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We study the stability of travelling wave solutions to dispersion generalized Benjamin--Ono type equations with the double power nonlinearities $|u|^{p-1}u + |u|^{q-1}u \ (1 < p < q < \infty)$. We prove that travelling wave solutions constructed by ground states are stable if $p$ is $L^{2}$-subcritical and speeds of waves are sufficiently small by showing the coercivity of the linearized operator around ground states, which is introduced by Grillakis-Shatah-Strauss (1987)., Comment: 21 pages

Surfaces with concentric $K$-contours and parallel $K$-contours in Euclidean $3$-space are defined. Crucial examples are presented and characterization of them are given., Comment: 10 pages, 4 figures

We study semilinear elliptic equations with the fractional Laplacian in $\mathbb{R}$. The equations with single power nonlinearities have been observed by Weinstein(1987), Frank--Lenzmann(2013) and so on. We focus on the equations with double power nonlinearities and consider the existence of ground states., Comment: 20 page. Abstract is corrected

We cut a general, static, spherically symmetric spacetime and paste its copy to make a wormhole with a thin shell of any barotropic fluid in general relativity. We show that the stability of the thin-shell wormhole is characterized by a set of circular photon orbits called an (anti)photon sphere in the original spacetime if a momentum flux passing through a throat is prohibited. Our result will be useful to classify the stability of the thin shell on the throat against linearized spherically symmetric perturbations., Comment: 8 papes, 2 figures, accepted for publication in Physical Review D

A system of two gravitating bodies floating around a restricted region of strong gravitational field is investigated. We consider two concentric spherically symmetric timelike shells spatially constrained by a perfectly reflecting inner and outer boundary. It is shown numerically that even when the gravitational radius of a contracting shell is larger than the radius of the inner boundary, energy transfer occurs due to the intersection with the other expanding shell before the contracting shell becomes a black hole, resulting nonlinearly stable motion. The system appears to be in a permanently stable periodic motion due to the repetition of forward and reverse energy transfer. The larger the specific energy of a shell, the more stable the motion is. In addition, the motion of the null shell as the fastest limit of the timelike shell is also investigated. Unlike the timelike shell, the motion of the two null shells reduces to exact recurrence equations. By analyzing the recurrence equations, we find the null shells also allow stable motions. Using the algebraic computation of the recurrence equations, we show numerical integration is not necessary for the nonlinear dynamics of the null shells in confined geometry., Comment: 19pages, 10figures. Acknowledgments improved

We show that kinematics of charged particles allows us to model the growth of particles' energy by consecutive particle-splits, once a spherical mirror as a perfectly reflective boundary is placed outside a charged black hole. We consider a charged version of the Penrose process, in which a charged particle decays into two fragments, one of them has negative energy and the other has positive energy that is larger than that of the parent particle. The confinement system with the mirror makes the particles' energy amplified each time a split of the parent particle occurs. Thus, the energy is a monotonically increasing function of time. However, the energy does not increase unboundedly, but rather asymptotes to a certain finite value, implying no instability of the system in this respect., Comment: 16 pages, 5 figures, typo corrected, miscalculation fixed, version accepted for PRD

In this paper we investigate equilibria of continuous differential equation models of network dynamics. The motivation comes from gene regulatory networks where each directed edge represents either down- or up-regulation, and is modeled by a sigmoidal nonlinear function. We show that the existence and stability of equilibria of a sigmoidal system is determined by a combinatorial analysis of the limiting switching system with piece-wise constant non-linearities. In addition, we describe a local decomposition of a switching system into a product of simpler cyclic feedback systems, where the cycles in each decomposition correspond to a particular subset of network loops., Comment: 29 pages, 3 figures, submitted to SIAM Journal on Applied Dynamical Systems

We show that a certain simply-stated notion of "analytic completeness" of the image of a real analytic map implies the map admits no analytic extension. We also give a useful criterion for that notion of analytic completeness by defining arc-properness of continuous maps, which can be considered as a very weak version of properness. As an application, we judge the analytic completeness of a certain class of constant mean curvature surfaces (the so-called "G-catenoids") or their analytic extensions in the de Sitter 3-space., Comment: 32 pages, 6 figures

High energy collision of massive bodies is investigated without fine tuning. We study multiple collisions of two spherical concentric shells in a gravitationally bound system and calculate the center of mass energy between the shells. We solve the equation of motions for two shells without imposing any fine tuning of the initial parameters. In this bound system, the shells collide many times and these motions are highly nontrivial due to chaotic behavior of the shells. Consequently, the center of mass energy for each collision varies nontrivially and even reach almost its theoretical upper limit. We confirm that a significant proportion of the theoretical limit is automatically achieved during multiple collisions without fine tuning. At the same time, we also study shell ejection from the system after some collisions. If the initial shell's energy is large enough, multiple collisions may cause one shell to accumulate energy so that it escapes to infinity, even though two shells are initially confined in the system. The ejection is caused by multiple collisions inducing nontrivial energy transfer between the shells. The relation between the maximum center of mass energy and the energy transfer causing the shell ejection is also discussed., Comment: 21 pages, 8 figures, 1 table. PRD accepted, typos corrected, references added, published version

We review recent works on the possibility for eternal existence of thin-shell wormholes on Einstein and Einstein-Gauss-Bonnet gravity. We introduce thin-shell wormholes that are categorized into a class of traversable wormhole solutions. After that, we discuss stable thin-shell wormholes with negative-tension branes in Reissner-Nordstr\"om-(anti) de Sitter spacetimes in $d$ dimensional Einstein gravity. Imposing $Z_2$ symmetry, we construct and classify traversable static thin-shell wormholes in spherical, planar and hyperbolic symmetries. It is found that the spherical wormholes are stable against spherically symmetric perturbations. It is also found that some classes of wormholes in planar and hyperbolic symmetries with a negative cosmological constant are stable against perturbations preserving symmetries. In most cases, stable wormholes are found with the appropriate combination of an electric charge and a negative cosmological constant. However, as special cases, there are stable wormholes even with a vanishing cosmological constant in spherical symmetry and with a vanishing electric charge in hyperbolic symmetry. Subsequently, the existence and dynamical stability of traversable thin-shell wormholes with electrically neutral negative-tension branes is discussed in Einstein-Gauss-Bonnet theory of gravitation. We consider radial perturbations against the shell for the solutions, which have the $Z_2$ symmetry. The effect of the Gauss-Bonnet term on the stability depends on the spacetime symmetry., Comment: 52pages, 17figures, 6tables. This article belongs to the Special Issue Recent Advances in Wormhole Physics (MDPI). This article is based on arXiv:1411.5454 [gr-qc] and arXiv:1506.08550 [gr-qc]

A Damour-Solodukhin wormhole with a metric which is similar to a Schwarzschild black hole seems to be a black hole mimicker since it is difficult to distinguish them by practical astrophysical observations. In this paper, we investigate a center-of-mass energy for the collision of two test particles in the Damour-Solodukhin wormhole spacetime. We show that the center-of-mass energy for the head-on collision of the particles is large if the difference between the metrics of the wormhole and the black hole is small. To deeply understand the high energy particle collision, we generalize the head-on collision to static, spherically symmetric black-hole-like wormholes., Comment: 21 pages, 1 figure, title changed, minor correction, references added, accepted for publication in Physical Review D

In the previous paper [Harada, Cardoso, and Miyata, Phys.\ Rev.\ D {\bf 99} (2019), 044039], it is shown that a hollow transmissive shell collapsing to an ultracompact object of radius very close to its horizon radius generally emits transient Hawking radiation followed by a couple of bursts separated each other by a long time interval. In the current paper, we expand the previous work in two independent directions: changing boundary conditions and specifying the equations of state of the matter. First, we introduce a perfectly reflective surface collapsing to an ultracompact object and find that this model also emits transient Hawking radiation that is followed only by a single burst. Second, we introduce two different collapse dynamics to an ultracompact object and specify the corresponding matter equations of state. We find that transient Hawking radiation is quite commonly seen in early times, while the subsequent bursts strongly depend on the boundary condition and the equation of state or the braking behavior of the surface., Comment: 30 pages, 12 figures

We investigate the effect of inhomogeneity on primordial black hole formation in the matter dominated era. In the gravitational collapse of an inhomogeneous density distribution, a black hole forms if apparent horizon prevents information of the central region of the configuration from leaking. Since information cannot propagate faster than the speed of light, we identify the threshold of the black hole formation by considering the finite speed for propagation of information. We show that the production probability $\beta_{inhom}(\sigma)$ of primordial black holes, where $\sigma$ is density fluctuation at horizon entry, is significantly enhanced from that derived in previous work in which the speed of propagation was effectively regarded as infinite. For $\sigma \ll 1$, we obtain $\beta_{inhom}\simeq 3.70 \sigma^{3/2}$, which is larger by about an order of magnitude than the probability derived in earlier work by assuming instantaneous propagation of information., Comment: 17pages, 7figures

We cut and paste two Banados-Teitelboim-Zanelli (BTZ) spacetimes at a throat by the Darmois-Israel method to construct a rotating wormhole with a thin shell filled with a barotropic fluid. The thin shell at the throat and both sides of the throat corotate. We investigate the linear stability of the thin shell of the rotating wormhole against radial perturbations. We show that the wormhole becomes more and more stable the larger its angular momentum is until the angular momentum reaches a critical value and that the behavior of a condition for stability significantly changes when the angular momentum exceeds the critical value. We find that the overcritical rotating wormhole has the radius of the thin shell, which is stable regardless of the equation of state for the barotropic fluid., Comment: 8 pages, 3 figures, title changed, minor correction, references added, accepted for publication in Physical Review D

A topological computation method, called the MGSTD method, is applied to time-series data obtained from meteorological measurement. The method gives decomposition of the dynamics into invariant sets and gradient-like transitions between them, by dividing the phase space into grids and representing the time-series as a combinatorial multi-valued map over the grids. Since the time-series is highly stochastic, the multi-valued map is statistically determined by taking preferable transitions between the grids into account. The time-series data are principal components of pressure pattern in troposphere and stratosphere in the northern hemisphere. The application yields some particular transitions between invariant sets, which leads to circular motion on the phase space spanned by the principal components. The Morse sets and the circular motion are consistent with the characteristic pressure patterns and the change between them that have been shown in preceding meteorological studies.

A strong gravity naked singular region can give important clues towards understanding classical as well as spontaneous nature of General Relativity. We propose here a model for energy emission from a naked singular region in a self-similar dust spacetime by gluing two self-similar dust solutions at the Cauchy horizon. The energy is defined and evaluated as a surface energy of a null hypersurface, the null shell. Also included are scenarios of spontaneous creation or disappearance of a singularity, end of inflation, black hole formation and bubble nucleation. Our examples investigated here explicitly show that one can model unlimitedly luminous and energetic objects in the framework of General Relativity., Comment: 18pages, 3 figures, accepted version

Kenmotsu's formula describes surfaces in Euclidean 3-space by their mean curvature functions and Gauss maps. In Lorentzian 3-space, Akutagawa-Nishikawa's formula and Magid's formula are Kenmotsu-type formulas for spacelike surfaces and for timelike surfaces, respectively. We apply them to a few problems concerning rotational or helicoidal surfaces with constant mean curvature. Before that, we show that the three formulas above can be written in a unified single equation., Comment: 27 pages, 13 figures

We investigate the relation between quadrics and their Christoffel duals on the one hand, and certain zero mean curvature surfaces and their Gauss maps on the other hand. To study the relation between timelike minimal surfaces and the Christoffel duals of 1-sheeted hyperboloids we introduce para-holomorphic elliptic functions. The curves of type change for real isothermic surfaces of mixed causal type turn out to be aligned with the real curvature line net., Comment: 19 pages, 6 figures

It is classically known that the only zero mean curvature entire graphs in the Euclidean 3-space are planes, by Bernstein's theorem. A surface in Lorentz-Minkowski 3-space $\boldsymbol{R}^3_1$ is called of mixed type if it changes causal type from space-like to time-like. In $\boldsymbol{R}^3_1$, Osamu Kobayashi found two zero mean curvature entire graphs of mixed type that are not planes. As far as the authors know, these two examples were the only known examples of entire zero mean curvature graphs of mixed type without singularities. In this paper, we construct several families of real analytic zero mean curvature entire graphs of mixed type in Lorentz-Minkowski $3$-space. The entire graphs mentioned above lie in one of these classes., Comment: 31 pages, 5 figures

The Jorge-Meeks $n$-noid ($n\ge 2$) is a complete minimal surface of genus zero with $n$ catenoidal ends in the Euclidean 3-space $\boldsymbol{R}^3$, which has $(2\pi/n)$-rotation symmetry with respect to its axis. In this paper, we show that the corresponding maximal surface $f_n$ in Lorentz-Minkowski 3-space $\boldsymbol{R}^3_1$ has an analytic extension $\tilde f_n$ as a properly embedded zero mean curvature surface. The extension changes type into a time-like (minimal) surface., Comment: 23 pages ; 17 figures

In this paper, we shall prove that space-like surfaces with bounded mean curvature functions in real analytic Lorentzian 3-manifolds can change their causality to time-like surfaces only if the mean curvature functions tend to zero. Moreover, we shall show the existence of such surfaces with non-vanishing mean curvature and investigate their properties., Comment: 12 pages, 2 figures

We introduce a novel approach to obtaining mathematically rigorous results on the global dynamics of ordinary differential equations. Motivated by models of regulatory networks, we construct a state transition graph from a piecewise affine ordinary differential equation. We use efficient graph algorithms to compute an associated Morse graph that codifies the recurrent and gradient-like dynamics. We prove that for 2-dimensional systems, the Morse graph defines a Morse decomposition for the dynamics of any smooth differential equation that is sufficiently close to the original piecewise affine ordinary differential equation.

Catenoids in de Sitter $3$-space $S^3_1$ belong to a certain class of space-like constant mean curvature one surfaces. In a previous work, the authors classified such catenoids, and found that two different classes of countably many exceptional elliptic catenoids are not realized as closed subsets in $S^3_1$. Here we show that such exceptional catenoids have closed analytic extensions in $S^3_1$ with interesting properties., Comment: 12 pages, 8 figures

The effect of the Gauss-Bonnet term on the existence and dynamical stability of thin-shell wormholes as negative tension branes is studied in the arbitrary dimensional spherically, planar, and hyperbolically symmetric spacetimes. We consider radial perturbations against the shell for the solutions which have the Z${}_2$ symmetry and admit the general relativistic limit. It is shown that the Gauss-Bonnet term shrinks the parameter region admitting static wormholes. The effect of the Gauss-Bonnet term on the stability depends on the spacetime symmetry. For planar symmetric wormholes, the Gauss-Bonnet term does not affect their stability. If the coupling constant is positive but small, the Gauss-Bonnet term tends to destabilize spherically symmetric wormholes, while it stabilizes hypebolically symmetric wormholes. The Gauss-Bonnet term can destabilize hypebolically symmetric wormholes as a non-perturbative effect, however, spherically symmetric wormholes cannot be stable., Comment: 37 pages, 6 figures

We study the homomorphism induced in homology by a closed correspondence between topological spaces, using projections from the graph of the correspondence to its domain and codomain. We provide assumptions under which the homomorphism induced by an outer approximation of a continuous map coincides with the homomorphism induced in homology by the map. In contrast to more classical results we do not require that the projection to the domain have acyclic preimages. Moreover, we show that it is possible to retrieve correct homological information from a correspondence even if some data is missing or perturbed. Finally, we describe an application to combinatorial maps that are either outer approximations of continuous maps or reconstructions of such maps from a finite set of data points.

We investigate negative tension branes as stable thin shell wormholes in Reissner-Nordstrom-(anti) de Sitter spacetimes in $d$ dimensional Einstein gravity. Imposing Z2 symmetry, we construct and classify traversable static thin shell wormholes in spherical, planar (or cylindrical) and hyperbolic symmetries. In spherical geometry, we find the higher dimensional counterpart of Barcelo and Visser's wormholes, which are stable against spherically symmetric perturbations. We also find the classes of thin shell wormholes in planar and hyperbolic symmetries with a negative cosmological constant, which are stable against perturbations preserving symmetries. In most cases, stable wormholes are found with the combination of an electric charge and a negative cosmological constant. However, as special cases, we find stable wormholes even with vanishing cosmological constant in spherical symmetry and with vanishing electric charge in hyperbolic symmetry., Comment: 24pages, 6 figures

Passive dynamic walking is a useful model for investigating the mechanical functions of the body that produce energy-efficient walking. The basin of attraction is very small and thin, and it has a fractal-like shape; this explains the difficulty in producing stable passive dynamic walking. The underlying mechanism that produces these geometric characteristics was not known. In this paper, we consider this from the viewpoint of dynamical systems theory, and we use the simplest walking model to clarify the mechanism that forms the basin of attraction for passive dynamic walking. We show that the intrinsic saddle-type hyperbolicity of the upright equilibrium point in the governing dynamics plays an important role in the geometrical characteristics of the basin of attraction; this contributes to our understanding of the stability mechanism of bipedal walking.

For particle collision near rapidly rotating Kerr black holes, the center-of-mass energy can be arbitrarily high if the angular momentum of either of the colliding particles is fine-tuned. Recently, it has been shown that particles which are produced by such a particle collision and escape to infinity cannot be very massive nor very energetic. For electrically charged black holes there is a similar phenomenon, where the center-of-mass energy for the collision of charged particles near the horizon can be arbitrarily high. One might expect that there would exist a similar bound on the energy and mass of particles that are produced by such a particle collision and escape to infinity. In this paper, however, we see that this expectation is not the case. We explicitly show that superheavy and highly energetic charged particles produced by the collision near maximally charged black holes can escape to infinity at least within classical theory if the backreaction and self-force of the particle can be neglected., Comment: 10 pages, minor correction, accepted for publication in Physical Review D

CMC-1 trinoids (i.e. constant mean curvature one immersed surface with three regular embedded ends) in hyperbolic 3-space H^3 are irreducible generically, and the irreducible ones have been classified. However, the reducible case has not yet been fully treated, so in this paper we give an explicit description of CMC-1 trinoids in H^3 that includes the reducible case., Comment: 10 pages, 3 figures

We prove several topological properties of linear Weingarten surfaces of Bryant type, as wave fronts in hyperbolic 3-space. For example, we show the orientability of such surfaces, and also co-orientability when they are not flat. Moreover, we show an explicit formula of the non-holomorphic hyperbolic Gauss map via another hyperbolic Gauss map which is holomorphic. Using this, we show the orientability and co-orientability of CMC-1 faces (i.e., constant mean curvature one surfaces with admissible singular points) in de Sitter 3-space. (CMC-1 faces might not be wave fronts in general, but belong to a class of linear Weingarten surfaces with singular points.) Since both linear Weingarten fronts and CMC-1 faces may have singular points, orientability and co-orientability are both nontrivial properties. We also remark on some properties of non-orientable maximal surfaces in Lorentz-Minkowski 3-space, comparing the corresponding properties of CMC-1 faces in de Sitter 3-space., Comment: 19 pages, 3 figures

We construct brane solutions in 6 dimensional Einstein-Skyrme systems. A class of baby skyrmion solutions realizes warped compactification of the extra dimensions and gravity localization on the brane for negative bulk cosmological constant. Coupling of the fermions with the brane skyrmions lead to the brane localized fermions. In terms of the level crossing picture, emergence of the massive localized modes as well as the zero mode are observed. Nonlinear nature of the skyrmions brings richer information for the fermions level structure. The level comprises doubly degenerate lowest plus single excited modes. The three generation of the fundamental fermions is based on this structure. The quark/lepton mass hierarchy is successfully obtained in terms of a slightly deformed baby-skyrmions with topological charge three., Comment: 16 pages, 17 figures. One figure added, some points clarified, references improved. Version accepted for publication

We construct brane solutions in six dimensional Einstein-Skyrme systems. A class of baby skyrmion solutions realize warped compactification of the extra dimensions and gravity localization on the brane for negative bulk cosmological constant. Coupling of the fermions with the brane skyrmions successfully lead to the brane localized fermions. The standard representation of the gamma matrices is used to obtain massive localized modes as well as the massless one. Nonlinear nature of the skyrmions brings richer information for the fermions level structure. In terms of the level crossing picture, emergence of the massive localized modes as well as the zero mode are observed., Comment: 11 pages, 10 figures, presented at Seventh Alexander Friedmann International Seminar on Gravitation and Cosmology, Joao Pessoa, Brasil, 29 June - 5 July, 2008

We shall consider about a 3-brane embedded in six-dimensional space-time with a negative bulk cosmological constant. The 3-brane is constructed by a topological soliton solution living in two-dimensional axially symmetric transverse subspace. Similar to most previous works of six-dimensional soliton models, our Maxwell gauged CP1 brane model can also achieve to localize gravity around the 3-brane. The CP1 field is described by a scalar doublet and derived from O(3) sigma model by projecting it onto two-dimensional complex space. In that sense, our framework is more effective than other solitonic brane models concerning with gauge theory. We shall also discuss about linear stability analysis for our new model by fluctuating all fields., Comment: 23 pages, 7 figures; references added

In this paper, we investigate the asymptotic behavior of regular ends of flat surfaces in the hyperbolic 3-space H^3. Galvez, Martinez and Milan showed that when the singular set does not accumulate at an end, the end is asymptotic to a rotationally symmetric flat surface. As a refinement of their result, we show that the asymptotic order (called "pitch" p) of the end determines the limiting shape, even when the singular set does accumulate at the end. If the singular set is bounded away from the end, we have -1

After Galvez, Martinez and Milan discovered a (Weierstrass-type) holomorphic representation formula for flat surfaces in hyperbolic 3-space, the first, third and fourth authors here gave a framework for complete flat fronts with singularities in H^3. In the present work we broaden the notion of completeness to weak completeness, and of front to p-front. As a front is a p-front and completeness implies weak completeness, the new framework and results here apply to a more general class of flat surfaces. This more general class contains the caustics of flat fronts -- shown also to be flat by Roitman (who gave a holomorphic representation formula for them) -- which are an important class of surfaces and are generally not complete but only weakly complete. Furthermore, although flat fronts have globally defined normals, caustics might not, making them flat fronts only locally, and hence only p-fronts. Using the new framework, we obtain characterizations for caustics., Comment: 26 pages, 5 figures

We investigate surfaces with constant harmonic-mean curvature one (HMC-1 surfaces) in hyperbolic three-space. We allow them to have certain kinds of singularities, and discuss some global properties. As well as flat surfaces and surfaces with constant mean curvature one (CMC-1 surfaces), HMC-1 surfaces belong to a certain class of Weingarten surfaces. From the viewpoint of parallel surfaces, CMC-1 surfaces and HMC-1 surfaces are representative among this class., Comment: 20 pages, 9 figures; Section 3 revised

It is well-known that the unit cotangent bundle of any Riemannian manifold has a canonical contact structure. A surface in a Riemannian 3-manifold is called a (wave) front if it is the projection of a Legendrian immersion into the unit cotangent bundle. We shall give easily-computable criteria for a singular point on a front to be a cuspidal edge or a swallowtail. Using this, we shall prove that generically flat fronts in the hyperbolic 3-space admit only cuspidal edges and swallowtails. Moreover, we will show that every complete flat front (which is not rotationally symmetric) has associated parallel surfaces whose singularities consist of only cuspidal edges and swallowtails., Comment: 35 pages, 13 figures

We shall investigate flat surfaces in hyperbolic 3-space with admissible singularities, called `flat fronts'. An Osserman-type inequality for complete flat fronts is shown. When equality holds in this inequality, we show that all the ends are embedded. Moreover, we shall give new examples for which equality holds., Comment: 22pages, 10 figures

For null curves in PSL(2,C), there exists a representation formula in terms of two meromorphic functions and their derivatives (Small's formula). In this paper, we give an elementary proof of Small's formula. Moreover, a similar formula for Legendrian curves in PSL(2,C) is given. As null curves in PSL(2,C) are related to mean curvature one surfaces in hyperbolic 3-space H^3, Legendrian curves are related to flat surfaces in H^3. So, as an application of Small-type formula for Lengendrian curves, we give new examples of flat surfaces in H^3., Comment: 13 pages, 9 figures

In the previous paper, Takahasi and the authors generalized the theory of minimal surfaces in Euclidean n-space to that of surfaces with holomorphic Gauss map in certain class of non-compact symmetric spaces. It also includes the theory of constant mean curvature one surfaces in hyperbolic 3-space. Moreover, a Chern-Osserman type inequality for such surfaces was shown. Though its equality condition is not solved yet, the authors have noticed that the equality condition of the original Chern-Osserman inequality itself is not found in any literature except for the case n=3, in spite of its importance. In this paper, a simple geometric condition for minimal surfaces that attains equality in the Chern-Osserman inequality is given. The authors hope it will be a useful reference for readers., Comment: 6 pages

We shall discuss the class of surfaces with holomorphic right Gauss maps in non-compact duals of compact semisimple Lie groups (e.g. SL(n,C)/SU(n)), which contains minimal surfaces in R^n and constant mean curvature 1 surfaces in H^3. A Weierstrass type representation formula, and a Chern-Osserman type inequality for such surfaces are given., Comment: 25 pages

We give the complete solution to the local diffeomorphism classification problem of generic singularities which appear in tangent surfaces, in as wider situations as possible. We interpret tangent geodesics as tangent lines whenever a (semi-)Riemannian metric, or, more generally, an affine connection is given in an ambient space of arbitrary dimension. Then, given an immersed curve, we define the tangent surface as the ruled surface by tangent geodesics to the curve. We apply the characterization of frontal singularities found by Kokubu, Rossman, Saji, Umehara, Yamada, and Fujimori, Saji, Umehara, Yamada, and found by the first author related to the procedure of openings of singularities., Comment: This paper is derived as the main extract of the results in the unpublished paper by the authors "G. Ishikawa, T. Yamashita, Affine connections and singularities of tangent surfaces to space curves, arXiv:1501.07341 [math.DG]

We give the complete solution to the local diffeomorphism classification problem of generic singularities which appear in tangent surfaces, in as wider situations as possible. We interpret tangent geodesics as tangent lines whenever a (semi-)Riemannian metric, or, more generally, an affine connection is given in an ambient space of arbitrary dimension. Then, given an immersed curve, or, more generally a directed curve or a frontal curve which has well-defined tangent directions along the curve, we define the tangent surface as the ruled surface by tangent geodesics to the curve. We apply the characterization of frontal singularities found by Kokubu, Rossman, Saji, Umehara, Yamada, and Fujimori, Saji, Umehara, Yamada, and found by the first author related to the procedure of openings of singularities., Comment: 24 pages, 0 figures