arXiv subjects

Naoya Ando

Publications and source records attributed to Naoya Ando.

14 recordsLinked to original sources

arXiv2025

An index formula for hemispheres of a $C^2$-regular convex closed surface in Euclidean $3$-space

Carath\'eodory's conjecture has long been regarded as one of the central problems in the classical theory of convex surfaces. In this paper, we establish an index formula for hemispheres of convex closed surfaces under $C^2$-regularity. The proof is based on studying a vertical section of the null hypersurfaces in Lorentz--Minkowski $4$-space associated with the originally given convex surface. As a consequence, the conjecture is affirmatively solved in the $C^2$-case.

math.DG↗

arXiv2025

Hopf differentials and curvature line flows on time-like CMC surfaces

We investigate the relationship between the Hopf differentials and the curvature line flows on time-like constant mean curvature (CMC) surfaces in Lorentzian 3-space forms. In particular, when the Hopf differential is non-degenerate, the index of a curvature line flow at an umbilic point depends precisely on the remainder of its order modulo four.

math.DG↗

arXiv2025

The twistor lifts of surfaces in 4-spaces

This is a survey of the twistor lifts of surfaces in $4$-dimensional spaces. In most part of this survey, the space is Euclidean $4$-space $E^4$. The definitions of the Gauss maps and the twistor lifts of surfaces in $E^4$ are given by orthogonal complex structures of $E^4$. Based on these definitions, we can understand holomorphicity of the Gauss maps of minimal surfaces in $E^4$ and isotropicity of such surfaces.

math.DG↗

arXiv2025

Classification of the topological holonomy groups in $SO(3)$

In this paper, we obtain classification of the topological holonomy groups in $SO(3)$. Such a group is given by one of the following: a finite group (such groups are classified by Klein); a commutative infinite group which is generated by one or two elements, and dense in a subgroup of $SO(3)$ isomorphic to $SO(2)$; a non-commutative infinite group generated by two elements of order $2$, $\infty$ such that these rotation axes are orthogonal; a non-commutative infinite group which is dense in $SO(3)$.

math.DG↗

arXiv2025

The equations of Gauss, Codazzi and Ricci of surfaces in 4-dimensional space forms

Let $N$ be a Riemannian, neutral or Lorentzian $4$-dimensional space form. In this paper, the expressions of the equations of Gauss, Codazzi and Ricci of a space-like or time-like surface in $N$ given in [7] are naturally understood in terms of the induced connection (of the complexification) of the two-fold exterior power of the pull-back bundle on the surface. Moreover, based on such expressions, we characterize several classes of surfaces related to the covariant derivatives of the twistor lifts and so on.

math.DG↗

arXiv2025

Surfaces with flat normal connection in 4-dimensional space forms

Let $N$ be a Riemannian, Lorentzian or neutral $4$-dimensional space form with constant sectional curvature $L_0$. In this paper, noticing the linearly dependent condition, we obtain characterizations of space-like surfaces in $N$ with flat normal connection and parallel normal vector fields. In addition, we obtain a generic characterization of space-like surfaces in $N$ with flat normal connection and $K\equiv L_0$ which do not admit any parallel normal vector fields. For time-like surfaces in $N$ with flat normal connection, we obtain analogous results.

math.DG↗

arXiv2024

The topological holonomy group and the complexity of horizontality

Based on [1], we study the complexity of horizontality in each twistor space $\hat{E}_{\varepsilon}$ associated with an oriented vector bundle $E$ of rank $4$ with a positive-definite metric over the $2$-torus $T^2$, and obtain classification of the topological holonomy groups in $SO(3)$. We observe that there exist many topological holonomy groups in $SO(3)$ generated by two finite order elements and equipped with noncommutative pairs which consist of infinite order elements. We find topological holonomy groups which are dense in $SO(4)$.

math.DG↗

arXiv2024

Nilpotent structures of oriented neutral vector bundles

In this paper, we study nilpotent structures of an oriented vector bundle $E$ of rank $4n$ with a neutral metric $h$ and an $h$-connection $\nabla$. We define $H$-nilpotent structures of $(E, h, \nabla )$ for a Lie subgroup $H$ of $SO(2n, 2n)$ related to neutral hyperK\"{a}hler structures. We observe that there exist a complex structure $I$ and paracomplex structures $J_1$, $J_2$ of $E$ such that $h$, $\nabla$, $I$, $J_1$, $J_2$ form a neutral hyperK\"{a}hler structure of $E$ if and only if there exists an $H$-nilpotent structure of $(E, h, \nabla )$.

math.DG↗

arXiv2024

Horizontality with infinite complexity in the twistor spaces on tori

We study the complexity of horizontality in the twistor space $\hat{E}$ associated with an oriented vector bundle $E$ of rank $4$ with a positive-definite metric over a torus. If the horizontality has finite complexity of degree $d>2$ for an element of a fiber of $\hat{E}$, then the complexity is expressed in terms of a finite subgroup of $SO(3)$ ([3]). In the present paper, we observe that if the horizontality has infinite complexity derived from one of the cases studied in [3], then the complexity is expressed by a dense subset of $S^2$.

math.DG↗

arXiv2023

The $SO(3,1)$-orbits in the light cone of the 2-fold exterior power of the Minkowski 4-space

Two special neutral hypersurfaces $\mathcal{L}_{\pm}$ in the light cone $L(\bigwedge^2 E^4_1 )$ studied in [1], [3] are $SO(3,1)$-orbits. In this paper, we see that each $SO(3,1)$-orbit in $L(\bigwedge^2 E^4_1 )$ is either a neutral hypersurface homothetic to one of $\mathcal{L}_{\pm}$ in $L(\bigwedge^2 E^4_1 )$ or a hypersurface with a two-dimensional involutive distribution where the induced metric is degenerate. The difference between these hypersurfaces can be understood in terms of the stabilizer and the $r$-slice of $L(\bigwedge^2 E^4_1 )$ for $r>0$.

math.DG↗

arXiv2023

Time-like surfaces with zero mean curvature vector in 4-dimensional neutral space forms

Let $M$ be a Lorentz surface and $F:M\rightarrow N$ a time-like and conformal immersion of $M$ into a 4-dimensional neutral space form $N$ with zero mean curvature vector. We see that the curvature $K$ of the induced metric on $M$ by $F$ is identically equal to the constant sectional curvature $L_0$ of $N$ if and only if the covariant derivatives of both of the time-like twistor lifts are zero or light-like. If $K\equiv L_0$, then the normal connection $\nabla^{\perp}$ of $F$ is flat, while the converse is not necessarily true. We see that a holomorphic paracomplex quartic differential $Q$ on $M$ defined by $F$ is zero or null if and only if the covariant derivative of at least one of the time-like twistor lifts is zero or light-like. In addition, we see that $K$ is identically equal to $L_0$ if and only if not only $\nabla^{\perp}$ is flat but also $Q$ is zero or null.

math.DG↗

arXiv2023

Sections of time-like twistor spaces with light-like or zero covariant derivatives

The conformal Gauss maps of time-like minimal surfaces in $E^3_1$ give sections of the time-like twistor spaces associated with the pull-back bundles such that the covariant derivatives are fully light-like, that is, these are either light-like or zero, and do not vanish at any point. For an oriented neutral $4n$-manifold $(M, h)$, if $J$ is an $h$-reversing almost paracomplex structure of $M$ such that $\nabla J$ is locally given by the tensor product of a nowhere zero 1-form and an almost nilpotent structure related to $J$, then we will see that $\nabla J$ is valued in a light-like $2n$-dimensional distribution $\mathcal{D}$ such that $(M , h, \mathcal{D} )$ is a Walker manifold and that the square norm $\parallel\!\nabla J\!\parallel^2$ of $\nabla J$ vanishes. We will obtain examples of $h$-reversing almost paracomplex structures of $E^{4n}_{2n}$ as above. In addition, we will obtain all the pairs of $h$-reversing almost paracomplex structures of $E^4_2$ such that each pair gives sections of the two time-like twistor spaces with fully light-like covariant derivatives.

math.DG↗

arXiv2023

Umbilics of surfaces in the Lorentz-Minkowski 3-space

In this paper, we prove several fundamental properties on umbilics of a space-like or time-like surface in the Lorentz-Minkowski space $L^3$. In particular, we show that the local behavior of the curvature line flows of the germ of a space-like surface in $L^3$ is essentially the same as that of a surface in Euclidean space. As a consequence, for each positive integer $m$, there exists a germ of a space-like surface with an isolated $C^{\infty}$-umbilic (resp. $C^1$-umbilic) of index $(3-m)/2$ (resp. $1+m/2$). We also show that the indices of isolated umbilics of time-like surfaces in $L^3$ that are not the accumulation points ofquasi-umbilics are always equal to zero. On the other hand, when quasi-umbilics accumulate, there exist countably many germs of time-like surfaces which admit an isolated umbilic with non-zero indices.

math.DG↗

arXiv2015

C^1-umbilics with arbitrarily high indices

In this paper, the existence of C^1-umbilics with arbitrarily high indices is shown. This implies that more than C^1-regularity is required to prove Loewner's conjecture.

math.DG↗