The theory of pseudo differential operators, discussed in § 1, is well suited for investigating various problems connected with elliptic differential equations. However, this theory fails to be adequate for studying equations of hyperbolic type, and one is then forced to examine a wider class of operators, the so-called Fourier integral operators (Egorov [1975], Hormander [1968, 1971, 1983, 1985], Kumano-go [1982], Shubin [1978], Taylor [1981], Treves [1980]).

Fourier Integral Operators

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2F60590931937BF33789B5A986069C6D/S0025557200107697a.pdf/div-class-title-span-class-italic-curves-and-singularities-span-by-j-w-bruce-and-p-j-giblin-pp-222-1984-isbn-0-521-24945-7-hardback-25-27091-x-paperback-8-95-cambridge-university-press-div.pdf

Curves and singularities, by J. W. Bruce and P. J. Giblin. Pp 222. 1984. ISBN 0-521-24945-7 (Hardback) £25, 27091-X (Paperback) £8·95 (Cambridge University Press)

We define new special curves in Euclidean 3-space which we call slant helices and conical geodesic curves Those notions are generalizations of the notion of cylindrical helices One of the results in this paper gives a classification of special developable surfaces under the condition of the existence of such a special curve as a geodesic As a result, we consider geometric invariants of space curves By using these invariants, we can estimate the order of contact with those special curves for general space curves All arguments in this paper are straight forward and classical However, there have been no papers which have investigated slant helices and conical geodesic curves so far as we know

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New Special Curves and Developable Surfaces

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 front if it is the projection of a Legendrian immersion into the unit cotangent bundle. We give easily computable criteria for a singular point on a front to be a cuspidal edge or a swallowtail. Using this, we prove that generically flat fronts in hyperbolic 3-space admit only cuspidal edges and swallowtails. We also show that any complete flat front (provided it is not rotationally symmetric) has associated parallel surfaces whose singularities consist of only cuspidal edges and swallowtails.

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Singularities of flat fronts in hyperbolic space

We shall introduce the singular curvature function on cuspidal edges of surfaces, which is related to the Gauss-Bonnet formula and which characterizes the shape of cuspidal edges Moreover, it is closely related to the behavior of the Gaussian curvature of a surface near cuspidal edges and swallowtails

http://annals.math.princeton.edu/wp-content/uploads/annals-v169-n2-p03.pdf

The geometry of fronts

We study generic properties of cylindrical helices and Bertrand curves as applications of singularity theory for plane curves and spherical curves.

Generic properties of helices and Bertrand curves

In this paper we adopt the hyperboloid in Minkowski space as the model of hyperbolic space. We define the hyperbolic Gauss map and the hyperbolic Gauss indicatrix of a hypersurface in hyperbolic space. The hyperbolic Gauss map has been introduced by Ch. Epstein [J. Reine Angew. Math. 372 (1986) 96–135] in the Poincare ball model, which is very useful for the study of constant mean curvature surfaces. However, it is very hard to perform the calculation because it has an intrinsic form. Here, we give an extrinsic definition and we study the singularities. In the study of the singularities of the hyperbolic Gauss map (indicatrix), we find that the hyperbolic Gauss indicatrix is much easier to calculate. We introduce the notion of hyperbolic Gauss–Kronecker curvature whose zero sets correspond to the singular set of the hyperbolic Gauss map (indicatrix). We also develop a local differential geometry of hypersurfaces concerning their contact with hyperhorospheres.2000 Mathematical Subject Classification: 53A25, 53A05, 58C27.

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Singularities of hyperbolic Gauss maps

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Singularities of Hyperbolic Gauss Maps

Differential Geometry from a Singularity Theory Viewpoint provides a new look at the fascinating and classical subject of the differential geometry of surfaces in Euclidean spaces. The book uses singularity theory to capture some key geometric features of surfaces. It describes the theory of contact and its link with the theory of caustics and wavefronts. It then uses the powerful techniques of these theories to deduce geometric information about surfaces embedded in 3, 4 and 5-dimensional Euclidean spaces. The book also includes recent work of the authors and their collaborators on the geometry of sub-manifolds in Minkowski spaces.

Shyuichi Izumiya

Papers

New Special Curves and Developable Surfaces

Generic properties of helices and Bertrand curves

Singularities of hyperbolic Gauss maps

Singularities of Hyperbolic Gauss Maps

Differential Geometry from a Singularity Theory Viewpoint