• Bulletin of the Korean Mathematical Society
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• v.50 no.4
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• pp.1109-1126
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• 2013

Let $\mathbb{M}^3_q(c)$ denote the 3-dimensional space form of index $q=0,1$, and constant curvature $c{\neq}0$. A curve ${\alpha}$ immersed in $\mathbb{M}^3_q(c)$ is said to be a Bertrand curve if there exists another curve ${\beta}$ and a one-to-one correspondence between ${\alpha}$ and ${\beta}$ such that both curves have common principal normal geodesics at corresponding points. We obtain characterizations for both the cases of non-null curves and null curves. For non-null curves our theorem formally agrees with the classical one: non-null Bertrand curves in $\mathbb{M}^3_q(c)$ correspond with curves for which there exist two constants ${\lambda}{\neq}0$ and ${\mu}$ such that ${\lambda}{\kappa}+{\mu}{\tau}=1$, where ${\kappa}$ and ${\tau}$ stand for the curvature and torsion of the curve. As a consequence, non-null helices in $\mathbb{M}^3_q(c)$ are the only twisted curves in $\mathbb{M}^3_q(c)$ having infinite non-null Bertrand conjugate curves. In the case of null curves in the 3-dimensional Lorentzian space forms, we show that a null curve is a Bertrand curve if and only if it has non-zero constant second Frenet curvature. In the particular case where null curves are parametrized by the pseudo-arc length parameter, null helices are the only null Bertrand curves.

• Kyungpook Mathematical Journal
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• v.58 no.2
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• pp.347-359
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• 2018

A Kenmotsu manifold $M^n({\phi},\;{\xi},\;{\eta},\;g)$, (n = 2m + 1 > 3) is called a generalized ${\phi}-recurrent$ if its curvature tensor R satisfies $${\phi}^2(({\nabla}_wR)(X,Y)Z)=A(W)R(X,Y)Z+B(W)G(X,Y)Z$$ for all $X,\;Y,\;Z,\;W{\in}{\chi}(M)$, where ${\nabla}$ denotes the operator of covariant differentiation with respect to the metric g, i.e. ${\nabla}$ is the Riemannian connection, A, B are non-vanishing 1-forms and G is given by G(X, Y)Z = g(Y, Z)X - g(X, Z)Y. In particular, if A = 0 = B then the manifold is called a ${\phi}-symmetric$. Now, a Kenmotsu manifold $M^n({\phi},\;{\xi},\;{\eta},\;g)$, (n = 2m + 1 > 3) is said to be generalized ${\phi}-Ricci$ recurrent if it satisfies $${\phi}^2(({\nabla}_wQ)(Y))=A(X)QY+B(X)Y$$ for any vector field $X,\;Y{\in}{\chi}(M)$, where Q is the Ricci operator, i.e., g(QX, Y) = S(X, Y) for all X, Y. In this paper, we study generalized ${\phi}-recurrent$ and generalized ${\phi}-Ricci$ recurrent Kenmotsu manifolds with respect to quarter-symmetric metric connection and obtain a necessary and sufficient condition of a generalized ${\phi}-recurrent$ Kenmotsu manifold with respect to quarter symmetric metric connection to be generalized Ricci recurrent Kenmotsu manifold with respect to quarter symmetric metric connection.

• Journal of Biomedical Engineering Research
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• v.43 no.6
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• pp.375-381
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• 2022

Brain-computer interface (BCI) is a technology that allows users with motor disturbance to control machines by brainwaves without a physical controller. Motor imagery (MI)-BCI is one of the popular BCI techniques, but it needs a long calibration time for users to perform a mental task that causes high fatigue to the users. MI is reported as showing a similar neural mechanism as motor execution (ME) and motor observation (MO). However, integrative investigations of these three tasks are rarely conducted. In this study, we propose a new paradigm that incorporates three tasks (MI, ME, and MO) and conducted a comparative analysis. For this study, we collected Electroencephalograms (EEG) of motor imagery/execution/observation from 28 healthy subjects and investigated alpha event-related (de)synchronization (ERD/ERS) and classification accuracy (left vs. right motor tasks). As result, we observed ERD and ERS in MI, MO and ME although the timing is different across tasks. In addition, the MI showed strong ERD on the contralateral hemisphere, while the MO showed strong ERD on the ipsilateral side. In the classification analysis using a Riemannian geometry-based classifier, we obtained classification accuracies as MO (66.34%), MI (60.06%) and ME (58.57%). We conclude that there are similarities and differences in fundamental neural mechanisms across the three motor tasks and that these results could be used to advance the current MI-BCI further by incorporating data from ME and MO.