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Yuichiro Sato

Publications and source records attributed to Yuichiro Sato.

7 recordsLinked to original sources
arXiv2025

Lorentzian homogeneous Ricci-flat metrics on almost abelian Lie groups

When the identity component of the full isometry group of a four-dimensional spacetime acts simply transitively, the unique Ricci-flat metric is the Petrov solution. This isometry group is almost abelian; that is, its Lie algebra contains an abelian ideal of codimension one. In this paper, we study Lorentzian left-invariant metrics on almost abelian Lie groups of dimension four or higher. In particular, we construct a Ricci-flat but non-flat metric that generalizes the Petrov solution to arbitrarily high dimensions. The generalized solution is geodesically complete and admits closed timelike curves. The construction of the closed timelike curves is new even in the four-dimensional Petrov solution, as it requires no identification of coordinates.

math.DG↗
arXiv2024

Educational Effects in Mathematics: Conditional Average Treatment Effect depending on the Number of Treatments

This study examines the educational effect of the Academic Support Center at Kogakuin University. Following the initial assessment, it was suggested that group bias had led to an underestimation of the Center's true impact. To address this issue, the authors applied the theory of causal inference. By using T-learner, the conditional average treatment effect (CATE) of the Center's face-to-face (F2F) personal assistance program was evaluated. Extending T-learner, the authors produced a new CATE function that depends on the number of treatments (F2F sessions) and used the estimated function to predict the CATE performance of F2F assistance.

stat.ME↗
arXiv2023

Spatially homogeneous solutions of vacuum Einstein equations in general dimensions

We study time-dependent compactification of extra dimensions. We assume that the spacetime is spatially homogeneous, and solve the vacuum Einstein equations without cosmological constant in more than three dimensions. We consider globally hyperbolic spacetimes in which almost Abelian Lie groups act on the spaces isometrically and simply transitively. We give left-invariant metrics on the spaces and solve Ricci-flat conditions of the spacetimes. In the four-dimensional case, our solutions correspond to the Bianchi type II solution. By our results and previous studies, all spatially homogeneous solutions whose spaces have zero-dimensional moduli spaces of left-invariant metrics are found. For the simplest solution, we show that each of the spatial dimensions cannot expand or contract simultaneously in the late-time it.

gr-qc↗
arXiv2022

Variational problems for integral invariants of the second fundamental form of a map between pseudo-Riemannian manifolds

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.

math.DG↗
arXiv2021

Totally umbilical submanifolds in pseudo-Riemannian space forms

A totally umbilical submanifold in pseudo-Riemannian manifolds is a fundamental notion, which is characterized by the condition that the second fundamental form is proportional to the metric. It is also a generalization of the notion of a totally geodesic submanifold. In this paper, we classify congruent classes of full totally umbilical submanifolds in non-flat pseudo-Riemannian space forms, and consider their moduli spaces. As a consequence, we show that some moduli spaces of isometric immersions between space forms which one of the same constant curvature are non-Hausdorff.

math.DG↗
arXiv2018

$d$-minimal surfaces in three-dimensional singular semi-Euclidean space $\mathbb{R}^{0,2,1}$

In this paper, we investigate surfaces in singular semi-Euclidean space $\mathbb{R}^{0,2,1}$ endowed with a degenerate metric. We define $d$-minimal surfaces, and give a representation formula of Weierstrass type. Moreover, we prove that $d$-minimal surfaces in $\mathbb{R}^{0,2,1}$ and spacelike flat zero mean curvature (ZMC) surfaces in four-dimensional Minkowski space $\mathbb{R}^{4}_{1}$ are in one-to-one correspondence.

math.DG↗