Papers

Peer-Reviewed Articles

1. On the fundamental group of semi-Riemannian manifolds with positive curvature tensor, Commun. Anal. Geom., vol. 29 (2021), no 5, 1255-1277.

Abstract: This paper presents an investigation of the relation between some positivity of the curvature and the finiteness of fundamental groups in semi- Riemannian geometry. We consider semi-Riemannian submersions π : (E, g) →(B, −g_B) under the condition with (B, g_B) Riemannian, the fiber closed Riemannian,and the horizontal distribution integrable. Then we prove that, if the lightlike geodesically complete or timelike geodesically complete semi- Riemannian manifold E has some positivity of curvature, then the fundamental group of the fiber is finite. Moreover we construct an example of semi-Riemannian submersions with some positivity of curvature, non-integrable horizontal distribution, and the finiteness of the fundamental group of the fiber.

2. (with Hajime Matsui) Natural Gradient Descent of Complex-Valued Neural Networks Invariant under Rotations, IEICE Trans. Fundamentals., vol. E102-A (2019), no 12, 1988-1996.

Abstract: The natural gradient descent is an optimization method for real-valued neural networks that was proposed from the viewpoint of information geometry. Here, we present an extension of the natural gradient descent to complex-valued neural networks. Our idea is to use the Hermitian extension of the Fisher information matrix. Moreover, we generalize the projected natural gradient (PRONG), which is a fast natural gradient descent algorithm, to complex-valued neural networks. We also consider the advantage of complex-valued neural networks over real-valued neural networks. A useful property of complex numbers in the complex plane is that the rotation is simply expressed by the multiplication. By focusing on this property, we construct the output function of complex-valued neural networks, which is invariant even if the input is changed to its rotated value. Then, our complex-valued neural network can learn rotated data without data augmentation. Finally, through simulation of online character recognition, we demonstrate the effectiveness of the proposed approach.

3. (with Yoshikazu Nagata) On a characterization of unbounded homogeneous domains with boundaries of light cone type, Tohoku Math. J. (2), 69 (2017), no 2, 161-181.

Abstract: We determine the automorphism groups of unbounded homogeneous domains with boundaries of light cone type. Furthermore we present a group- theoretic characterization of one of the domains. As a corollary we prove the non-existence of compact quotients of the homogeneous domain. We also give a counterexample of the characterization.

4. On the fundamental group of a complete globally hyperbolic Lorentzian manifold with a lower bound for the curvature tensor, Differ. Geom. Appl., 41 (2015), 33–38.

Abstract: In this paper, we study the fundamental group of a certain class of globally hyperbolic Lorentzian manifolds with a positive curvature tensor. We prove that the fundamental group of lightlike geodesically complete parametrized Lorentzian products is finite under the conditions of a positive curvature tensor and the fiber compact.

5. Properly discontinuous isometric group actions on inhomogeneous Lorentzian manifolds, Geom. Dedicata, 168 (2014), no. 1, 359–368.

Abstract: If a homogeneous space G/H is acted properly discontinuously upon by a subgroup Γ of G via the left action, the quotient space Γ\G/H is called a Clifford–Klein form. In Calabi and Markus (Ann Math (2) 75: 63–76, 1962) proved that there is no infinite subgroup of the Lorentz group O(n + 1, 1) whose left action on the de Sitter space O(n + 1, 1)/O(n, 1) is properly discontinuous. It follows that a compact Clifford–Klein form of the de Sitter space never exists. In the present paper, we provide a new extension of the theorem of E. Calabi and L. Markus to a certain class of Lorentzian manifolds that are not necessarily homogeneous by using the techniques of differential geometry.