# Solving Quasiparticle Band Spectra of Real Solids using Neural-Network
Quantum States

Nobuyuki Yoshioka<EMAIL_ADDRESS>Theoretical Quantum Physics
Laboratory, RIKEN Cluster for Pioneering Research (CPR), Wako-shi, Saitama
351-0198, Japan  Department of Applied Physics, University of Tokyo, 7-3-1
Hongo, Bunkyo-ku, Tokyo 113-8656, Japan  Wataru Mizukami  Center for Quantum
Information and Quantum Biology, Institute for Open and Transdisciplinary
Research Initiatives, Osaka University, Osaka 560-8531, Japan  JST, PRESTO,
Kawaguchi, Saitama 332-0012, Japan  Graduate School of Engineering Science,
Osaka University, Toyonaka, Osaka 560-8531, Japan  Franco Nori  Theoretical
Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research (CPR), Wako-
shi, Saitama 351-0198, Japan  Physics Department, University of Michigan, Ann
Arbor, Michigan 48109-1040, USA

###### Abstract

Abstract.— Establishing a predictive ab initio method for solid systems is one
of the fundamental goals in condensed matter physics and computational
materials science. The central challenge is how to encode a highly-complex
quantum-many-body wave function compactly. Here, we demonstrate that
artificial neural networks, known for their overwhelming expressibility in the
context of machine learning, are excellent tool for first-principles
calculations of extended periodic materials. We show that the ground-state
energies in real solids in one-, two-, and three-dimensional systems are
simulated precisely, reaching their chemical accuracy. The highlight of our
work is that the quasiparticle band spectra, which are both essential and
peculiar to solid-state systems, can be efficiently extracted with a
computational technique designed to exploit the low-lying energy structure
from neural networks. This work opens up a path to elucidate the intriguing
and complex many-body phenomena in solid-state systems.

##  I Introduction

Artificial neural networks (ANNs) are a class of expressive mathematical
models originally designed to imitate the high computing power of the human
brain. Driven by the outstanding success over existing data processing methods
in the field of machine intelligence Krizhevsky  _et al._ (2012); Goodfellow
_et al._ (2016); Silver  _et al._ (2016), ANNs have been used in a wide range
of applications, from physical science Carleo  _et al._ (2019a); Das Sarma
_et al._ (2019); Carrasquilla (2020); Carrasquilla and Melko (2017); Yoshioka
_et al._ (2018), medical diagnosis, to astronomical observations. Remarkable
among numerous factors underlying their performance is their ability to
perform efficient feature extraction from high-dimensional data.

Figure 1:  Schematic illustration of the relationship between the formal
computational complexity and accuracy in various first-principles calculation
methods for solid systems. Our goal is to demonstrate that the variational
calculation using neural-network-based ansatz can readily describe both weakly
and strongly correlated electronic structures with moderate number of
variational parameters, i.e., computational cost. We denote the full
configuration interaction (FCI) method by the black square, whereas the
Hartree–Fock (HF) and post-HF calculation methods are indicated by blue
squares: the second-order Møller–Plesset perturbation theory (MP2), the
coupled-cluster singles and doubles (CCSD), and CCSD with perturbative triple
excitations (CCSD(T)). Also, the green squares indicate methods based on the
Density Functional Theory (DFT): the DFT and DFT-based Random Phase
Approximation (RPA). The number of orbitals at each $k$-point is denoted as
$N$ and the total number of $k$-points as $N_{k}$. Note that this is a
qualitative (approximate) illustration which will vary from case to case.

As universal approximators, ANNs have a rich expressive power, which can also
be exemplified by encoding complicated quantum correlations Schuld  _et al._
(2014). Reference Carleo and Troyer (2017) showed that ANNs, employed as a
quantum many-body wavefunction ansatz, can solve strongly correlated lattice
systems at state-of-the-art level. Such quantum state ansatze, often referred
to as neural quantum states (NQS), capture quantum entanglement that even
scales extensively Deng  _et al._ (2017a). The use of such a powerful non-
linear parametrization has been keenly investigated in the quantum physics
community: both equilibrium Nomura  _et al._ (2017); Deng  _et al._ (2017b)
and out-of-equilibrium Yoshioka and Hamazaki (2019); Hartmann and Carleo
(2019); Nagy and Savona (2019); Vicentini  _et al._ (2019) properties,
extension of the network structure Gao and Duan (2017); Choo  _et al._ (2018);
Levine  _et al._ (2019), and quantum tomography Torlai  _et al._ (2018);
Torlai and Melko (2018); Melkani  _et al._ (2020); Ahmed  _et al._ (2020).
Meanwhile, we point out that the application of ANNs to fermionic systems is
much less explored, despite their practical significance, such as the modeling
of real materials and the experimental realizability in quantum simulators
Georgescu  _et al._ (2014); Mazurenko  _et al._ (2017). The proof of concept
for small molecular systems was first presented by Choo et al. Choo  _et al._
(2020) which applied the ANNs to solve the many-body Schrödinger equation
governed by the second-quantized Hamiltonian for molecular orbits. Few
implementations have been further performed to simulate the electronic
structures using ANNs Yang  _et al._ (2020); Pfau  _et al._ (2020); Hermann
_et al._ (2020); Spencer  _et al._ (2020). Thus, a crucial question remains to
be answered: are ANNs powerful enough to represent the electronic structures
of real solid materials? This is related to one of the fundamental problems in
condensed matter physics and computational materials science; namely,
establishing a predictive ab initio method for solids or surfaces. In
particular, it must be demonstrated that the ANNs are capable of investigating
the thermodynamic limit.

We stress that no current first-principles method can take into account both
weak and strong electron correlations compactly and sufficiently. For
instance, it is well known that the accuracy of the de facto standard method,
density functional theory (DFT), is semi-quantitative and it is very difficult
to improve significantly Medvedev  _et al._ (2017); Mardirossian and Head-
Gordon (2017). Many-body-wave-function-based methodologies are, in contrast,
systematically improvable. Such techniques, mainly based on coupled cluster
(CC) theory (or many-body perturbation theory) Shavitt and Bartlett (2009),
have been successful for the electronic states of molecules. This has
encouraged the application of quantum chemical methods to solid state physics
Gruber  _et al._ (2018); Zhang and Grüneis (2019). However, methods such as CC
specialize in describing weak electronic correlations, and only work well for
electronic states where the mean-field approximation is valid.

Methods for dealing with strongly correlated electrons, called multi-reference
theory, also exists in quantum chemistry Roos  _et al._ (2016); but these
assume that the number of strongly correlated electrons is small. Such a
condition usually holds in the case of molecules, because the number of
strongly correlated electrons is often localized and limited. In contrast,
there can be a large number of moderately or strongly correlated electrons in
solid-state systems, owing to their high symmetry and dense structure. Based
on its success in spin systems, it is natural to expect that the NQS have the
potential to compactly describe a variety of electron correlations appearing
in first-principles calculations of solids with a moderate computational cost
(See Fig. 1 for a schematic diagram of the hierarchy of quantum chemical
methods Hirata  _et al._ (2004); Zgid and Chan (2011); Liao and Grüneis
(2016); McClain  _et al._ (2017); Booth  _et al._ (2013)).

In this work, we demonstrate that neural-network-based many-body wave
functions can readily simulate the essense of first-principles calculations
for extended periodic materials: the ground-state and excited-state
properties. The second-quantized fermionic Hamiltonian is transformed into a
spin representation, such that the problematic sign structure of fermions,
which usually imposes severe limits on the numerical accuracy, is naturally
encoded. Employing the variational Monte Carlo (VMC)-based stochastic
optimization, we show that the thermodynamic limit of a one-dimensional system
can be simulated within chemical accuracy. For real solids in both two and
three dimensions, the static electronic correlation in the minimal active
space is compactly represented by the NQS. Our work’s main contribution is
that multiple excited states, forming quasiparticle band spectra, are computed
by constructing an effective Hamiltonian in the truncated Hilbert space. To
the best of our knowledge we offer the first demonstration that the NQS can be
applied to simulate low-lying eigenstates in the identical-quantum-number
sector.

##  II Results

###  II.1 Second quantization representation of solid systems

To alleviate the notorious difficulty of simulating the many-body problem of
solid systems, we employ a linear combination of the single-particle basis.
Namely, we construct crystalline orbitals (COs) using the solution of the
crystalline Hartree–Fock (HF) equation Del Re  _et al._ (1967); Andre (1969).
The second-quantization form of the many-body fermionic Hamiltonian is

$\displaystyle H$ $\displaystyle=$ $\displaystyle\sum_{pq}\sum_{{\bf
k}}t_{pq}^{{\bf k}}c_{p{\bf k}}^{\dagger}c_{q{\bf k}}$ (1) $\displaystyle+$
$\displaystyle\frac{1}{2}\sum_{pqrs}\sum_{{\bf k}_{p}{\bf k}_{q}{\bf
k}_{r}{\bf k}_{s}}^{\prime}v_{pqrs}^{{\bf k}_{p}{\bf k}_{q}{\bf k}_{r}{\bf
k}_{s}}c_{p{\bf k}_{p}}^{\dagger}c_{q{\bf k}_{q}}c_{r{\bf
k}_{r}}^{\dagger}c_{s{\bf k}_{s}},$

where $c_{p{\bf k}}$ ($c_{p{\bf k}}^{\dagger}$) denotes the annihilation
(creation) operator of an electron on the $p$-th CO with crystal momentum
${\bf k}$. Here, the anticommutation relation $\\{c_{p{\bf k}_{p}},c_{q{\bf
k}_{q}}^{\dagger}\\}=\delta_{pq}\delta_{{\bf k}_{p}{\bf k}_{q}}$ is imposed,
and one-body (two-body) integrals are given as $t_{pq}^{{\bf k}}$
($v_{pqrs}^{{\bf k}_{p}{\bf k}_{q}{\bf k}_{r}{\bf k}_{s}}$). For simplicity,
hereafter we denote the suffix as $\mu:=(p{\bf k})$. While the general
framework of the crystalline HF equation is common with that for molecular
systems, it must be noted that the contribution from the reciprocal lattice
vector ${\bf G}=0$ requires extra numerical care owing to the divergence of
the exchange integrals. In this work, we employ the crystalline Gaussian-based
atomic functions as the single-particle basis. The Gaussian density fitting
technique is applied to efficiently compute the two-body integrals Sun  _et
al._ (2017).

The summation in the first term of Eq. (1) is taken over a uniform grid, which
is typically obtained by shifting the $k$’s obeying the Monkhorst–Pack rule
Monkhorst and Pack (1976). Note that the number $N_{k}$ of sampled $k$-points
can be arbitrary. The primed summation in the second term satisfies the
conservation of crystal momentum, which follows from translational invariance:

$\displaystyle{\bf k}_{p}+{\bf k}_{r}-{\bf k}_{q}-{\bf k}_{s}\in\mathcal{G},$
(2)

where $\mathcal{G}$ is the set of reciprocal lattice vectors. With the number
of COs at each $k$-point denoted as $N$, the total number of terms in Eq. (1)
is given as $\mathcal{O}(N^{4}N_{k}^{3})$.

Figure 2: Solving the ground state of the linear hydrogen chain using the
minimal STO-3G basis set. (a) The potential energy curve calculated by the
restricted Boltzmann machine (RBM) agrees with the full configuration
interaction (FCI) method within chemical accuracy (1.6 mHa) for any atom
separation $d_{H}$. This indicates that the RBM states are capable of
describing both the weakly and strongly interacting regimes, where gold-
standard techniques, such as coupled-cluster singles and doubles (CCSD) shown
by the yellow line and CCSD with perturbative triple excitations (CCSD(T)) in
black line, break down. The results by restricted Hartree–Fock (RHF) and
second-order Møller–Plesset perturbation theory are indicated by blue and gray
lines, respectively. A unit cell consists of four hydrogen atoms placed at
even intervals, and two $k$-points are sampled from a uniform grid. (b)
Finite-size scaling of the ground state energy up to $N_{k}=18$ and its
deviation from the FCI, $\Delta E$, at near-equilibrium $d_{H}=2$. The results
show excellent agreement with conventional methods even in the thermodynamic
limit $N_{k}\rightarrow\infty$. Here, the unit cell consists of a single
hydrogen atom, and hence the maximum number of spin-orbitals considered here
is 36. The error bars denote the standard deviation of the estimation by the
Monte Carlo sampling.

To solve the fermionic many-body Hamiltonian (1), we must explicitly impose
the antisymmetric sign structure in the quantum state. Here, we map the
Hamiltonian into the spin-1/2 representation such that the sign structure is
encoded in the operators rather than the quantum states, as Choo et al. Choo
_et al._ (2020) considered in their application of the NQS to small molecules.
The Jordan–Wigner (JW) transformation Jordan and Wigner (1928) defines the
relation of fermionic and spin operators as
$c_{\mu}^{(\dagger)}=(-1)^{\mu-1}\prod_{\nu<\mu}\sigma_{\nu}^{z}\sigma_{\mu}^{+(-)}$,
where $\sigma_{\mu}^{+(-)}$ is the raising (lowering) operator of the $\mu$-th
spin. Such a mapping yields a non-local spin Hamiltonian

$\displaystyle H=\sum_{Q}c_{Q}P_{Q},$ (3)

where $P_{Q}\in\bigotimes_{\mu}\\{I,X,Y,Z\\}$ is a product of Pauli matrices
for a corresponding Pauli string $Q$.

Let us make two remarks on the application of JW transformation. First, the
use of the fermion-to-spin transformation for stochastic variational
calculations was initially considered in the context of near-term quantum
computers Peruzzo  _et al._ (2014), including the application to real solids
Liu  _et al._ (2020); Manrique  _et al._ (2020); Yoshioka  _et al._ (2020),
while the spin-to-fermion mapping has been long applied in condensed matter
and statistical physics community, e.g., to solve exactly soluble quantum spin
models. Second, the JW transformation merely generates the spin operator
representation of the Hamiltonian (1) and does not alter the computational
basis. The evaluation of physical observables in the Monte Carlo approach by
the occupation-number basis of the fermionic representation is identical to
that by the spin computational basis of the spin representation. This is not
the case when we apply other transformations developed in quantum information,
such as the Bravyi–Kitaev transformation Bravyi and Kitaev (2002).

###  II.2 Ground states in the thermodynamic limit

In general, it is classically intractable to solve for the ground state of the
many-body Hamiltonian defined in Eq. (1) or (3). Here we alternatively rely on
a variational method that exemplifies the expressive power of neural networks.
Namely, a neural network is used as a variational many-body wave function
ansatz. It is optimized so that the expectation value of the energy, estimated
via the Monte Carlo simulation, is minimized by approximating the imaginary-
time evolution. Such a technique, called variation Monte Carlo (VMC), has been
successfully applied to condensed-matter systems McMillan (1965); Kolorenč and
Mitas (2011); Sorella  _et al._ (2002); Misawa and Imada (2014) and quantum
chemistry problems Hammond  _et al._ (1994); Foulkes  _et al._ (2001), leading
to state-of-the-art numerical analysis on strongly correlated phenomena. The
choice of the variational ansatz plays a key role for the accuracy, which, as
has been pointed out by Carleo and Troyer Carleo and Troyer (2017), can be
significantly improved by using neural networks.

Let us briefly review the general protocol of VMC for simulating ground-states
in many-body spin systems using the quantum-state ansatz based on the
restricted Boltzmann machine (RBM) Smolensky (MIT Press, Cambridge, 1986).
First, we introduce the quantum many-body wave function expressed as follows
Carleo and Troyer (2017),

$\displaystyle\ket{\Psi^{\rm RBM}_{\theta}}$ $\displaystyle=$
$\displaystyle\frac{1}{Z}\sum_{{\bm{\sigma}}}\Psi^{\rm
RBM}_{\theta}({\bm{\sigma}})\ket{{\bm{\sigma}}},$
$\displaystyle\Psi_{\theta}^{\rm RBM}({\bm{\sigma}})$ $\displaystyle=$
$\displaystyle\sum_{h}\exp(W_{\mu\nu}\sigma_{\mu}h_{\nu}+\sum_{\mu}a_{\mu}\sigma_{\mu}+\sum_{\nu}b_{\nu}h_{\nu}),$
(4)

where $\Psi^{\rm RBM}_{\theta}(\sigma)$ is the unnormalized amplitude for a
spin configuration $\sigma\in\\{-1,+1\\}^{N_{v}}$ where $N_{v}=NN_{k}$ is the
total number of spin orbitals and $Z=\sqrt{\sum_{\sigma}|\Psi^{\rm
RBM}_{\theta}(\sigma)|^{2}}$ is the normalization factor. We denote the set of
complex variational parameters as $\theta=\\{W_{\mu\nu},a_{\mu},b_{\nu}\\}$,
where the interaction $W_{\mu\nu}$ denotes the virtual coupling between the
spin $\sigma_{\mu}$ and the auxilliary degrees of freedom, or the hidden spin
$h_{\nu}$. One-body terms $a_{\mu}$ and $b_{\nu}$ are also introduced to
enhance the expressive power of the RBM state. In the present work, we find
that the it suffices to take the total number of the hidden spin as
$N_{h}=N_{v}$, and therefore the number of the complex variational parameters
is $(N_{v}^{2}+2N_{v})$ in total. The all-to-all connectivity between $\sigma$
and $h$ allows the RBM state to capture complicated quantum correlations such
as topological orders Deng  _et al._ (2017b); Glasser  _et al._ (2018), spin-
liquid behaviours Choo  _et al._ (2019); Ferrari  _et al._ (2019); Nomura and
Imada (2020), and electronic structures in small molecular systems Choo  _et
al._ (2020); Yang  _et al._ (2020).

Using the RBM state (II.2) as the many-body variational ansatz, the ground
state problem is solved in the variational Monte Carlo framework. In
particular, we rely on the stochastic reconfiguration technique Sorella (2001)
to approximate the imaginary-time evolution as

$\displaystyle\ket{\Psi_{GS}}\propto\lim_{\tau\rightarrow\infty}e^{-\tau
H}\ket{\Psi_{0}}\sim\ket{\Psi^{\rm
RBM}_{\theta_{0}+\sum_{k}\Delta\theta_{k}}},$ (5)

where the parameter update at the $k$-th step $\Delta\theta_{k}$ is given by
the Monte Carlo simulation, and the initial state $\ket{\Psi_{0}}$ is taken as
the HF state in our simulation. Detailed information on the implementation and
optimization techniques is provided in “Methods”.

Figure 3: Potential energy curves for 2D and 3D real solids calculated by
neural networks.  The ground state energy is computed for various lattice
constants in the vicinity of equilibrium values. (a) Graphene on a honeycomb
lattice solved using the cc-pVDZ basis set. The smallest active space is taken
at each $2\times 2$ $\Gamma$-centered $k$-point, and hence 16 spin orbitals in
total. (b) LiH with the rocksalt structure solved using the STO-3G basis. The
smallest active space is taken at each $2\times 2\times 2$ $\Gamma$-centered
$k$-point, and hence 32 spin orbitals in total. The result obtained for the
RBM state (green triangle) shows remarkable agreement either with the full
configuration interaction (FCI) method, achieving an error within chemical
accuracy (1.6 mHa). The red, blue dotted, gray, yellow, and black dashed lines
denote the results by the FCI method, restricted Hartree-Fock (RHF) method,
second-order Møller–Plesset perturbation theory (MP2), coupled-cluster singles
and doubles (CCSD), and CCSD with perturbative triple excitations (CCSD(T)),
respectively.

As a first demonstration, we provide the potential energy curve for a one-
dimensional system whose electronic correlation varies drastically as the
geometry is changed. Concretely, we consider a linear hydrogen chain with
homogeneous atom separation $d_{H}$ in a minimal basis set (STO-3G) Motta  _et
al._ (2017, 2020). Figure 2(a) presents the result of the calculation using
the RBM state as well as the second-order Møller–Plesset perturbation theory
(MP2) Sun and Bartlett (1996), the coupled-cluster singles and doubles (CCSD)
Hirata  _et al._ (2001); McClain  _et al._ (2017), and CCSD with perturbative
triple excitations (CCSD(T)) Grüneis  _et al._ (2011), which is considered as
the gold-standard in modern quantum chemistry. While the weakly correlated
regime at near-equilibrium is simulated quite well by all the conventional
methods, we see that they start to collapse as the correlation grows at the
intermediate $d_{H}$ regime, not to mention the Mott-insulating large $d_{H}$
regime. In sharp contrast, the RBM state precisely describes the electronic
correlation and achieves chemical accuracy at any atom separation $d_{H}$.
Here, two $k$-points are sampled from each unit cell which contains four
hydrogen atoms so that the interactions between nearby sites are reflected
explicitly on the model.

To further illustrate the RBM state’s power and reliability, we calculate the
energy in the thermodynamic limit by extrapolating $N_{k}\rightarrow\infty$ in
a system with a single atom per unit cell. The numerical result at near-
equilibrium ($d_{H}=2.0a_{B}$) is shown in Fig. 2(b). We confirm the excellent
agreement with conventional methods by comparing the result with the FCI for
$N_{k}\leq 8$ and CCSD for $10\leq N_{k}\leq 18$. Clearly, the thermodynamic
limit is simulated precisely as well as the finite-size system.

Next, we provide the demonstration in both 2D and 3D real solids: graphene and
the lithium hydride (LiH) crystal in the rocksalt structure. Here, we restrict
the active space per each $k$-point to its highest occupied CO and lowest
unoccupied CO. The results for graphene [Fig. 3(a)] and the crystalline LiH
[Fig. 3(b)] are both in remarkable agreement with the FCI or CCSD(T). Clearly,
the RBM ansatz gives a quantitatively accurate description which may allow
crystal structure determinations of weakly to moderately correlated real solid
systems.

Figure 4:  Quasiparticle band spectra from multiple excited-state calculation.
(a) Schematic diagram of the trans-polyacetylene (C2H2)n. The cyan and gray
spheres indicate the carbon and hydrogen atoms, respectively.  (b) Three
quasiparticle bands below and above the Fermi energy. Here, the yellow lines
and black dashed lines indicate results obtained from the equation-of-motion
coupled cluster (EOM-CC) formalism; CCSD and CCSD(T) stand for the unperturbed
EOM-CCSD and the perturbed EOM-CCSD(T)(a)*methods, respectively. The blue
dotted lines denote the restricted Hartree–Fock (RHF) method. (c) A zoom-in of
the first conduction band, which is computed from the electron attatchment
(EA) energy. It is clearly shown, from the energy differences against the EA-
EOM-CCSD(T)(a)* method, that the results by the RBM (green triangle) are
comparable or better than the unperturbed EA-EOM-CCSD method. In all
calculations, a single $k$-point is taken under the minimal basis set (STO-3G)
and hence 24 spin orbitals are taken into account. The size of the unit cell
is taken as 2.451 Å.

###  II.3 Quasiparticle band structure from the one-particle excitation

Interest beyond the ground-state electronic structures in solids is diverse:
the response against electromagnetic fields, impurity effects, phononic
dispersions, and so on. Here, we focus on the band structure, which is a
peculiar yet fundamental property that characterizes solid systems. We stress
that variational calculations for the lowest band gap, which can be
experimentally measured from photoemissions, are already few, not to mention
the simulation of the band spectra based on stochastic methods Ma  _et al._
(2013). Furthermore, to the best of our knowledge, there is no NQS simulation
of excited states in the identical sector of quantum numbers except the first
excited state Choo  _et al._ (2018). This motivates us to perform the first
attempt to calculate multiple low-lying states and deepen our understanding on
the representability of the NQS beyond the well-studied regimes.

In general, the calculation of band structures is based on the assumption that
the system is weakly to moderately correlated. In other words, the mean-field
approximation is qualitatively valid, so that one-particle excitations
dominate the low-lying spectrum. By employing such a picture in a quantum
many-body context, we can also simulate the band structure via quasiparticle
excitations. We take a similar approach here and compute the band structure
from the single-particle linear-response behavior of the ground state.

Let us construct an appropriately truncated Hilbert space which captures the
low-lying states in a stochastic manner. It is justified from the above
argument that we consider a subspace spanned by a set of non-orthonormal bases
$\\{R_{\alpha}\ket{\Psi_{\rm GS}}\\}$, where $R_{\alpha}$ denotes the
$\alpha$-th single-particle excitation operator. Here, the valence
(conduction) bands are obtained from the ionization (electron attachment)
operators $\\{c_{p{\bf k}_{p}}\\}$ ($\\{c^{\dagger}_{p{\bf k}_{p}}\\}$), which
allows us to compute the quasiparticle band with an additional computational
cost of $\mathcal{O}(N_{v}^{3})$. Though it is possible to include higher
order excitation operators, here we avoid them from the viewpoint of
computational cost and size inconsitency.  It can be shown that the
diagonalization of the effective Hamiltonian given the non-orthonormal basis
is done by the following generalized eigenvalue equation McClean  _et al._
(2017),

$\widetilde{H}C=SCE,$ (6)

where $E={\rm diag}(E_{1},...,E_{N_{v}})$ denote the eigenvalues and $C$ is an
array of eigenvectors. The matrix elements of the non-hermitian matrix
$\widetilde{H}$ and the metric $S$ are estimated via the Monte Carlo sampling
as expectation values:

$\displaystyle\widetilde{H}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\braket{\Psi^{\rm
RBM}_{\rm\theta^{*}}}{R^{\dagger}_{\alpha}HR_{\beta}}{\Psi^{\rm
RBM}_{\rm\theta^{*}}},$ (7) $\displaystyle S_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\braket{\Psi^{\rm
RBM}_{\rm\theta^{*}}}{R^{\dagger}_{\alpha}R_{\beta}}{\Psi^{\rm
RBM}_{\rm\theta^{*}}},$ (8)

where the ground state is now replaced by the RBM ansatz
$\ket{\Psi_{\theta^{*}}^{\rm RBM}}$, with the optimized variational parameter
$\theta^{*}$. In the field of quantum chemistry, this procedure is referred to
as the internally-contracted multireference configuration interaction Werner
and Reinsch (1982); Werner and Knowles (1988).

To enhance the numerical reliability, we incorporate the effect of orbital
relaxation by estimating the band gap from the extended Koopmans’ theorem Day
_et al._ (1974); Smith and Day (1975); Morrell  _et al._ (1975). The energies
are shifted so that the first valence and conduction bands coincide with the
energy difference $\Delta E^{IP}$ and $\Delta E^{EA}$ as

$\displaystyle\begin{cases}\Delta E^{IP}&=E_{GS}^{N_{v}}-E_{GS}^{N_{v}-1},\\\
\Delta E^{EA}&=E_{GS}^{N_{v}+1}-E_{GS}^{N_{v}},\end{cases}$ (9)

where $E_{GS}^{n}$ is the energy of the RBM optimized in the particle-number
sector $n$ (See “Methods”).

We provide a demonstration for the quasiparticle band structure of the
polyacetylene [Fig. 4(a)] using the STO-3G basis sets. The result is compared
with a variant of the equation-of-motion coupled cluster theories (EOM-CC):
ionization-potential (electron-attached) EOM-CC (IP-EOM-CC, EA-EOM-CC) which
considers up to 2-hole and 1-particle (2-particle and 1-hole) excitations
McClain  _et al._ (2017). The agreement with EOM-CCSD(T)(a)* Matthews and
Stanton (2016) is very good for the first valence and conduction bands, while
it becomes slightly worse for higher excitations. As is shown in Fig. 4(b),
the first conduction band is simulated almost within chemical accuracy, which
is partly due to the cancellation of the optimization errors induced by Eq.
(9). Meanwhile, Fig. 4(c) indicates that errors in the higher excitations can
be an order of magnitude larger in the worst case, which cannot be explained
merely from the variational simulation error. Rather, it can be understood as
a systematic error originating in the insufficiency of the truncated Hilbert
space; there is a trade-off between the computational cost and the accuracy.
Systematic improvement can be expected from using higher-order excitation
operators, e.g., two-electron excitation operators $\\{c^{\dagger}_{p{\bf
k}_{p}}c_{q{\bf k}_{q}}\\}$ for the lowest energy state in the particle-number
sectors $(N_{v}\pm 1)$.

##  III Conclusion

We have shown that a shallow neural network with a moderate number of
variational parameters allows us to perform the essence of first-principles
calculations in solid systems, i.e., the ground-state property and the
quasiparticle band spectra. In the weakly to moderately correlated regions of
the linear hydrogen chain, we have demonstrated that even the thermodynamic
limit can be simulated using the RBM state. The representability of the RBM is
also exhibited in the strongly correlated regions, where the standard
approaches break down. We have furthermore shown that the electronic
structures of real solids in both 2D and 3D can be described accurately.
Furthermore, we have successfully obtained the quasiparticle band spectra of a
polymer in the linear response regime. To the best of our knowledge, this is
the first demonstration proving that NQS are capable of computing multiple
excited states, in addition to precise ground-state simulations that reach
their chemical accuracy.

Numerous future directions can be envisioned. We remark the following three
points. First is the extension towards the complete basis limit. While we have
here focused on relatively simple basis sets, the quantitative prediction and
comparison with experiments would necessarily require larger basis sets.
Working in the continuum space is a possibility, but the calculation would be
much more involved than in molecular systems. Second is the systematic
improvement of the calculations for excited states. It is intriguing to
investigate the quantitative performance; whether higher-order subspace
expansions can be efficiently implemented, how the accuracy is compared to
other excited-state calculation framework such as the equation-of-motion and
time-dependent linear response Mussard  _et al._ (2018), and so on. Third is
the behaviour of physical observables. One may want to know the
optical/magnetoelectric/thermal responses, so that experimental results can be
directly compared. If the system is either quasi-static or static, those
properties can be evaluated as derivatives of the energy with respect to an
external perturbation (e.g., electric field) Pulay (2014).

The main bottleneck that prevents the simulation by the NQS in larger systems
is the sampling efficiency. As mentioned by Choo et al. for the case of RBM
Choo  _et al._ (2020), and as known before in the VMC community, accurate
calculations for relatively weak electronic correlations in the HF basis
requires increasingly larger number of Monte Carlo samplings, because the
amplitudes for multi-electron excitations are small. One may consider applying
efficient sampling techniques, such as parallel tempering, heat-bath
configuration interaction Holmes  _et al._ (2016), or even employ non-HF
bases.

##  IV Data Availability

The data that support the findings of this study are available from the
corresponding author upon request.

##  V Code Availability

Codes written for and used in this study is available from the corresponding
author upon reasonable request.

##  VI Acknowledgements

We thank Kenny Choo, Antonio Mezzacappo, and James Spencer for fruitful
discussions. This work was supported by MEXT Quantum Leap Flagship Program
(MEXT Q-LEAP) Grant Number JPMXS0118067394 and JPMXS0120319794. N.Y. is
supported by the Japan Science and Technology Agency (JST) (via the Q-LEAP
program). W.M. wishes to thank Japan Society for the Promotion of Science
(JSPS) KAKENHI No. 18K14181 and JST PRESTO No. JPMJPR191A. F.N. is supported
in part by: NTT Research, Army Research Office (ARO) (Grant No.
W911NF-18-1-0358), Japan Science and Technology Agency (JST) (via the CREST
Grant No. JPMJCR1676), Japan Society for the Promotion of Science (JSPS) (via
the KAKENHI Grant No. JP20H00134 and the JSPS-RFBR Grant No. JPJSBP120194828),
the Asian Office of Aerospace Research and Development (AOARD) (via Grant No.
FA2386-20-1-4069), and the Foundational Questions Institute Fund (FQXi) via
Grant No. FQXi-IAF19-06. Numerical calculations were performed using
OpenFermion McClean  _et al._ (2020), PySCF (v1.7.1) Sun  _et al._ (2018), and
NetKet Carleo  _et al._ (2019b). Some calculations were performed using the
supercomputer systems in RIKEN (HOKUSAI GreatWave), the Institute of Solid
State Physics at the University of Tokyo, and in the Research Institute for
Information Technology (RIIT) at Kyushu University, Japan.

##  VII Author contributions

N.Y. and W.M. conceived the project and contributed equally to the numerical
simulations. W.M. and F.N. supervised the research. All authors discussed the
results and contributed to writing the paper.

##  VIII Competing interests

The authors declare no competing interests.

##  IX Methods

###  IX.1 Stochastic imaginary-time evolution by variational Monte Carlo

Given an initial state $\ket{\Psi_{0}}$ whose overlap with the true ground
state is nonzero (and desirably not exponentially small), the ground state
$\ket{\Psi_{\rm GS}}$ can be simulated as

$\displaystyle\ket{\Psi_{GS}}\propto\lim_{N\rightarrow\infty}\lim_{\eta\rightarrow
0}\left(\prod_{k=1}^{N}e^{-\eta H}\right)\ket{\Psi_{0}},$ (10)

where $H$ is the Hamiltonian of the system and $\eta$ is a ”learning rate”
that determines the step of the imaginary-time evolution. The exact simulation
of Eq. (10) for generic quantum many-body systems becomes exponentially
inefficient as the system size grows. Hence, we approximate the quantum state
by a variational ansatz $\ket{\Psi_{\theta}}$ and consider the update rule of
the parameters $\theta$ such that Eq. (10) is realized approximately.

There are numerous variational principles that dictate the parameter updates.
Here, we choose the stochastic reconfiguration method Amari  _et al._ (1992);
Sorella (2001), which uses the Fubini-Study metric $\mathcal{F}$ to measure
the difference between the exact and variational imaginary-time evolution.
Given a set of variational parameter $\theta$, the update $\delta\theta$ is
determined as

$\displaystyle\delta\theta$ $\displaystyle=$ $\displaystyle\mathop{\rm
arg~{}min}\limits_{\Delta}\left(\mathcal{F}\left[e^{-\eta\hat{H}}\ket{\Psi_{\theta}},\ket{\Psi_{\theta+\Delta}}\right]\right)$
(11) $\displaystyle=$ $\displaystyle-\eta g^{-1}f$

where
$\mathcal{F}[\ket{\psi},\ket{\phi}]=\arccos(\sqrt{\braket{\psi}{\phi}\braket{\phi}{\psi}/\braket{\psi}{\psi}\braket{\phi}{\phi}})$
and elements of the generic force $f_{i}$ and the geometric tensor $g_{ij}$
are given as

$\displaystyle f_{i}$ $\displaystyle=$
$\displaystyle\frac{\braket{\partial_{i}\Psi_{\theta}}{H}{\Psi_{\theta}}}{\braket{\Psi_{\theta}}{\Psi_{\theta}}}-\frac{\braket{\partial_{i}\Psi_{\theta}}{\Psi_{\theta}}}{\braket{\Psi_{\theta}}{\Psi_{\theta}}}\frac{\braket{\Psi_{\theta}}{H}{\Psi_{\theta}}}{\braket{\Psi_{\theta}}{\Psi_{\theta}}},$
(12) $\displaystyle g_{ij}$ $\displaystyle=$
$\displaystyle\frac{\braket{\partial_{i}\Psi_{\theta}}{\partial_{j}\Psi_{\theta}}}{\braket{\Psi_{\theta}}{\Psi_{\theta}}}-\frac{\braket{\partial_{i}\Psi_{\theta}}{\Psi_{\theta}}}{\braket{\Psi_{\theta}}{\Psi_{\theta}}}\frac{\braket{\Psi_{\theta}}{\partial_{j}\Psi_{\theta}}}{\braket{\Psi_{\theta}}{\Psi_{\theta}}},$
(13)

where $\partial_{i}$ is the derivative with respect to the $i$-th element of
the parameter $\theta_{i}$.  It is noteworthy that the geometric tensor $g$ is
the extension of the Fisher information to quantum states. The stochastic
gradient method based on $g$, or the Fisher information, was independently
developed in the machine learning community Amari  _et al._ (1992), and is
frequently referred to as the natural gradient method.

Note that both $f$ and $g$ can be estimated efficiently using Monte Carlo
sampling. Indeed, any physical observable $O$ can be estimated for a quantum
state $\ket{\Psi}$ as

$\displaystyle\braket{O}=\frac{\braket{\Psi}{O}{\Psi}}{\braket{\Psi}{\Psi}}=\frac{\sum_{\sigma}|\Psi(\sigma)|^{2}O_{\rm
loc}(\sigma)}{\sum_{\sigma}|\Psi(\sigma)|^{2}}=\sum_{\sigma}p(\sigma)O_{\rm
loc}(\sigma),$ (14)

where $O_{\rm
loc}(\sigma)=\sum_{\sigma^{\prime}}\frac{\Psi(\sigma^{\prime})}{\Psi(\sigma)}\braket{\sigma}{O}{\sigma^{\prime}}$
is introduced to enable the simulation of the expectation value from classical
sampling over the probability distribution
$p(\sigma)=|\Psi(\sigma)|^{2}/\sum_{\sigma}|\Psi(\sigma)|^{2}$. Using the
Metropolis–Hastings algorithm with particle number conservation, we typically
sample $\mathcal{O}(10^{5})$ to $\mathcal{O}(10^{7})$ spin configurations to
estimate $p(\sigma)$. Each configuration is drawn every 10 to 20 Monte Carlo
steps so that the autocorrelation, and hence the sampling error, is
sufficiently small when the optimization converges.

Three technical remarks are in order.  First, we take the initial state
$\ket{\Psi_{0}}(=\ket{\Psi_{\theta_{0}}^{\rm RBM}})$ as the HF state such that
the overlap with the ground state is non-zero. Small noise is added to avoid
the gradient vanishing problem, which arises when the parameters of the RBM
state are tuned to express any computational basis exactly.  Second, to
stabilize the optimization, small number $\epsilon$ is uniformly added to the
diagonal elements of $g$ as $g_{ii}\rightarrow g_{ii}+\epsilon$. While large
$\epsilon$ is beneficial in early iterations, it is necessary to decrease it,
or otherwise one may result in undesirable local minima. Therefore, $\epsilon$
is initially set as $\mathcal{O}(10^{-2})$ and gradually decreased to
$\mathcal{O}(10^{-3})$ after several hundred steps. Third, we find that it is
crucial to adopt an appropriate scheduling of $\eta$ to speed up the
optimization and, more importantly, avoid local minima. In the present work,
we exclusively employ the RMSProp method Hinton  _et al._ (2012), which
adaptively modifies $\eta$ according to the magnitude of the gradient.

###  IX.2 Energy corrections by the extended Koopmans’ theorem

In Fig. 5, we visualize the effect of the corrections to the energy bands by
the extended Koopmans’ theorem, which are defined in Eq. (9) in the main text
as

$\displaystyle\begin{cases}\Delta E^{IP}&=E_{GS}^{N_{v}}-E_{GS}^{N_{v}-1},\\\
\Delta E^{EA}&=E_{GS}^{N_{v}+1}-E_{GS}^{N_{v}},\end{cases}$

where $E_{GS}^{n}$ is the energy of the RBM optimized in the particle-number
sector $n$. Here, panels (a) and (b) indicate the first conduction and valence
bands, respectively. In both bands, we observe a systematic deviation, which
we attribute to the lack of orbital relaxation effect caused by the removal or
addition of a single electron. The order of the correction $\Delta E\sim 0.05$
Ha is comparable to that of the electronic correlation ($\sim 0.1$ Ha).

Figure 5:  The effect of energy correction to quasiparticle bands. Here, we
display the (a) lowest conduction band and (b) highest valence band. The
unfilled green triangle denotes the raw values obtained by solving Eq. (6)
defined in the main text, and the filled ones indicate the values corrected by
Eq. (9) following the extended Koopmans’ theorem. The blue dotted lines,
yellow lines, and black dashd lines indicate the result by restricted Hartree-
Fock method, coupled-cluster equation-of-motion formalism with singles and
doubles (EOM-CCSD), and EOM-CCSD with perturbative triple excitation (EOM-
CCSD(T)(a)*), respectively.

##  X References

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# Geometric inequalities for CR-submanifolds

Mirjana Djorić 111 University of Belgrade, Faculty of Mathematics, Studentski
trg 16, 11000 Belgrade, Serbia
e-mail<EMAIL_ADDRESS>and Vladimir Rovenski 222University of Haifa,
Department of Mathematics, Mount Carmel, 3498838, Haifa, Israel
e-mail<EMAIL_ADDRESS>
###### Abstract

We study two kinds of curvature invariants of Riemannian manifold equipped
with a distribution (for example, a CR-submanifold of an almost Hermitian
manifold) related to sets of pairwise orthogonal subspaces of the
distribution: one is similar to Chen’s $\delta$-invariants and another kind of
invariants is based on the mutual curvature of the subspaces. We compare Chen-
type invariants with the mutual curvature invariants and prove geometric
inequalities with intermediate mean curvature squared for CR-submanifolds in
almost Hermitian spaces. In the case of a set of complex planes, we study
curvature invariants based on the concept of holomorphic bisectional
curvature. As applications, we give consequences of the absence of some
$D$-minimal CR-submanifolds in almost Hermitian manifolds.

Keywords: almost Hermitian manifold; CR-submanifold; curvature invariant

Mathematics Subject Classifications (2010) 53C12; 53C15; 53C42

##  1 Introduction

The development of the extrinsic geometry of submanifolds and foliations (see
[10]) led to the following problem (see, for example, [3]): find a simple
optimal connection between the intrinsic and extrinsic invariants of a
submanifold in a Riemannian manifold; in particular, in space forms.

B.Y. Chen introduced the concept of $\delta$-curvature invariants for
Riemannian manifolds in the 1990s and proved an optimal inequality for a
submanifold, including $\delta$-curvature invariants and the square of mean
curvature, e.g., [3]. The case of equality led to the notion of “ideal
immersions” in Euclidean space, that is isometric immersions with the smallest
possible tension. Chen invariants are obtained from the scalar curvature by
discarding some of the sectional curvatures.

In [8, 9], we introduced invariants of a Riemannian manifold, which are
related to the mutual curvature of noncomplementary pairwise orthogonal
subspaces of the tangent bundle, and proved geometrical inequalities for
Riemannian submanifolds with applications to foliations.

In the article, we study two kinds of curvature invariants of a Riemannian
manifold equipped with a complex distribution (for example, a CR-submanifold
of an almost Hermitian manifold) related to sets of pairwise orthogonal
subspaces of the distribution: one is similar to Chen’s $\delta$-invariants
and another kind of invariants is based on the mutual curvature of the
subspaces.

In this paper, we compare Chen-type invariants with the mutual curvature
invariants and prove geometric inequalities with intermediate mean curvature
squared for CR-submanifolds in almost Hermitian spaces. In the case of a set
of complex planes, we study curvature invariants based on the holomorphic
bisectional curvature of two planes. As applications, we give consequences of
the absence of some $D$-minimal CR-submanifolds in almost Hermitian manifolds.

The paper is organized as follows. After reporting in Section 2 some basic
information about the curvature invariants of a manifold with a distribution,
in Section 3, we study geometric inequalities for CR-submanifolds in almost
Hermitian manifolds.

##  2 Curvature invariants of a manifold with a distribution

In this section, we recall two kinds of curvature invariants of a manifold
with a distribution (Chen-type invariants and invariants based on the mutual
curvature, see [8, 9]), and for the complex distribution we define invariants
based on the holomorphic bisectional curvature.

Let $(M^{d+m},g)\ (d,m>0)$ be a Riemannian manifold with a $d$-dimensional
distribution $D$. Denote by $\nabla$ the Levi-Civita connection of $g$ and
$R_{X,Y}=\nabla_{X}\nabla_{Y}-\nabla_{Y}\nabla_{X}-\nabla_{[X,\,Y]}$ the
curvature tensor, where $X,Y$ are any vector fields on the tangent bundle
$TM$.

The scalar curvature $\tau$ (function on $M$) is the trace of the Ricci tensor
$\operatorname{Ric}_{X,Y}=\operatorname{trace}(Z\mapsto R_{Z,X}\,Y)$. Some
authors, for example, [3, 4], define the scalar curvature as half of “trace
Ricci”.

######  Example 1.

Let $g$ be an admissible metric for an almost contact structure
$(\varphi,\xi,\eta)$ on $M^{2n+1}$,

$\displaystyle g(\varphi X,\varphi
Y)=g(X,Y)-\eta(X)\,\eta(Y),\quad\eta(\xi)=1,\quad X,Y\in\mathfrak{X}_{M},$

see [2], where $\varphi$ is a $(1,1)$-tensor, $\xi$ is a unit vector field
(called Reeb vector field) and $\eta$ is a 1-form. Then $d=2n$ and
$D=\ker\eta$ is a $2n$-dimensional contact distribution on $M^{2n+1}$.

Next, we define curvature invariants related with $D$, which for $D=TM$ are
reduced to Chen’s $\delta$-invariants, for example, [3, Section 13.2].

######  Definition 1 ([9]).

Define Chen’s type curvature invariants $\delta_{D}$ and $\hat{\delta}_{D}$ by

$\displaystyle
2\,\delta_{D}(n_{1},\ldots,n_{k})(x)=\tau_{D}(x)-\min\\{\tau(V_{1})+\ldots+\tau(V_{k})\\},$
$\displaystyle
2\,\hat{\delta}_{D}(n_{1},\ldots,n_{k})(x)=\tau_{D}(x)-\max\\{\tau(V_{1})+\ldots+\tau(V_{k})\\},$
(1)

where $\tau(V_{i})=\operatorname{trace}_{g}\operatorname{Ric}|_{V_{i}}$,
$\tau_{D}(x)=\operatorname{trace}_{g}\operatorname{Ric}|_{D_{x}}$, and
$V_{1},\ldots,V_{k}$ run over all $k\geq 0$ mutually orthogonal subspaces of
$D_{x}$ at a point $x\in M$ such that $\dim V_{i}=n_{i}\ (0\leq i\leq k)$.

For example, $2\,\delta_{D}=\tau_{D}$ when $k=0$, and
$2\,\delta_{D}(n_{1})(x)=\tau_{D}(x)-\min\tau(V_{1})$ when $k=1$.

######  Remark 1.

In (1), we can use max and min (instead of sup and inf, see [3]) because the
set “all mutually orthogonal subspaces $V_{1},\ldots,V_{k}$ at a point $x\in
M$ such that …” is compact.

Let $\\{e_{i}\\}$ be an orthonormal frame of a subspace
$V=\bigoplus_{\,i=1}^{\,k}V_{i}$ of $T_{x}M$ such that

$\\{e_{1},\ldots,e_{n_{1}}\\}\subset
V_{1},\quad\ldots\quad\\{e_{n_{k-1}+1},\ldots,e_{n_{k}}\\}\subset V_{k}.$

For $k\geq 2$, the mutual curvature of a set $\\{V_{1},\ldots,V_{k}\\}$ is
defined by

${\rm S}_{\rm m}(V_{1},\ldots,V_{k})=\sum\nolimits_{\,i<j}{\rm S}_{\rm
m}(V_{i},V_{j}),$ (2)

where ${\rm S}_{\rm m}(V_{i},V_{j})=\sum\limits_{\,n_{i-1}<a\,\leq n_{i},\
n_{j-1}<b\leq n_{j}}g(R_{e_{a},{e}_{b}}\,e_{b},{e}_{a})$ is the mutual
curvature of a pair $(V_{i},V_{j})$.

Note that ${\rm S}_{\rm m}(V_{1},\ldots,V_{k})$ does not depend on the choice
of frames. We obtain

$\tau(V)=2\,{\rm S}_{\rm
m}(V_{1},\ldots,V_{k})+\sum\nolimits_{\,i=1}^{k}\tau(V_{i}),$ (3)

where $\tau(V)=\operatorname{trace}_{g}\operatorname{Ric}|_{\,V}$ is the trace
of the Ricci tensor on a subspace $V=\bigoplus_{\,i=1}^{\,k}V_{i}$.

For example, if all subspaces $V_{i}$ are one-dimensional, then $2\,{\rm
S}_{\rm m}(V_{1},\ldots,V_{k})=\tau(V)$.

We also introduce the curvature invariants based on the concept of mutual
curvature, see [9].

######  Definition 2 ([9]).

Define the mutual curvature invariants of a Riemannian manifold $(M^{d+m},g)$
equipped with a $d$-dimensional distribution $D$ by

$\delta^{+}_{{\rm m},D}(n_{1},\ldots,n_{k})(x)=\max{\rm S}_{\rm
m}(V_{1},\ldots,V_{k}),\quad\delta^{-}_{{\rm
m},D}(n_{1},\ldots,n_{k})(x)=\min{\rm S}_{\rm m}(V_{1},\ldots,V_{k}),$ (4)

where $V_{1},\ldots,V_{k}$ run over all $k\geq 2$ mutually orthogonal
subspaces of $D_{x}$ such that $\dim V_{i}=n_{i}$ $(2\leq i\leq k)$. For
$D=TM$, we get the mutual curvature invariants $\delta^{\pm}_{{\rm
m}}=\delta^{\pm}_{{\rm m},TM}$.

The invariants in (1) and (4) are related by the following inequalities.

######  Proposition 1.

For $k\geq 2$ and $n_{1}+\ldots+n_{k}<d$, the following inequalities are true:

$\displaystyle\delta^{+}_{{\rm
m},{D}}(n_{1},\ldots,n_{k})\geq\delta_{D}(n_{1},\ldots,n_{k})-\delta_{D}(n_{1}+\ldots+n_{k})\,,$
$\displaystyle\delta^{-}_{{\rm
m},{D}}(n_{1},\ldots,n_{k})\leq\hat{\delta}_{D}(n_{1},\ldots,n_{k})-\hat{\delta}_{D}(n_{1}+\ldots+n_{k})\,,$

and if $n_{1}+\ldots+n_{k}=d$, then

$\hat{\delta}_{D}(n_{1},\ldots,n_{k})=\delta^{-}_{{\rm
m},D}(n_{1},\ldots,n_{k})\leq\delta^{+}_{{\rm
m},D}(n_{1},\ldots,n_{k})=\delta_{D}(n_{1},\ldots,\linebreak n_{k}).$

If the sectional curvature $K$ along $D$ satisfies $c\leq K\leq C$ and
$\sum\nolimits_{\,i=1}^{k}n_{i}=s\leq d$, then

$\displaystyle\frac{c}{2}\,(s^{2}-\sum\nolimits_{\,i}n_{i}^{2})=c\sum\nolimits_{\,i<j}n_{i}\,n_{j}\leq\delta^{-}_{{\rm
m},D}(n_{1},\ldots n_{k})$ $\displaystyle\leq\delta^{+}_{{\rm
m},D}(n_{1},\ldots n_{k})\leq
C\sum\nolimits_{\,i<j}n_{i}\,n_{j}=\frac{C}{2}(s^{2}-\sum\nolimits_{\,i}n_{i}^{2})\,.$
(5)

###### Proof.

This is similar to the proof of Proposition 2 in [9]. ∎

######  Corollary 1.

If $(M^{d+m},D,g)$ has non-negative sectional curvature of planes tangent to
$D$, then

$\hat{\delta}_{D}(n_{1},\ldots,n_{k})\leq\delta^{-}_{{\rm
m},D}(n_{1},\ldots,n_{k})\leq\delta^{+}_{{\rm
m},D}(n_{1},\ldots,n_{k})\leq\delta_{D}(n_{1},\ldots,n_{k}),$

and if this sectional curvature is nonpositive, then the above inequalities
are opposite.

Given two $J$-invariant planes $\sigma$ and $\sigma^{\prime}$ (2-dimensional
subspaces) in $T_{x}M$ of an almost Hermitian manifold $(M,J,g)$, and unit
vectors $X\in\sigma$ and $Y\in\sigma^{\prime}$, Goldberg and Kobayashi [6]
defined the holomorphic bisectional curvature $K_{H}(\sigma,\sigma^{\prime})$
by

$K_{H}(\sigma,\sigma^{\prime})=R(X,JX,Y,JY).$ (6)

This definition depends on $\sigma$ and $\sigma^{\prime}$ only, and for
$\sigma=\sigma^{\prime}$ gives the holomorphic sectional curvature. For a set
of $J$-invariant planes in a complex distribution $D$ of real dimension $d\geq
4$, we introduce curvature invariants based on the holomorphic bisectional
curvature.

######  Definition 3.

Let $D$ be a $d$-dimensional complex distribution of a Riemannian manifold
$(M,g)$, i.e., there is a skew-symmetric (1,1)-tensor $J:D\to D$ such that

$J^{\,2}X=-X,\quad g(JX,JY)=g(X,Y),\quad X,Y\in D.$

The holomorphic mutual curvature invariants $\delta^{\pm}_{H,D}(k)\ (1<k\leq
d/2)$ are defined by

$\delta^{+}_{H,D}(k)(x)=\max{\rm
S}_{H}(\sigma_{1},\ldots,\sigma_{k}),\quad\delta^{-}_{H,D}(k)(x)=\min{\rm
S}_{H}(\sigma_{1},\ldots,\sigma_{k}),$

where $\sigma_{1},\ldots,\sigma_{k}$ run over all $k$ mutually orthogonal
$J$-invariant planes of $D_{x}$ at a point $x\in M$, and ${\rm
S}_{H}(\sigma_{1},\ldots,\sigma_{k})$ is defined using (6) by

${\rm
S}_{H}(\sigma_{1},\ldots,\sigma_{k})=\sum\nolimits_{\,i<j}K_{H}(\sigma_{i},\sigma_{j}).$
(7)

For $D=TM$, i.e., for an almost Hermitian manifold, we get the holomorphic
mutual curvature invariants $\delta^{\pm}_{H}(k):=\delta^{\pm}_{H,TM}(k)$.

######  Lemma 1.

Let $\\{\sigma_{1},\ldots,\sigma_{k}\\}\ (2\leq k\leq d/2)$ be mutually
orthogonal $J$-invariant planes of a complex distribution $D_{x}$ at a point
$x\in M$. Then

$2\,{\rm S}_{H}(\sigma_{1},\ldots,\sigma_{k})={\rm S}_{\rm
m}(\sigma_{1},\ldots,\sigma_{k}).$ (8)

###### Proof.

By the Bianchi identity, they get

$K_{H}(\sigma,\sigma^{\prime})=R(X,Y,X,Y)+R(X,JY,X,JY).$ (9)

Replacing $X$ by $JX$ in (9), we get
$K_{H}(\sigma,\sigma^{\prime})=R(JX,Y,JX,Y)+R(JX,JY,JX,JY)$. Thus, ${\rm
S}_{\rm m}(\sigma,\sigma^{\prime})=2\,K_{H}(\sigma,\sigma^{\prime})$. From
this, (2) and (7), the equality (8) follows. ∎

######  Corollary 2.

The following inequalities are true for $2\leq k\leq d/2$:

$2\,\delta^{+}_{H,D}(k)\leq\delta^{+}_{{\rm
m},D}(\underbrace{2,\ldots,2}_{k}),\quad
2\,\delta^{-}_{H,D}(k)\geq\delta^{-}_{{\rm
m},D}(\underbrace{2,\ldots,2}_{k}).$

##  3 CR-submanifolds in almost Hermitian manifolds

In this section, we prove geometric inequalities for CR-submanifolds.

An even-dimensional Riemannian manifold $(\bar{M},\bar{g})$ equipped with a
skew-symmetric (1,1)-tensor $\bar{J}$ such that $\bar{J}^{\,2}X=-X$ and
$\bar{g}(\bar{J}X,\bar{J}Y)=\bar{g}(X,Y)$ for all $X,Y\in TM$, is called an
almost Hermitian manifold, see [7]. We will put a top “bar” for objects
related to $\bar{M}$.

A submanifold $M^{d+m}\ (d,m>0)$ of an almost Hermitian manifold
$(\bar{M},\bar{J},\bar{g})$ is called a CR-submanifold if $D=\bar{J}(TM)\cap
TM$ is a complex distribution (the maximal $\bar{J}$-invariant subbundle) of
constant real dimension $d$, see [5, Definition 7.2].

A more rigid definition is given in [1]: a real submanifold $M^{d+m}\ (d,m>0)$
of an almost Hermitian manifold $(\bar{M},\bar{J},\bar{g})$ is called a CR-
submanifold if there exists on $M$ a totally real distribution $D^{\bot}$
(i.e., $\bar{J}(D^{\bot})\subset T^{\bot}M$) whose orthogonal complement $D$
(i.e., $TM=D\oplus D^{\bot}$) is a complex distribution (i.e., $\bar{J}(D)=D$)
of constant real dimension $d$. Both definitions above give the same thing
when the dimension of $D$ is maximum, that is, $D^{\perp}$ is one-dimensional.
The main examples are real hypersurfaces, although there are others.

######  Example 2.

Let $(M^{d+1},g)$ be a CR-submanifold of an almost Hermitian manifold
$(\bar{M},\bar{J},\bar{g})$ with a $d$-dimensional complex distribution
$D=\bar{J}(TM)\cap TM$. Then $M^{d+1}$ admits an almost contact metric
structure $(\varphi,\xi,\eta,g)$, where $\varphi=\bar{J}|_{D}$ and $\xi$ is a
unit vector field orthogonal to $D$.

Let ${h}:TM\times TM\to TM^{\bot}$ be the second fundamental form of the
submanifold $(M,g)$ of the Riemannian manifold $(\bar{M},\bar{g})$. Recall the
Gauss equation for $M\subset\bar{M}$, e.g., [3]:

$\bar{g}(\bar{R}_{Y,Z}\,U,X)=g(R_{Y,Z}\,U,X)+g({h}(Y,U),{h}(Z,X))-g({h}(Z,U),{h}(Y,X)),$
(10)

where $U,X,Y,Z\in TM$ and $\bar{R}$ and $R$ are the curvature tensors of
$(\bar{M},\bar{g})$ and $(M,g)$, respectively. The mean curvature vector field
of a subspace $V\subset T_{x}M$ is given by $H_{V}=\sum_{i}h(e_{i},e_{i})$,
where $e_{i}$ is an orthonormal basis of $V$. In short form, we will write
$H_{i}$ instead of $H_{V_{i}}$, and $H$ when $V=T_{x}M$. For a CR-submanifold
$(M^{d+m},g)$, set

${\cal H}_{D_{x}}(s)=\max\\{\,\|{H}_{V}\,\|:\ V\subset D_{x},\ \dim V=s>0\\}.$

If $s=d$, then ${\cal H}_{D_{x}}(d)=\|H_{D_{x}}\|$, where $H_{D_{x}}$ is the
mean curvature vector of $D_{x}$. Note that for $s<d$, the equality ${\cal
H}_{D_{x}}(s)=0$ implies $h\,|_{D_{x}}=0$.

A CR-submanifold $(M,g)$ in an almost Hermitian space
$(\bar{M},\bar{J},\bar{g})$ is called ${D}$-minimal (where $D=\bar{J}(TM)\cap
TM$) if ${H}_{D}\equiv 0$. A CR-submanifold $(M^{d+m},g)$ is called mixed
totally geodesic on $V=\bigoplus_{\,i=1}^{\,k}V_{i}\subset D$ if ${h}(X,Y)=0$
for all $X\in{V}_{i},\,Y\in{V}_{j}$ and $i\neq j$. Note that

$\delta^{+}_{{\rm m},{D}\,}(n_{1},\ldots,n_{k})\leq\delta^{+}_{{\rm
m}}(n_{1},\ldots,n_{k}),$

and, for $s<d$, by the condition ${\cal H}_{{D}}(s)=0$, $M$ is totally
geodesic on ${D}$, i.e., $h\,|_{{D}_{x}}=0$.

######  Theorem 1.

Let $(M^{d+m},g)$ be a CR-submanifold of an almost Hermitian manifold
$(\bar{M},\bar{J},\bar{g})$, and $D=\bar{J}(TM)\cap TM$. For any natural
numbers $n_{1},\ldots,n_{k}$ such that $\sum_{\,i}n_{i}=s\leq d$, we obtain

$\delta^{+}_{{\rm m},D}(n_{1},\ldots,n_{k})\leq\bar{\delta}^{+}_{\rm
m}(n_{1},\ldots,n_{k})+\frac{k-1}{2\,k}\left\\{\begin{array}[]{cc}{\cal
H}_{D}(s)^{2},&{\rm if}\ s<d,\\\ \|\,{H}_{D}\,\|^{2},&{\rm if}\
s=d,\end{array}\right.$ (11)

where $\bar{\delta}^{+}_{\rm m}(n_{1},\ldots,n_{k})$ are defined for
$(\bar{M},\bar{g})$ similarly to $\delta^{+}_{\rm m}(n_{1},\ldots,n_{k})$ for
$(M,g)$, see Definition 2. The equality in (11) holds at a point $x\in
M^{d+m}$ if and only if there exist mutually orthogonal subspaces
${V}_{1},\ldots,{V}_{k}$ of $D_{x}$ with $\sum_{\,i}n_{i}=s$ such that
$M^{d+m}$ is mixed totally geodesic on $V=\bigoplus_{\,i=1}^{\,k}V_{i}$,
${H}_{1}=\ldots={H}_{k}$, $\|H_{V}\|={\cal H}_{D_{x}}(s)$ and $\bar{\rm
S}_{\rm m}({V}_{1},\ldots,{V}_{k})=\bar{\delta}^{+}_{\rm
m}(n_{1},\ldots,n_{k})(x)$.

###### Proof.

Taking a trace of the Gauss equation (10) for the submanifold $M^{d+m}$ along
$V$ and $V_{i}$ yields the equalities

$\displaystyle\bar{\tau}(V)-\tau(V)=\|{h}_{V}\|^{2}-\|{H}_{V}\|^{2},\quad\bar{\tau}({V}_{i})-\tau({V}_{i})=\|{h}_{i}\|^{2}-\|{H}_{i}\|^{2},$
(12)

where $\bar{\tau}(V)$, $\bar{\tau}({V}_{i})$ and $\tau(V)$, $\tau({V}_{i})$
are the scalar curvatures of subspaces $V=\bigoplus_{\,i=1}^{\,k}V_{i}$ and
${V}_{i}$ for the curvature tensors $\bar{R}$ and $R$, respectively, at the
point $x\in M$.

Assume that ${H}_{V}\neq 0$ is valid on an open set $U\subset M$. We
complement an adapted local orthonormal frame $\\{e_{1},\ldots,e_{d}\\}$ of
$D$ over $U$ with a vector field $e_{d+1}$ parallel to ${H}_{V}$. Using
$H_{V}=\sum_{\,i=1}^{\,k}H_{i}$ and the inequality
$a_{1}^{2}+\ldots+a_{k}^{2}\geq\frac{1}{k}\,(a_{1}+\ldots+a_{k})^{2}$ for
$a_{i}=\bar{g}({H}_{i},e_{d+1})$, we find

$\sum\nolimits_{\,i}\|{H}_{i}\|^{2}\geq\sum\nolimits_{\,i}\bar{g}({H}_{i},e_{d+1})^{2}\geq\frac{1}{k}\,\|{H}_{V}\|^{2},$
(13)

and the equality holds if and only if ${H}_{1}=\ldots={H}_{k}$. The inequality
(13) is trivially satisfied for ${H}_{V}=0$; hence, it is valid on $M$. Set
$\|{h}^{\rm
mix}_{ij}\|^{2}=\sum_{\,e_{a}\in{V}_{i},\,e_{b}\in{V}_{j}}\|{h}(e_{a},e_{b})\|^{2}$
for $i\neq j$. Note that

$\|{h}_{V}\|^{2}=\sum\nolimits_{\,i}\|{h}_{i}\|^{2}+\sum\nolimits_{\,i<j}\|{h}^{\rm
mix}_{ij}\|^{2}\geq\sum\nolimits_{\,i}\|{h}_{i}\|^{2},$ (14)

and the equality holds if and only if $\|{h}^{\rm mix}_{ij}\|^{2}=0\
(\forall\,i<j)$, i.e., $M^{d+m}$ is mixed totally geodesic along $V$. By
(12)–(14) and the following equalities, see (3):

$\bar{\tau}(V)=2\,\bar{\rm S}_{\rm
m}({V}_{1},\ldots,{V}_{k})+\sum\nolimits_{\,i}\bar{\tau}({V}_{i}),\quad\tau(V)=2\,{\rm
S}_{\rm m}({V}_{1},\ldots,{V}_{k})+\sum\nolimits_{\,i}\tau({V}_{i}),$

we obtain

$\displaystyle 2\,{\rm S}_{\rm m}({V}_{1},\ldots,{V}_{k})=2\,\bar{\rm S}_{\rm
m}({V}_{1},\ldots,{V}_{k})+\sum\nolimits_{\,i}\big{(}\bar{\tau}({V}_{i})-\tau({V}_{i})\big{)}+\|{H}_{V}\|^{2}-\|{h}_{V}\|^{2}$
$\displaystyle\leq 2\,\bar{\delta}^{+}_{{\rm
m},D}(n_{1},\ldots,n_{k})-(\|{h}_{V}\|^{2}-\sum\nolimits_{\,i}\|{h}_{i}\|^{2})+(\|{H}_{V}\|^{2}-\sum\nolimits_{\,i}\|{H}_{i}\|^{2})$
$\displaystyle\leq 2\,\bar{\delta}^{+}_{{\rm
m},D}(n_{1},\ldots,n_{k})+\frac{k-1}{k}\,{\cal H}_{D}(s)^{2},$

and the equality holds in the second line if and only if $\bar{\rm S}_{\rm
m}({V}_{1},\ldots,{V}_{k})=\bar{\delta}^{+}_{{\rm m},D}(n_{1},\ldots,n_{k})$
and $\|H_{V}\|={\cal H}_{x}(s)$ at each point $x\in M$. This proves (11) for
$s<d$. The case $\sum_{\,i}n_{i}=d$ of (11) can be proved similarly. ∎

######  Remark 2.

For a CR-submanifold $(M^{d+m},g)$ in an almost Hermitian space
$(\bar{M},\bar{J},\bar{g})$ with sectional curvature bounded above by
$c\in\mathbb{R}$, for $\sum_{\,i}n_{i}=s\leq d$ from (1) and (11), we obtain

$\delta^{+}_{{\rm
m},D}(n_{1},\ldots,n_{k})\leq\left\\{\begin{array}[]{cc}\frac{c}{2}\,(s^{2}-\sum\nolimits_{\,i}n_{i}^{2})+\frac{k-1}{2\,k}\,{\cal
H}_{D}(s)^{2},&{\rm if}\ s<d,\\\
\frac{c}{2}\,(d^{2}-\sum\nolimits_{\,i}n_{i}^{2})+\frac{k-1}{2\,k}\,\|{H}_{D}\,\|^{2},&{\rm
if}\ s=d.\end{array}\right.$ (15)

For $s=d$, the RHS of (15) coincides with the RHS of Eqn. (13.43) in [3] for
$\sum_{\,i}n_{i}=d$.

As real hypersurfaces of almost Hermitian manifolds are the main examples of
CR-submanifolds and important objects in the study of geometrical
inequalities, we reformulate Theorem 1 especially for this case.

######  Corollary 3.

Let $(M^{2n+1},g)$ be a hypersurface with a contact distribution
$D=\bar{J}(TM)\cap TM$ of an almost Hermitian manifold
$(\bar{M},\bar{J},\bar{g})$. For any natural numbers $n_{1},\ldots,n_{k}$ such
that $\sum_{\,i}n_{i}=s\leq 2n$, we obtain the inequality

$\delta^{+}_{{\rm m},D}(n_{1},\ldots,n_{k})\leq\bar{\delta}^{+}_{\rm
m}(n_{1},\ldots,n_{k})+\frac{k-1}{2\,k}\left\\{\begin{array}[]{cc}{\cal
H}_{D}(s)^{2},&{\rm if}\ s<2n,\\\ \|\,{H}_{D}\,\|^{2},&{\rm if}\
s=2n,\end{array}\right.$ (16)

where $\bar{\delta}^{+}_{\rm m}(n_{1},\ldots,n_{k})$ are defined for
$(\bar{M},\bar{g})$ similarly to $\delta^{+}_{\rm m}(n_{1},\ldots,n_{k})$ for
$(M,g)$. The equality in (16) holds at a point $x\in M^{2n+1}$ if and only if
there exist mutually orthogonal subspaces ${V}_{1},\ldots,{V}_{k}$ of $D_{x}$
with $\sum_{\,i}n_{i}=s$ such that $M^{2n+1}$ is mixed totally geodesic on
$V=\bigoplus_{\,i=1}^{\,k}V_{i}$, ${H}_{1}=\ldots={H}_{k}$, $\|H_{V}\|={\cal
H}_{D_{x}}(s)$ and $\bar{\rm S}_{\rm
m}({V}_{1},\ldots,{V}_{k})=\bar{\delta}^{+}_{\rm m}(n_{1},\ldots,n_{k})(x)$.

For any $k$-tuple $(n_{1},\ldots,n_{k})$ with $\sum_{\,i}n_{i}=s\leq d$,
define the normalized $\delta_{{\rm m},D}$-curvature by

$\Delta_{{\rm m},D}(n_{1},\ldots,n_{k})=\frac{2\,k}{k-1}\,\delta^{+}_{{\rm
m},D}(n_{1},\ldots,n_{k}),$

and put $\bar{\Delta}_{{\rm m},D}:=\max\Delta_{{\rm
m},D}(n_{1},\ldots,n_{k})$, where $\sum_{\,i}n_{i}=s$.

Theorem 1 provides the following (compare with the maximum principle in [3, p.
268]).

######  Proposition 2.

If a CR-submanifold $(M^{d+m},g)$ of $\mathbb{C}^{l}$ satisfies ${\cal
H}_{D}(s)^{2}=\Delta_{{\rm m},D}(n_{1},\ldots,n_{k})$ for some $k$-tuple
$(n_{1},\ldots,n_{k})$ with $\sum_{\,i}n_{i}=s\leq d$, then for all
$(m_{1},\ldots,m_{k})$ with $\sum_{\,i}m_{i}=s$, we get

$\Delta_{{\rm m},D}(n_{1},\ldots,n_{k})\geq\Delta_{{\rm
m},D}(m_{1},\ldots,m_{k}).$

###### Proof.

By the conditions, $\Delta_{{\rm m},D}(n_{1},\ldots,n_{k})=\bar{\Delta}_{{\rm
m},D}(s)$. Since $\Delta_{{\rm m},D}(m_{1},\ldots,m_{k})\leq{\cal
H}_{D}(s)^{2}$, we obtain the inequality $\Delta_{{\rm
m},D}(m_{1},\ldots,m_{k})\leq\Delta_{{\rm m},D}(n_{1},\ldots,n_{k})$, which
completes the proof. ∎

######  Corollary 4.

For every CR-submanifold $(M^{d+m},g)$ in $\mathbb{C}^{l}$, we have ${\cal
H}_{D}(s)^{2}\geq\bar{\Delta}_{{\rm m},D}(s)$ for any $s<d$, and
$\|{H}_{D}\,\|^{2}\geq\bar{\Delta}_{{\rm m},D}$.

######  Remark 3.

The case of equality in Corollary 4 is of special interest. Such extremal CR-
immersions in $\mathbb{C}^{l}$ can be compared to “ideal immersions”
introduced by Chen’s for real space forms in terms of $\delta$-invariants,
e.g., [3, Definition 13.3].

The theory of $\delta_{D}$-invariants (1) of CR-submanifolds can be developed
similarly to the theory of Chen’s $\delta$-invariants of a Riemannian
submanifolds. Similarly to [3, Theorem 13.5], we get

######  Theorem 2.

Let $(M^{d+m},g)$ be a CR-submanifold of an almost Hermitian manifold
$(\bar{M},\bar{J},\bar{g})$ with sectional curvature bounded above by
$c\in\mathbb{R}$. For each $k$-tuple $(n_{1},\ldots,n_{k})$ such that
$\sum_{\,i}n_{i}\leq d$, we obtain

$\delta_{D}(n_{1},\ldots,n_{k})\leq\frac{d^{2}(d+k-1-\sum_{\,i}n_{i})}{2(d+k-\sum_{\,i}n_{i})}\|H_{D}\|^{2}+\frac{c}{2}\,[d(d-1)-\sum\nolimits_{\,i}n_{i}(n_{i}-1)].$
(17)

The case of equality in (17) is of special interest: extremal CR-submanifolds
in almost Hermitian spaces in terms of $\delta_{D}$-invariants are an analogue
of Chen’s “ideal immersions”.

Set $\delta^{+}_{{\rm m},D}(k){=}\max\delta^{+}_{{\rm
m},D}(n_{1},\ldots,n_{k})$ and $\delta^{-}_{{\rm
m},D}(k){=}\min\delta^{-}_{{\rm m},D}(n_{1},\ldots,n_{k})$, where
$\sum_{\,i}n_{i}\leq d$. The $\bar{\delta}^{+}_{{\rm m}}(k+1)$ are defined for
$(\bar{M},\bar{g})$ similarly to $\delta^{+}_{{\rm m}}(k+1)$ for $(M,g)$.

The following geometric inequality supplements (16).

######  Theorem 3.

Let $(M^{d+m},g)$ be a CR-submanifold of an almost Hermitian manifold
$(\bar{M},\bar{J},\bar{g})$. For any $k\geq 2$, we obtain

$\displaystyle\delta^{-}_{{\rm
m},D}(k)\leq\frac{k-1}{2\,k(k+1)}\,\|{H}_{D}\,\|^{2}+\bar{\delta}^{+}_{{\rm
m}}(k+1)\,.$ (18)

The equality in (18) holds at a point $x\in M^{d+m}$ if and only if there
exist mutually orthogonal subspaces ${V}_{1},\ldots,{V}_{k+1}$ of $D_{x}$ with
$\sum_{\,i=1}^{k+1}n_{i}=d$ such that $M^{d+m}$ is mixed totally geodesic,
${H}_{1}=\ldots={H}_{k+1}$, $\bar{\rm S}_{\rm
m}({V}_{1},\ldots,{V}_{k+1})=\bar{\delta}^{+}_{{\rm m}}(n_{1},\ldots,n_{k+1})$
and ${\rm S}_{\rm
m}(V_{1},\ldots,\widehat{V}_{i},\ldots,V_{k+1})=\delta^{-}_{{\rm m},D}(k)$ for
any $i=1,\ldots,k+1$, where $\widehat{V}_{i}$ means removing the space $V_{i}$
from the set $\\{V_{1},\ldots,V_{k+1}\\}$.

###### Proof.

Let $V_{k+1}$ be the orthogonal complement to $V=\bigoplus_{\,i=1}^{\,k}V_{i}$
in $D_{x}$. Note that

$\sum\nolimits_{\,i}{\rm S}_{\rm
m}(V_{1},\ldots,\widehat{V}_{i},\ldots,V_{k+1})=(k+1)\,{\rm S}_{\rm
m}(V_{1},\ldots,V_{k+1})\,.$

We also obtain $\delta^{-}_{{\rm m},D}(k)\leq\delta^{-}_{{\rm
m},D}(n_{1},\ldots,\widehat{n}_{i},\ldots,n_{k+1})\leq{\rm S}_{{\rm
m},D}(V_{1},\ldots,\widehat{V}_{i},\ldots,V_{k+1})$ for any $i=1,\ldots,k+1$.
Thus, $\delta^{-}_{{\rm m},D}(k)\leq{\rm S}_{\rm m}(V_{1},\ldots,V_{k+1})$,
and using (16) for $\sum_{\,i}n_{i}=d$ gives (18). ∎

From Theorems 1 and 3, we obtain the assertions on the absence of some CR-
submanifolds.

######  Corollary 5.

There are no $D$-minimal CR-submanifolds in $\mathbb{C}^{l}$ with any of the
following properties:

(a) $\delta^{+}_{{\rm m},D}(n_{1},\ldots,n_{k})>0$ for some
$(n_{1},\ldots,n_{k})$ with $\sum_{\,i}n_{i}=d$,

(b) $\delta^{-}_{{\rm m},D}(k)>0$ for some $k\geq 2$.

Next, we use the fact that the tangent distribution $TM$ is the sum
$TM={D}\oplus{D}^{\bot}$ of two mutually orthogonal distributions $D$ and
$D^{\bot}$ of ranks $d$ and $m$. Let $x\in M$ and $\\{e_{i}\\}$ on $(M,g)$ be
an adapted orthonormal frame, i.e., $\\{e_{1},\ldots,e_{d}\\}\subset{{D}(x)},\
\\{e_{d+1},\ldots,e_{d+m}\\}\subset{{D}^{\bot}(x)}$. The mutual curvature of
$({D},{D}^{\bot})$ is a function ${\rm S}_{\rm m}({D},{D}^{\bot})$ on $M$,
given at $x\in M$ by

${\rm S}_{\rm m}({D}(x),{D}^{\bot}(x))=\sum\nolimits_{\,1\leq a\,\leq d,\
d<b\leq d+m}K(e_{a},{e}_{b}).$

In this case, ${\rm S}_{\rm m}({D},{D}^{\bot})$ is the mixed scalar curvature;
see [8]. A CR-submanifold $(M,g)$ is called mixed totally geodesic on a pair
$({D},D^{\bot})$ if ${h}(X,Y)=0$ for all $X\in{D},\ Y\in{D}^{\bot}$.

######  Theorem 4.

Let $(M^{d+m},g;{D},{D}^{\bot})$ be a CR-submanifold of an almost Hermitian
manifold $(\bar{M},\bar{J},\bar{g})$. Then the following inequality holds:

${\rm S}_{\rm
m}({D},{D}^{\bot})\leq\frac{1}{2}\,\|H\|^{2}+\bar{\delta}^{+}_{\rm m}(d,m)\,.$
(19)

The equality in (19) holds at a point $x\in M$ if and only if $M^{d+m}$ is
mixed totally geodesic, ${H}_{D}(x)={H}_{D^{\bot}}(x)$ and $\bar{\rm S}_{\rm
m}({D}(x),{D}^{\bot}(x))=\bar{\delta}^{+}_{\rm m}(d,m)(x)$.

###### Proof.

The proof of the first claim is similar to the proof of Theorem 1. The proof
of the second assertion follows directly from the cases of equality, as in the
proof of Theorem 1. ∎

######  Example 3 (see [8]).

Consider distributions ${D},D^{\bot}$ on a domain $M$ on a unit sphere
$S^{d+m}(1)$ in $\mathbb{C}^{l}$; thus, $\bar{\delta}^{+}_{\rm m}(d,m)=0$.
Using coordinate charts, we can take integrable distributions ${D},D^{\bot}$,
and $M$ is diffeomorphic to the product of two manifolds.

Let $d=2$ and $m=1$; then, $\|{H}\,\|^{2}=9$ and ${\rm S}_{\rm
m}({D},{D}^{\bot})=2$. Hence, (19) reduces to the inequality $2<9/4$. Note
that ${H}_{D}=\frac{1}{3}\,{H}\not=\frac{2}{3}\,{H}={H}_{D^{\bot}}$.

Let $d=m=2$ and locally $M\subset S^{4}(1)$ be diffeomorphic to
$\mathbb{C}\times\mathbb{C}$. Then, $\|{H}\,\|^{2}=16$,
${H}_{D}={H}_{D^{\bot}}$, ${\rm S}_{\rm m}({D},{D}^{\bot})=4$ and (19) reduces
to the equality $4=16/4$.

######  Corollary 6.

A CR-submanifold $(M,g)$ in $\mathbb{C}^{l}$ with ${\rm S}_{\rm
m}({D},{D}^{\bot})>0$ cannot be ${D}$-minimal.

The following theorem states an inequality with holomorphic bisectional
curvature invariants.

######  Theorem 5.

Let $(M^{d+m},g)$ be a CR-submanifold of an almost Hermitian manifold
$(\bar{M},\bar{J},\bar{g})$. For any natural number $k\in[2,d/2]$, we obtain

$\delta^{+}_{H,D}(k)\leq\bar{\delta}^{+}_{H}(k)+\frac{k-1}{4\,k}\left\\{\begin{array}[]{cc}{\cal
H}_{D}(2k)^{2},&{\rm if}\ 2k<d,\\\ \|\,{H}_{D}\,\|^{2},&{\rm if}\
2k=d,\end{array}\right.$ (20)

where $\bar{\delta}^{+}_{H}(k)$ are defined for $(\bar{M},\bar{g})$ similarly
to $\delta^{+}_{H}(k)$ for $(M,g)$, see Definition 3. The equality in (20)
holds at a point $x\in M^{d+m}$ if and only if there exist mutually orthogonal
$J$-invariant planes $\\{\sigma_{1},\ldots,\sigma_{k}\\}$ of $D_{x}$ such that
$M^{d+m}$ is mixed totally geodesic on $V=\bigoplus_{\,i=1}^{\,k}\sigma_{i}$,
${H}_{1}=\ldots={H}_{k}$, $\|H_{V}\|={\cal H}_{D_{x}}(2k)$ and $\bar{\rm
S}_{H}(\sigma_{1},\ldots,\sigma_{k})=\bar{\delta}^{+}_{H}(k)(x)$.

###### Proof.

This is similar to the proof of Theorem 1. ∎

######  Corollary 7.

Let $(M^{d+1},g)$ be a hypersurface of an almost Hermitian manifold
$(\bar{M},\bar{J},\bar{g})$. Then (20) is true for any natural number
$k\in[2,d/2]$.

Using $\delta(2,\ldots,2)$-invariants, Chen classified in [3, Section 15.7]
extremal real hypersurfaces of Kählerian space forms. Similarly, we would like
to study the extreme case of Corollary 7 when $(\bar{M},\bar{J},\bar{g})$ is a
Kählerian space form.

From Theorem 5 we obtain the assertion on the absence of some CR-submanifolds.

######  Corollary 8.

A CR-submanifold $(M,g)$ in $\mathbb{C}^{l}$ satisfying
$\delta^{+}_{H,D}(d/2)>0$ cannot be $D$-minimal.

##  4 Conclusions

We studied the question of finding a simple optimal connection between the
intrinsic and extrinsic invariants of a manifold equipped with a distribution,
which are CR-submanifolds of almost Hermitian manifolds. The main contribution
of the paper are the concepts of curvature invariants of a manifold equipped
with a distribution, e.g., a CR-submanifold, based on the mutual curvature of
several pairwise orthogonal subspaces of a contact distribution $D$. We used
these curvature invariants and Chen-type curvature invariants to prove new
geometric inequalities involving the squared intermediate mean curvature for
hypersurfaces of almost Hermitian manifolds. In the case of a set of complex
planes, we study curvature invariants based on the concept of holomorphic
bisectional curvature. Consequences of the absence of some $D$-minimal
hypersurfaces in almost Hermitian manifolds were given.

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