arXiv subjects
Publications and source records attributed to Takashi Sakai.
We determine the polar and the maximal antipodal set $P$ for the outer 3-symmetric space $\mathbb{S}^7 \times \mathbb{S}^7 = \mathrm{Spin}_8/G_2$ where the 3-symmetric structure is given by the triality automorphism $τ$ on $\mathrm{Spin}_8$. It turns out that $P$ has three elements. The 3-symmetric structure extends to a (non-abelian) $S_3$-structure which we also investigate.
In this paper, we introduce the notion of generalized $s$-manifolds, which is a generalization of symmetric spaces. Then we give a method to construct generalized $s$-manifolds and present some typical examples. We study polars and antipodal sets of generalized $s$-manifolds aiming to extend the results on compact symmetric spaces due to Chen and Nagano.
We give a necessary and sufficient condition for two real flag manifolds, which are not necessarily congruent, in a complex flag manifold to intersect transversally in terms of the symmetric triad. Then we show that the intersection of two real flag manifolds is antipodal. As an application, we prove that any real flag manifold in a complex flag manifold is a globally tight Lagrangian submanifold.
We study variational problems for integral invariants, which are defined as integrations of invariant functions of the second fundamental form, of a smooth map between pseudo-Riemannian manifolds. We derive the first variational formulae for integral invariants defined from invariant homogeneous polynomials of degree two. Among these integral invariants, we show that the Euler-Lagrange equation of the Chern-Federer energy functional is reduced to a second order PDE. Then we give some examples of Chern-Federer submanifolds in Riemannian space forms.
The human dosimetry for electromagnetic field exposure is an essential task to develop exposure guidelines/standards for human safety as well as product safety assessment. At frequencies from a few hundred Hz to 10 MHz, the adverse effect to be protected is the stimulation of the peripheral nervous system. The in situ electric field in the skin is used as a surrogate of nerve activation. In the low-frequency dosimetry, a high but inaccurate in situ electric field has been reported at positions where a skin-to-skin contact exists, whose relation to the stimulation is controversial. One of the reasons for high electric fields may be attributable to the current resolution of anatomical models. In this study, we first evaluate the stimulation threshold at postures of skin-to-skin contact experimentally for different hand/finger positions to represent skin touching/non-touching scenarios. We confirm that the skin-to-skin contact does not lower the threshold current of magnetic stimulation devices needed to induce pain. Second, a new method is proposed for hand modeling to configure different finger positions using static hand models with similar postures to the experiments. We compute the in situ electric field at skin-to-skin contact for the different hand posture scenarios that indicate an excessive raise of the electric field in skin-to-skin regions that is not justified by the experiments. The comparison suggests that a high in situ electric field in the skin would be caused by poor modeling of the skin layers, which is not enough to represent in a resolution of the order of a millimeter. This skin-to-skin contact should not be considered to set the restriction in the international exposure guidelines/standards as well as product safety assessment.
We classify $R$-spaces that admit a certain natural $Γ$-symmetric structure. We further determine the maximal antipodal sets of these structures.
We give a necessary and sufficient condition for orbits of commutative Hermann actions and actions of the direct product of two symmetric subgroups on compact Lie groups to be biharmonic in terms of symmetric triad with multiplicities. By this criterion, we determine all the proper biharmonic submanifolds in irreducible symmetric spaces of compact type which are singular orbits of commutative Hermann actions of cohomogeneity two. Also, in compact simple Lie groups, we determine all the biharmonic hypersurfaces which are regular orbits of actions of the direct product of two symmetric subgroups which are associated to commutative Hermann actions of cohomogeneity one.
In this paper, by constructing area-nonincreasing retractions, we prove area-minimizing properties of some cones over minimal embeddings of R-spaces.
In this paper, we study biharmonic hypersurfaces in Einstein manifolds. Then, we determine all the biharmonic hypersurfaces in irreducible symmetric spaces of compact type which are regular orbits of commutative Hermann actions of cohomogeneity one.
In this paper we calculate the Lagrangian Floer homology $HF(L_0, L_1 : {\mathbb Z}_2)$ of a pair of real forms $(L_0,L_1)$ in a monotone Hermitian symmetric space $M$ of compact type in the case where $L_0$ is not necessarily congruent to $L_1$. In particular, we have a generalization of the Arnold-Givental inequality in the case where $M$ is irreducible. As its application, we prove that the totally geodesic Lagrangian sphere in the complex hyperquadric is globally volume minimizing under Hamiltonian deformations.
We determine all tight Lagrangian surfaces in $S^2 \times S^2$. In particular, globally tight Lagrangian surfaces in $S^2 \times S^2$ are nothing but real forms.
In this paper we study tangentially degeneracy of the orbits of s-representations in the sphere. We show that an orbit of an s-representation is tangentially degenerate if and only if it is through a long root, or a short root of restricted root system of type G_2. Moreover these orbits provide many new examples of tangentially degenerate submanifolds which satisfy the Ferus equality.
We introduce the notion of a weakly reflective submanifold, which is an austere submanifold with a certain global condition, and study its fundamental properties. Using these, we determine weakly reflective orbits and austere orbits of s-representations.
We prove that the product of equators $S^{1} \times S^{1}$ in $S^{2} \times S^{2}$ is globally volume minimizing under Hamiltonian deformations.
A body of theory is completely different between relativity theory and quantum mechanics. Most targeting physical phenomena are different between them as well. Despite that both theories describe our time-space world, no one has succeeded in arriving at one of these theories by using the other one, deductively. Fusion of these two theories produced quantum field theory. However this theory impresses us as a theory that was made by merging the two theories by force. So this theory cannot indicate the reason why both relativity theory and quantum mechanics exist in this world at the same time. Here I show you that quantum mechanics can be led by the equation of Minkowski's metrics, which stands up in the relativistic world of space and time. This is the first report proving mathematically that quantum mechanics can be born reductively in our relativistic world.