# Calculation of Coulomb breakup cross sections using a new Coulomb dynamical
polarization potential

H. M. Maridi<EMAIL_ADDRESS>Heavy Ion Laboratory, University of
Warsaw, ul. Pasteura 5a, 02-093, Warsaw, Poland  Physics Department, Faculty
of Applied Science, Taiz University, Taiz, Yemen  K. Rusek  Heavy Ion
Laboratory, University of Warsaw, ul. Pasteura 5a, 02-093, Warsaw, Poland  N.
Keeley  National Centre for Nuclear Research, ul. Andrzeja Sołtana 7, 05-400
Otwock, Poland

###### Abstract

A new method for calculating the Coulomb breakup of unstable neutron-rich
isotopes at high energies is presented. The calculations employ the eikonal
approximation and use a new Coulomb dynamical polarization potential (CDPP),
calculated by solving the Schrödinger equation for the entire motion of the
exotic projectile as a two-body cluster structure using the adiabatic
approximation and incorporating excitations to the continuum. Calculations for
some exotic isotopes are compared with Coulomb dissociation cross section data
and found to be in good agreement.

##  I  Introduction

Recently Mar21 , we presented a new expression for the Coulomb dynamical
polarization potential (CDPP) which was obtained by solving the Schrödinger
equation for the internal motion of an exotic neutron-rich projectile
(considered as a two-body deuteronlike cluster structure) incident on a heavy
target nucleus using the adiabatic approximation. In this work we generalize
this CDPP to include both excited states of the core and excitations to the
continuum. This generalized CDPP is then used to calculate the differential
Coulomb dissociation (CD) cross section as a function of relative excitation
energy. These calculations start from the continuity equation using the
imaginary part of the CDPP. The eikonal approximation is then used to
calculate the CD cross section for a number of weakly-bound neutron-rich
exotic nuclei at high incident energies.

##  II  Theory

###  II.1  Continuum based CDPP

In Ref. Mar21  we gave the formalism for the scattering of a weakly-bound two-
body projectile (p) consisting of a core in its ground state plus a cluster of
$n$ valence neutrons from a heavy-ion target and the CDPP was obtained. If the
core is in an excited state, this expression for the CDPP can be easily
generalized by replacing the binding energy of the valence neutron or neutron
cluster with respect to the charged core of the projectile, $\varepsilon_{0}$,
with an effective separation energy
$\varepsilon^{*}_{0}=\varepsilon_{0}+\varepsilon_{I^{\pi}_{c}}$, where
$\varepsilon_{I^{\pi}_{c}}$ is the excitation energy of the core state of
spin-parity $I^{\pi}_{c}$. To solve the Schrödinger equations of the system
and obtain the CDPP one may use the adiabatic approximation $\Psi({\bf
r,R})\approx\psi({\bf R})\phi({\bf r,R})$, where $\psi({\bf R})$ refers to the
wave function of the center of mass and $\phi({\bf r,R})$ to that of the
relative motion of the projectile; ${\bf R}$ and ${\bf r}$ are the coordinates
of the projectile-target and the projectile valence-core systems,
respectively. By making the same approximations as in Ref. Mar21  the real and
imaginary parts of the CDPP can be given as:

$\displaystyle\delta V(R)$ $\displaystyle=$
$\displaystyle\varepsilon_{0}^{*}\left[\frac{QG_{0}F_{0}+Q^{2}G_{0}F_{0}G_{0}^{\prime}F_{0}^{\prime}+Q^{2}F_{0}^{2}F_{0}^{\prime
2}}{F_{0}^{4}+G_{0}^{2}F_{0}^{2}}-1\right]$ $\displaystyle\delta W(R)$
$\displaystyle=$
$\displaystyle\varepsilon_{0}^{*}\left[\frac{Q^{2}F_{0}F_{0}^{\prime}-QF_{0}^{2}}{F_{0}^{4}+G_{0}^{2}F_{0}^{2}}\right]$
(1)

where $F_{0}$ and $G_{0}$ are the regular and irregular Coulomb functions in
$\rho=k(R)R$ and $Q(R)=(\mu_{p}/m_{c})(k(R)/\kappa_{0})$ with
$\kappa_{0}=\sqrt{-2\mu_{p}\varepsilon_{0}^{*}/\hbar^{2}}$ where $\mu_{p}$ is
the core-valence reduced mass, $m_{c}$ the mass of the charged core, and

$k(R)\approx{\sqrt{\frac{2m_{c}^{2}}{\mu_{p}{\hbar}^{2}}(V_{C}({R})+\varepsilon_{0}^{*})}}$
(2)

is the wave number of the charged core in the field of the target that is
associated with the wave function of the internal motion of the projectile,
$\phi(r,R)$. It depends parametrically on the Coulomb potential between the
projectile and target, $V_{C}({R})$ and is different from the wave number of
the center-of-mass motion of the system that describes the motion of the
projectile along the Rutherford trajectory,
$K=\sqrt{2\mu(E-\varepsilon_{0}^{*})/\hbar^{2}}$ where $E$ is the incident
energy of the projectile and $\mu$ is the reduced mass of projectile-target
system. This CDPP (II.1) depends on the structure of the system but does not
depend on the incident energy of the projectile.

Excitations of the projectile to the continuum can be included by adding the
continuum energy $\varepsilon$, a continuous variable, to
$\varepsilon_{0}^{*}$ so that $k$ now becomes a function of $\varepsilon$:

$k(R,\varepsilon)\approx{\sqrt{\frac{2m_{c}^{2}}{\mu_{p}{\hbar}^{2}}(V_{C}({R})+\varepsilon_{0}^{*}+\varepsilon)}},$
(3)

making the same approximations as before, and we accordingly obtain the
Continuum based CDPP, $\delta U(R,\varepsilon)=\delta V(R,\varepsilon)+i\delta
W(R,\varepsilon)$:

$\displaystyle\delta V(R,\varepsilon)$ $\displaystyle=$
$\displaystyle\varepsilon_{0}^{*}\left[\frac{QG_{0}F_{0}+Q^{2}G_{0}F_{0}G_{0}^{\prime}F_{0}^{\prime}+Q^{2}F_{0}^{2}F_{0}^{\prime
2}}{F_{0}^{4}+G_{0}^{2}F_{0}^{2}}\right]$
$\displaystyle-\varepsilon_{0}^{*}-\varepsilon$ $\displaystyle\delta
W(R,\varepsilon)$ $\displaystyle=$
$\displaystyle\varepsilon_{0}^{*}\left[\frac{Q^{2}F_{0}F_{0}^{\prime}-QF_{0}^{2}}{F_{0}^{4}+G_{0}^{2}F_{0}^{2}}\right]$
(4)

where $Q$ is now also a function of $\varepsilon$ as well as $R$.

We now consider an $E\lambda$-transition ($E1,E2,...$) from a bound state with
angular momentum $l_{0}$ to the $l$-wave continuum, with $l=l_{0}+\lambda$
where ${\lambda}$ is the transition multipolarity. The initial bound state
${J_{0}^{\pi}}$ is described by the wave function of the valence neutron in
the $l_{0}j_{0}$ orbital relative to the core $I^{\pi}_{c}$,
$\phi^{I^{\pi}_{c}}_{l_{0}j_{0}}({\bf r})$, whereas the final state
(continuum) is represented by the wave function $\phi_{\varepsilon lj}({\bf
r})$, where $\varepsilon$ in the subscript denotes the dependence on the
continuum energy. Here we include the notation $I^{\pi}_{c}$ to indicate that
the core is in state $I^{\pi}_{c}$. Then, we define a new CDPP that describes
this transition as

$\delta U^{{E\lambda,I^{\pi}_{c}}}_{l_{0}j_{0}\rightarrow\varepsilon lj}({\bf
R})=\int d\varepsilon\left\langle\phi_{\varepsilon lj}({\bf
r},\varepsilon)\left|\delta U^{I^{\pi}_{c}}({\bf
R,r},\varepsilon)\right|\phi^{I^{\pi}_{c}}_{l_{0}j_{0}}({\bf r})\right\rangle$
(5)

where $\delta U^{I^{\pi}_{c}}({\bf R,r},\varepsilon)$ can be represented by
expansion in Legendre polynomials as

$\delta U^{I^{\pi}_{c}}({\bf R,r},\varepsilon)=\sum_{\lambda}\delta
U^{I^{\pi}_{c}}({R},\varepsilon)P_{\lambda}(\cos(\theta_{r})).$ (6)

Let us define the excitation energy distribution of the $E\lambda$-transition
${l_{0}j_{0}\rightarrow\varepsilon lj}$ as

$\rho^{E\lambda,I^{\pi}_{c}}_{l_{0}j_{0}\rightarrow\varepsilon
lj}(\varepsilon)=\left\langle\phi_{\varepsilon lj}({\bf
r},\varepsilon)\left|P_{\lambda}(\cos(\theta_{r}))\right|\phi^{I^{\pi}_{c}}_{l_{0}j_{0}}({\bf
r})\right\rangle$ (7)

so that we may write

$\delta U^{E\lambda,I^{\pi}_{c}}_{l_{0}j_{0}\rightarrow\varepsilon
lj}({R})=\int
d\varepsilon\rho^{E\lambda,I^{\pi}_{c}}_{l_{0}j_{0}\rightarrow\varepsilon
lj}(\varepsilon)\delta U^{I^{\pi}_{c}}({R},\varepsilon).$ (8)

The radial wave function of the bound state is given by
$\phi^{I^{\pi}_{c}}_{l_{0}j_{0}}({r})=u_{l_{0}j_{0}}({r})/r$, where the
single-particle wave function $u_{l_{0}j_{0}}({r})$ is usually calculated
using a single-particle potential model, see for example Ref. Xu94 , by
adjusting the Woods-Saxon potential parameters to reproduce the experimental
neutron separation energy, taking the excitation energy of the core into
account where appropriate. Typical values of the potential radius parameter
$r_{0}$ range from 1.15 to 1.25 fm and the diffuseness $a$ from 0.5 to 0.7 fm.
It is, however, important to note that the extracted spectroscopic factor can
differ by 20$\%$ or more depending on the choice of parameters Pra03 . We
choose here the potential parameters ($r_{0}=1.15$ fm, $a=0.5$ fm) suggested
by Sauvan et al. Sau00  for light neutron-rich nuclei. Note that the potential
includes a Thomas form spin-orbit term with a strength fixed at 7.0 MeV for
all the cases studied here.

The final state (continuum) wave function is usually calculated assuming the
plane wave approximation (expanded in spherical Bessel functions and Legendre
polynomials) with the radial wave function normalized as $\phi_{\varepsilon
lj}(r)=\sqrt{2\mu_{p}\kappa/\hbar^{2}\pi}j_{l}(\kappa r)=u_{\varepsilon
lj}({r})/r$, where $\kappa=\sqrt{2\mu_{p}\varepsilon}/\hbar$ with relative
(continuum) energy $\varepsilon\equiv\varepsilon_{rel}$, and $l$ is the
orbital angular momentum of the final state.

Now, by considering the expansions of the bound-state and the continuum plane
wave functions, Eq. (7) becomes

$\displaystyle\rho^{E\lambda,I^{\pi}_{c}}_{l_{0}j_{0}\rightarrow\varepsilon
lj}(\varepsilon)$ $\displaystyle=$ $\displaystyle\int d{\bf
r}\phi_{\varepsilon lj}({\bf
r},\varepsilon)P_{\lambda}(\cos(\theta_{r}))\phi^{I^{\pi}_{c}}_{l_{0}j_{0}}({\bf
r})$ (9) $\displaystyle=$ $\displaystyle
4\pi\sqrt{\frac{2l+1}{2\lambda+1}}\int
d\Omega_{r}Y_{l_{0}0}(\Omega_{r})Y_{\lambda 0}(\Omega_{r})Y_{l0}(\Omega_{r})$
$\displaystyle\times$ $\displaystyle\int_{0}^{\infty}dru_{\varepsilon
lj}\left(r\right)u_{l_{0}j_{0}}\left(r\right)$ $\displaystyle=$
$\displaystyle\sqrt{4\pi(2l_{0}+1)}\left\langle l_{0}0\lambda
0|l0\right\rangle^{2}$ $\displaystyle\times$
$\displaystyle\int_{0}^{\infty}dr\ u_{\varepsilon
lj}\left(r\right)u_{l_{0}j_{0}}\left(r\right).$

Instead of the single-particle wave functions, the asymptotic form for overlap
integrals is often used

$\displaystyle
u_{l_{0}j_{0}}({r})=C_{l_{0}j_{0}}W_{-\eta_{0},l_{0}+1/2}(2\kappa_{0}r)$ (10)

where $W_{-\eta_{0},l_{0}+1/2}$ is the Whittaker function and $C_{l_{0}j_{0}}$
is the asymptotic normalization coefficient (ANC). The quantity
$\kappa_{0}=\sqrt{-2\mu_{p}\varepsilon_{0}^{*}}/\hbar$ where
$\varepsilon_{0}^{*}=\varepsilon_{0}+\varepsilon_{I^{\pi}_{c}}$ is the
effective separation energy of the valence nucleon(s) with respect to the
different core states ${I^{\pi}_{c}}$ of excitation energies
$\varepsilon_{I^{\pi}_{c}}$. When the valence cluster is formed exclusively of
neutron(s) $\eta_{0}=Z_{c}Z_{v}e^{2}\mu_{p}/\hbar^{2}\kappa_{0}=0$ and
$W_{0,{l_{0}+\scriptscriptstyle\frac{1}{2}}}(2\kappa_{0}r)=\frac{2\kappa_{0}r}{\pi}k_{l_{0}}(\kappa_{0}r)$
where $k_{l_{0}}$ is a modified spherical Bessel function and
$h^{(1)}_{l_{0}}$ is a spherical Hankel function of the first kind. Then
$u_{l_{0}j_{0}}(r)=C_{l_{0}j_{0}}\frac{2\kappa_{0}r}{\pi}k_{l_{0}}(\kappa_{0}r)$.
Now by using the Whittaker function for the bound-state wave function, Eq. (9)
becomes

$\displaystyle\rho^{E\lambda,I^{\pi}_{c}}_{l_{0}j_{0}\rightarrow\varepsilon
lj}(\varepsilon)$ $\displaystyle=$
$\displaystyle\sqrt{4\pi(2l_{0}+1)}\left\langle l_{0}0\lambda
0|l0\right\rangle^{2}\int_{0}^{\infty}dr\ u_{\varepsilon
lj}\left(r\right)u_{l_{0}j_{0}}\left(r\right)$ (11) $\displaystyle=$
$\displaystyle C_{l_{0}j_{0}}\sqrt{4\pi(2l_{0}+1)}\left\langle l_{0}0\lambda
0|l0\right\rangle^{2}{2\kappa_{0}\over\pi}\sqrt{2\mu_{p}\kappa\over\hbar^{2}\pi}\int_{0}^{\infty}d{r}r^{2}k_{l_{0}}({\kappa_{0}r})j_{l}({\kappa
r})$ $\displaystyle=$ $\displaystyle
C_{l_{0}j_{0}}\sqrt{4\pi(2l_{0}+1)}\left\langle l_{0}0\lambda
0|l0\right\rangle^{2}\sqrt{2\mu_{p}\kappa_{0}\over\hbar^{2}\pi}\frac{\kappa^{{\ell}+1/2}}{\kappa_{0}^{{\ell}+5/2}}\frac{\Gamma(\frac{\ell+\ell_{0}+3}{2})\Gamma(\frac{\ell-\ell_{0}+2}{2})}{\Gamma(\ell+\frac{3}{2})}$
$\displaystyle\times$
$\displaystyle{{}_{2}F_{1}}\left(\frac{\ell+\ell_{0}+3}{2},\frac{\ell-\ell_{0}+2}{2},\ell+\frac{3}{2},-\frac{\kappa^{2}}{\kappa_{0}^{2}}\right).$

In the particular case when the initial bound state has $l_{0}=0$, the radial
wave function can be given analytically as the Yukawa form
$\phi^{I^{\pi}_{c}}_{l_{0}j_{0}}({r})=\frac{u_{l_{0}j_{0}}({r})}{r}=C_{s_{1/2}}\frac{e^{-\kappa_{0}r}}{r}$,
so that for the $E1$ transition to the $p$-wave continuum, we obtain the
following analytical function:

$\rho^{{E1},I^{\pi}_{c}}_{s\rightarrow
p}(\varepsilon)=C_{s_{1/2}}\sqrt{\frac{8\mu_{p}\kappa}{\hbar^{2}}}\frac{1}{\kappa^{2}}\left(\frac{-\kappa_{0}\kappa}{\kappa_{0}^{2}+\kappa^{2}}+\tan^{-1}\left(\frac{\kappa}{\kappa_{0}}\right)\right)$
(12)

which will be especially useful since the valence neutron(s) are in a relative
$s$-state in the ground state of many light exotic projectiles of interest.

###  II.2  Coulomb dissociation using the CDPP

This new CDPP (8) has many potential applications to the scattering and
reactions of weakly-bound neutron-rich nuclei. Here we use it to calculate
Coulomb dissociation, keeping other applications for future publications.
Coulomb breakup or dissociation can take place when a high energy (several
hundred MeV/nucleon) projectile impinges on a heavy target and is excited by
absorbing virtual photons from the time-dependent Coulomb field. Under these
circumstances the electromagnetic excitation is dominated by dipole excitation
Pra03 . The corresponding differential cross section for dipole excitation
decomposes into an incoherent sum of components corresponding to the different
core states populated by neutron removal. For each core state the cross
section further decomposes into an incoherent sum over contributions from the
different allowed angular momenta of the valence neutron in its initial state.
In these calculations the projectile is assumed to have a core plus valence
neutron structure. Choosing the lowest core states there are various ways to
couple the spins of the core and the valence neutron to the total angular
momentum and parity of the projectile Hei17 . Note that for heavy targets the
core is assumed to act as a spectator so that the projectile core remains in
its initial state after removal of a neutron Han03 .

We start from the usual formula for the absorption cross section derived from
the continuity equation

$\sigma_{\mathrm{abs}}=-\frac{2}{\hbar v}\left\langle\psi^{(+)}_{K}({\bf
R})\left|W({R})\right|\psi^{(+)}_{K}({\bf R})\right\rangle$ (13)

where $W({R})$ is the imaginary potential, $\psi^{(+)}_{K}$ the usual
distorted wave function, $v$ the relative velocity, and $K$ the wavenumber in
the center of mass system. By considering the imaginary CDPP (5) that
describes the ${E\lambda}$-transitions we can write the total absorption due
to Coulomb dissociation and excitation by taking a summation over the
${E\lambda}$-transitions, the different core states $I^{\pi}_{c}$, and
contributions from different angular momenta $l_{0}j_{0}$ of the valence
neutron in its initial state for each core state:

$\displaystyle\sigma_{\mathrm{CD}}$ $\displaystyle=$
$\displaystyle-\frac{2}{\hbar v}\left\langle\psi^{(+)}_{K}({\bf
R})\left|\delta W({R})\right|\psi^{(+)}_{K}({\bf R})\right\rangle$ (14)
$\displaystyle=$ $\displaystyle-\frac{2}{\hbar v}\int
d\varepsilon\sum_{I^{\pi}_{c}}\sum_{\lambda}\sum_{l_{0}j_{0}}\sum_{lj}\rho^{E\lambda,I^{\pi}_{c}}_{l_{0}j_{0}\rightarrow\varepsilon
lj}(\varepsilon)\left\langle\psi^{(+)}_{K}({\bf R})\left|\delta
W^{I^{\pi}_{c}}({R},\varepsilon)\right|\psi^{(+)}_{K}({\bf R})\right\rangle.$

Then, the differential cross section for the $E\lambda$-transition
$l_{0}j_{0}\rightarrow\varepsilon lj$ with the core in state $I^{\pi}_{c}$ is

$\frac{d\sigma^{I^{\pi}_{c}}}{d\varepsilon}(E\lambda,{l_{0}j_{0}\rightarrow\varepsilon
lj})=-\frac{2}{\hbar
v}\rho^{E\lambda,I^{\pi}_{c}}_{l_{0}j_{0}\rightarrow\varepsilon
lj}(\varepsilon)\left\langle\psi^{(+)}_{K}({\bf R})\left|\delta
W^{I^{\pi}_{c}}({R},\varepsilon)\right|\psi^{(+)}_{K}({\bf R})\right\rangle,$
(15)

which is similar to the well-known Hussein–McVoy formula and the similar
formulae for the inclusive breakup Hus85 ; Ram21 . Introducing

$\displaystyle\hat{\sigma}^{I^{\pi}_{c}}(\varepsilon)$ $\displaystyle=$
$\displaystyle-\frac{2}{\hbar v}\left\langle\psi^{(+)}_{K}({\bf
R})\left|\delta W^{I^{\pi}_{c}}({\bf R},\varepsilon)\right|\psi^{(+)}_{K}({\bf
R})\right\rangle$ (16)

as the total absorption cross section for the $I^{\pi}_{c}$ core state, the
breakup cross section for the core in state $I^{\pi}_{c}$ and the
$E\lambda$-transition from ${l_{0}j_{0}}$ is

$\sigma^{I^{\pi}_{c}}_{E\lambda,l_{0}j_{0}}=\sum_{lj}\int_{0}^{\infty}d\varepsilon\rho^{E\lambda,I^{\pi}_{c}}_{l_{0}j_{0}\rightarrow\varepsilon
lj}(\varepsilon)\hat{\sigma}^{I^{\pi}_{c}}(\varepsilon).$ (17)

Equations (15) and (17) can be generalized to account for excitations of the
target. An imaginary nuclear potential can be added to account for any other
reaction channels. From Equation (17) and using the virtual photon method
Ber88  one may obtain the photo-absorption cross section $\sigma_{\gamma n}$,
the radiative neutron capture cross section $\sigma_{n\gamma}$, the dipole
response function $dB/d\varepsilon$, and the double differential cross section
$d^{2}\sigma/d\Omega d\varepsilon$. These calculations can also be generalized
to consider transitions between two bound states by replacing the continuum
wave function $u_{\varepsilon lj}(r)$ with a bound state wave function
$u_{lj}(r)$ similar to $u_{l_{0}j_{0}}(r)$.

Equation (16) can be solved by the method of partial wave expansion and the
complete scattering amplitude is obtained by summing over all the partial wave
scattering amplitudes. At high energies, i.e. hundreds of MeV/nucleon, the
wave function will oscillate rapidly and the calculation of scattering wave
functions for each partial wave becomes more complicated. In the eikonal
approximation and the optical limit of the Glauber theory Gla59 , the wave
function of the projectile-target system can be written in terms of the total
potential, $V+iW$, as

$\psi_{K}({\bf R})=\exp\left(iKz+\frac{1}{i\hbar
v}\int_{-\infty}^{z}(V(\mathbf{b},{z}^{\prime})+iW(\mathbf{b},{z}^{\prime}))d{z}^{\prime}\right)$
(18)

where ${\bf R}=(x,y,z)=(\mathbf{b},z)$ and ${\bf b}$ is the impact parameter.
Equation (16) may then easily be written as

$\displaystyle\hat{\sigma}^{I^{\pi}_{c}}(\varepsilon)$ $\displaystyle=$
$\displaystyle\int d\mathbf{b}\left[1-e^{\frac{2}{\hbar
v}\int_{-\infty}^{\infty}\delta W^{I^{\pi}_{c}}({\bf
b},\acute{z},\varepsilon)d\acute{z}}\right]$ (19)

where $d\mathbf{b}=2\pi bdb$. At this point we should include the correction
of the impact parameter due to Coulomb deflection of the particle trajectory:
$b^{\prime}=a_{0}+\sqrt{a_{0}^{2}+b^{2}}$, where
$a_{0}=Z_{p}Z_{t}e^{2}/mv^{2}$ is half the distance of closest approach in a
head-on collision of point charged particles. The relativistic correction to
the kinematics can be appropriately made if one replaces the quantity $a_{0}$
with $a_{0}=Z_{p}Z_{t}e^{2}/\gamma mv^{2}$, where
$\gamma=1/\sqrt{1-v^{2}/c^{2}}$ is the Lorentz factor Ber03 . Since the phase
in Eq. (19) is integrated over $z$ it is invariant under Lorentz
transformations, so no further relativistic corrections to the dynamical
equations are required Mos19 .

##  III  Application to some reactions

We now apply our model to calculate the Coulomb dissociation cross sections
for some exotic projectiles (11Be, 15C, 17C) incident on a lead target at a
few hundreds of MeV/nucleon. Our results using both the single particle (s.p.)
wave function $u_{l_{0}j_{0}}$ and the analytical formula (11) which assumes a
Whittaker wave function for the bound states are summarized in Table 1. All of
these calculations consider only the $E1$ transition to the continuum. In
addition, we define $R_{\sigma}={d\sigma_{\mathrm{exp}}\over
d\varepsilon}/{d\sigma_{\mathrm{th}}\over d\varepsilon}$ which is the ratio of
the experimental to the theoretical differential cross section using the best
fit in the energy range of the data. This ratio can be compared with the
theoretical and empirical spectroscopic factors listed in the last two columns
of Table 1. The experimental data are known to be affected by the experimental
resolution, so, using the method of Ref. Mor20 , the calculated cross sections
were convoluted with the detector response function obtained from simulated
spectra. Since the Coulomb dissociation data of 11Be, 15C, and 17C were all
measured at GSI and their experiments employed a similar setup Pal03 ; Pra03
we use the energy resolution obtained in Ref. Pal03  for all calculations. The
cross section is calculated by integration up to a continuum energy of 10 MeV
for most cases.

Table 1:  Calculated Coulomb dissociation cross sections for several exotic projectiles (with a core-neutron structure) incident on lead targets for different core excitation states using single-particle (s.p.) and Whittaker functions for the bound-state wave functions $u_{l_{0}j_{0}}$. The ratios $R_{\sigma}$ (see text) are compared with the spectroscopic factors extracted from experiment and from Shell-model calculations. Proj. | E/n | $J^{\pi}$ | Core | $I^{\pi}_{c}$ | ${l_{0}j_{0}}$ |  $\sigma_{\mathrm{exp}}$ (mb) |  $\sigma_{th}$ (mb) | $R_{\sigma}$ | Spectroscopic factors
---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  | s.p. | Whitt. | s.p. | Whitt. | Shell model |  experiment
11Be | 520 | $1/2^{+}$ |  10Be | $0^{+}$ | $s_{1/2}$ | 605$\pm$30 Pal03  | 1563 | 1649 | 0.40 | 0.40 | 0.74 Aum00  |  0.36, 0.60 Tim99 ; 0.61(5) Pal03 ; 0.72(4) Fuk04 ; 1.0(2) Nak94
|  |  |  | $2^{+}$ | $d_{5/2}$ |  | 34.4 | 58.2 |  |  |  |
15C | 605 | $1/2^{+}$ |  14C | $0^{+}$ | $s_{1/2}$ | 324$\pm$15 Pra03  | 570 | 643 | 0.52 | 0.48 | 0.98 Ter04 , 0.83 Sau00  |  0.91(6), 0.72(5) Nak09 ; 0.73(5), 0.97(8) Pra03
|  |  |  | $1^{-}$ | $p_{1/2}$ | 36$\pm$3 Pra03  | 8.2 | 17.1 |  |  | 1.03 Sau00  |  1.3(1) Pra03
|  |  |  |  | $p_{3/2}$ |  | 8.3 | 17.9 |  |  | 0.16 Sau00  |
|  |  |  | $0^{-}$ | $p_{1/2}$ |  | 4.8 | 10.5 |  |  | 0.46 Sau00 ; Ter04  |
|  |  |  | $2^{-}$ | $p_{3/2}$ |  | 3.9 | 8.8 |  |  | 0.016 Ter04  |
17C | 496 | $3/2^{+}$ |  16C | $0^{+}$ | $d_{3/2}$ |  $9^{+15}_{-9}$ Pra03  | 607 | 854 | 0.015 | 0.011 | 0.035 Sau00  |
|  |  |  | $2^{+}$ | $s_{1/2}$ | 62$\pm$7 Pra03  | 118.9 | 145.5 | 0.28 | 0.23 | 0.16Sau00 ; Mad01  |  0.23(8),26(14) Pra03
|  |  |  |  | $d_{5/2}$ |  | 96.8 | 162.1 | 0.23 | 0.14 | 1.41 Sau00 , 1.44 Mad01  |  0.6(4),1.6(6) Pra03
|  |  |  | $0^{+}$ | $d_{3/2}$ | 25$\pm$7 Pra03  | 33.9 | 60.5 |  |  |  |
|  |  |  | $4^{+}$ | $d_{5/2}$ |  | 15.5 | 30.1 |  |  | 0.76 Sau04 ; Mad01  |
Figure 1:  Coulomb dissociation cross section for a 11Be projectile incident
on a lead target at 520 MeV/nucleon. The solid symbols represent the data of
Ref. Pal03  and the open symbols the data after removal of the nuclear
contribution. The solid and short-dashed lines represent calculations using
Whittaker and single-particle wave functions for the bound state wave function
$u_{l_{0}j_{0}}$, respectively. The calculations have been convoluted with the
experimental resolution. See text for details.

We take as our first example the Coulomb dissociation of 11Be. The ground
state of 11Be is $1/2^{+}$, so the $0^{+}$ ground state of the 10Be core is
coupled to an $s$-wave valence neutron and the $2^{+}$ first excited state of
the core (3.368 MeV) is coupled to a $d$-wave valence neutron. The
spectroscopic factors for the $\left|{}^{10}\mathrm{Be}(0^{+})\otimes\nu
2s_{1/2}\right\rangle$ single-particle configuration obtained from shell model
calculations and deuteron stripping and pick-up reactions range from about
$0.4$–$0.8$, see for example Aum00  and references therein. An analysis of the
Coulomb dissociation of a 11Be projectile with an energy of 520 MeV/nucleon
impinging on lead and carbon targets gives a spectroscopic factor for the
$\left|{}^{10}\mathrm{Be}(0^{+})\otimes\nu 2s_{1/2}\right\rangle$ single-
particle configuration of 0.61(5) Pal03 . These data Pal03  include both
Coulomb and nuclear contributions and the Coulomb breakup spectrum can be
extracted by subtracting the nuclear contribution estimated using the data
taken with the carbon target in the following way

$\frac{d\sigma_{CD}}{d\varepsilon}=\frac{d\sigma}{d\varepsilon}(\mathrm{Pb})-\Gamma\frac{d\sigma}{d\varepsilon}(\mathrm{C})$
(20)

where $\Gamma$ is a scaling factor. Assuming the peripheral nature of the
nuclear excitation, $\Gamma$ can be taken as the ratio of the sum of the radii
of the target and the projectile which is 1.8 Fuk04 ; Nak94 . Other estimates
of $\Gamma$ for this reaction may be found in Refs. Fuk04 ; Nak94 ; Pal03 .
The effect of the nuclear contribution on this reaction has also recently been
studied in Ref. Mor20  by comparing the Equivalent Photon Method (EPM) and
continuum discretized coupled channels (CDCC) calculations.

The analysis of the data using the present formalism, with the Yukawa and s.p.
wave functions, gives us calculated differential cross sections in good
agreement with the data, as shown in Fig. 1, and a cross section ratio of
about 0.4 is obtained, which is similar to the spectroscopic factor of Ref.
Tim99 . It is clear that the contribution from the
$\left|{}^{10}\mathrm{Be}(2^{+})\otimes\nu 1d_{5/2}\right\rangle$
configuration is small. We expect that the contribution from the higher
$1^{-}$ and $2^{-}$ core states with $p$-wave neutrons will also be small.

Figure 2:  The same as Fig. 1 but for a 605 MeV/nucleon 15C projectile. The
data are from Ref. Pra03 . The calculations have been convoluted with the
experimental resolution.

The one-neutron separation energy of 15C is 1.218 MeV and its ground-state
spin-parity is $1/2^{+}$, so we suppose a $2s_{1/2}$ neutron coupled to a
14C($0^{+}$) core. Figure 2 presents calculations of the Coulomb breakup of
15C incident on a Pb target at 605 MeV/nucleon Pra03 . The peak of the
relative energy spectrum occurs at a value of 0.7 MeV. The original data Pra03
that include both Coulomb and nuclear contributions as well as the data
obtained by subtracting the nuclear contribution are plotted by the filled
dots and open squares, respectively. The calculated total Coulomb dissociation
cross section for 15C$\rightarrow$14C+$n$+$\gamma$ is about 570 mb using the
s.p. wave function and 643 mb using the Yukawa wave function for the
$|^{14}\mathrm{C}(0^{+})\otimes\nu 2s_{1/2}>$ configuration, and the
corresponding ratios of the experimental to calculated cross sections are
about 0.52 and 0.48, respectively. These ratios are compared with the
spectroscopic factors calculated by the shell model Ter04  or extracted from
Coulomb dissociation experiments Nak09 ; Pra03  in Table 1. There are many
excited states in 14C at energies between 6–7 MeV and previous calculations
with the core in these states, see Ref. Pra03 , gave small cross sections
compared to the observed value of 36(3) mb. For the 6.09 MeV ($1^{-}$), 6.90
MeV ($0^{-}$), and 7.34 MeV ($2^{-}$) core excited states the possible neutron
orbitals are $p_{1/2}$ or $p_{3/2}$. Our calculation using the s.p. wave
functions gives 4.2, 1.1, and 1 mb for dipole transitions to the $d$-continuum
and 12.4, 3.7, and 3 mb for transitions to the $s$-continuum for the $1^{-}$,
$0^{-}$, and $2^{-}$ core states, respectively. The calculation using the
Whittaker wave functions gives 4.7, 1.3, and 1.1 mb for dipole transitions to
the $d$-continuum and 30.2, 9.2, and 7.8 mb for transitions to the
$s$-continuum for the $1^{-}$, $0^{-}$, and $2^{-}$ core states, respectively.
If we multiply these cross sections by the corresponding spectroscopic factors
from shell model calculations Sau00 ; Ter04  this gives us values of 12.1 mb
and 25.4 mb from the calculations using s.p. and Whittaker functions,
respectively, whereas the experimental value is 36(3) mb Pra03 . The remaining
cross section may correspond to the 14C core in the other three states at 6–7
MeV.

Figure 3:  Coulomb dissociation cross sections for a 496 MeV/nucleon 17C
projectile incident on a lead target. The data are from Ref. Pra03 . The
calculations have been convoluted with the experimental resolution.

Calculations for different core excitations were performed for the 17C Coulomb
breakup since this yields the 16C core mainly in excited states. The
differential Coulomb dissociation cross section for
${}^{17}\mathrm{C}\rightarrow{{}^{16}\mathrm{C}}(2^{+})+n$ is reported in Ref.
Pra03 . As shown in Fig. 3, the data are well reproduced by our calculation
using a Yukawa wave function for the $|^{16}\mathrm{C}(2^{+})\otimes\nu
2s_{1/2}>$ configuration with an integrated cross section of 60 mb, similar to
the reported one of 62(7) mb. The peak position is also well reproduced. The
calculations using the s.p. wave functions and Whittaker wave function can fit
the data with cross section ratios of about 0.28 and 0.23 for the $s_{1/2}$
and 0.23 and 0.14 for $d_{5/2}$-orbits, respectively. The contribution of both
orbits to the cross section of the 16C($2^{+}$) excited state is not known
experimentally, only the total cross section of 62(7) mb. There are thus many
combinations which can fit the data, and the cross section ratio for $s_{1/2}$
or $d_{5/2}$ can be varied from zero to 0.5–0.6. As reported in Ref. Pra03 ,
the measured cross sections show that 64(9) % of the cross section corresponds
to the ($2^{+}$, 1.766 MeV) core state, 27(9) % to higher core excited states
at 3–4 MeV and a small part of the cross section leaves the core in its
$0^{+}$ ground state Pra03 . As shown in Table 1, our calculation using the
s.p. wave functions for the core in its ground state with a $1d_{3/2}$ valence
neutron gives a cross section of 153 mb for the transition to the $f$-wave and
454 mb for the transition to the $p$-wave continuum to give a total of 607 mb,
and the resulting ratio is very small (about 0.015), close to the value
expected from a shell model calculation Mad01 . Calculations using the
Whittaker wave functions give a cross section of 854 mb with ratio of 0.011.
For the higher states, the spectroscopic factor is unknown for the $1d_{3/2}$
valence neutron plus the ($0^{+}$, 3.027 MeV) core state and is 0.76 for the
$d_{5/2}$ valence neutron in the ($4^{+}$, 4.142 MeV) core state; our
calculations using s.p. and Whittaker functions give about 15 mb and 30 mb for
the cross section of $4^{+}$ state, respectively, whereas the measured value
is 25(7) mb Pra03 .

##  IV  Summary and conclusions

In summary, we have presented a new method to calculate the Coulomb
dissociation of exotic nuclei using an extended version of a recent model of
the CDPP taking into account excited states of the core and excitation to the
continuum. Breakup cross sections for two-body exotic projectiles may be
calculated using this new version of the CDPP which depends on the relative
excitation energy of the continuum. The method was used to calculate the
differential and integrated Coulomb breakup cross sections for several exotic
neutron-rich nuclei incident on lead targets at a few hundreds of MeV/nucleon.
The calculations at these high energies were performed using the eikonal
approximation and the results are in good agreement with the data. The
calculations could easily be generalized and applied to calculate neutron
removal and gamma capture cross sections. The new, extended CDPP may also be
applied to low energy scattering and reaction data via the optical model or
distorted wave Born approximation or coupled channel formalisms and has many
potential applications.

## Acknowledgments

We thank Professor Antonio Moro for his help in calculating the energy
convolution of the calculated cross sections. This work was funded by the
Polish National Agency for Academic Exchange (NAWA) within the Ulam Programme
under grant agreement No. PPN/ULM/2019/1/00189/U/00001.

## References

  * (1) H. M. Maridi, K. Rusek, and N. Keeley, Phys. Rev.C 104, 024614 (2021).
  * (2) H. M. Xu, C. A. Gagliardi, R. E. Tribble, A. M. Mukhamedzhanov, and N. K. Timofeyuk, Phys. Rev. Lett. 73, 2027 (1994).
  * (3) U. Datta Pramanik et al., Physics Letters B 551, 63 (2003).
  * (4) E. Sauvan et al., Phys. Lett. B 491 1 (2000).
  * (5) M. Heine et al., Phys. Rev. C 95, 014613 (2017).
  * (6) P. G. Hansen and J. A. Tostevin, Annu. Rev. Nucl. Part. Sci. 53, 219 (2003).
  * (7) M. S. Hussein and K. W. McVoy, Nucl. Phys. A 445, 124 (1985).
  * (8) M. Gómez-Ramos, J. Gómez-Camacho, Jin Lei, and A. M. Moro, Eur. Phys. J. A 57, 57 (2021).
  * (9) C. A. Bertulani and G. Baur, Phys. Rep. 163, 299 (1988).
  * (10) R. J. Glauber, in _Lectures in Theoretical Physics_ edited by W.E. Brittin and G.L. Dunham (Interscience, New York, 1959), Vol. 1, p. 315.
  * (11) C. A. Bertulani, C. M. Campbell, and T. Glasmacher, Comp. Phys. Comm. 152, 317 (2003).
  * (12) L. Moschini and P.Capel, Phys. Lett. B 790, 367 (2019).
  * (13) R. Palit, P. Adrich, T. Aumann, K. Boretzky, B. V. Carlson, D. Cortina _et al._ , Phys. Rev. C 68, 034318 (2003).
  * (14) T. Aumann, A. Navin, D. P. Balamuth, D. Bazin, B. Blank, B. A. Brown et al., Phys. Rev. Lett. 84, 35 (2000).
  * (15) N. K. Timofeyuk and R. C. Johnson, Phys. Rev. C 59, 1545 (1999).
  * (16) N. Fukuda, T. Nakamura, N. Aoi, N. Imai, M. Ishihara, T. Kobayashi _et al._ , Phys. Rev. C 70, 054606 (2004).
  * (17) T. Nakamura _et al._ , Phys. Lett. B 331, 296 (1994).
  * (18) J. R. Terry, D. Bazin, B. A. Brown, J. Enders, T. Glasmacher, P. G. Hansen, B. M. Sherrill, and J. A. Tostevin, Phys. Rev. C 69, 054306 (2004).
  * (19) T. Nakamura, N. Fukuda, N. Aoi, N. Imai, M. Ishihara, H. Iwasaki et al., Phys. Rev. C 79, 035805 (2009).
  * (20) V. Maddalena, T. Aumann, D. Bazin, B. A. Brown, J. A. Caggiano, B. Davids et al., Phys. Rev. C 63, 024613 (2001).
  * (21) E. Sauvan, F. Carstoiu, N. A. Orr, J. S. Winfield, M. Freer, J. C. Angélique et al., Phys. Rev. C 69, 044603 (2004).
  * (22) A. M. Moro, J. A. Lay, and J. Gómez-Camacho, Phys. Lett. B 811, 135959 (2020).


# On Locally $\phi$-semisymmetric Kenmotsu Manifolds

Absos Ali Shaikh  Department of Mathematics, Aligarh Muslim University,
Aligarh, Uttar Pradesh, INDIA<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Ali Akbar  Department of Mathematics,
Rampurhat College, Birbhum, West Bengal 731224, INDIA
<EMAIL_ADDRESS>

MSC 2010 Classifications:53C25, 53D15. Keywords and phrases: Kenmotsu
manifold, locally $\phi$\- symmetric, $\phi$-semisymmetric, manifold of
constant curvature.

Abstract The object of the present paper is to study the locally $\phi$\-
semisymmetric Kenmotsu manifolds along with the characterization of such
notion.

##  1\. Introduction

Let $M$ be an n-dimensional, $n\geq 3$, connected smooth Riemannian manifold
endowed with the Riemannian metric $g$. Let $\nabla$, $R$, $S$ and $r$ be the
Levi-Civita connection, curvature tensor, Ricci tensor and the scalar
curvature of $M$ respectively. The manifold $M$ is called locally symmetric
due to Cartan ([2], [3]) if the local geodesic symmetry at $p\in M$ is an
isometry, which is equivalent to the fact that $\nabla R=0$. Generalizing the
concept of local symmetry, the notion of semisymmetric manifold was introduced
by Cartan [4] and fully classified by Szabo ([11], [12], [13]). The manifold
$M$ is said to be semisymmetric if $(R(U,V).R)(X,Y)Z=0$, for all vector fields
$X$, $Y$, $Z$, $U$, $V$ on $M$, where $R(U,V)$ is considered as the derivation
of the tensor algebra at each point of $M$.

In 1977 Takahashi [14] introduced the notion of local $\phi$\- symmetry on a
Sasakian manifold. A Sasakian manifold is said to be locally $\phi$-symmetric
if

(1.1) $\phi^{2}((\nabla_{W}R)(X,Y)Z)=0,$

for all horizontal vector fields $X$, $Y$, $Z$, $W$ on $M$ that is all vector
fields orthogonal to $\xi$, where $\phi$ is the structure tensor of the
manifold $M$. The concept of local $\phi$\- symmetry on various structures and
their generalizations or extension are studied in ( [6], [7], [8], [9]). By
extending the notion of semisymmetry and generalizing the concept of local
$\phi$\- symmetry of Takahashi [14], the first author and his coauthor
introduced [10] the notion of local $\phi$-semisymmetry on a Sasakian
manifold. A Sasakian manifold $M$, $n\geq 3$, is said to be locally
$\phi$-semisymmetric if

(1.2) $\phi^{2}((R(U,V).R)(X,Y)Z)=0,$

for all horizontal vector fields $X$, $Y$, $Z$, $U$, $V$ on $M$. In the
present paper we study locally $\phi$-semisymmetric Kenmotsu manifolds. The
paper is organized as follows:

In section 2 some rudimentary facts and curvature related properties of
Kenmotsu manifolds are discussed. In section 3 we study locally
$\phi$-semisymmetric Kenmotsu manifolds and obtained the characterization of
such notion.

##  2\. Preliminaries

Let $M$ be a $(2n+1)$-dimensional connected smooth manifold endowed with an
almost contact metric structure $(\phi,\xi,\eta,g),$ where $\phi$ is a tensor
field of type $(1,1),$ $\xi$ is a vector field, $\eta$ is an $1$-form and $g$
is a Riemannian metric on $M$ such that [1]

(2.1) $\phi^{2}X=-X+\eta(X)\xi,\hskip 10.0pt\eta(\xi)=1.$ (2.2) $g(\phi X,\phi
Y)=g(X,Y)-\eta(X)\eta(Y)\hskip 10.0pt$

for all vector fields X, Y on $M$.
Then we have [1]

(2.3) $\phi\xi=0,\hskip 10.0pt\eta(\phi X)=0,\hskip 10.0pt\eta(X)=g(X,\xi).$
(2.4) $g(\phi X,X)=0.$ (2.5) $g(\phi X,Y)=-g(X,\phi Y)$

for all vector fields X, Y on $M$.
If

(2.6) $(\nabla_{X}\phi)Y=-g(X,\phi Y)\xi-\eta(Y)\phi X,$ (2.7)
$\nabla_{X}\xi=X-\eta(X)\xi,$

holds on $M$, then it is called a Kenmotsu manifold [5].
In a Kenmotsu manifold the following relations hold [5]

(2.8) $(\nabla_{X}\eta)Y=g(X,Y)-\eta(X)\eta(Y),$ (2.9)
$\eta(R(X,Y)Z)=g(X,Z)\eta(Y)-g(Y,Z)\eta(X),$ (2.10)
$R(X,Y)\xi=\eta(X)Y-\eta(Y)X,$ (2.11) $R(X,\xi)Z=g(X,Z)\xi-\eta(Z)X,$ (2.12)
$R(X,\xi)\xi=\eta(X)\xi-X,$ (2.13) $S(X,\xi)=-2n\eta(X),$ (2.14)
$(\nabla_{W}R)(X,Y)\xi=g(X,W)Y-g(Y,W)X-R(X,Y)W,$ (2.15)
$(\nabla_{W}R)(X,\xi)Z=g(X,Z)W-g(W,Z)X-R(X,W)Z,$

for all vector fields $X$, $Y$, $Z$ and $W$ on $M$.
In a Kenmotsu manifold we also have [5]

(2.16) $\begin{array}[]{rcl}R(X,Y)\phi W&=&g(Y,W)\phi X-g(X,W)\phi Y+g(X,\phi
W)Y-g(Y,\phi W)X\\\ &+&\phi R(X,Y)W.\end{array}$

Applying $\phi$ and using (2.1) we get from (2.16)

(2.17) $\begin{array}[]{rcl}\phi R(X,Y)\phi W&=&-g(Y,W)X+g(X,W)Y+g(X,\phi
W)\phi Y-g(Y,\phi W)\phi X\\\ &-&R(X,Y)W.\end{array}$

In view of (2.17) we obtain from (2.14)

(2.18) $(\nabla_{W}R)(X,Y)\xi=g(Y,\phi W)\phi X-g(X,\phi W)\phi Y+\phi
R(X,Y)\phi W.$

##  3\.  Locally $\phi$-semisymmetric Kenmotsu Manifolds

######  Definition 3.1.

A Kenmotsu manifold $M$ is said to be locally $\phi$-semisymmetric if

(3.1) $\phi^{2}((R(U,V).R)(X,Y)Z)=0,$

for all horizontal vector fields $X$, $Y$, $Z$, $U$, $V$ on $M$.

First we suppose that $M$ is a Kenmotsu manifold such that

(3.2) $\phi^{2}((R(U,V).R)(X,Y)\xi)=0,$

for all horizontal vector fields $X$, $Y$, $U$ and $V$ on $M$.

Differentiating (2.18) covariantly with respect to a horizontal vector field
$U$, we get

(3.3) $\begin{array}[]{rcl}&&(\nabla_{U}\nabla_{V}R)(X,Y)\xi\\\ &=&[g(X,\phi
V)g(U,\phi Y)-g(Y,\phi V)g(U,\phi X)+g(\phi U,R(X,Y)\phi V)]\xi\\\
&+&\phi(\nabla_{U}R)(X,Y)\phi V.\end{array}$

Using (2.16) we obtain from (3.3)

(3.4)
$\begin{array}[]{rcl}(\nabla_{U}\nabla_{V}R)(X,Y)\xi&=&[g(Y,V)g(U,X)-g(X,V)g(U,Y)+g(R(X,Y)V,U)]\xi\\\
&+&\phi(\nabla_{U}R)(X,Y)\phi V.\end{array}$

Interchanging $U$ and $V$ on (3.4) we get

(3.5)
$\begin{array}[]{rcl}(\nabla_{V}\nabla_{U}R)(X,Y)\xi&=&[g(Y,U)g(V,X)-g(X,U)g(V,Y)+g(R(X,Y)U,V)]\xi\\\
&+&\phi(\nabla_{V}R)(X,Y)\phi U.\end{array}$

From (3.4) and (3.5) it follows that

(3.6)
$\begin{array}[]{rcl}(R(U,V).R)(X,Y)\xi&=&2[g(Y,V)g(U,X)-g(X,V)g(U,Y)-R(X,Y,U,V)]\xi\\\
&+&\phi\\{(\nabla_{U}R)(X,Y)\phi V-(\nabla_{V}R)(X,Y)\phi U\\}.\end{array}$

Again from (3.2) we have

(3.7) $(R(U,V).R)(X,Y)\xi=0,$

From (3.6) and (3.7) we have

(3.8) $\begin{array}[]{rcl}&&2[g(Y,V)g(U,X)-g(X,V)g(U,Y)-R(X,Y,U,V)]\xi\\\
&+&\phi\\{(\nabla_{U}R)(X,Y)\phi V-(\nabla_{V}R)(X,Y)\phi U\\}\\\
&=&0.\end{array}$

Applying $\phi$ on (3.8) and using (2.16), (2.18) and (2.3) we get

(3.9) $(\nabla_{U}R)(X,Y)\phi V-(\nabla_{V}R)(X,Y)\phi U=0.$

In view of (3.8) and (3.9) we get

(3.10) $R(X,Y,U,V)=g(Y,V)g(U,Y)-g(X,V)g(U,Y),$ (3.11)
$R(X,Y,U,V)=-\\{g(X,V)g(U,Y)-g(Y,V)g(U,X)\\},$

for all horizontal vector fields $X$, $Y$, $U$ and $V$ on $M$. Hence M is of
constant $\phi$-holomorphic sectional curvature -1 and hence of constant
curvature -1. This leads to the following:

######  Theorem 3.1.

If a Kenmotsu manifold $M$ satisfies the condition
$\phi^{2}((R(U,V).R)(X,Y)\xi)=0$, for all horizontal vector fields $X$, $Y$,
$Z$, $U$ and $V$ on $M$, then $M$ is a manifold of constant curvature -1.

We consider a Kenmotsu manifold which is locally $\phi$-semisymmetric. Then
from (3.1) we have

(3.12) $(R(U,V).R)(X,Y)Z=g((R(U,V).R)(X,Y)Z,\xi)\xi,$

from which we get

(3.13) $(R(U,V).R)(X,Y)Z=-g((R(U,V).R)(X,Y)\xi,Z)\xi$

for all horizontal vector fields $X$, $Y$, $Z$, $U$, $V$ on $M$.

Now taking inner product on both side of (3.6) with a horizontal vector field
$Z$, we obtain

(3.14) $g((R(U,V).R)(X,Y)\xi,Z)=g(\phi(\nabla_{U}R)(X,Y)\phi
V,Z)-g(\phi(\nabla_{V}R)(X,Y)\phi U,Z).$

Using (2.5) and (3.13) we get from (3.14)

(3.15) $(R(U,V).R)(X,Y)Z=[g((\nabla_{U}R)(X,Y)\phi V,\phi
Z)-g((\nabla_{V}R)(X,Y)\phi U,\phi Z)]\xi$

Differentiating (2.16) covariantly with respect to a horizontal vector field
$V$, we get

(3.16) $\begin{array}[]{rcl}&&(\nabla_{V}R)(X,Y)\phi Z\\\ &=&[-g(Y,Z)g(V,\phi
X)+g(X,Z)g(V,\phi Y)-g(V,R(X,Y)Z)]\xi\\\
&+&\phi(\nabla_{V}R)(X,Y)Z.\end{array}$

Taking inner product on both sides of (3.16) with a horizontal vector field
$U$, we obtain

(3.17) $g\\{(\nabla_{V}R)(X,Y)\phi Z,U\\}=g\\{\phi(\nabla_{V}R)(X,Y)Z,U\\}.$

Using (2.5) we get from above

(3.18) $g\\{(\nabla_{V}R)(X,Y)\phi Z,U\\}=-g\\{(\nabla_{V}R)(X,Y)Z,\phi U\\}.$

In view of (3.18) we obtain from (3.15)

(3.19)
$(R(U,V).R)(X,Y)Z=[-g((\nabla_{U}R)(X,Y)V,\phi^{2}Z)+g((\nabla_{V}R)(X,Y)U,\phi^{2}Z)]\xi,$

which implies that

(3.20)
$(R(U,V).R)(X,Y)Z=[g((\nabla_{U}R)(X,Y)V,Z)-g((\nabla_{V}R)(X,Y)U,Z)]\xi,$

i.e.

(3.21) $(R(U,V).R)(X,Y)Z=[-(\nabla_{U}R)(X,Y,Z,V)+(\nabla_{V}R)(X,Y,Z,U)]\xi,$

for any horizontal vector field $X$, $Y$, $Z$, $U$, $V$ on $M$. Hence we can
state the following:

######  Theorem 3.2.

A Kenmotsu manifold $M$, $n\geq 3$, is locally $\phi$-semisymmetric if and
only if the relation (3.21) holds for all horizontal vector fields $X$, $Y$,
$Z$, $U$, $V$ on $M$.

##  4\.  Characterization of Locally $\phi$-semisymmetric Kenmotsu Manifolds

In this section we investigate the condition of local $\phi$-semisymmetry of a
Kenmotsu manifold for arbitrary vector fields on $M$. To find this we need the
following results.

######  Lemma 4.1.

For any horizontal vector field $X$, $Y$ and $Z$ on a Kenmotsu manifold $M$,
we have

(4.1) $(\nabla_{\xi}R)(X,Y)Z=(\ell_{\xi}R)(X,Y)Z+2R(X,Y)Z.$

###### Proof.

Let $X^{\ast}$, $Y^{\ast}$ and $Z^{\ast}$ be $\xi$\- invariant horizontal
vector field extensions on $X$, $Y$ and $Z$ respectively. Since $X^{\ast}$ is
$\xi$\- invariant of $X$, we get by using (2.7)

(4.2) $\nabla_{\xi}X^{\ast}=\nabla_{X}{{}^{\ast}}\xi=X^{\ast}$

Now making use of invariance of $X^{\ast}$, $Y^{\ast}$ and $Z^{\ast}$ by $\xi$
and using (4.2) we get

(4.3)
$\begin{array}[]{rcl}(\ell_{\xi}R)(X^{\ast},Y^{\ast})Z^{\ast}&=&[\xi,R(X^{\ast},Y^{\ast})Z^{\ast}]\\\
&=&\nabla_{\xi}(R(X^{\ast},Y^{\ast})Z^{\ast})-\nabla_{R(X^{\ast},Y^{\ast})Z^{\ast}}\xi\\\
&=&(\nabla_{\xi}R)(X^{\ast},Y^{\ast})Z^{\ast}+R(\nabla_{\xi}X^{\ast},Y^{\ast})Z^{\ast}+R(X^{\ast},\nabla_{\xi}Y^{\ast})Z^{\ast}\\\
&+&R(X^{\ast},Y^{\ast})\nabla_{\xi}Z^{\ast}-R(X^{\ast},Y^{\ast})Z^{\ast}\\\
&=&(\nabla_{\xi}R)(X^{\ast},Y^{\ast})Z^{\ast}+R(X^{\ast},Y^{\ast})Z^{\ast}+R(X^{\ast},Y^{\ast})Z^{\ast}\\\
&+&R(X^{\ast},Y^{\ast})Z^{\ast}-R(X^{\ast},Y^{\ast})Z^{\ast}\\\
&=&(\nabla_{\xi}R)(X^{\ast},Y^{\ast})Z^{\ast}+2R(X^{\ast},Y^{\ast})Z^{\ast}\end{array}$

Hence we get the conclusion.∎

######  Lemma 4.2.

For any vector field $X$, $Y$ and $Z$ on a Kenmotsu manifold $M$ we have

(4.4)
$\begin{array}[]{rcl}R(\phi^{2}X,\phi^{2}Y)\phi^{2}Z&=&-R(X,Y)Z+\eta(Z)\\{\eta(X)Y-\eta(Y)X\\}\\\
&+&\\{\eta(Y)g(X,Z)-\eta(X)g(Y,Z)\\}\xi\end{array}$

Now lemma (4.1) and lemma (4.2) together imply the following:

######  Lemma 4.3.

For any vector field $X$, $Y$, $Z$ and $U$ on a Kenmotsu manifold $M$, we have

(4.5)
$\begin{array}[]{rcl}&&(\nabla_{\phi^{2}U}R)(\phi^{2}X,\phi^{2}Y)\phi^{2}Z\\\
&=&(\nabla_{U}R)(X,Y)Z-\eta(X)H_{1}(Y,U)Z+\eta(Y)H_{1}(X,U)Z+\eta(Z)H_{1}(X,Y)U\\\
&+&\eta(U)[\eta(Z)\\{\eta(X)\ell_{\xi}Y-\eta(Y)\ell_{\xi}X\\}-(\ell_{\xi}R)(X,Y)Z]\\\
&+&2\eta(U)[R(X,Y)Z-\eta(Z)\\{\eta(X)Y-\eta(Y)X\\}\\\
&-&\\{\eta(Y)g(X,Z)-\eta(X)g(Y,Z)\\}\xi].\end{array}$

where the tensor field $H_{1}$ of type (1, 3) is given by

(4.6) $H_{1}(X,Y)Z=R(X,Y)Z-g(X,Z)Y+g(Y,Z)X,$

for all vector fields $X$, $Y$, $Z$ on $M$.

Now let $X$, $Y$, $Z$, $U$, $V$ be arbitrary vector fields on $M$.
Now we compute $(R(\phi^{2}U,\phi^{2}V).R)(\phi^{2}X,\phi^{2}Y)\phi^{2}Z$ in
two different ways. Firstly from (3.21), (2.1) and (4.5) we get

(4.7)
$\begin{array}[]{rcl}&&(R(\phi^{2}U,\phi^{2}V).R)(\phi^{2}X,\phi^{2}Y)\phi^{2}Z\\\
&=&\\{(\nabla_{U}R)(X,Y,Z,V)-(\nabla_{V}R)(X,Y,Z,U)\\}\xi\\\
&+&\\{\eta(U)\eta\\{(\nabla_{V}R)(X,Y)Z)\\}-\eta(V)\eta\\{(\nabla_{U}R)(X,Y)Z)\\}\\}\xi\\\
&-&\eta(X)\\{H(Y,U,Z,V)-H(Y,V,Z,U)\\}\xi\\\
&+&\eta(Y)\\{H(X,U,Z,V)-H(X,V,Z,U)\\}\xi\\\
&+&\eta(Z)\\{H(X,Y,U,V)-H(X,Y,V,U)\\}\xi\\\
&+&\eta(X)\eta(Z)\\{\eta(U)g(\ell_{\xi}Y,V)-\eta(V)g(\ell_{\xi}Y,U)\\}\xi\\\
&-&\eta(Y)\eta(Z)\\{\eta(U)g(\ell_{\xi}X,V)-\eta(V)g(\ell_{\xi}X,U)\\}\xi\\\
&+&2\\{\eta(U)R(X,Y,Z,V)-\eta(V)R(X,Y,Z,U)\\}\xi\\\
&+&2\eta(Z)\eta(V)\\{\eta(X)g(Y,U)-\eta(Y)g(X,U)\\}\xi\\\
&-&2\eta(Z)\eta(U)\\{\eta(X)g(Y,V)-\eta(Y)g(X,V)\\}\xi,\end{array}$

where $H(X,Y,Z,U)=g(H_{1}(X,Y)Z,U)$ and the tensor field $H_{1}$ of type (1,
3) is given by (4.6)
Secondly we have

(4.8)
$\begin{array}[]{rcl}&&(R(\phi^{2}U,\phi^{2}V).R)(\phi^{2}X,\phi^{2}Y)\phi^{2}Z=R(\phi^{2}U,\phi^{2}V)R(\phi^{2}X,\phi^{2}Y)\phi^{2}Z\\\
&-&R(R(\phi^{2}U,\phi^{2}V)\phi^{2}X,\phi^{2}Y)\phi^{2}Z-R(\phi^{2}X,R(\phi^{2}U,\phi^{2}V)\phi^{2}Y)\phi^{2}Z\\\
&-&R(\phi^{2}X,\phi^{2}Y)R(\phi^{2}U,\phi^{2}V)\phi^{2}Z.\end{array}$

By straightforward calculation from (4.8) we get

(4.9)
$\begin{array}[]{rcl}&&(R(\phi^{2}U,\phi^{2}V).R)(\phi^{2}X,\phi^{2}Y)\phi^{2}Z\\\
&=&-(R(U,V).R)(X,Y)Z\\\
&+&\eta(X)\\{\eta(V)H_{1}(U,Y)Z-\eta(U)H_{1}(V,Y)Z\\}\\\
&+&\eta(Y)\\{\eta(V)H_{1}(X,U)Z-\eta(U)H_{1}(X,V)Z\\}\\\
&+&\eta(Z)\\{\eta(V)H_{1}(X,Y)U-\eta(U)H_{1}(X,Y)V\\}\\\
&+&\\{\eta(V)g(H(X,Y,Z,U)-\eta(U)g(H(X,Y,Z,V)\\}\xi,\end{array}$

where $H(X,Y,Z,U)=g(H_{1}(X,Y)Z,U)$ and the tensor field $H_{1}$ of type (1,
3) is given by (4.6)
From (4.7) and (4.9) we obtain

(4.10) $\begin{array}[]{rcl}&&(R(U,V).R)(X,Y)Z\\\
&=&[-(\nabla_{U}R)(X,Y,Z,V)+(\nabla_{V}R)(X,Y,Z,U)]\xi\\\
&+&[\eta(V)\eta\\{(\nabla_{U}R)(X,Y)Z)\\}-\eta(U)\eta\\{(\nabla_{V}R)(X,Y)Z)\\}]\xi\\\
&+&\eta(X)[\\{H(Y,U,Z,V)-H(Y,V,Z,U)\\}\xi+\eta(V)H_{1}(U,Y)Z-\eta(U)H_{1}(V,Y)Z]\\\
&-&\eta(Y)[\\{H(X,U,Z,V)-H(X,V,Z,U)\\}\xi-\eta(V)H_{1}(X,U)Z+\eta(U)H_{1}(X,V)Z]\\\
&-&\eta(Z)[\\{H(X,Y,U,V)-H(X,Y,V,U)\\}\xi+\eta(V)H_{1}(X,U)Z-\eta(U)H_{1}(X,V)Z]\\\
&+&\\{\eta(V)H(X,Y,Z,U)-\eta(U)H(X,Y,Z,V)\\}\xi\\\
&-&2\\{\eta(U)R(X,Y,Z,V)-\eta(V)R(X,Y,Z,U)\\}\xi\\\
&+&.\\{\eta(U)(\ell_{\xi}R)(X,Y,Z,V)-\eta(V)(\ell_{\xi}R)(X,Y,Z,U)\\}\xi\\\
&-&\eta(Z)\eta(X)\\{\eta(U)g(\ell_{\xi}Y,V)-\eta(V)g(\ell_{\xi}Y,U)\\}\xi\\\
&+&\eta(Z)\eta(Y)\\{\eta(U)g(\ell_{\xi}X,V)-\eta(V)g(\ell_{\xi}X,U)\\}\xi\\\
&-&2\eta(Z)\eta(V)\\{\eta(X)g(Y,U)-\eta(Y)g(X,U)\\}\xi\\\
&+&2\eta(Z)\eta(U)\\{\eta(X)g(Y,V)-\eta(Y)g(X,V)\\}\xi.\end{array}$

Thus in a locally $\phi$-semisymmetric Kenmotsu manifold the relation (4.10)
holds for all arbitrary vector fields $X$, $Y$, $Z$, $U$, $V$ on $M$. Next if
the relation (4.10) holds in a Kenmotsu manifold, then for any horizontal
vector field $X$, $Y$, $Z$, $U$, $V$ on $M$, we get the relation (3.21) and
hence the manifold is locally $\phi$-semisymmetric.
Thus we can state the following:

######  Theorem 4.1.

A Kenmotsu manifold $M$ is locally $\phi$-semisymmetric if and only if the
relation (4.10) holds for any arbitrary vector field $X$, $Y$, $Z$, $U$, $V$
on $M$.

## References

  * [1] Blair, D. E., Contact manifolds in Riemannian geometry. Lecture Notes in Math. No. 509. Springer 1976.
  * [2] Cartan, E., Sur une class remarquable despace de Riemann, I, Bull. de la Soc. Math. de France , 54(1926), 214-216.
  * [3] Cartan, E., Sur une class remarquable despace de Riemann, II, Bull. de la Soc. Math. de France , 55(1927), 114-134.
  * [4] Cartan, E., Lecons sur la geometric des espaces de Riemann, 2nd ed., Paris 1946.
  * [5] Kenmotsu, K., A class of almost contact Riemannian manifolds, Tohoku Math. J., 24(1972), 93-102.
  * [6] Shaikh, A. A., Baishya, K. K., On $\phi$-Symmetric LP- Sasakian manifolds, Yokohama Math. J., 52(2005), 97-112.
  * [7] Shaikh, A. A., Baishya, K. K. and Eyasmin, S., On $\phi$-recurrent generalized ($k,\mu$)-contact metric manifolds, Lobachevski J. Math., 27(2007), 3-13.
  * [8] Shaikh, A. A., Basu, T. and Eyasmin, S., On locally $\phi$-symmetric $(LCS)_{n}$-manifolds, Int. J. of Pure and Appl. Math., 41(8)(2007), 1161-1170.
  * [9] Shaikh, A. A., Basu, T. and Eyasmin, S., On the existence of $\phi$-recurrent $(LCS)_{n}$-manifolds, Extracta Mathematica, 23(1)(2008) , 71-83.
  * [10] Shaikh, A. A., Mondal, C.K. and Ahmad, H., On locally $\phi$-semisymmetric Sasakian manifolds, arxive: 1302. 2139v3 [math.DG] 11 Feb 2017.
  * [11] Szabó, Z. I., Structure theorems on Riemannian spaces satisfying $R(X,Y).R=0$, I, The local version, J. Diff. Geom. 17(1982), 531-582.
  * [12] Szabó, Z. I., Structure theorems on Riemannian spaces satisfying $R(X,Y).R=0$, II, Global version, Geom. Dedicata, 19(1983), 65-108.
  * [13] Szabó , Z. I., Classification and construction of complete hypersurfaces satisfying $R(X,Y).R=0$, Acta. Sci. Math., 47(1984), 321-348.
  * [14] Takahashi. T., Sasakian $\phi$-symmetric spaces, Tohoku Math. J., 29(1977), 91-113.


# Triharmonic curves in the $3$-dimensional Sol space

S. Montaldo  Università degli Studi di Cagliari
Dipartimento di Matematica e Informatica
Via Ospedale 72
09124 Cagliari, Italia<EMAIL_ADDRESS>and  A. Ratto  Università degli
Studi di Cagliari
Dipartimento di Matematica e Informatica
Via Ospedale 72
09124 Cagliari, Italia<EMAIL_ADDRESS>
###### Abstract.

The main aim of this paper is to study triharmonic curves in the
$3$-dimensional homogeneous space Sol. In the first part of the paper we shall
obtain a complete classification of proper triharmonic curves with constant
geodesic curvature and torsion. In the final section we shall show that these
triharmonic curves form a constant angle with a suitable Killing field of
constant length along the curve.

###### Key words and phrases:

Triharmonic curves, $3$-dimensional Sol space, helices, Killing fields

###### 2020 Mathematics Subject Classification:

Primary: 58E20. Secondary: 53A04, 53C30.

The authors S.M. and A.R. are members of the Italian National Group
G.N.S.A.G.A. of INdAM. The work was partially supported by the Project ISI-
HOMOS funded by Fondazione di Sardegna and by PNRR e.INS Ecosystem of
Innovation for Next Generation Sardinia (CUP F53C22000430001, codice MUR
ECS00000038)

##  1\. Introduction

Harmonic maps are the critical points of the energy functional

(1.1) $E(\varphi)=\frac{1}{2}\int_{M}\,|d\varphi|^{2}\,dV\,,$

where $\varphi:M\to N$ is a smooth map between two Riemannian manifolds
$(M,g)$ and $(N,h)$. We refer to [7, 8] for an introduction to this important
topic.

The study of polyharmonic maps of order $r$, shortly $r$-harmonic maps, was
first proposed by Eells and Lemaire in [7]. These maps are defined as the
critical points of the functionals

$E_{r}(\varphi)=\frac{1}{2}\int_{M}\,|(d+d^{*})^{r}\varphi|^{2}\,dV$

which represent a version of order $r$ of the classical energy (1.1). We refer
to [3] for a detailed discussion on the definition of the functionals
$E_{r}(\varphi)$.

In this context, the most widely studied case is $r=2$, where we have the so-
called bienergy functional and its critical points are known as biharmonic
maps. At present, a very ample literature on biharmonic maps is available (see
[18] and references therein).

More recently, several authors worked intensively on the case $r\geq 3$. For
instance, we refer to [3, 5, 6, 9, 11, 12, 13, 14, 16, 17] for background and
results.

In this paper we shall focus on the study of $3$-harmonic curves in the
$3$-dimensional solvable Lie group Sol.

More generally, let $\gamma:I\subset{\mathbb{R}}\to N$ denote a smooth curve
parametrized by the arc length $s$ and let $T$ denote its unit tangent vector.
Then $\gamma$ is $r$-harmonic if and only if (see [2, 11, 15])

(1.2)
$\nabla^{2r-1}_{T}T+\sum_{\ell=0}^{r-2}(-1)^{\ell}R\left(\nabla^{2r-3-\ell}_{T}T,\nabla^{\ell}_{T}T\right)T=0\,,$

where
$\nabla^{0}_{T}T=T,\,\nabla^{k}_{T}T=\nabla_{T}\left(\nabla^{k-1}_{T}T\right)$
and $R$ is the Riemannian curvature operator of $N$.

We point out that any geodesic is trivially $r$-harmonic for all $r\geq 1$.
Therefore, we say that $\gamma$ is a proper $r$-harmonic curve if it is
$r$-harmonic and not a geodesic.

The Chen-Maeta conjecture states that any $r$-harmonic submanifolds of
${\mathbb{R}}^{n}$ is minimal. In this direction, Maeta (see [11]) proved that
the conjecture is true for curves. Note that, in order to avoid trivialities
arising from re-parametrization, it is convenient to restrict the attention to
curves parametrized by the arc length, and we shall adopt this assumption
throughout the whole paper.

By contrast, when the target is an $n$-dimensional sphere, examples of proper
$r$-harmonic curves do exist. We refer to the paper of Branding [2] and
references therein for an illustration of the case of $r$-harmonic curves into
space forms.

An investigation of triharmonic curves in $3$-dimensional homogeneous spaces
with a $4$-dimensional group of isometries was carried out in [15].

As a successive step, in this paper we undertake the study of triharmonic
curves in the $3$-dimensional homogeneous geometry with isometries group of
dimension $3$, i.e., the $3$-dimensional solvable Lie group Sol. A study of
triharmonic curves in the $3$-dimensional solvable Lie group Sol was initiated
in [19], where the author gives the explicit relevant differential equations.

It is natural to start this investigation by considering a special class of
geometrically significant curves, the so-called helices.

Classically, in the Euclidean $3$-dimensional space ${\mathbb{R}}^{3}$, a
general helix is a curve characterized by the property that its tangent vector
makes a constant angle with a fixed straight line which represents the axis of
the helix.

A famous result, stated by M. A. Lancret in $1802$ and first proved by B. de
Saint Venant in $1845$, is: A necessary and sufficient condition that a curve
be a general helix is that the ratio of curvature to torsion be constant.

Helices arise in nanosprings, carbon nanotubes and the double helix shape is
commonly associated with the structure of DNA.

In a Riemannian geometric context, the definition of a general helix is more
delicate and some new interesting phenomena appear (see [1]). For instance,
for general helices according to [1], in the $3$-dimensional sphere
${\mathbb{S}}^{3}$ the ratio of the curvature and the torsion is not
necessarily constant.

In the first part of the paper we follow the approach and the notation of [15]
and say that a Frenet curve $\gamma(s)$ is a helix if its geodesic curvature
$\kappa(s)$ and its torsion $\tau(s)$ are constant. In Theorem 3.2 we shall
obtain a complete classification of proper $3$-harmonic helices in the
$3$-dimensional Sol space.

Our paper is organised as follows. In Section 2 we recall some basic facts
about curves and the geometry of Sol. The main aim of this section is to make
this paper reasonably self-contained.

In Section 3 we shall state our main result. Next, proofs will be given in
Section 4.

Finally, in Section 5 we shall follow [1] and consider a more general
definition of a helix. This approach involves Killing fields which have
constant length along $\gamma$. In this context, we shall show that the proper
$3$-harmonic curve determined in Theorem 3.2 is an integral curve of one of
these Killing fields.

##  2\. Preliminaries

We adopt the following notation and sign convention for the Riemannian
curvature tensor field $R$:

${R}(X,Y)Z={\nabla}_{X}{\nabla}_{Y}Z-{\nabla}_{Y}{\nabla}_{X}Z-{\nabla}_{[X,Y]}Z\,,\quad{R}(X,Y,Z,W)=\langle{R}(X,Y)W,Z\rangle\,,$

where $X,Y,Z,W$ are vector fields on $N$.

Let Sol $=(\mathbb{R}^{3},g_{Sol})$ denote the $3$-dimensional Riemannian
manifold given by ${\mathbb{R}}^{3}$ endowed with the metric tensor
$g_{Sol}=e^{2z}{\rm d}x^{2}+e^{-2z}{\rm d}y^{2}+{\rm d}z^{2}$ with respect to
the standard Cartesian coordinates $(x,y,z)$.

The space Sol is a solvable Lie group and the metric $g_{Sol}$ is left-
invariant with respect to the group operation

$(x,y,z).(x^{\prime},y^{\prime},z^{\prime})=\left(e^{z}x^{\prime}+x,e^{-z}y^{\prime}+y,z^{\prime}+z\right)\,.$

The group of isometries of Sol has dimension $3$ and the connected component
of the identity is generated by the following three families:

(2.1) $\begin{array}[]{l}(x,y,z)\to(x+c,y,z)\\\ (x,y,z)\to(x,y+c,z)\\\
(x,y,z)\to(e^{-c}x,e^{c}y,z+c)\,.\end{array}$

For future use, we point out that also

(2.2) $\begin{array}[]{l}(x,y,z)\to(-x,y,z)\\\
(x,y,z)\to(x,-y,z)\,.\end{array}$

are isometries of Sol. Moreover, one can easily check that a global
orthonormal frame field on Sol is

$E_{1}=e^{-z}\frac{\partial}{\partial x},\;E_{2}=e^{z}\frac{\partial}{\partial
y},\;E_{3}=\frac{\partial}{\partial z}.$

With respect to this orthonormal frame field, the Lie brackets and the Levi-
Civita connection can be easily computed as:

$[E_{1},E_{2}]=0,\;[E_{2},E_{3}]=-E_{2},\;[E_{1},E_{3}]=E_{1},$ (2.3)
$\begin{array}[]{lll}\nabla_{E_{1}}E_{1}=-E_{3},&\nabla_{E_{1}}E_{2}=0,&\nabla_{E_{1}}E_{3}=E_{1}\\\
\nabla_{E_{2}}E_{1}=0,&\nabla_{E_{2}}E_{2}=E_{3},&\nabla_{E_{2}}E_{3}=-E_{2}\\\
\nabla_{E_{3}}E_{1}=0,&\nabla_{E_{3}}E_{2}=0,&\nabla_{E_{3}}E_{3}=0.\\\
\end{array}$

A further computation gives

(2.4)
$\begin{array}[]{lll}R(E_{1},E_{2})E_{1}=-E_{2},&R(E_{1},E_{3})E_{1}=E_{3},&R(E_{1},E_{2})E_{2}=E_{1},\\\
R(E_{2},E_{3})E_{2}=E_{3},&R(E_{1},E_{3})E_{3}=-E_{1},&R(E_{2},E_{3})E_{3}=-E_{2},\end{array}$

and from this it is easy to deduce

(2.5) $\begin{array}[]{lll}R_{1212}=1,\\\ R_{1313}=-1,\\\
R_{2323}=-1.\end{array}$

Now, let $\gamma(s)=(x(s),y(s),z(s))$ be a Frenet curve in Sol parametrized by
arc length. Then

(2.6)
$T={x^{\prime}}e^{z}E_{1}+{y^{\prime}}e^{-z}E_{2}+{z^{\prime}}E_{3}=\left(T_{1},T_{2},T_{3}\right)$

and we can associate to $\gamma$ the classical Frenet equations

$\displaystyle\nabla_{T}T$ $\displaystyle=$ $\displaystyle\kappa\,N$ (2.7)
$\displaystyle\nabla_{T}N$ $\displaystyle=$ $\displaystyle-\kappa\,T+\tau\,B$
$\displaystyle\nabla_{T}B$ $\displaystyle=$ $\displaystyle-\tau\,N\,,$

where $\\{T,N,B\\}$ is a global orthonormal frame field along $\gamma$ called
the Frenet frame field and the functions $\kappa=\kappa(s)>0$, $\tau=\tau(s)$
are called the geodesic curvature and the torsion of $\gamma$ respectively.

Next, let
$V=V_{1}(s)E_{1}+V_{2}(s)E_{2}+V_{3}(s)E_{3}=\left(V_{1}(s),V_{2}(s),V_{3}(s)\right)$
be any vector field along $\gamma$. Then a computation using (2.3) and (2.6)
shows that

(2.8)
$\nabla_{T}V=\left({V_{1}}^{\prime}+T_{1}V_{3},{V_{2}}^{\prime}-T_{2}V_{3},{V_{3}}^{\prime}-T_{1}V_{1}+T_{2}V_{2}\right)\,.$

For our purposes, it will be useful to know the explicit expression of
$\kappa(s)$ and $\tau(s)$ for a Frenet curve $\gamma(s)=(x(s),y(s),z(s))$.
Observing that

$\kappa^{2}(s)=\langle\nabla_{T}T,\nabla_{T}T\rangle\,\quad{\rm
and}\quad\tau(s)=-\langle\nabla_{T}B,N\rangle$

a long, straightforward computation using (2.8) gives:

(2.9)
$\kappa^{2}(s)=e^{2z}\left(x^{\prime\prime}+2x^{\prime}z^{\prime}\right)^{2}+e^{-2z}\left(y^{\prime\prime}-2y^{\prime}z^{\prime}\right)^{2}+\left(z^{\prime\prime}-e^{2z}x^{\prime
2}+e^{-2z}y^{\prime 2}\right)^{2}$

and

(2.10) $\tau(s)=\frac{{\mathcal{A}}}{{\mathcal{B}}},$

where

$\displaystyle{\mathcal{A}}$ $\displaystyle=$ $\displaystyle 2e^{8z}x^{\prime
5}y^{\prime}+2x^{\prime}y^{\prime 5}+e^{6z}x^{\prime}(3x^{\prime\prime
2}y^{\prime}+x^{\prime
2}(y^{(3)}-6y^{\prime\prime}z^{\prime}+8y^{\prime}(2z^{\prime
2}-z^{\prime\prime}))$
$\displaystyle+x^{\prime}(y^{\prime}(14x^{\prime\prime}z^{\prime}-x^{(3)})-3x^{\prime\prime}y^{\prime\prime}))+e^{2z}y^{\prime}(x^{\prime}(3y^{\prime\prime
2}+8y^{\prime 2}(z^{\prime\prime}+2z^{\prime 2})$
$\displaystyle-y^{\prime}(y^{(3)}+14y^{\prime\prime}z^{\prime}))+y^{\prime}(y^{\prime}(x^{(3)}+6x^{\prime\prime}z^{\prime})-3x^{\prime\prime}y^{\prime\prime}))$
$\displaystyle+e^{4z}(-4x^{\prime 3}y^{\prime
3}+x^{\prime}(-y^{(3)}z^{\prime\prime}+2y^{(3)}z^{\prime
2}+z^{(3)}y^{\prime\prime}-8y^{\prime\prime}z^{\prime
3}+y^{\prime}(6z^{\prime\prime 2}+8z^{\prime 4}-4z^{(3)}z^{\prime}))$
$\displaystyle+y^{\prime}(x^{(3)}z^{\prime\prime}+2x^{(3)}z^{\prime
2}-z^{(3)}x^{\prime\prime}+8x^{\prime\prime}z^{\prime
3})+z^{\prime}(y^{(3)}x^{\prime\prime}-y^{\prime\prime}(x^{(3)}+6x^{\prime\prime}z^{\prime})))$

and

$\displaystyle{\mathcal{B}}$ $\displaystyle=$ $\displaystyle
e^{4z}(z^{\prime\prime 2}-2x^{\prime 2}y^{\prime 2})+e^{8z}x^{\prime
4}+e^{6z}((x^{\prime\prime}+2x^{\prime}z^{\prime})^{2}-2x^{\prime
2}z^{\prime\prime})$ $\displaystyle+y^{\prime 4}+e^{2z}(2y^{\prime
2}z^{\prime\prime}+(y^{\prime\prime}-2y^{\prime}z^{\prime})^{2})\,.$

As a special case of (1.2) we see that $\gamma$ is triharmonic if it satisfies

(2.11)
$\nabla^{5}_{T}T+R\left(\nabla^{3}_{T}T,T\right)T-R\left(\nabla^{2}_{T}T,\nabla_{T}T\right)T=0\,.$

In the following proposition we compute explicitly the left side of (2.11) and
we obtain:

######  Proposition 2.1.

Let $\gamma(s)$ be a Frenet curve in Sol parametrized by arc length. Then
$\gamma$ is proper triharmonic if and only if the following three equations
hold:

(2.12)
$5\left[2\kappa^{3}\kappa^{\prime}+\kappa\left(\tau^{2}\kappa^{\prime}-\kappa^{(3)}\right)-2\kappa^{\prime}\kappa^{\prime\prime}+\kappa^{2}\tau\tau^{\prime}\right]=0\,;$
(2.13)
$\begin{array}[]{l}-2\kappa^{2}\big{(}{B_{3}}{T_{3}}\tau+5\kappa^{\prime\prime}\big{)}+\kappa\big{[}-2{B_{3}}{N_{3}}\tau^{\prime}-15\kappa^{\prime
2}+\big{(}1-2{B_{3}}^{2}\big{)}\tau^{2}-4\tau\tau^{\prime\prime}-3\tau^{\prime
2}+\tau^{4}\big{]}\\\ \\\
-4\tau\kappa^{\prime}\big{(}{B_{3}}{N_{3}}+3\tau^{\prime}\big{)}+\kappa^{(4)}-2\kappa^{\prime\prime}\big{(}1-{B_{3}}^{2}+3\tau^{2}\big{)}+\kappa^{\prime\prime}+2\kappa^{3}\big{(}1-2{B_{3}}^{2}+\tau^{2}\big{)}+\kappa^{5}=0\,;\end{array}$
(2.14)
$\begin{array}[]{l}-2\kappa^{\prime\prime}({B_{3}}{N_{3}}-3\tau^{\prime})+\kappa\big{(}2\tau^{2}({B_{3}}{N_{3}}-3\tau^{\prime})+(2{N_{3}}^{2}-1)\tau^{\prime}+\tau^{(3)}\big{)}+\kappa^{3}(4{B_{3}}{N_{3}}-\tau^{\prime})\\\
\\\
+\kappa^{2}\tau(2{N_{3}}{T_{3}}-9\kappa^{\prime})+4\kappa^{\prime}\tau^{\prime\prime}-4\tau^{3}\kappa^{\prime}+2\tau\big{(}2\kappa^{(3)}+(2{N_{3}}^{2}-1)\kappa^{\prime}\big{)}=0\,.\end{array}$

The left sides of (2.12), (2.13) and (2.14) are precisely the component of the
$3$-tension field with respect to the Frenet frame field. The proof of
Proposition 2.1 will be carried out in Section 4.

##  3\. Main Theorem

A simple look at Proposition 2.1, together with the general formulae (2.9) and
(2.10), suggest that a complete study of proper triharmonic curves in Sol may
be a rather difficult task. Therefore, as a starting point, we restrict our
investigation to a suitable family of geometrically significant curves.

######  Definition 3.1.

We say that a Frenet curve $\gamma(s)$ in Sol is a Frenet helix (shortly, a
helix) if its geodesic curvature $\kappa(s)$ and its torsion $\tau(s)$ are
both constant, say $\kappa(s)=a>0$, $\tau(s)=b$.

We prove a complete classification of proper triharmonic Frenet helices in
Sol:

######  Theorem 3.2.

There exist a proper triharmonic Frenet helix $\gamma$ in Sol if and only if
$\kappa(s)=a=1/2$ and $\tau(s)=b=\pm 1/2$. Moreover, up to isometries of Sol,
the parametrization of such proper triharmonic helix when $b=1/2$ is

(3.1)
$\gamma(s)=\frac{1}{\sqrt{2}}\left(-e^{-\frac{s}{\sqrt{2}}},\,e^{\frac{s}{\sqrt{2}}}\,,\,\,s\right)\,.$

######  Remark 3.3.

In [4] the authors proved that there is no proper biharmonic curve in Sol. By
contrast, Theorem 3.2 shows that proper triharmonic curves in Sol do exist.

##  4\. Proofs

###### Proof of Proposition 2.1.

We want to compute the $3$-tension field, i.e., the left side of (2.11)
expressing it with respect to the Frenet frame field $\\{T,N,B\\}$.

We shall find that, when $r=3$, the left-hand side of (2.12), (2.13) and
(2.14) are precisely the components of the left-hand side of (1.2) along $T$,
$N$ and $B$ respectively.

We shall use again the notation
$\big{(}v_{1},v_{2},v_{3}\big{)}=v_{1}T+v_{2}N+v_{3}B$. First, a simple
computation gives:

$\displaystyle\nabla_{T}T$ $\displaystyle=$
$\displaystyle\big{(}0,\kappa,0\big{)}$ $\displaystyle\nabla_{T}^{2}T$
$\displaystyle=$
$\displaystyle\big{(}-\kappa^{2},\kappa^{\prime},\kappa\tau\big{)}$
$\displaystyle\nabla_{T}^{3}T$ $\displaystyle=$
$\displaystyle\big{(}-3\kappa\kappa^{\prime},-\kappa^{3}-\kappa\tau^{2}+\kappa^{\prime\prime},2\kappa^{\prime}\tau+\kappa\tau^{\prime}\big{)}$
$\displaystyle\nabla_{T}^{5}T$ $\displaystyle=$
$\displaystyle\Big{(}5\big{(}2\kappa^{3}\kappa^{\prime}+\kappa\big{(}\tau^{2}\kappa^{\prime}-\kappa^{(3)}\big{)}-2\kappa^{\prime}\kappa^{\prime\prime}+\kappa^{2}\tau\tau^{\prime}\big{)},$
$\displaystyle\,\,\,\,\kappa^{(4)}-6\tau^{2}\kappa^{\prime\prime}-10\kappa^{2}\kappa^{\prime\prime}-12\tau\kappa^{\prime}\tau^{\prime}+\kappa\left(\tau^{4}-3\left(5\kappa^{\prime
2}+\tau^{\prime 2}\right)-4\tau\tau^{\prime\prime}\right)$
$\displaystyle+2\kappa^{3}\tau^{2}+\kappa^{5},$ $\displaystyle
4\kappa^{(3)}\tau+6\kappa^{\prime\prime}\tau^{\prime}+4\kappa^{\prime}\tau^{\prime\prime}-9\kappa^{2}\tau\kappa^{\prime}-4\tau^{3}\kappa^{\prime}-\kappa^{3}\tau^{\prime}+\kappa\big{(}\tau^{(3)}-6\tau^{2}\tau^{\prime}\big{)}\Big{)}.$

Next, using (2.4) and (2.5) we compute

(4.2) $\begin{array}[]{lll}R(T,N,T,N)&=&-1+2B_{3}^{2}\\\
R(T,N,T,B)&=&-2N_{3}B_{3}\\\ R(B,T,B,T)&=&-1+2N_{3}^{2}\\\
R(B,N,N,T)&=&2T_{3}B_{3}\\\ R(B,N,B,T)&=&-2T_{3}N_{3}\,.\end{array}$

Finally, a computation using (4) and (4.2) into (2.11) leads us to complete
the proof of this proposition. ∎

###### Proof of Theorem 3.2.

First, we establish a technical Lemma:

######  Lemma 4.1.

Let $\gamma$ be a proper triharmonic Frenet helix in Sol. Then

  * (i)

$N_{3}\equiv 0$ along $\gamma$.

  * (ii)

Both $T_{3}$ and $B_{3}$ are constant along $\gamma$.

###### Proof of Lemma 4.1.

(i) We use Proposition 2.1. Direct substitution shows that (2.12) always holds
when $\kappa(s)\equiv a$ and $\tau(s)\equiv b$. As for equations (2.13) and
(2.14), they become

(4.3)
$a\Big{[}a^{4}-2abB_{3}T_{3}+2a^{2}(1+b^{2}-2B_{3}^{2})+b^{2}(1+b^{2}-2B_{3}^{2})\Big{]}=0$

and

(4.4) $2aN_{3}\big{(}2a^{2}B_{3}+b^{2}B_{3}+abT_{3}\big{)}=0$

respectively. Now, let us assume that $N_{3}(s_{0})\neq 0$ for some $s_{0}$.
Then, near $s_{0}$, we must have

$B_{3}=-\frac{abT_{3}}{2a^{2}+b^{2}}\,.$

Inserting this condition into the left side of (4.3) and simplifying we find

$a\Big{[}a^{4}+b^{2}+b^{4}+2a^{2}(1+b^{2})\Big{]}=0\,,$

which is clearly impossible since $a>0$. Then $N_{3}\equiv 0$.

(ii) Since $\\{T,N,B\\}$ is an orthonormal frame field along $\gamma$ we
deduce that $T_{3}^{2}+N_{3}^{2}+B_{3}^{2}=1$. Thus, since $N_{3}=0$, we must
have

$T_{3}=\pm\sqrt{1-B_{3}^{2}}\,.$

Replacing this into (4.3) we obtain

$a^{4}+2a^{2}(1+b^{2}-2B_{3}^{2})+b^{2}(1+b^{2}-2B_{3}^{2})\mp
2abB_{3}\sqrt{1-B_{3}^{2}}=0\,.$

Then it is easy to deduce that

$4a^{2}b^{2}B_{3}^{2}(-1+B_{3}^{2})+\Big{(}a^{4}+2a^{2}(1+b^{2}-2B_{3}^{2})+b^{2}(1+b^{2}-2B_{3}^{2})\Big{)}^{2}=0$

along $\gamma$. It follows that $B_{3}$ is a root of a not identically zero
polynomial with constant coefficients. Therefore, $B_{3}$ is constant along
$\gamma$ and so is $T_{3}$. ∎

######  Remark 4.2.

Part (i) of Lemma 4.1 was obtained by similar methods in [19], but to help the
reader we have inserted here its proof.

As a consequence of Lemma 4.1, we can assume that $N_{3}=0$ along $\gamma$.
According to (4.3), we have to understand when there exists a Frenet helix
$\gamma(s)=(x(s),y(s),z(s))$ such that

(4.5) $T_{3}^{2}+B_{3}^{2}=1$

and

(4.6)
$a^{4}+2a^{2}(1+b^{2}-2B_{3}^{2})+b^{2}(1+b^{2}-2B_{3}^{2})-2abB_{3}\sqrt{1-B_{3}^{2}}=0$

with $T_{3}=\sqrt{1-B_{3}^{2}}$ constant along $\gamma$, or

(4.7)
$a^{4}+2a^{2}(1+b^{2}-2B_{3}^{2})+b^{2}(1+b^{2}-2B_{3}^{2})+2abB_{3}\sqrt{1-B_{3}^{2}}=0$

with $T_{3}=-\sqrt{1-B_{3}^{2}}$.

A routine computation shows that for any helix $\gamma$ we have:

(4.8) $\begin{array}[]{lll}{\rm(i)}&T_{3}=&z^{\prime}\\\ &&\\\
{\rm(ii)}&N_{3}=&-\displaystyle{\frac{1}{a}\big{(}e^{2z}x^{\prime
2}-e^{-2z}y^{\prime 2}-z^{\prime\prime}\big{)}}\\\ &&\\\
{\rm(iii)}&B_{3}=&\displaystyle{\frac{1}{a}\big{(}-y^{\prime}x^{\prime\prime}+x^{\prime}(-4y^{\prime}z^{\prime}+y^{\prime\prime})\big{)}}\end{array}$

Now, since $N_{3}=0$ and $T_{3}$ is constant, we deduce from (4.8)(ii) that
either

(4.9)
$z(s)=z_{1}(s)=\frac{1}{2}\log\left(\frac{y^{\prime}}{x^{\prime}}\right)$

or

(4.10)
$z(s)=z_{2}(s)=\frac{1}{2}\log\left(-\frac{y^{\prime}}{x^{\prime}}\right)$

with the obvious sign restrictions on $x^{\prime},y^{\prime}$. Let us first
assume that $z(s)=z_{1}(s)$. Then

$T_{3}=z_{1}^{\prime}=\frac{1}{2}\left(-\frac{x^{\prime\prime}}{x^{\prime}}+\frac{y^{\prime\prime}}{y^{\prime}}\right)=c_{1}\quad\quad(|c_{1}|\leq
1)\,.$

Integration of this condition yields

(4.11) $y^{\prime}=c_{2}e^{2c_{1}s}x^{\prime}\quad(c_{2}>0)\,.$

Next, using (2.6), (4.9) and imposing $\langle T,T\rangle=1$ we deduce

$c_{1}^{2}+2c_{2}e^{2c_{1}s}x^{\prime 2}=1$

along $\gamma$. Integration of this equality tells us that either

(4.12)
$x(s)=x_{1}(s)=-\frac{\sqrt{(1-c_{1}^{2})}}{c_{1}\sqrt{2c_{2}}}e^{-c_{1}s}+c_{x}$

or

(4.13)
$x(s)=\tilde{x}_{1}(s)=\frac{\sqrt{(1-c_{1}^{2})}}{c_{1}\sqrt{2c_{2}}}e^{-c_{1}s}+c_{x}\,.$

First, let us assume that we are in the case (4.12). Then direct substitution
into (4.11) and integration enables us to deduce the explicit expression for
$y(s)$ and, using (4.9), for $z(s)$ as well. The final output is:

$y(s)=y_{1}(s)=\frac{\sqrt{(1-c_{1}^{2})c_{2}}}{c_{1}\sqrt{2}}e^{c_{1}s}+c_{y}\,,\quad
z_{1}(s)=\frac{1}{2}\log\left(c_{2}e^{2c_{1}s}\right)\,.$

Now, using the explicit expression $(x_{1}(s),y_{1}(s),z_{1}(s))$ into (2.9)
and (2.10) we find that we must have:

$a^{2}=c_{1}^{2}-c_{1}^{4}\,,\quad b=1-c_{1}^{2}\,.$

Finally, using these values for $a,b$ into (4.6), (4.7), together with
$B_{3}=\pm\sqrt{1-c_{1}^{2}}$, it is straightforward to check that the only
acceptable values are $c_{1}=\pm 1/\sqrt{2}$. Both these values for $c_{1}$
give $a=1/2$ and $b=1/2$. If we now choose $x=\tilde{x}_{1}$ the analysis is
similar. We find

$y(s)=\tilde{y}_{1}(s)=-\frac{\sqrt{(1-c_{1}^{2})c_{2}}}{c_{1}\sqrt{2}}e^{c_{1}s}+c_{y}\,,\quad
z_{1}(s)=\frac{1}{2}\log\left(c_{2}e^{2c_{1}s}\right)$

and the conclusion is as before. By way of summary, the only possible
solutions associated to the case $z(s)=z_{1}(s)$ are

$\gamma(s)=\left(x_{1},y_{1},z_{1}\right)$

where

(4.14)
$\begin{array}[]{l}x_{1}=-\frac{e^{-\frac{s}{\sqrt{2}}}}{\sqrt{{2c_{2}}}}+c_{x}\,,\quad
y_{1}=\frac{e^{\frac{s}{\sqrt{2}}}\sqrt{c_{2}}}{\sqrt{2}}+c_{y}\,,\quad
z_{1}(s)=\frac{1}{2}\log\left(c_{2}e^{\sqrt{2}s}\right)\\\ {\rm or}\\\
x_{1}=\frac{e^{\frac{s}{\sqrt{2}}}}{\sqrt{{2c_{2}}}}+c_{x}\,,\quad
y_{1}=-\frac{e^{\frac{-s}{\sqrt{2}}}\sqrt{c_{2}}}{\sqrt{2}}+c_{y}\,,\quad
z_{1}(s)=\frac{1}{2}\log\left(c_{2}e^{-\sqrt{2}s}\right)\\\ {\rm or}\\\
x_{1}=\frac{e^{-\frac{s}{\sqrt{2}}}}{\sqrt{{2c_{2}}}}+c_{x}\,,\quad
y_{1}=-\frac{e^{\frac{s}{\sqrt{2}}}\sqrt{c_{2}}}{\sqrt{2}}+c_{y}\,,\quad
z_{1}(s)=\frac{1}{2}\log\left(c_{2}e^{\sqrt{2}s}\right)\\\ {\rm or}\\\
x_{1}=-\frac{e^{\frac{s}{\sqrt{2}}}}{\sqrt{{2c_{2}}}}+c_{x}\,,\quad
y_{1}=\frac{e^{\frac{-s}{\sqrt{2}}}\sqrt{c_{2}}}{\sqrt{2}}+c_{y}\,,\quad
z_{1}(s)=\frac{1}{2}\log\left(c_{2}e^{-\sqrt{2}s}\right)\end{array}$

where $c_{1},c_{2},c_{x},c_{y}$ are integration constants with $c_{2}>0$.
Taking into account (2.1) and (2.2), we conclude that, up to isometries, the
only proper triharmonic curve associated to the case $z(s)=z_{1}(s)$ is (3.1),
which is the first parametrization of (4.14) with $c_{x}=0,c_{y}=0,c_{2}=1$.

Next, if we repeat the same analysis with $z(s)=z_{2}(s)$ as in (4.10), we
find that

$c_{1}^{2}-c_{1}^{4}=a^{2}\,,\quad b=-1+c_{1}^{2}\,.$

Again, the only acceptable values are $c_{1}=\pm 1/\sqrt{2}$. Both these
values for $c_{1}$ give $a=1/2$ and $b=-1/2$. The parametrization of $\gamma$
in this case, up to isometries, is

$\gamma(s)=\frac{1}{\sqrt{2}}\left(e^{-\frac{s}{\sqrt{2}}},\,e^{\frac{s}{\sqrt{2}}}\,,\,\,s\right)$

which, up to the isometry $(x,y,z)\to(-x,y,z)$, coincides with (3.1) and so
the proof is completed. ∎

##  5\. General helices

As we pointed out in the introduction, the most suitable definition of a
general helix in a Riemannian geometric context is a delicate matter and may
depend on the specific situation. Definition 3.1 above seemed to us the most
appropriate for a first approach to the study of triharmonic curves in Sol.

In this section we study the connection between the proper triharmonic Frenet
helices provided by Theorem 3.2 and the notion of a general helix according to
the paper [1] of Barros. In this article the author, along the lines of
previous work by Langer and Singer (see [10]), proposes the following
definition:

######  Definition 5.1.

A smooth, regular curve $\gamma(s)$ into a $3$-dimensional space form will be
called a general helix if there exists a Killing vector field $\mathcal{V}(s)$
with constant length along $\gamma$ and such that the angle between
$\mathcal{V}$ and the unit tangent $T$ is a non-zero constant along $\gamma$.
We will say that $\mathcal{V}$ is an axis of the general helix.

Using this definition, Barros proved a version of Lancret Theorem for both
${\mathbb{H}}^{3}$ and ${\mathbb{S}}^{3}$. The spherical case is the most
interesting because of the appearance of general helices whose geodesic
curvature and torsion are related by a rule of the type $\tau(s)=\kappa(s)\pm
1$ (see [1]).

Now, we discuss this notion of a general helix in our context.

First, we recall that a base for the Killing fields on Sol is
$\left\\{\mathcal{V}_{1},\mathcal{V}_{2},\mathcal{V}_{3}\right\\}$, where

(5.1)
$\mathcal{V}_{1}=e^{z}E_{1}\,,\quad\mathcal{V}_{2}=e^{-z}E_{2}\,,\quad\mathcal{V}_{3}=-xe^{z}E_{1}+ye^{z}E_{2}+E_{3}\,.$

Our result is:

######  Proposition 5.2.

(i) There exists no proper triharmonic curve $\gamma(s)$ in Sol such that both
$|\mathcal{V}_{1}|$ and $\beta$ are constant along $\gamma$, where $\beta$
denotes the angle between $\mathcal{V}_{1}$ and the unit tangent vector field
$T$. The same is true for $\mathcal{V}_{2}$.

(ii) Let $\gamma(s)$ be a proper triharmonic curve in Sol as in Theorem 3.2.
Then $|\mathcal{V}_{3}|=\sqrt{2}$ and $\beta=0$ along $\gamma$, where $\beta$
is the angle between $\mathcal{V}_{3}$ and $T$. Then $\gamma$ is an integral
curve of $\mathcal{V}_{3}/\sqrt{2}$.

###### Proof.

(i) We argue by contradiction. We observe that
$|\mathcal{V}_{1}|^{2}=e^{2z(s)}$. Therefore, we must have $z(s)=c$ along
$\gamma$. Then explicit integration of the conditions $\langle T,T\rangle=1$
and $\langle T,\mathcal{V}_{1}\rangle=|\mathcal{V}_{1}|\cos\beta$ tells us
that $\gamma(s)$ must be of the type

$\gamma(s)=(\cos\beta e^{-c}s+c_{x},\sin\beta e^{c}s+c_{y},c)\,.$

Then a computation using (2.9) and (2.10) shows that, for this type of curve,
$\kappa(s)=\sqrt{\cos^{2}(2\beta)}$, $\tau(s)=\sin(2\beta)$. Moreover,
equations (2.12) and (2.14) are verified. But (2.13) becomes equivalent to

$\sqrt{\cos^{2}(2\beta)}(5+\cos(4\beta))=0\,.$

This is a contradiction since $\cos(2\beta)=0$ is not acceptable because
$\gamma$ is not a geodesic. The case of $\mathcal{V}_{2}$ is analogous.

(ii) We use the parametrization (3.1) of $\gamma$. Computing we find that
along $\gamma$

$|\mathcal{V}_{3}|^{2}=x^{2}e^{2z}+y^{2}e^{-2z}+1=2\,;\quad|\mathcal{V}_{3}|\cos\beta=\langle
T,\mathcal{V}_{3}\rangle=-xx^{\prime}e^{2z}+yy^{\prime}e^{-2z}+z^{\prime}=\sqrt{2}$

so that $|\mathcal{V}_{3}|=\sqrt{2}$ and $\beta=0$. Then $\gamma$ is an
integral curve of $\mathcal{V}_{3}/\sqrt{2}$, as required to end the proof. ∎

## References

  * [1] M. Barros, General helices and a theorem of Lancret, Proc. Amer. Math. Soc., 125 (1997), pp. 1503–1509.
  * [2] V. Branding, On polyharmonic helices in space forms, Arch. Math. (Basel), 120 (2023), pp. 213–225.
  * [3] V. Branding, S. Montaldo, C. Oniciuc, and A. Ratto, Higher order energy functionals, Adv. Math., 370 (2020), pp. 107236, 60.
  * [4] R. Caddeo, S. Montaldo, C. Oniciuc, and P. Piu, The Euler-Lagrange method for biharmonic curves, Mediterr. J. Math., 3 (2006), pp. 449–465.
  * [5] H. Chen and Z. Guan, Triharmonic CMC hypersurfaces in space forms with at most 3 distinct principal curvatures, Results Math., 77 (2022), pp. Paper No. 155, 17.
  * [6] , Triharmonic CMC hypersurfaces in $\mathbb{R}^{5}(c)$, Manuscripta Math., 172 (2023), pp. 209–220.
  * [7] J. Eells and L. Lemaire, Selected topics in harmonic maps, vol. 50 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1983.
  * [8] , Two reports on harmonic maps, World Scientific Publishing Co., Inc., River Edge, NJ, 1995.
  * [9] Y. Fu and D. Yang, On triharmonic hypersurfaces in space forms, J. Geom. Anal., 33 (2023), pp. Paper No. 263, 19.
  * [10] J. Langer and D. A. Singer, The total squared curvature of closed curves, J. Differential Geom., 20 (1984), pp. 1–22.
  * [11] S. Maeta, $k$-harmonic maps into a Riemannian manifold with constant sectional curvature, Proc. Amer. Math. Soc., 140 (2012), pp. 1835–1847.
  * [12] , Construction of triharmonic maps, Houston J. Math., 41 (2015), pp. 433–444.
  * [13] S. Maeta, N. Nakauchi, and H. Urakawa, Triharmonic isometric immersions into a manifold of non-positively constant curvature, Monatsh. Math., 177 (2015), pp. 551–567.
  * [14] S. Montaldo, C. Oniciuc, and A. Ratto, Polyharmonic hypersurfaces into space forms, Israel J. Math., 249 (2022), pp. 343–374.
  * [15] S. Montaldo and A. Pámpano, Triharmonic curves in 3-dimensional homogeneous spaces, Mediterr. J. Math., 18 (2021), pp. Paper No. 198, 17.
  * [16] S. Montaldo and A. Ratto, New examples of $r$-harmonic immersions into the sphere, J. Math. Anal. Appl., 458 (2018), pp. 849–859.
  * [17] , Proper $r$-harmonic submanifolds into ellipsoids and rotation hypersurfaces, Nonlinear Anal., 172 (2018), pp. 59–72.
  * [18] Y.-L. Ou and B.-Y. Chen, Biharmonic submanifolds and biharmonic maps in Riemannian geometry, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, [2020] ©2020.
  * [19] B. Senoussi, Triharmonic curves in $\rm SOL_{3}$ space, Palest. J. Math., 11 (2022), pp. 264–273.