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We introduce the sphere fibration for real map germs with radial discriminant and we address the problem of its equivalence with the Milnor-Hamm tube fibration. Under natural conditions, we prove the existence of open book structures with singularities and solve the equivalence problem., updated references before publication

Let $N$ (resp., $U$) be a manifold (resp., an open subset of $\mathbb{R}^m$). Let $f:N\to U$ and $F:U\to \mathbb{R}^\ell$ be an immersion and a $C^{\infty}$ mapping, respectively. Generally, the composition $F\circ f$ does not necessarily yield a mapping transverse to a given subfiber-bundle of $J^1(N,\mathbb{R}^\ell)$. Nevertheless, in this paper, for any $\mathcal{A}^1$-invariant fiber, we show that composing generic linearly perturbed mappings of $F$ and the given immersion $f$ yields a mapping transverse to the subfiber-bundle of $J^1(N,\mathbb{R}^\ell)$ with the given fiber. Moreover, we show a specialized transversality theorem on crossings of compositions of generic linearly perturbed mappings of a given mapping $F:U\to \mathbb{R}^\ell$ and a given injection $f:N\to U$. Furthermore, applications of the two main theorems are given., 19 pages. Accepted for publication in Journal of the Mathematical Society of Japan. The paper is a generalization of arXiv:1610.02880. As the appendix, the statement of the main theorem in arXiv:1607.03220 (the proof of it is in arXiv:1607.03220) is introduced in the final section of this paper

We construct constant negative Gaussian curvature tori with one family of planar curvature lines in Euclidean 3-space. We show that these tori are wave fronts. The singularities of these tori are studied.

Given a null-cobordant oriented framed link $L$ in a closed oriented $3$--manifold $M$, we determine those links in $M \setminus L$ which can be realized as the singular point set of a generic map $M \to \mathbb{R}^2$ that has $L$ as an oriented framed regular fiber. Then, we study the linking behavior between the singular point set and regular fibers for generic maps of $M$ into $\mathbb{R}^2$., 13 pages, 8 figures

We show that the module of lowerable vector fields for a finitely ${\cal L}$-determined multigerm is finitely generated in a constructive way.

We consider the local geometry of a generic 1-parameter family of smooth curves in the real plane for which one member of the family has parallel tangents at two inflexion points. We study the equidistants of this family, that is the loci of points at a fixed ratio along chords joining points with parallel tangents, as a 2-parameter family depending on the value of the fixed ratio and on the parameter in the family of curves. Codimension 2 singularities of type ‘gull’ arise in this way and are in general versally unfolded by the two parameters. We also calculate the family of duals of the equidistants; here it is necessary to view them as bifurcation sets of bigerms and they evolve through ‘moth’ and ‘nib’ singularities also encountered in 1-parameter families of symmetry sets in the plane. Finally we show that certain sub-families of the 2- parameter family of equidistants can be classified by reduction to a normal form.

We construct, for $m\geq 6$ and $2n\leq m$, closed manifolds $M^{m}$ with finite nonzero $\varphi(M^{m},S^{n}$), where $\varphi(M,N)$ denotes the minimum number of critical points of a smooth map $M\to N$. We also give some explicit families of examples for even $m\geq 6, n=3$, taking advantage of the Lie group structure on $S^3$. Moreover, there are infinitely many such examples with $\varphi(M^{m},S^{n})=1$. Eventually we compute the signature of the manifolds $M^{2n}$ occurring for even $n$., Comment: Michigan Math. J., to appear, 21p

This is a half survey article on the recent development of the theory of graph-like Legendrian unfoldings which is the sequel to the previous surveys. The notion of big Legendrian submanifolds was introduced by Zakalyukin for describing the wave front propagations. Graph-like Legendrian unfoldings belong to a special class of big Legendrian submanifolds. In particular, natural equivalence relations among graph-like Legendrian unfoldings are introduced and geometric properties of these equivalence relations are investigated. Although this is a survey article, some new original results and proofs for some implicitly known results are given.

A simplified trisection is a trisection map on a 4-manifold such that, in its critical value set, there is no double point and cusps only appear in triples on innermost fold circles. We give a necessary and sufficient condition for a 3-tuple of systems of simple closed curves in a surface to be a diagram of a simplified trisection in terms of mapping class groups. As an application of this criterion, we show that trisections of spun 4-manifolds due to Meier are diffeomorphic (as trisections) to simplified ones. Baykur and Saeki recently gave an algorithmic construction of a simplified trisection from a directed broken Lefschetz fibration. We also give an algorithm to obtain a diagram of a simplified trisection derived from their construction., 18 pages, many figures in color

We consider bundle homomorphisms between tangent distributions and vector bundles of the same rank. We study the conditions for fundamental singularities when the bundle homomorphism is induced from a Morin map. When the tangent distribution is the contact structure, we characterize singularities of the bundle homomorphism by using the Hamilton vector fields., 11 pages 5 figures

We consider the problems of sufficiency of jets relative to a given closed set. In the non-relative case, criteria for r-jets to be V-sufficient and C^0-sufficient in C^r mappings or C^{r+1} mappings have been obtained. In particular, it is shown that V-sufficiency and C^0-sufficiency in C^r functions or C^{r+1} functions are equivalent. In this paper we discuss characterisations of V-sufficiency and C^0-sufficiency in the relative case, corresponding to the above non-relative results. Applying the results obtained in the relative case, we construct examples of polynomial functions whose relative r-jets are V-sufficient in C^r functions and C^{r+1} functions but not C^0-sufficient in C^r functions and C^{r+1} functions, respectively. In addition, we give characterisations of relative finite V-determinacy and also relative finite C^r contact determinacy., 34 pages, no figure

The Lojasiewicz inequalities for real analytic functions on Euclidean space were first proved by Stanislaw Lojasiewicz (1965) using methods of semianalytic and subanalytic sets, arguments later simplified by Bierstone and Milman (1988). In this article, we first give an elementary geometric, coordinate-based proof of the Lojasiewicz inequalities in the special case where the function is $C^1$ with simple normal crossings. We then prove, partly following Bierstone and Milman (1997) and using resolution of singularities for real analytic varieties, that the gradient inequality for an arbitrary real or complex analytic function follows from the special case where it has simple normal crossings. In addition, we prove the Lojasiewicz inequalities when a function is $C^N$ and generalized Morse-Bott of order $N \geq 3$; we gave an elementary proof of the Lojasiewicz inequalities when a function is $C^2$ and Morse-Bott in arXiv:1708.09775v4 (finite-dimensional case) and arXiv:1706.09349 (infinite-dimensional case)., Comment: 28 pages, incorporating final galley proof corrections

Cuspidal edges and swallowtails are typical non-degenerate singular points on wave fronts in the Euclidean $3$-space. Their first fundamental forms belong to a class of positive semi-definite metrics called "Kossowski metrics". A point where a Kossowski metric is not positive definite is called a singular point or a semi-definite point of the metric. Kossowski proved that real analytic Kossowski metric germs at their non-parabolic singular points(the definition of "non-parabolic singular point" is stated in the introduction here) can be realized as wave front germs (Kossowski's realization theorem). On the other hand, in a previous work with K. Saji, the third and the fourth authors introduced the notion of "coherent tangent bundle". Moreover, the authors, with M. Hasegawa and K. Saji, proved that a Kossowski metric canonically induces an associated coherent tangent bundle. In this paper, we shall explain Kossowski's realization theorem from the viewpoint of coherent tangent bundles. Moreover, as refinements of it, we give a criterion that a given Kossowski metric can be realized as the induced metric of a germ of cuspidal edge (resp. swallowtail or cuspidal cross cap). Several applications of these criteria are given. Also, some remaining problems on isometric deformations of singularities of analytic maps are given at the end of this paper., Comment: 34 pages, 2 figures

In the 3-dimensional Lorentz-Minkowski space we prove that the sign of the Gaussian curvature of any timelike minimal surface is determined by the degeneracy and the orientations of the two null curves that generate the surface. Moreover, we also investigate the behavior of the Gaussian curvature near singular points of a timelike minimal surface which admits some kind of singular points., Comment: 21 pages, 5 figures

By Hartman--Nirenberg's theorem, any complete flat hypersurface in Euclidean space must be a cylinder over a plane curve. However, if we admit some singularities, there are many non-trivial examples. Flat fronts are flat hypersurfaces with admissible singularities. Murata--Umehara gave a representation formula for complete flat fronts with non-empty singular set in Euclidean $3$-space, and proved the four vertex type theorem. In this paper, we prove that, unlike the case of $n=2$, there do not exist any complete flat fronts with non-empty singular set in Euclidean $(n+1)$-space $(n\geq 3)$., Comment: 8 pages

We prove that the asymptotic completion of a developable Möbius strip in Euclidean three-space must have at least one singular point other than cuspidal edge singularities. Moreover, if the strip is generated by a closed geodesic, then the number of such singular points is at least three. These lower bounds are both sharp.

We study the geometry of the cuspidal edge $M$ in $\mathbb R^3$ derived from its contact with planes and lines (referred to as flat geometry). The contact of $M$ with planes is measured by the singularities of the height functions on $M$. We classify submersions on a model of $M$ by diffeomorphisms and recover the contact of $M$ with planes from that classification. The contact of $M$ with lines is measured by the singularities of orthogonal projections of $M$. We list the generic singularities of the projections and obtain the generic deformations of the apparent contour (profile) when the direction of projection varies locally in $S^2$. We also relate the singularities of the height functions and of the projections to some geometric invariants of the cuspidal edge., 32 pages, 13 figures

The holonomic approximation lemma of Eliashberg and Mishachev is a powerful tool in the philosophy of the $h-$principle. By carefully keeping track of the quantitative geometry behind the holonomic approximation process, we establish several refinements of this lemma. Gromov's idea from convex integration of working one pure partial derivative at a time is central to the discussion. We give applications of our results to flexible symplectic and contact topology., 30 pages, 18 figures. Comments welcome!

Two cross caps in Euclidean $3$-space are said to be formally isometric if their Taylor expansions of the first fundamental forms coincide by taking a suitable local coordinate system. For a given $C^\infty$ cross cap $f$, we give a method to find all cross caps which are formally isometric to $f$. As an application, we give a countable family of intrinsic invariants of cross caps which recognizes formal isometry classes completely., Comment: 24 pages, 5 figures

We study the reflectional symmetry of a generically embedded $2$-dimensional surface $M$ in the hyperbolic or de Sitter $3$-dimensional spaces. This symmetry is picked up by the singularities of folding maps that are defined and studied here. We also define the evolute and symmetry set of $M$ and prove duality results that relate them to the bifurcation sets of the family of folding maps.

In the previous paper (arXiv:0804.0701), the authors gave criteria for A_{k+1}-type singularities on wave fronts. Using them, we show in this paper that there is a duality between singular points and inflection points on wave fronts in the projective space. As an application, we show that the algebraic sum of 2-inflection points (i.e. godron points) on an immersed surface in the real projective space is equal to the Euler number of M_-. Here M^2 is a compact orientable 2-manifold, and M_-$is the open subset of M^2 where the Hessian of f takes negative values. This is a generalization of Bleecker and Wilson's formula for immersed surfaces in the affine 3-space., Comment: 12 pages, one figure; to appear in Osaka Journal of Mathematics

Let [math] be a smooth manifold of dimension [math] . We will describe the group of all cobordism classes of smooth maps of [math] –dimensional closed manifolds into [math] with singularities of given class (including all fold singularities if [math] ) in terms of certain stable homotopy groups by applying the homotopy principle on the existence level, which is assumed to hold for those smooth maps. It will enable us to construct an explicit classifying space for this cobordism group in the dimensions [math] and [math] .

Let $M^2$ be an oriented 2-manifold and $f:M^2\to R^3$ a $C^\infty$-map. A point $p\in M^2$ is called a singular point if $f$ is not an immersion at $p$. The map $f$ is called a front (or wave front), if there exists a unit $C^\infty$-vector field $\nu$ such that the image of each tangent vector $df(X)$ $(X\in TM^2)$ is perpendicular to $\nu$, and the pair $(f,\nu)$ gives an immersion into $R^3\times S^2$. In our previous paper, we gave an intrinsic formulation of wave fronts in $R^3$. In this paper, we shall investigate the behavior of cuspidal edges near corank one singular points and establish Gauss-Bonnet-type formulas under the intrinsic formulation., Comment: 20 pages, 12 figures

In this paper we study fundamental directional properties of sets under the assumption of condition (SSP) (introduced in a previous paper). We show several transversality theorems in the singular case and an (SSP)-structure preserving theorem. As an illustration, our transversality results are used to prove several facts concerning complex analytic varieties. The (SSP)-property is most suitable for understanding transversality in the Lipschitz category. This property is shared by a large class of sets, in particular by subanalytic sets or by definable sets in an o-minimal structure., 28 pages, 3 figures included

In this paper, we first classify singular fibers of proper $C^\infty$ stable maps of 3-dimensional manifolds with boundary into surfaces. Then, we compute the cohomology groups of the associated universal complex of singular fibers, and obtain certain cobordism invariants for Morse functions on compact surfaces with boundary., Comment: 18 pages with 7 figures

We study the geometry of surfaces in $\mathbb{R}^3$ with corank 1 singularities. At a singular point we define the curvature parabola using the first and second fundamental forms of the surface, which contains all the local second order geometrical information about the surface. The curvature parabola is used to introduce the concepts of asymptotic directions and umbilic curvature, which are related to contact properties of the surface with planes and spheres.

Donaldson showed that every closed symplectic 4-manifold can be given the structure of a topological Lefschetz pencil. Gay and Kirby showed that every closed 4-manifold has a trisection. In this paper we relate these two structure theorems, showing how to construct a trisection directly from a topological Lefschetz pencil. This trisection is such that each of the three sectors is a regular neighborhood of a regular fiber of the pencil. This is a 4-dimensional analog of the following trivial 3-dimensional result: For every open book decomposition of a 3-manifold M, there is a decomposition of M into three handlebodies, each of which is a regular neighborhood of a page., Comment: 9 pages, 9 figures

Any stable map from a surface to the plane has an associated graph. In the case of the sphere, such graphs are of tree type [Hacon et al. 03]. We characterize the trees that can occur as graphs of fold maps from the sphere to the plane. In order to do so, we first determine the sets of integers that may occur as winding numbers for the branch sets of these maps.