Kentaro Saji

Geometric invariants of cuspidal edges

We give a normal form of the cuspidal edge which uses only diffeomorphisms on the source and isometries on the target. Using this normal form, we study differential geometric invariants of cuspidal edges which determine them up to order three. We also clarify relations between these invariants.

The duality between singular points and inflection points on wave fronts

In the previous paper, the authors gave criteria for AkC1-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 M2 is a compact orientable 2-manifold, and M is the open subset of M2 where the Hessian of f takes negative values. This is a generalization of Bleecker and Wilson’s formula [3] for immersed surfaces in the affine 3-space.

Intrinsic properties of surfaces with singularities

In this paper, we give two classes of positive semi-definite metrics on 2-manifolds. The one is called a class of Kossowski metrics and the other is called a class of Whitney metrics: The pull-back metrics of wave fronts which admit only cuspidal edges and swallowtails in R3 are Kossowski metrics, and the pull-back metrics of surfaces consisting only of cross cap singularities are Whitney metrics. Since the singular sets of Kossowski metrics are the union of regular curves on the domains of definitions, and Whitney metrics admit only isolated singularities, these two classes of metrics are disjoint. In this paper, we give several characterizations of intrinsic invariants of cuspidal edges and cross caps in these classes of metrics. Moreover, we prove Gauss–Bonnet type formulas for Kossowski metrics and for Whitney metrics on compact 2-manifolds.

Behavior of Gaussian Curvature and Mean Curvature Near Non-degenerate Singular Points on Wave Fronts

We define cuspidal curvature

Singularities of Blaschke normal maps of convex surfaces

\kappa_c

Schwarz maps for the hypergeometric differential equation

(resp. normalized cuspidal curvature

On the geometry of folded cuspidal edges

\mu_c

BIFURCATIONS OF VORONOI DIAGRAMS AND ITS APPLICATION TO BRAID THEORY

) along cuspidal edges (resp. at swallowtail singularity) in Riemannian

Singularities of flat fronts in hyperbolic space

3

The geometry of fronts

-manifolds, and show that it gives a coefficient of the divergent term of the mean curvature function. Moreover, we show that the product