frac{m^{f}_{j}}{v}\times\left((\xi^{f}_{h})_{ij}\bar{f}_{Li}f_{Rj}h+(\xi^{f}_{H})_{ij}\bar{f}_{Li}f_{Rj}H-i(\xi^{f}_{A})_{ij}\bar{f}_{Li}f_{Rj}A\right)$
$\displaystyle+\frac{\sqrt{2}}{v}\sum_{k=1}^{3}\bar{u}_{i}\left[\left(m^{u}_{i}(\xi^{u*}_{A})_{ki}V_{kj}P_{L}+V_{ik}(\xi^{d}_{A})_{kj}m^{d}_{j}P_{R}\right)\right]d_{j}H^{+}$
$\displaystyle+\frac{\sqrt{2}}{v}\bar{\nu}_{i}(\xi^{\ell}_{A})_{ij}m^{\ell}_{j}P_{R}\ell_{j}H^{+}+H.c.\,$
(3)

Here the reduced Yukawa couplings $(\xi^{f,\ell}_{\phi})_{ij}$ are given in
Table 1, in terms of the mixing angles $\alpha$ and $\beta$ , and of the free
parameters111The free parameters $\chi_{ij}^{f,\ell}$ are tested at the
current B physics constraints (more details can be found in Refs [7]).
$\chi_{ij}^{f,\ell}$.

$\phi$ | $(\xi^{u}_{\phi})_{ij}$ | $(\xi^{d}_{\phi})_{ij}$ | $(\xi^{\ell}_{\phi})_{ij}$
---|---|---|---
$h$ |  $\frac{c_{\alpha}}{s_{\beta}}\delta_{ij}-\frac{c_{\beta-\alpha}}{\sqrt{2}s_{\beta}}\sqrt{\frac{m^{u}_{i}}{m^{u}_{j}}}\chi^{u}_{ij}$ |  $-\frac{s_{\alpha}}{c_{\beta}}\delta_{ij}+\frac{c_{\beta-\alpha}}{\sqrt{2}c_{\beta}}\sqrt{\frac{m^{d}_{i}}{m^{d}_{j}}}\chi^{d}_{ij}$ |  $-\frac{s_{\alpha}}{c_{\beta}}\delta_{ij}+\frac{c_{\beta-\alpha}}{\sqrt{2}c_{\beta}}\sqrt{\frac{m^{\ell}_{i}}{m^{\ell}_{j}}}\chi^{\ell}_{ij}$
$H$ | $\frac{s_{\alpha}}{s_{\beta}}\delta_{ij}+\frac{s_{\beta-\alpha}}{\sqrt{2}s_{\beta}}\sqrt{\frac{m^{u}_{i}}{m^{u}_{j}}}\chi^{u}_{ij}$ | $\frac{c_{\alpha}}{c_{\beta}}\delta_{ij}-\frac{s_{\beta-\alpha}}{\sqrt{2}c_{\beta}}\sqrt{\frac{m^{d}_{i}}{m^{d}_{j}}}\chi^{d}_{ij}$ | $\frac{c_{\alpha}}{c_{\beta}}\delta_{ij}-\frac{s_{\beta-\alpha}}{\sqrt{2}c_{\beta}}\sqrt{\frac{m^{\ell}_{i}}{m^{\ell}_{j}}}\chi^{\ell}_{ij}$
$A$ | $\frac{1}{t_{\beta}}\delta_{ij}-\frac{1}{\sqrt{2}s_{\beta}}\sqrt{\frac{m^{u}_{i}}{m^{u}_{j}}}\chi^{u}_{ij}$ | $t_{\beta}\delta_{ij}-\frac{1}{\sqrt{2}c_{\beta}}\sqrt{\frac{m^{d}_{i}}{m^{d}_{j}}}\chi^{d}_{ij}$ | $t_{\beta}\delta_{ij}-\frac{1}{\sqrt{2}c_{\beta}}\sqrt{\frac{m^{\ell}_{i}}{m^{\ell}_{j}}}\chi^{\ell}_{ij}$
Table 1: Yukawa couplings of the $h$, $H$, and $A$ bosons to the quarks and
leptons in the 2HDM Type-III.

##  3 Explaining the excesses with 2HDM Type-III

In this section, we investigate whether the 2HDM Type-III can explain
simultaneously the excesses observed by both LEP and CMS at 96 GeV. The
evaluation of the signal strengths for these excesses was done in the narrow
width approximation where we assume that the cross section ratio can be
expressed via the normalized coupling:

$\displaystyle\mu_{\mathrm{LEP}}^{\mathrm{bb}}$
$\displaystyle=\frac{\sigma_{\rm 2HDM}(e^{+}e^{-}\to Zh)}{\sigma_{\rm
SM}(e^{+}e^{-}\to Zh)}\cdot\frac{{\cal BR}_{\rm 2HDM}(h\to b\bar{b})}{{\cal
BR}_{\rm SM}(h_{\rm SM}\to
b\bar{b})}=\left|c_{hZZ}\right|^{2}\times\frac{{\cal BR}_{\rm 2HDM}(h\to
b\bar{b})}{{\cal BR}_{\rm SM}(h_{\rm SM}\to b\bar{b})}=0.6\pm 0.2.$ (4)
$\displaystyle\mu_{\rm CMS}^{\mathrm{\gamma\gamma}}$
$\displaystyle=\frac{\sigma_{\rm 2HDM}(gg\to h)}{\sigma_{\rm SM}(gg\to h_{\rm
SM})}\cdot\frac{{\cal BR}_{\rm 2HDM}(h\to\gamma\gamma)}{{\cal BR}_{\rm
SM}(h_{\rm SM}\to\gamma\gamma)}=\left|c_{htt}\right|^{2}\times\frac{{\cal
BR}_{\rm 2HDM}(h\to\gamma\gamma)}{{\cal BR}_{\rm SM}(h_{\rm
SM}\to\gamma\gamma)}=0.117\pm 0.057.$ (5)

In order to analyse whether a simultaneous fit to the observed excesses is
possible, we perform a $\chi^{2}$ analysis where the latter can be defined by
the measured central values $\mu^{\mathrm{exp}}$ and the 1$\sigma$
uncertainties $\Delta\mu^{\mathrm{exp}}$ of the signal rates related to the
two excesses as mentioned in Eq. 4 and Eq. 5.

$\displaystyle\chi^{2}_{96}=\left(\frac{\mu_{\mathrm{LEP}}^{\mathrm{bb}}-\mu_{\mathrm{LEP}}^{\mathrm{bb,exp}}}{\Delta\mu_{\rm
CMS}^{\mathrm{\gamma\gamma,exp}}}\right)^{2}+\left(\frac{\mu_{\rm
CMS}^{\mathrm{\gamma\gamma}}-\mu_{\rm
CMS}^{\mathrm{\gamma\gamma,exp}}}{\Delta\mu_{\mathrm{LEP}}^{\mathrm{bb,exp}}}\right)^{2}.$
(6)

##  4 Numerical Results

Before presenting our result, we describe how theoretical self-consistency
requirements and experimental measurements were used to constrain the
parameter space of the 2HDM Type-III scenario that we are aiming for. The
theoretical requirements are perturbativity of the scalar quartic couplings,
vacuum stability and the tree-level perturbative unitarity conditions for
various scattering amplitudes of gauge and Higgs boson states (All these
constraints are tested via the public code 2HDMC-1.8.0 [8]). Moreover, on the
experimental side, we consider constraints from electroweak precision
observables (EWPO) in terms of the oblique parameters $S$, $T$, and $U$,
measurements at the LHC of the properties of the newly discovered Higgs boson
(HiggsSignal-2.6.0[9]), Cross-section limits from LEP, Tevatron and the LHC
(HiggsBouns-5.9.0[10]) and finally constraints from flavor physics [11]. For
more details we refer to Ref. [7]. Then we performed a systematic scan over
the following parameter ranges.

$\displaystyle\hskip 19.91684ptm_{h}\in[80,~{}110]\ \text{GeV},\ \ m_{H}=125\
\text{GeV},\sin(\beta-\alpha)\in[-0.5,~{}-0.1],\ \ m_{A}\in[70,~{}90]\
\text{GeV},$ $\displaystyle m_{H^{\pm}}\in[140,~{}180]\ \text{GeV},\ \
\tan\beta\in[1.1,~{}1.5],~{}~{}m_{12}^{2}=m_{h}^{2}\tan\beta/(1+\tan^{2}\beta).\hskip
32.72049pt$ (7)

In Fig. 1 we present the surviving points over the ($\mu_{\rm
CMS},\mu_{\mathrm{LEP}}$) plane, where the colour code indicates
$\Delta\chi^{2}_{125}$. The dashed and solid lines correspond to the 1$\sigma$
and 2$\sigma$ ellipses, respectively, and the pink star indicates the best fit
point. All the points shown in the figure have $\Delta\chi^{2}_{125}\leq 12$.
It is interesting to note, that there are many points (in red) with
$\Delta\chi_{125}^{2}\leq 2.33$ within the 1$\sigma$ ellipse. Also, our best
fit point is near the centre of the ellipses. Overall, the 2HDM Type-III is
fully capable of capturing the excess of 96 GeV. Fig. 2 shows the predicted
rate for $\sigma(pp\to h\to\gamma\gamma)$ (left) in our 2HDM Type-III and its
ratio to the SM results, $\sigma(pp\to h\to\gamma\gamma)/{\rm SM}$, for each
parameter point, in combination with the expected and observed upper limits
from the CMS analysis [3]. Its clearly visible in both panels that many points
could indeed contribute to the excess observed by CMS in the
$h\to\gamma\gamma$ final state.

Figure 1: The signal strengths $\mu_{\rm CMS}$ and $\mu_{\mathrm{LEP}}$
following the scan described in the text. The dashed and solid black lines
indicate the $1\sigma$ and $2\sigma$ ranges of $\chi^{2}_{96}$, respectively.
The pink star corresponds to the best fit point in the $h$ mass range [94, 98]
GeV. The colour code indicates the $\Delta\chi^{2}_{125}$ values. Figure 2:
Surviving points, following the discussed theoretical and experimental
constraints, superimposed on the results of the CMS $8+13$ TeV low-mass di-
photon analysis [3]. The dashed line corresponds to the expected upper limit
on $\sigma\times{\cal BR}(h\rightarrow\gamma\gamma)$ (left) and
$\sigma\times{\cal BR}(h\rightarrow\gamma\gamma)/\sigma^{SM}\times{\cal
BR}^{SM}(h\rightarrow\gamma\gamma)$ (right) at $95\%$ C.L., with $1$ and $2$
sigma errors in green and yellow, respectively. The solid line is the observed
upper limit at $95\%$ C.L.

##  5 Conclusion

In this contribution, we have shown how excesses observed by the CMS and LEP
Collaborations could possibly be attributed to a Higgs boson produced by $gg$
fusion and decaying into $\gamma\gamma$ and $b\bar{b}$ with mass around 96
GeV, we have identified the regions of parameter space where the light
$\mathcal{CP}$-even state, $h$, can fit such excesses while being fully
compliant with the required signal strengths measured at both colliders. This
has been accomplished after considering all the up-to-date theoretical and
experimental constraints.

##  6 Acknowledgments

SM acknowledges support from the STFC Consolidated Grant ST/L000296/1 and is
partially financed through the NExT Institute. The work of SS is supported in
full by the NExT Institute. The work of RB and MB is supported by the Moroccan
Ministry of Higher Education and Scientific Research MESRSFC and CNRST Project
PPR/2015/6.

## References

  * [1] ATLAS collaboration, G. Aad et al., Phys. Lett. B716 (2012) 1-29.
  * [2] CMS collaboration, S. Chatrchyan et al., Phys. Lett. B716 (2012) 30-61.
  * [3] CMS Collaboration, A. M. Sirunyan et al., Phys. Lett. B793 (2019) 320-347.
  * [4] R. Barate et al. [LEP Working Group for Higgs boson searches, ALEPH, DELPHI, L3 and OPAL], Phys. Lett. B565 (2003), 61-75.
  * [5] G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, M. Sher and J. P. Silva, Phys. Rept. 516 (2012) 1-102.
  * [6] T. P. Cheng and M. Sher, Phys. Rev. D35 (1987) 3484.
  * [7] R. Benbrik, M. Boukidi, S. Moretti and S. Semlali, Phys. Lett. B832 (2022), 137245.
  * [8] D. Eriksson, J. Rathsman and O. Stal, Comput. Phys. Commun. 181 (2010) 189-205, arXiv:0902.0851 [hep-ph].
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# Geometry and topology of maximal antipodal sets and related topics

Bang-Yen Chen  Department of Mathematics, Michigan State University, East
Lansing, MI 48824, USA<EMAIL_ADDRESS>
###### Abstract.

Maximal antipodal sets of Riemannian manifolds were introduced by the author
and T. Nagano in [Un invariant géométrique riemannien, C. R. Acad. Sci. Paris
Sér. I Math. 295 (1982), no. 5, 389–391]. Since then maximal antipodal sets
have been studied by many mathematicians and they shown that maximal antipodal
sets are related to several important areas in mathematics. The main purpose
of this paper is thus to present a comprehensive survey on geometry and
topology of maximal antipodal sets and also on their applications to several
related topics.
Mathematics Subject Classification (2020): 22-02, 53-02, 58-02, 00-02.
Key words: Great antipodal set, maximal antipodal set, Borel-Serre problem,
2-rank, 2-number, $(M_{+},M_{-})$-theory, real form, symmetric $R$-manifold,
flag manifold.

##  1\. Maximal antipodal sets and two-numbers

The notion of maximal antipodal sets of Riemannian manifolds was introduced in
[27]. The primeval concept of maximal antipodal sets is the notion of
antipodal points on a circle. For a circle $S^{1}$ in the Euclidean plane
$\mathbb{E}^{2}$, the antipodal point $q$ of a point $p\in S^{1}$ is the point
on $S^{1}$ which is diametrically opposite to $p$.

In Riemannian geometry, a geodesic in a Riemannian manifold $M$ is a curve
which yields locally the shortest distance between any two nearby points.
Since a closed geodesic in a Riemannian manifold $M$ is isometric to a planar
circle, antipodal points can be defined for every closed geodesic in $M$,
i.e., a point $q$ in a closed geodesic is called an antipodal point of another
point $p$ on the closed geodesic if the distance $d(p,q)$ between $p$ and $q$
on the two arcs connecting $p$ and $q$ are equal. For simplicity, a closed
geodesic in a Riemannian manifold is also called a circle in this article.

A subset $S$ of a Riemannian manifold $M$ is said to be an antipodal set if
any two points in $S$ are antipodal on some circle of $M$. An antipodal set in
a connected Riemannian manifold $M$ is called a maximal antipodal set if it
doesn’t lie in any antipodal set as a proper subset. The supremum of the
cardinality of all maximal antipodal set of $M$ is called the two-number of
$M$, denote by $\\#_{2}M$. If an antipodal set $S$ of $M$ satisfies
$\\#S=\\#_{2}M$, then $S$ is called a great antipodal set or a 2-set.

If a Riemannian manifold $M$ contains no closed geodesics, we put
$\\#_{2}M=0$. On the other hand, for any compact Riemannian manifold $M$ we
have

(1.1) $\displaystyle\\#_{2}M\geq 2,$

since every compact connected Riemannian manifold contains at least one close
geodesic (see [59]). Clearly, inequality (1.1) is sharp, because
$\\#_{2}S^{n}=2$ for every standard $n$-sphere $S^{n}$. A clear proof of the
finiteness of the 2-number for compact connect Riemannian manifolds was given
by M. S. Tanaka and H. Tasaki [93].

##  2\. $(M_{+},M_{-})$-theory for compact symmetric spaces

###  2.1. E. Cartan’s classification of irreducible compact symmetric spaces

In 1926, Èlie Cartan achieved his classification of symmetric spaces by
reducing the problem to the classification of simple Lie algebras over real
field ${\mathbb{R}}$, a problem which Cartan himself solved earlier in 1914.

È. Cartan proved that simply-connected irreducible symmetric spaces of compact
type consist of the following four families:

  * (1)

Classical simple Lie groups $SO(n),SU(n),Sp(n)$.

  * (2)

Exceptional simple Lie groups $E_{6},E_{7},E_{8},F_{4},G_{2}$.

  * (3)

Eight classes of classical symmetric spaces
$AI(n),AII(n),AIII(p,q),BDI(p,q),BDII(n)$,

$DIII(n),CI(p),CII(p,q)$ corresponding to the classical groups
$SO(n),SU(n),Sp(n)$.

  * (4)

Twelve exceptional symmetric spaces $EI,EII,EIII,EIV,EV,EVI,EVII,EVIII,EIX$,

$FI,FII$ and $GI$ corresponding to the exceptional simple groups
$E_{6},E_{7},E_{8},F_{4},G_{2}$.

Since the discovery by É Cartan, symmetric spaces, a distinguished class of
Riemannian manifolds, attracted the attention of numerous mathematicians from
various fields such as differential geometry, algebraic topology,
representation theory and harmonic analysis.

###  2.2. $(M_{+},M_{-})$-theory

A geometric new approach to compact symmetric spaces, the
$(M_{+},M_{-})$-theory, was developed by the author and T. Nagano during the
1970–1980s (see [19, 24, 25, 26, 27, 28]). This theory was also known today as
the Chen–Nagano theory in some literatures (see, e.g., [90, 91]).

The fundamental principle of this theory is that a pair of a polar and the
corresponding meridian determines a compact Riemannian symmetric space. One of
advantages of this theory is that it is useful for inductive arguments on
polars or meridians.

Assume that $o$ is a point of a compact symmetric space, say $M=G/K$. A
connected component of the fixed point set $F(s_{o},M)\smallsetminus\\{o\\}$
of the symmetry $s_{o}$ at $o$ is called a polar of $o$. We denote it by
$M_{+}$ or $M_{+}(p)$ if $M_{+}$ contains a point $p$.

The following propositions from [26] is quite useful (also [19, page 15]).

######  Proposition 2.1.

Let $M=G/K$ be a compact symmetric space. Then, for each antipodal point $p$
of $o\in M$, the isotropy subgroup $K$ at $o$ acts transitively on the polar
$M_{+}(p)$. Further, we have $K(p)=M_{+}(p)$ and $K(p)$ is connected. Hence,
$M_{+}(p)=K/K_{p}$, where $K_{p}=\\{k\in K:k(p)=p\\}$.

If a polar consists of a single point, then it is called a pole.

######  Proposition 2.2.

Under the hypothesis of Proposition 2.1, the normal space to $M_{+}(p)$ at
$p\in M$ is the tangent space of a connected complete totally geodesic
submanifold $M_{-}(p)$. Thus we have

(2.1) $\displaystyle\dim M_{+}(p)+\dim M_{-}(p)=\dim M.$

We call $theM_{-}(p)$ the meridian of $o$ at $p$. For meridians, we have

######  Proposition 2.3.

For each antipodal point $p$ of $o$ in a compact symmetric space $M$, we have

  * (1)

$rk(M_{-}(p))=rk(M)$ and

  * (2)

$M_{-}(p)$ is a connected component of the fixed point set $F(s_{p}\circ
s_{o},M)$ of $s_{p}\circ s_{o}$ through $p$.

For a compact symmetric space $M$, polars and meridians of $M$ are totally
geodesic submanifolds; in fact, they are compact symmetric spaces as well.
Polars and meridians have been determined for every compact connected
irreducible Riemannian symmetric space (see [19, 28, 63, 64]).

One of the most important properties of polars and meridians is that $M$ is
determined globally by any pair of $(M_{+}(p),M_{-}(p))$.

Besides polars and meridians of symmetric spaces of compact type, there exist
another important totally geodesic submanifolds called centrosomes which are
defined as follows.

######  Definition 2.4.

Let $o$ be a point of a compact connected Riemannian symmetric space $M$. If
$p$ is a pole of $o\in M$, then the centrosome $C(o,p)$ of $\\{o,p\\}$ is the
set consisting of the midpoints of all geodesics in $M$ joining $o$ and $p$. A
connected component of a centrosome is called a centriole.

######  Remark 2.5.

P. Quast described in [68] all centrioles in irreducible simply-connected
compact symmetric spaces of compact type in terms of the root system of the
ambient space. He also investigated geometric properties of centrioles in
[68].

A connected component of the centrosome $C(o,p)$ is a totally geodesic
submanifold of $M$. Centrosomes play some important roles in topology as well.
For example, centrosomes were used by J. M. Burns to compute homotopy of
compact symmetric spaces in [17].

The following result from [28] characterizes poles in compact symmetric spaces
(see also [20]).

######  Proposition 2.6.

The following six conditions are equivalent to each other for two distinct
points $o,p$ of a connected compact symmetric space $M=G_{M}/K_{G}$.

  * (i)

$p$ is a pole of $o\in M$;

  * (ii)

$s_{p}=s_{o}$;

  * (iii)

$\\{p\\}$ is a polar of $o\in M$;

  * (iv)

there is a double covering totally geodesic immersion
$\pi=\pi_{\\{o,p\\}}:M\to M^{\prime\prime}$ with $\pi(p)=\pi(o)$;

  * (v)

$p$ is a point in the orbit $F(\sigma,G_{M})(o)$ of the group
$F(\sigma,G_{M})$ through $o$, where $\sigma={\rm ad}(s_{o})$;

  * (vi)

the isotropy subgroup of $SG_{M}$ at $p$ is that, $SK_{G}$ $($of $SG_{M}$ at
$o)$, where $SG_{M}$ is the group generated by $G_{M}$ and the symmetries;
$SG_{M}/G_{M}$ is a group of order $\leq 2$.

For compact symmetric spaces, the author and T. Nagano proved the following
result.

######  Proposition 2.7.

[28] For any compact symmetric space $M$, the two-number $\\#_{2}M$ is equal
to the maximal possible cardinality $\\#(A_{2}M)$ of a subset $A_{2}M$ of $M$
such that the point symmetry $s_{x}$ fixes every point of $A_{2}M$ for every
$x\in A_{2}M$.

Proposition 2.7 can be regarded as an alternative definition of 2-number for
symmetric spaces of compact type.

######  Remark 2.8.

T. Nagano and M. Sumi [65] proved that the root system $R(M_{-})$ of a
meridian $M_{-}\neq M$ is obtained from the Dynkin diagram of the root system
$R(M)$ of the compact symmetric space $M$. Furthermore, they have determined
in [65] all maximal totally geodesic spheres in $SU(n)$ by means of the
$(M_{+},M_{-})$-theory.

##  3\. Descriptions of great antipodal sets

In 1988, maximal antipodal sets of compact symmetric spaces were determined
and used by the author and T. Nagano in [28] for determining 2-numbers of
irreducible compact symmetries and also for simple Lie groups. In addition,
they explicitly described antipodal sets in many compact Riemannian symmetric
spaces, but did not mention maximal antipodal sets for oriented real
Grassmannian manifolds in [28].

On the other hand, there are quite many works done in recent years related to
great antipodal sets. In particular, many authors have provided detailed
descriptions of great antipodal sets for many symmetric spaces of compact
type. In this section, we will present some of their works in this respect.

###  3.1. Great antipodal sets of classical Lie groups

In [96], M. S. Tanaka and H. Tasaki provided an explicit classification of
maximal antipodal subgroups of compact classical Lie groups and of their
factors by cyclic central subgroups. Their constructions of these subgroups
are based on $D[4]$, i.e., the dihedral group of order 8 or the automorphism
group of a square in the plane. Also, maximum cardinalities of arbitrary
antipodal sets in these compact Lie groups and factors are also calculated by
them.

In [98], M. S. Takano, H. Tasaki and O. Yasukura provided the classification
of maximal antipodal subgroups of compact exceptional Lie group $G_{2}$ and
compact symmetric space $GI=G_{2}/SO(4)$ an explicit description of them by
regarding as the automorphism group of the octonions $\mathcal{O}$. They also
presented in [98] the classification of maximal antipodal sets of $GI$.
Furthermore, they pointed out a relation between maximal antipodal sets of and
those of the oriented Grassmannian manifold
$\widetilde{G}_{3}(\mathbb{R}^{7})$ by using the identification of with the
set of associative 3-dimensional subspaces in ${\rm Im}(\mathcal{O})$, which
is a totally geodesic submanifold of $\widetilde{G}_{3}(\mathbb{R}^{7})$.

###  3.2. Great antipodal sets of exceptional Lie groups and exceptional
symmetric spaces

M. S. Tanaka, H. Tasaki and O. Yasukura [99] explicitly described maximal
antipodal sets of compact symmetric spaces related to $G_{2}$ by realizing it
as the automorphism group of the octonions ${\mathcal{O}}$. Applying their
explicit descriptions, they provided a close relation between maximal
antipodal sets of the associative Grassmannian of the octonions and the Fano
plane.

In [81], Y. Sasaki provided an explicit classification of congruent classes of
maximal antipodal sets of $F_{4}$ by using Jordan algebra
$H_{3}({\mathcal{O}})$. Moreover, he explicitly classified congruent classes
of maximal antipodal sets of $FI$.

Y. Sasaki classified in [81] congruent classes of maximal antipodal sets of
the exceptional Lie group$E_{6}$ and compact symmetric spaces of type
$EI,EII,EIII,EIV$ related to $E_{6}$ . Moreover, he gave realizations of these
compact symmetric spaces by using some subalgebras of the complex exceptional
Jordan algebra. In this realization, he explicitly described congruent classes
of maximal antipodal sets of these compact symmetric spaces.

###  3.3. Great antipodal sets of symmetric spaces of compact type

In [89], H. Tasaki described antipodal sets in oriented real Grassmannian
manifolds, $\widetilde{G}_{k}(\mathbb{R}^{n})$. For a set $X$, the sets

$P_{k}(X)=\\{\alpha\subset X:\\#\alpha=k\\}$

are defined. The sets $P_{k}(n)=P_{k}(1,\ldots,n)$ were used to classify
maximal antipodal sets of $\widetilde{G}_{k}(\mathbb{R}^{n})$ for each $k\leq
4$. Furthermore, some arguments using for $k\leq 4$ are generalized to
construct some maximal antipodal subsets for higher $k$. In another article
[91], H. Tasaki shown that great antipodal sets of
$\widetilde{G}_{5}(\mathbb{R}^{n})$ are unique up to isometries of
$\widetilde{G}_{5}(\mathbb{R}^{n})$ for $n\geq 87$.

In [97], M. S. Tanaka and H. Tasaki classified and explicitly described
maximal antipodal sets of some compact classical symmetric spaces and those of
their quotient spaces by making use of suitable embeddings of these symmetric
spaces into certain compact classical Lie groups. They also provided the
cardinalities of maximal antipodal sets and they determined the maximum of the
cardinalities and maximal antipodal sets whose cardinalities attain the
maximum.

Very recently, J. Yu provided in [106] explicit classification of maximal
antipodal sets in any irreducible compact symmetric space except for spin and
half spin groups, and some quotient symmetric spaces associated to them.

###  3.4. Expansion of antipodal sets and homogeneous antipodal sets

Y. Sasaki [78] introduced the notion of connectedness of antipodal sets. Using
connectedness, he defined a subgroup $G_{W}$ of the isometry group of a
compact symmetric space $M$. He also constructed a method to build a bigger
antipodal set from a given antipodal set via the subgroup $G_{W}$.

An antipodal set $A\subset M$ is called homogeneous if there exists a subgroup
of the isometry group of $M$ acting on $A$ transitively. In [78], Y. Sasaki
proved that the connectedness is a sufficient condition that a maximal
antipodal set is homogeneous.

######  Remark 3.1.

For further results on maximal antipodal sets of compact symmetric spaces, we
refer to [12, 56, 78, 79, 102, 104].

##  4\. Links between two-numbers and topology

Two-numbers link closely with topology. In this section, we provide several
results in this respect.

###  4.1. Two-numbers and Euler numbers

In [28], the author and T. Nagano proved the following very simple link
between 2-number and Euler number.

######  Theorem 4.1.

For any compact symmetric space $M$, we have

(4.1) $\displaystyle\\#_{2}M\geq\chi(M),$

where $\chi(M)$ denotes the Euler number of $M$.

The proof of this theorem was based on the $(M_{+},M_{-})$-theory in
conjunction with a result of H. Hopf on fixed point sets and a result of Hopf
and H. Samelson in [42].

For any compact Hermitian symmetric space of semisimple type, the author and
Nagano proved the following.

######  Theorem 4.2.

[28] For any compact hermitian symmetric space $M$ of semi-simple type, we
have

(4.2) $\displaystyle\\#_{2}M=\chi(M)=1+\sum\\#_{2}M_{+}.$

The proof of this theorem based heavily on the $(M_{+},M_{-})$-theory as well
as the Lefschetz fixed point theorem in the version of M. F. Atiyah and I. M.
Singer [6].

The following result is an immediate consequence of Theorem 4.2.

######  Corollary 4.3.

For every complete totally geodesic hermitian subspace $B$ of a semi-simple
hermitian symmetric space $M$, we have $\chi(M)\geq\chi(B).$

###  4.2. Two-numbers and covering maps

The author and T. Nagano discovered in [28] the following links between
2-numbers and covering maps for compact symmetric spaces.

For double coverings we have

######  Theorem 4.4.

If $M$ is a double covering of $M^{\prime\prime}$, then $\\#_{2}M\leq
2\\#_{2}(M^{\prime\prime}).$

######  Remark 4.5.

The inequality in Theorem 4.4 is sharp, because the equality case holds for
the group manifold $M=SO(2m)$ with $m>2$.

For $k$-fold coverings with odd $k$, we have

######  Theorem 4.6.

Let $\phi:\widetilde{M}\to M$ is a $k$-fold covering between two compact
symmetric spaces. Then $\\#_{2}\widetilde{M}=\\#_{2}(M)$ whenever $k$ is odd.

###  4.3. A link between two-number and projective rank in algebraic geometry

A. Fauntleroy [35] defined projective rank, denoted by $Pr(M)$, of a compact
Hermitian symmetric space $M$ as the maximal complex dimension of totally
geodesic complex projective spaces of $M$.

C. U. Sánchez and A. Guinta [77] proved the following link between the two-
number and the projective rank for irreducible Hermitian symmetric spaces of
compact type.

######  Theorem 4.7.

$Pr(M)\cdot rk(M)\leq\\#_{2}(M)$ for any irreducible Hermitian symmetric space
of compact type.

###  4.4. Holomorphic two-number of compact hermitian symmetric space

C. U. Sánchez defined in [76] the notion of holomorphic two-number,
$\\#_{2}^{H}(M)$, for a compact connected Hermitian manifold $M$ as the
maximal possible cardinality of a subset $A_{2}$ such that for every pair of
points $x$ and $y$ of $A_{2}$, there exists a totally geodesic complex curve
of genus 0 in $M$ on which $x$ and $y$ are antipodal to each other.

C. U. Sánchez proved the following.

######  Theorem 4.8.

[76] $\\#_{2}^{H}M=\\#_{2}(M)\,$ for every compact hermitian symmetric space.

By combining Theorem 4.8 with our equality $\\#_{2}M=\chi(M)$ from Theorem
7.2, one obtains the following.

######  Corollary 4.9.

[76] $\\#_{2}^{H}(M)$ $=\chi(M)$ for every compact hermitian symmetric space
of semi-simple type.

###  4.5. Symmetric $R$-spaces

A R-space (or a real flag manifold) is an orbit of the isotropy representation
of a symmetric space $G/K$ of compact type, where $G$ is a connected
semisimple Lie group. The notion of a symmetric $R$-spaces was introduced
independent by T. Nagano [62] and M. Takeuchi [85] in 1965. By definition
symmetric $R$-spaces are compact symmetric spaces which are at the same time
$R$-spaces. In fact, symmetric $R$-spaces admit a transitive action of a
centre-free non-compact semisimple Lie group and the corresponding stabilizer
of a point is a certain maximal parabolic subgroup.

A symmetric space $M$of compact type is said to have a cubic lattice if a
maximal torus is isometric to the quotient of ${\mathbb{E}}^{r}$ by a lattice
of ${\mathbb{E}}^{r}$ generated by an orthogonal basis of the same length. O.
Loos [57] gave another intrinsic characterization of symmetric $R$-spaces
among all compact symmetric spaces with the property that the unit lattice of
the maximal torus of the compact symmetric space (with respect to a canonical
metric) is a cubic lattice. Loos’ proof was based on the correspondence
between the symmetric $R$-spaces and compact Jordan triple systems.

S. Kobayashi and T. Nagano classified symmetric $R$-spaces in [48]. The class
of symmetric $R$-spaces consists of the following seven families:

  * (a)

All hermitian symmetric spaces of compact type

  * (b)

Grassmann manifolds $O(p+q)/O(p)\times O(q),Sp(p+q)/Sp(p)\times Sp(q)$

  * (c)

The classical groups $SO(m),\,U(m),\,Sp(m)$,

  * (d)

$U(2m)/Sp(m),\,U(m)/O(m)$,

  * (e)

$(SO(p+1)\times SO(q+1))/S(O(p)\times O(q))$, where $S(O(p)\times O(q))$ is
the subgroup of

$SO(p+1)\times SO(q+1)$ consisting of matrices of the form:

$\begin{pmatrix}\epsilon&0&&\\\ 0&A&&\\\ &&\epsilon&0\\\
&&0&B\end{pmatrix},\quad\epsilon=\pm 1,\;\;A\in O(p),\;\;B\in O(q),$

  * (f)

Cayley projective plane $FII={\mathcal{O}}P^{2}$, and

  * (g)

The three exceptional spaces $EIII=E_{6}/Spin(10)\times
SO(2),EVII=E_{7}/E_{6}\times SO(2),$

and $EIV=E_{6}/F_{4}.$

R. Bott [15] used symmetric $R$-spaces to prove his famous periodicity theorem
for the stable homotopy of classical Lie groups.

Bott’s original results may be succinctly summarized as

######  Theorem 4.10.

[16] The homotopy groups of the classical groups are periodic:

$\pi_{i}(U)=\pi_{i+2}(U),\;\pi_{i}(O)=\pi_{i+4}(Sp),\;\pi_{i}(Sp)=\pi_{i+4}(O)$

for $i=0,1,\cdots$, where $U$ is the direct limit defined by
$U=\cup_{k=1}^{\infty}U(k)$ and similarly for $O$ and $Sp$.

######  Remark 4.11.

The second and third of these isomorphisms given in Theorem 4.8 imply the
following 8-fold periodicity:
$\pi_{i}(O)=\pi_{i+8}(O),\;\pi_{i}(Sp)=\pi_{i+8}(Sp),\;\;i=0,1,\cdots.$

Also, M. S. Tanaka and H. Tasaki proved in 2013 the following

######  Theorem 4.12.

[93] Let $M$ be a symmetric $R$-space. Then

  * (A)

Every antipodal set is contained in a great antipodal set.

  * (B)

Any two great antipodal sets are congruent, where two subsets are congruent if
they are

transformed to each other by an element of the identity component of the
isometry group.

######  Remark 4.13.

Tanaka and Tasaki [93] also proved that there exists an antipodal set of the
adjoint group of $SU(4)$ which does not satisfy condition (A) of Theorem 4.12.
Notice that the adjoint group of $SU(4)$ is a compact symmetric space, but not
one of symmetric $R$-spaces.

###  4.6. Intrinsically and extrinsically reflective submanifolds

A connected component $N$ of the fixed point set of an involutive isometry
$\sigma$ of a Riemannian manifold $M$ is called a reflective submanifold. This
isometry $\sigma$ is called the reflection of $M$ through $N$. Both polars and
meridians of a compact symmetric spaces are reflective submanifolds.

A totally geodesic submanifold $M\subset\widetilde{M}$ of a submanifold
$\widetilde{M}\subset\mathbb{E}^{m}$ of a Euclidean $m$-space $\mathbb{E}^{m}$
is called extrinsically reflective, if $M$ is a connected component of the
intersection of $\widetilde{M}$ with the fixed set of an involutive isometry
of $\mathbb{E}^{m}$ that leaves $\widetilde{M}$ invariant.

A connected submanifold $P\subset\mathbb{E}^{m}$ of $\mathbb{E}^{m}$ is called
an extrinsically symmetric space if for all $x\in P$ the submanifold $P$ is
invariant under the reflections $\rho_{x}\in{\rm Isom}(\mathbb{E}^{m})$
through the affine normal space of $x+T^{\perp}_{x}(P)$ of $P$ at the point
$x$, where $T^{\perp}_{x}(P)$ denotes the normal space of $M$ in
$\mathbb{E}^{m}$ at $x$.

In [34], J.-H. Eschenburg, P. Quast and M. S. Tanaka proved the following.

######  Proposition 4.14.

Every extrinsically reflective submanifold $M\subset P$ of an extrinsically
symmetric space $P\subset\mathbb{E}^{m}$ is extrinsically symmetric in
$\mathbb{E}^{m}$.

######  Proposition 4.15.

Any meridian $P_{-}$ of a compact extrinsically symmetric space
$P\subset\mathbb{E}^{m}$ is itself extrinsically symmetric in
$\mathbb{E}^{m}$.

######  Proposition 4.16.

Every reflective submanifold of a compact extrinsically symmetric space is
actually extrinsically reflective, and thus extrinsically symmetric. In other
words, any reflective submanifold of a symmetric $R$-space is a symmetric
$R$-space.

A connected submanifold $S$ of Riemannian manifold $M$ is called
(geodetically) convex if any shortest geodesic segment in $S$ is still
shortest in $M$.

In 2012, P. Quast and M. S. Tanaka proved the following.

######  Theorem 4.17.

[69] Every reflective submanifold of a symmetric $R$-space is convex.

###  4.7. Links between two-numbers and homology

Analogous to Theorem 4.2, M. Takeuchi proved in [87] the following.

######  Theorem 4.18.

[87] For any compact hermitian symmetric space $M$ of semi-simple type, we
have

(4.3) $\displaystyle\\#_{2}M=\chi(M)=1+\sum\\#_{2}M_{+}.$

The $i$-th Betti number of a manifold $M$ with coefficients in
$\mathbb{Z}_{2}$ is the rank of the $i$-th homology group
$H_{i}(M,\mathbb{Z}_{2})$. For any symmetric $R$-space, M. Takeuchi also
proved the following.

######  Theorem 4.19.

[87] For any symmetric $R$-space $M$, we have

(4.4) $\displaystyle\\#_{2}M=\sum_{i\geq 0}b_{i}(M,\mathbb{Z}_{2}),$

where $b_{i}(M,\mathbb{Z}_{2})$ is the i-th Betti number of $M$ with
coefficients in $\mathbb{Z}_{2}$.

M. Takeuchi proved this theorem by applying Theorem 4.18 and a result of
Chen–Nagano from [28] in conjunction with an earlier result of Takeuchi in
[85].

##  5\. Antipodal sets and Borsuk-Ulam’s theorem

The following result in algebraic topology is well-known.

The Borsuk–Ulam Antipodal Theorem. [14] Every continuous function from an
$n$-sphere $S^{n}$ into the Euclidean $k$-space $\mathbb{E}^{k}$ with $k\leq
n$ maps some pair of antipodal points to the same point.

Obviously, Borsuk-Ulam’s theorem fails for $k>n$, because $S^{n}$ can be
embedded in $\mathbb{E}^{n+1}$. It is well-known that Borsuk-Ulam’s theorem
has numerous applications. For instance, H. Steinlein provided in [83] a list
of 457 publications involving various generalizations and/or applications of
the Borsuk-Ulam theorem.

A continuous function $f:M\to{\mathbb{R}}$ of a compact symmetric space
$M=G/K$ is called isotropic if it is invariant under the action of the
isotropic subgroup $K$.

In [22], the author proved some Borsuk-Ulam’s type theorems involving maximal
antipodal sets of compact symmetric spaces as follows.

######  Theorem 5.1.

Let $f:M\to{\mathbb{R}}$ be an isotropic continuous function from a compact
symmetric space. Then $f$ maps a great antipodal set of $M$ to the same point
in ${\mathbb{R}}$, whenever $M$ is one of the following spaces: Spheres; the
projective spaces
${\mathbb{F}}P^{n}\,({\mathbb{F}}={\mathbb{R}},{\mathbb{C}},{\mathbb{H}})$;
the Cayley plane $FI\hskip-1.4457ptI$; the exceptional spaces
$EIV;EIV^{*};GI$; and the exceptional Lie group $G_{2}$. Hence,
$f:M\to{\mathbb{R}}$ maps a great antipodal set with $\\#_{2}M$ elements to
the same point in ${\mathbb{R}}$.

The following example illustrates that the isotropic condition on $f$ in
Theorem 5.1 is necessary.

######  Example 5.2.

Let ${\mathbb{R}}P^{2}$ denote the real projective plane of curvature one.
Then there exists a canonical double covering map
$\pi:S^{2}(1)\to{\mathbb{R}}P^{2}.$ Assume that $S^{2}(1)$ is the unit sphere
in $\mathbb{E}^{3}$ centered at the origin of $\mathbb{E}^{3}$. For each
continuous function $f:{\mathbb{R}}P^{2}\to{\mathbb{R}}$, the lift
$\hat{f}:S^{2}(1)\to{\mathbb{R}}$ of $f$ is an even function via the double
covering $\pi$, so that $\hat{f}(-{\bf x})=\hat{f}({\bf x})$ for any ${\bf
x}=(x,y,z)\in S^{2}(1).$

Conversely, for any continuous even function $h:S^{2}(1)\to\mathbb{R}$ of
$S^{2}(1)$, $h$ induces a continuous function
$\check{h}:{\mathbb{R}}P^{2}\to\mathbb{R}$ of ${\mathbb{R}}P^{2}$. Let
$h=(x-y)^{2}$. Then $h$ induces a function
$\check{h}:{\mathbb{R}}P^{2}\to\mathbb{R}$ which does not map any maximal
antipodal set of ${\mathbb{R}}P^{2}$ to the same point in $\mathbb{R}$.

Every compact symmetric spaces $M$ in the list of Theorem 5.1 admits only a
polar for $o\in M$. Now, let us assume that $M$ is a compact symmetric space
with multiple polars, said $M_{+}^{1},M_{+}^{2},\ldots,M_{+}^{i}$ for $o\in
M$. In this case, let $\hat{M}_{+}$ denote a polar of $o\in M$ which has the
maximal 2-number among all polars of $o$. For such compact symmetric spaces,
we have the following result.

######  Theorem 5.3.

[22] Let $f:M\to{\mathbb{R}}$ be an isotropic continuous function of a compact
symmetric space $M$. If $M$ admits more than one polar, then $f$ maps an
antipodal set of $M$ consisting of $1+\\#_{2}\hat{M}_{+}$ points of $M$ to the
same point in ${\mathbb{R}}$.

In particular, we have the following.

######  Theorem 5.4.

[22] If $f:E_{8}\to{\mathbb{R}}$ is an isotropic continuous function, then $f$
maps an antipodal set of $E_{8}$ with $392$ elements to the same point in
${\mathbb{R}}$.

######  Theorem 5.5.

[22] If $f:FI\to{\mathbb{R}}$ is an isotropic continuous function of $FI$,
then $f$ maps an antipodal set of $FI$ with $24$ elements to the same point in
${\mathbb{R}}$.

##  6\. Two-rank of Borel and Serre

The 2-rank of a compact Lie group $G$ was introduced by A. Borel and J.-P.
Serre [13]. The 2-rank of $G$, denoted by $r_{2}G$, is the maximal possible
rank of the elementary 2-subgroup of $G$.

Borel and Serre [13] proved the following:

  * (a)

$rk(G)\leq r_{2}(G)\leq 2rk(G)$ and

  * (b)

$G$ has 2-torsion if $rk(G)<r_{2}(G)$,

where $rk(G)$ denotes the ordinary rank of $G$.

In [13], Borel and Serre were able to determine the 2-rank of simply-connected
simple Lie groups $SO(n),Sp(n),U(n),G_{2}$ and $F_{4}$. In addition, they
proved that the exceptional Lie groups $G_{2},F_{4}$ and $E_{8}$ have
2-torsion. On the other hand, they pointed out in [13, page 139] that they
were unable to determine the 2-rank for the exceptional simple Lie groups
$E_{6}$ and $E_{7}$.

After A. Borel and J.-P. Serre’s paper, 2-ranks have been investigated by many
mathematicians. For instance, it was shown that the 2-ranks have some links
with commutative algebra. Here, we provide two of such links.

(i) Assume that $F$ is either a field or the rational integer ring
$\mathbb{Z}$. Let $A=\sum_{i\geq 0}A_{i}$ be a graded commutative $F$-algebra
in sense of J. Milnor and J. Moore [61]. If $A$ is connected, then it admits a
unique augmentation $\varepsilon:A\to F$.

Put $\bar{A}={\rm Ker}\,\varepsilon$. The $\bar{A}$ is called the augmentation
ideal of $A$. A sequence of elements $\\{x_{1},\ldots,x_{n}\in\bar{A}\\}$ is
said to be a simple system of generators if $\\{x_{1}^{\epsilon_{1}}\cdots
x_{n}^{\epsilon_{n}}:\epsilon_{i}=0\hbox{ or }1\\}$ is a module base of $A$.
For a compact connected Lie group $G$, let us denote by $s(G)$ the number of
generators of a simple system of the $\mathbb{Z}_{2}$-cohomology
$H^{*}(G,{\mathbb{Z}}_{2})$ of $G$.

In [49], A. Kono proved the following.

######  Theorem 6.1.

[49] If $G$ is a connected compact Lie group, then the following three
conditions are equivalent:

  * (1)

$s(G)\leq r_{2}G$;

  * (2)

$s(G)=r_{2}G$;

  * (3)

$H^{*}(G,{\mathbb{Z}}_{2})$ is generated by universally transgressive
elements.

To prove Theorem 6.1, A. Kono applied P. May’s spectral sequence [60], S.
Eilenberg and J. C. Moore’s spectral sequence [33] and also D. Quillen’s
result from [72].

In [49], Kano also described some properties of compact Lie groups satisfying
condition (3) in Theorem 6.1 and provided some applications.

(ii) The Krull dimension of a ring $R$ is the supremum of the number of strict
inclusions in a chain of prime ideals, i.e., we say that a strict chain of
inclusions of prime ideals of the form:

${\mathfrak{p}}_{0}\subsetneq{\mathfrak{p}}_{1}\subsetneq\cdots\subsetneq{\mathfrak{p}}_{n}$

is of length $n$; i.e., it is counting the number of strict inclusions. Given
a prime ideal ${\mathfrak{p}}\subset R$, the height of ${\mathfrak{p}}$ is
defined to be the supremum of the set

$\\{n\in{\mathbb{N}}:{\mathfrak{p}}\hbox{ is the supremum of a strict chain of
length $n$}\\},$

and the Krull dimension is the supremum of the heights of all of its primes.

Let $G$ be a compact Lie group. We put $H_{G}^{*}=H^{*}(BG;{\mathbb{Z}}_{2}),$
where $BG$ denotes the classifying space for $G$. Let $N_{G}^{*}\subset
H_{G}^{*}$ be the ideal of nilpotent elements. Then
$H_{G}^{*}/N_{G}^{*}=H_{G}^{\\#}$ is a finitely generated commutative algebra.

In [72], D. Quillen investigated the relationship between the finitely
generated commutative algebra $H_{G}^{\\#}$ and the structure of the Lie group
$G$. He proved that the Krull dimension of $H_{G}^{\\#}$ is equal to the
2-rank of $G$ (under some suitable assumptions). Quillen proved the result by
calculating the mod 2 cohomology ring of extra special 2-groups. Quillen’s
result gave rise to an affirmative answer to a conjecture of M. F. Atiyah
posed in [3], and a conjecture of R. G. Swan given in [84].

##  7\. Applications of maximal antipodal sets to Borel-Serre’s problem

If $G$ is a connected compact Lie group, then by assigning $s_{x}(y)=xy^{-1}x$
to every point $x\in G$, we have $s^{2}_{x}=id_{G}$ to each point $x$. Thus,
$G$ is a compact symmetric space with respect to a bi-invariant Riemannian
metric.

###  7.1. Links between two-numbers and 2-ranks

The author and Nagano proved the following link between the 2-rank and the
two-number of a connected compact Lie group.

######  Theorem 7.1.

[27] Let $G$ be a connected compact Lie group. Then we have

(7.1) $\\#_{2}G=2^{r_{2}G}.$

For products of two compact Lie groups, we have the following result from [28,
Lemma 1.7].

######  Theorem 7.2.

[27] Let $G_{1}$ and $G_{2}$ be connected compact Lie groups. Then

(7.2) $\\#_{2}(G_{1}\times G_{2})=2^{r_{2}G_{1}+r_{2}G_{2}}.$

###  7.2. 2-ranks of classical groups

Applying Theorems 7.1, Theorem 7.2 and $(M_{+},M_{-})$-theory, the author and
Nagano were able to determine the 2-ranks of all compact connected simple Lie
groups in [28]. Therefore, we have settled the problem of Borel-Serre for the
determination of 2-ranks of all compact connected simple Lie groups.

For classical groups we have:

######  Theorem 7.3.

Let $U(n)/{\mathbb{Z}}\mu$ be the quotient group of the unitary group $U(n)$
by the cyclic normal subgroup ${\mathbb{Z}}\mu$ of order $\mu$. Then we have

(7.3) $r_{2}(U(n)/{\mathbb{Z}}\mu)=\begin{cases}n+1&\text{if $\mu$ is even and
$n=2$ or $4$;}\\\ n&\text{otherwise.}\end{cases}$

######  Theorem 7.4.

For $SU(n)/{\mathbb{Z}}\mu$, we have

(7.4) $r_{2}(SU(n)/{\mathbb{Z}}\mu)=\begin{cases}n+1&\text{for
$(n,\mu)=(4,2)$;}\\\ n&\text{for $(n,\mu)=(2,2)$ or $(4,4)$;}\\\ n-1&\text{for
the other cases.}\end{cases}$

######  Theorem 7.5.

One has $r_{2}(SO(n))=n-1$ and, for $SO(n)^{*}$, we have

(7.5) $r_{2}(SO(n)^{*})=\begin{cases}4&\text{for $n=4$;}\\\ n-2&\text{for $n$
even $>4$}.\end{cases}$

######  Theorem 7.6.

Let $O(n)^{*}=O(n)/\\{\pm 1\\}$. We have

(a) $r_{2}(O(n))=n$;

(b) $r_{2}(O(n)^{*})$ is $n$ if $n$ is 2 or 4, while it is $n-1$ otherwise.

######  Theorem 7.7.

One has $r_{2}(Sp(n))=n$, and, for $Sp(n)^{*}$, we have

(7.6) $r_{2}(Sp(n)^{*})=\begin{cases}n+2&\text{for $n=2$ or $4$}\\\
n+1&\text{otherwise}.\end{cases}$

Thus we also have

(7.7) $r_{2}(Sp(n)^{*})=r_{2}(U(n)/{\mathbb{Z}}_{2})+1$

for every $n$.

###  7.3. 2-ranks of spinors, semi-spinors and $Pin(n)$

For $Spin(n)$ we have the following two results.

######  Theorem 7.8.

We have

$r_{2}(Spin(n))=\begin{cases}r+1&\text{if $\,n\equiv-1,0$ or {\rm 1 (mod
8)}}\\\ r&\text{otherwise},\end{cases}$

where $r$ is the rank of $Spin(n)$, $r=[\frac{n}{2}]$.

######  Theorem 7.9.

(PERIODICITY) For $n\geq 0$, One has

$r_{2}(Spin(n+8))=r_{2}(Spin(n))+4$

$Pin(n)$ was discovered by M. F. Atiyah, R. Bott and A. Shapiro while they
studied Clifford modules in [5]. For the group $Pin(n)$ we have

######  Theorem 7.10.

For $Pin(n)$ with $n\geq 0$, one has

$r_{2}(Pin(n))=r_{2}(Spin(n+1)).$

For the semi-spinor group $SO(4m)^{\\#}=Spin(4m)/\\{1,e_{((4m))}\\}$, we have:

######  Theorem 7.11.

We have

$r_{2}(SO(4m)^{\\#})=\begin{cases}3&\text{if $m=1$}\\\ 6&\text{if $m=2$,}\\\
r+1&\text{if $m$ is even $>2$, }\\\ r&\text{if $m$ is odd $>1$},\end{cases}$

where $r=2m$ is the rank of $SO(4m)^{\\#}$.

######  Remark 7.12.

The 2-rank of $Spin(16)$ and of $SO(16)^{\\#}$ were obtained in [1]
independently by J. F. Adams. His method of proof was completely different
from ours given in [28].

###  7.4. 2-ranks of exceptional groups

For exceptional Lie groups we have the following.

######  Theorem 7.13.

One has $r_{2}E^{*}_{6}=6$.

######  Theorem 7.14.

One has

$r_{2}G_{2}=3,\;\;r_{2}F_{4}=5,\;\;r_{2}E_{6}=6,\;\;r_{2}E_{7}=7,\;\;r_{2}E_{8}=9$

for simply-connected exceptional simple Lie groups $G_{2},F_{4},E_{6},E_{7}$
and $E_{8}$.

######  Remark 7.15.

$r_{2}G_{2}=3$ and $r_{2}F_{4}=5$ were proved by Borel and Serre in [13].

##  8\. Antipodal sets and real forms

Let $\psi$ be an involutive anti-holomorphic isometry of a Hermitian symmetric
space $M$ of compact type so that we have $\psi_{*}J=-J\psi_{*}$, where $J$ is
the almost complex structure of $M$. Then the fixed point set

$F(\psi,M)=\\{p\in M:\psi(p)=p\\}$

is called a real form of $M$ which is a connected totally geodesic Lagrangian
submanifold $M$. The classification of real forms of an irreducible Hermitian
symmetric space of compact type have been obtained by D. S. P Leung [55] and
M. Takeuchi [86].

M. S. Tanaka and H. Tasaki proved in [94] that a real form of a Hermitian
symmetric space $M$ of compact type is a product of real forms of irreducible
factors of $M$ and diagonal real forms determined from irreducible factors of
$M$.

The following result implies that any two real forms in any Hermitian
symmetric space of compact type have a non-empty intersection.

######  Proposition 8.1.

[29, 100]. Let $M$ be a compact Kähler manifold whose holomorphic sectional
curvatures are positive. If $L_{1}$ and $L_{2}$ are totally geodesic compact
Lagrangian submanifolds of $M$, then $L_{1}\cap L_{2}\neq\emptyset$.

The following theorem of M. Takeuchi characterized real forms as symmetric
$R$-spaces.

######  Theorem 8.2.

[86] Every real form of a Hermitian symmetric space of compact type is a
symmetric $R$-space. Conversely, every symmetric $R$-space is realized as a
real form of a Hermitian symmetric space of compact type. The correspondence
is one-to-one.

M. S. Tanaka and H. Tasaki [92] studied the intersection of two real forms in
a Hermitian symmetric space of compact type. They proved the following four
results.

######  Theorem 8.3.

[92] Let $M$ be a Hermitian symmetric space of compact type. If two real forms
$L_{1}$ and $L_{2}$ of $M$ intersect transversally, then $L_{1}\cap L_{2}$ is
an antipodal set of $L_{1}$ and $L_{2}$.

######  Theorem 8.4.

[92] Let $M$ be a Hermitian symmetric space of compact type and let
$L_{1},L_{2},L^{\prime}_{1},L^{\prime}_{2}$ be real forms of $M$ such that
$L_{1},L^{\prime}_{1}$ are congruent and $L_{2},L^{\prime}_{2}$ are congruent.
If $L_{1},L_{2}$ intersect transversally and if
$L^{\prime}_{1},L^{\prime}_{2}$ intersect transversally, then $\\#(L_{1}\cap
L_{2})=\\#(L^{\prime}_{1}\cap L^{\prime}_{2})$.

######  Theorem 8.5.

[92] Let $L_{1},L_{2}$ be real forms of a Hermitian symmetric space of compact
type whose intersection is discrete. Then $L_{1}\cap L_{2}$ is an antipodal
set in $L_{1}$ and $L_{2}$. Moreover, if $L_{1}$ and $L_{2}$ are congruent,
then $L_{1}\cap L_{2}$ is a great antipodal set. Thus $\\#(L_{1}\cap
L_{2})=\\#_{2}L_{1}=\\#_{2}L_{2}.$

######  Theorem 8.6.

[92] Let $M$ be an irreducible Hermitian symmetric space of compact type and
let $L_{1},L_{2}$ be real forms of $M$ with $\\#_{2}L_{1}\leq\\#_{2}L_{2}$ and
we assume that $L_{1}\cap L_{2}$ is discrete. Then

  * (a)

If $M=G_{2m}(\mathbb{C}^{4m})\,(m\geq 2)$, $L_{1}$ is congruent to
$G_{m}(\mathbb{H}^{2m})$, $L_{2}$ is congruent to $U(2m)$, and

$\\#(L_{1}\cap L_{2})=2^{m}<\binom{2m}{m}=\\#_{2}L_{1}<2^{2m}=\\#_{2}L_{2}.$

  * (b)

Otherwise, $\\#(L_{1}\cap L_{2})=\\#_{2}L_{1}(\leq\\#_{2}L_{2}).$

Y.-G. Oh defined in [66] the notion of global tightness of Lagrangian
submanifolds in a Hermitian symmetric space; namely, a Lagrangian submanifold
$L$ of a Hermitian symmetric space $M$ is called globally tight if $L$
satisfies

$\\#(L\cap g\cdot L)=\dim H_{*}(L,\mathbb{Z}_{2})$

for any isometry $g$ of $M$ such that $L$ intersects $g\cdot L$ transversally.

H. Tasaki proved the following.

######  Theorem 8.7.

[100] In the complex hyperquadric, the intersection of two real forms is an
antipodal set whose cardinality attains the smaller 2-number of the two real
forms. In particular, every real form in the complex hyperquadric is a
globally tight Lagrangian submanifold.

###  8.1. Fixed point sets and holomorphic isometries

In [44], O. Ikawa, M. S. Tanaka and H. Tasaki discovered a necessary and
sufficient condition for the fixed point set of a holomorphic isometry of a
Hermitian symmetric space of compact type to be discrete. They also shown that
the discrete fixed point set is an antipodal set. Further, they derived a
necessary and sufficient condition that the intersection of two real forms in
a Hermitian symmetric space of compact type is discrete. Moreover, they
discussed some relations between the intersection of two real forms and the
fixed point set of a certain holomorphic isometry by the use of the symmetric
triads.

######  Remark 8.8.

For further results on real forms, we refer to [43, 89, 90].

##  9\. Application to Lagrangian Floer homology

Suppose that $(M,\omega)$ is a symplectic manifold, i.e., $M$ is a manifold
equipped with a closed nondegenerate 2-form $\omega$. Let $L$ be a Lagrangian
submanifold in $M$. For a pair of closed Lagrangian submanifolds
$(L_{0},L_{1})$ of $M$, one can define Lagrangian Floer homology
$HF(L_{0},L_{1}:\mathbb{Z}_{2})$ with coefficient $\mathbb{Z}_{2}$ under some
appropriate topological conditions.

A. Floer [37] defined in 1988 the homology when $\pi_{2}(M,L_{i})=0$, $i=0,1$.
He proved that it is isomorphic to the singular homology group
$H_{*}(L_{0},\mathbb{Z}_{2})$ of $L_{0}$ in the case where $L_{0}$ is
Hamiltonian isotopic to $L_{1}$. As a result, Floer solved affirmatively the
so called Arnold conjecture for Lagrangian intersections in that case (see [2,
37]). Symplectic Floer homology is invariant under Hamiltonian isotropy of the
symplectomorphism. Denote by $Hamilt(M,\omega)$ the set of all Hamiltonian
diffeomorphisms of $M$.

In 1989, A. Givental [38] proposed the following conjecture that generalized
the results of Floer and himself.

Arnold-Givental Conjecture. Let $(M,\omega)$ be a symplectic manifold and
$\psi:M\to M$ be an anti-symplectic involution of $M$. Suppose that the fixed
point set $L=F(M,\psi)$ is compact and nonempty. Then, for any $\phi\in
Hamilt(M,\omega)$ such that the Lagrangian submanifold $L$ and its image
$\phi(L)$ intersect transversally, the inequality

(9.1) $\displaystyle\\#(L\cap\phi(L))\geq\ b(L,\mathbb{Z}_{2})$

holds, where $b(L,\mathbb{Z}_{2})=\sum_{i\geq 0}b_{i}(L,\mathbb{Z}_{2})$ is
the total Betti number of $L$ with $\mathbb{Z}_{2}$ coefficient.

In [45], H. Iriyeh, T. Sakai and H. Tasaki computed Lagrangian Floer homology
$HF(L_{0},L_{1};{\mathbb{Z}}_{2})$ for a pair of real forms $(L_{0},L_{1})$ in
a monotone Hermitian symmetric space $M$ of compact type in the case where
$L_{0}$ is not necessarily congruent to $L_{1}$. In particular, they
established a generalization of the Arnold-Givental inequality (9.1) in the
case where $M$ is irreducible. As an application, H. Iriyeh, T. Sakai and H.
Tasaki established the following result.

######  Theorem 9.1.

[45] Every totally geodesic Lagrangian sphere in the complex hyperquadric is
globally volume minimizing under Hamiltonian deformations.

##  10\. Application to theories of designs and codes

The theory of designs is the part of combinatorial mathematics that deals with
the existence, construction and properties of systems of finite sets whose
arrangements satisfy generalized concepts of balance and/or symmetry.

###  10.1. Codes and designs

Codes and designs on association schemes are important research themes in
combinatorics. In 1973, P. Delsarte [30] gave linear programming bounds for
cardinalities of codes and designs on commutative association schemes in terms
of eigen-matrices.

After Delsarte’s work, the theory of designs on spheres was introduced in 1977
by P. Delsarte, J. M. Goethals and J. J. Seide in [31] as an analogy of
Delsarte theory. The main tool in their works is the addition formula for
polynomials; polynomials associated with metric or cometric association
schemes, or the Gegenbauer polynomials with spheres. As a result, Delsarte’s
bounds were established in terms of spherical Fourier transforms. For a survey
on the studies of codes and designs on spheres, we refer to [9].

Compact symmetric spaces of rank one are natural and significant examples of
the Delsarte spaces or the polynomial spaces for continuous metric spaces. The
theory of designs on rank one compact symmetric spaces was also investigated
by S. G. Hoggar [41] in details. E. Bannai and S. G. Hoggar also studied on
rank one compact symmetric spaces in [10].

For other compact symmetric spaces, codes, designs and Delsarte’s bounds have
been studied by many researchers. For examples, studied by C. Bachoc, R.
Coulangeon and G. Nebe [8] and C. Bachoc, E. Bannai and R. Coulangeon [7] on
real Grassmannian manifolds; by A. Roy [73] on complex Grassmannian manifolds;
and by A. Roy and A. J. Scott [74] on unitary groups.

In [53], H. Kurihara and T. Okuda provided a definition of codes and designs
on general compact symmetric spaces. They also established in [53] a general
formulation of Delsare’s bounds on compact symmetric spaces.

###  10.2. Great antipodal sets and designs on complex Grassmannian manifolds

For compact symmetric spaces of higher rank, H. Kurihara and T. Okuda [52]
obtained a characterization of maximal antipodal sets of complex Grassmannians
in term of certain designs (more precisely,
${\mathcal{E}}\cup{\mathcal{F}}$-designs) with the smallest cardinalities. In
particular, Kurihara and Okuda’s main result in [52] implies the following.

######  Theorem 10.1.

[52] A great antipodal set of a complex Grassmannian manifold is an
${\mathcal{E}}$-design with the smallest cardinality.

###  10.3. Cubature formulas for great antipodal sets on complex Grassmannian
manifolds

In [67], H. Kurihara and T. Okuda established a formulation of Delsarte theory
for finite subsets of compact symmetric spaces. As its application, they
proved that great antipodal subsets of complex Grassmannian manifolds give
rise to cubature formulas for certain functional spaces.

###  10.4. Great antipodal sets and designs on unitary groups

Put $[n]=\\{1,2,...,n\\}$ and let $2^{[n]}$ denote the power set of $[n]$. Let
$Q$ be the $n$-ary Cartesian product of the two elements set $\\{1,-1\\}$.
Then the Hamming cube graph $Q_{n}$ of degree $n$ is the graph with the vertex
set $Q$ and two vertices are adjacent whenever they differ in precisely one
coordinate.

In [51] H. Kurihara investigated a relation between great antipodal sets on
unitary group $U(n)$ and design theory on $U(n)$. In [51], he also established
a beautiful relationship between a great antipodal set on $U(n)$ and a Hamming
cube graph $Q_{n}$.

###  10.5. Application to coding theory

Coding theory studies properties of codes and their respective fitness for
specific applications. The main purpose of codes is to be able to recover the
original content of a transmitted message by correcting errors that have
entered the message during transmission. This capability is useful in
maintaining the integrity of computer networks, communication systems, compact
disk recording, etc.

A $p$-group $H$ is called extra special if its center ${\mathbb{Z}}$ is cyclic
of order $p$, and the quotient $H/{\mathbb{Z}}$ is a non-trivial elementary
abelian $p$-group. In 1989, J. A. Wood [105] investigated the equivalence
between the diagonal extra-special 2-group of spinor $Spin(n)$ and the self-
orthogonal linear binary codes of algebraic coding theory. In Wood’s article
[105], Theorem 7.8 and Theorem 7.9 were mentioned and used.

##  11\. $k$-symmetric spaces, $\Gamma$-symmetric spaces, $k$-number and flag
manifolds

###  11.1. $k$-symmetric spaces and $\Gamma$-symmetric spaces

Since the 1960s, generalizations of symmetric spaces have been proposed in
various directions. In 1967, A. J. Ledger [54] initiated the study of
$s$-manifolds. These are Riemannian manifolds $M$ which admit at each point
$x\in M$ a symmetry $s_{x}$ with $x$ as an isolated fixed point. A
$k$-symmetric structure is called regular [50] if it satisfies

(11.1)
$\displaystyle\theta_{x}\circ\theta_{y}=\theta_{z}\circ\theta_{z},\;\;z=\theta_{x}(y).$

If $s_{x}$ is of finite order $k$, a regular $s$-manifold is called a
$k$-symmetric space (see [49]).

As a further generalization of Riemannian symmetric spaces, P. Lutz [58]
introduced in 1981 $\Gamma$-symmetric space, where $\Gamma$ is a finite
abelian group. These are manifolds $M$ which admit the following structure: To
each point $x\in M$ one assigns in a suitable way a group $\Gamma_{x}$
isomorphic to $\Gamma$ which acts effectively on $M$ with $x$ as an isolated
fixed point. If $\Gamma$ is isomorphic to $\mathbb{Z}_{2}$, then a
$\Gamma$-symmetric space is just a Riemannian symmetric space.

Every complex flag manifold can be regarded as an $R$-space. Let $U$ be a
compact connected semisimple centerless Lie group and let $\mathfrak{u}$ be
the Lie algebra of $U$. Then the complex flag manifold of $U$ is the orbit of
the adjoint action of $U$ on $\mathfrak{u}$. Take $M=Ad(U)Y$ for $Y\neq 0$ in
$U$ and let
${\mathfrak{g}}={\mathfrak{u}}_{\mathbb{C}}={\mathfrak{u}}+i{\mathfrak{u}}$.
Then there exists a Cartan decomposition of the realization
${\mathfrak{g}}_{\mathbb{R}}$ of ${\mathfrak{g}}$ and one may consider $M$ as
the orbit of $iY$ in $i{\mathfrak{u}}$ by the adjoint action of $U$.

###  11.2. Maximal antipodal sets and $\Gamma$-symmetric $R$-spaces

In 2020, P. Quast and T. Sakai [70] extended the definition of antipodal sets
of compact symmetric spaces naturally extends to $\Gamma$-symmetric spaces as
follows (see also [71]).

######  Definition 11.1.

[71] Let $\Gamma$ be a finite abelian group, and let
$\mu=\\{\mu^{\gamma}\\}_{\gamma\in\Gamma}$ be a $\Gamma$-symmetric structure
on a manifold $M$. A subset $A$ of the $\Gamma$-symmetric space $(M,\mu)$ is
called antipodal if $\gamma_{x}(y)=y$ for all $x,y\in A$ and for all
$\gamma\in\Gamma$. An antipodal set $A$ of $(M,\mu)$ is called maximal if $A$
is not a proper subset of another antipodal set of $M$. The supremum of the
cardinalities of antipodal sets of $(M,\mu)$ is called the antipodal number
denoted by $\\#_{\Gamma}M$. An antipodal set $A$ of $(M,\mu)$ is called great
if the cardinality of $A$ is equal to $\\#_{\Gamma}M$.

In [70], P. Quast and T. Sakai defined the induced natural $\Gamma$-symmetric
structure on $R$-spaces. Further, they determined the maximal antipodal sets
of $R$-spaces with respect to the induced natural $\Gamma$-symmetric
structures. In particular, they shown that any two maximal antipodal sets of a
$R$-space with respect to an induced natural $\Gamma$-symmetric structures are
conjugate.

###  11.3. A link between real flag manifolds and complex flag manifolds

In 1997, C. U. Sánchez proved the following.

######  Proposition 11.2.

[76] If $M$ is a real flag manifold, then there exists a complex flag manifold
$M_{\mathbb{C}}$ such that $M$ is isometrically imbedded in $M_{\mathbb{C}}$.
If $M$ is a symmetric $R$-space, then $M_{\mathbb{C}}$ is a hermitian
symmetric space and the isometric imbedding is totally geodesic. If $M$ is
already a complex flag manifold, then $M_{\mathbb{C}}=M$.

######  Remark 11.3.

H. Iriyeh, T. Sakai and H. Tasaki [46, 47] proved that the intersection of
real flag manifolds in the complex flag manifold consisting of sequences of
complex subspaces in a complex vector space is an antipodal set, which is a
generalization of that in a Hermitian symmetric space of compact type.

###  11.4. $k$-number, index number and complex flag manifold

For a complex flag manifold $M_{\mathbb{C}}$, there exists a positive integer
$k_{0}=k_{o}(M_{\mathbb{C}})\geq 2$ such that, for each integer $k\geq k_{0}$,
there exists a $k$-symmetric structure [50] on $M_{\mathbb{C}}$, i.e., for
each point $x\in M_{\mathbb{C}}$ there exists an isometry $\theta_{x}$ such
that $\theta_{x}^{k}=id$ with $x$ as an isolated fixed point.

Analogous to Proposition 2.7 for 2-number of compact symmetric spaces, C. U.
Sánchez defined $k$-number, denoted by $\\#_{k}(M_{\mathbb{C}})$, of a complex
flag manifold $M_{\mathbb{C}}$ as the maximal possible cardinality of the
$k$-sets $A_{k}\subset M_{\mathbb{C}}$ which satisfies the property that for
each point $x\in A_{k}$ the corresponding $k$-symmetry at $x$ fixes every
point in $A_{k}$.

As an extension of Theorem 4.19 of Takeuchi, C. U. Sánchez proved the
following.

######  Theorem 11.4.

[75] For each complex flag manifold $M_{\mathbb{C}}$, we have
$\\#_{k}(M_{\mathbb{C}})=\dim H^{*}(M_{\mathbb{C}},\mathbb{Z}_{2})$,

Applying Proposition 11.2, Sánchez [76] defined in 1997 the index number of a
real flag manifold $M$, denoted by $\\#_{I}M$, as the maximal possible
cardinality of the $p$-sets $A_{p}M$ with $p$ a prime number, in terms of
fixed points of symmetries of the complex flag manifolds restricted to the
real one.

For index number of a real flag manifold, C. U. Sánchez obtained the following
result in 1997.

######  Theorem 11.5.

[76] Let $M$ be a real flag manifold. Then $\\#_{I}M=b(M,\mathbb{Z}_{2}).$

###  11.5. $k$-number and generalized flag manifolds

Let $G$ be a compact connected semisimple Lie group. Then the homogeneous
spaces one obtains as orbits of $G$ under the adjoint representation on the
Lie algebra of $G$ are also called generalized flag manifolds. It is known
that every generalized flag manifold admits $k$-symmetric structure.

In a similar way, Sánchez [76] also proved the following.

######  Theorem 11.6.

If $M$ is a generalized flag manifold, then $\\#_{k}(M)=\chi(M)$ for any
$k$-symmetric structure on $M$.

###  11.6. $k$-number and generalized flag manifolds

Let $G$ be a compact connected semisimple Lie group. Then the homogeneous
spaces one obtains as orbits of $G$ under the adjoint representation on the
Lie algebra of $G$ are also called generalized flag manifolds. It is known
that every generalized flag manifold admits $k$-symmetric structure.

In a similar way, C. U. Sánchez also proved the following.

######  Theorem 11.7.

If $M$ is a generalized flag manifold, then any of its $k$-symmetric structure
on $M$ satisfies $\\#_{k}(M)=\chi(M).$

###  11.7. $k$-number and $k$-symmetric submanifolds

Let $M\subset{\mathbb{E}}^{m}$ be submanifold of ${\mathbb{E}}^{m}$. If $M$
satisfies

  * (a)

For each $x\in M$, there is an isometry
$\sigma_{x}:{\mathbb{E}}^{m}\to{\mathbb{E}}^{m}$ such that
$\sigma_{x}^{k}=id_{M}$, $\sigma_{x}(x)=x$, and $\sigma_{x}|_{T^{\perp}_{x}M}$
= identity on $T^{\perp}_{x}M$;

  * (b)

$\sigma_{x}(M)\subset M$; and

  * (c)

Let $\theta_{x}=\sigma_{x}|_{M}$. The collection $\\{\theta_{x},\,x\in M\\}$
defines on $M$ a Riemannian regular $s$-structure of order $k$,

then $M$ is called an extrinsic $k$-symmetric submanifold (see [36]).

C. U. Sánchez [75] proved the following result.

######  Theorem 11.8.

If $M\subset\mathbb{E}^{m}$ is an extrinsic $k$-symmetric submanifold, then
$\\#_{k}(M)=b(M,\mathbb{Z}_{p})$ for any prime number $p\geq 2$ which divides
$k$.

###  11.8. Morse functions and great antipodal sets on $G_{2}/SO(4)$

In [82], Y. Sasaki constructed $\mathbb{Z}_{2}$-perfect Morse functions of
$GI=G_{2}/SO(4)$ whose set of all critical points is a great antipodal set of
$GI$. Consequently, he provided a reason why the 2-number $\\#_{2}(GI)$
matches the Betti-number of the $Z_{2}$-coefficient homology group of $GI$.

##  12\. 2-number, index number and CW complex structure

The following conjecture was posed first time in author’s 1987 report [19].

Conjecture 1.  For any a compact symmetric space $M$, $\\#_{2}M$ is equal to
the smallest number of cells that are needed for a CW complex structure on
$M$.

Related to this conjecture, J. Berndt, S. Console and A. Fino proved the
following.

######  Theorem 12.1.

[11] The index number $\\#_{I}M$ is equal to the smallest number of cells that
are needed for a CW complex structure for each real flag manifold $M$.

In the proof of this theorem, the authors have applied the convexity theorems
of M. F. Atiyah [4], V. Guillemin and S. Sternberg’s result in [39] for
symplectic manifolds with a hamiltonian torus action as well as a
generalization of J. J. Duistermaat’s result in [32] for fixed point set of
antisymplectic involutions.

The next result was also proved by Berndt, Console and Fino in [11].

######  Theorem 12.2.

The index number $\\#_{I}M$ of a real flag manifold $M$ satisfies
$\\#_{I}M=\chi(M)\,({\rm mod\,2}).$

In [28], the author and T. Nagano made the following:

Conjecture 2.  $\\#_{2}M=\chi(M)\,({\rm mod\,2})$ holds for every irreducible
compact symmetric space $M$.

It was known that the total Betti numbers of a simply-connected compact
symmetric space $M$ satisfies $b(M;{\bf R})\leq\\#_{2}M$ (see [19, page 54]).
Professor T. Nagano asked the following open problem.

Problem. $b(M;{\bf R})<\\#_{2}M$ $\Longrightarrow$ $M$ has 2-torsion?

As far as I know, this problem remains open till now.

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