We obtain a complete topological classification of $k$-folding map-germs on generic surfaces in $\mathbb {R}^3$, discover new robust features of surfaces and recover, in a unified way, many of the robust features that were obtained previously by considering the contact of a surface with lines, planes or spheres.

Let $f\,{:}\,(\mathbb R^n,0)\to (\mathbb R,0)$ be an analytic function germ with non-isolated singularities and let $F\,{:}\, (\mathbb{R}^{1+n},0) \to (\mathbb{R},0)$ be a 1-parameter deformation of f. Let $ f_t ^{-1}(0) \cap B_\epsilon^n$ , $0 < \vert t \vert \ll \epsilon$ , be the “generalized” Milnor fiber of the deformation F. Under some conditions on F, we give a topological degree formula for the Euler characteristic of this fiber. This generalizes a result of Fukui.

In order to investigate envelopes for singular surfaces, we introduce one- and two-parameter families of framed surfaces and the basic invariants, respectively. By using the basic invariants, the existence and uniqueness theorems of one- and two-parameter families of framed surfaces are given. Then we define envelopes of one- and two-parameter families of framed surfaces and give the existence conditions of envelopes which are called envelope theorems. As an application of the envelope theorems, we show that the projections of singular solutions of completely integrable first-order partial differential equations are envelopes.

We study frames in ℝ3 and mapping from a surface M in ℝ3 to the space of frames. We consider in detail mapping frames determined by a unit tangent principal or asymptotic direction field U and the normal field N. We obtain their generic local singularities as well as the generic singularities of the direction field itself. We show, for instance, that the cross-cap singularities of the principal frame map occur precisely at the intersection points of the parabolic and subparabilic curves of different colours. We study the images of the asymptotic and principal foliations on the unit sphere by their associated unit direction fields. We show that these curves are solutions of certain first order differential equations and point out a duality in the unit sphere between some of their configurations.

We consider a smooth curve with singular points in the Euclidean space. As a smooth curve with singular points, we have introduced a framed curve or a framed immersion. A framed immersion is a smooth curve with a moving frame and the pair is an immersion. We define an evolute and a focal surface of a framed immersion in the Euclidean space. The evolutes and focal surfaces of framed immersions are generalizations of each object of regular space curves. We give relationships between singularities of the evolutes and of the focal surfaces. Moreover, we consider properties of the evolutes, focal surfaces and repeated evolutes.

The detection of the bifurcation set of polynomial mapping ℝn → ℝp, n ⩾ p, in more than two variables remains an unsolved problem. In this note we provide a solution for n = p + 1 ⩾ 3.

As an application of the theory of graph-like Legendrian unfoldings, relations of the hidden structures of caustics, and wave front propagations are revealed.

We study orthogonal projections of generic embedded hypersurfaces in ℝ4 with boundary to 2-spaces. Therefore, we classify simple map germs from ℝ3 to the plane of codimension less than or equal to 4 with the source containing a distinguished plane which is preserved by coordinate changes. We also go into some detail on their geometrical properties in order to recognize the cases of codimension less than or equal to 1.

In this paper we provide lower bounds for the dimension of various critical sets, and we point out some differential maps with high dimensional critical sets.

The paper is an addendum to D. Andrica and L. Funar, ‘On smooth maps with finitely many critical points’, J. London Math. Soc. (2) 69 (2004) 783–800.

The minimum number of critical points of a small codimension smooth map between two manifolds is computed. Some partial results for the case of higher codimension when the manifolds are spheres are also given.

We establish the correspondence between Euclidean differential geometry of submanifolds in $\mathbb{R}^n$ and projective differential geometry of submanifolds in $\mathbb{R}^{n +1 }$ under stereographic projection to quadrics of revolution (and to a more general class of quadrics called quasi-revolution quadrics). V. D. Sedykh found a relation between Lagrangian and Legendrian singularities by stereographic projection to a sphere in Euclidean space. We generalise Sedykh's results in several directions, for instance:

(1) such a relation holds for any stereographic projection of a hyperplane to a quasi-revolution quadric in $\mathbb{R}^n \times \mathbb{R}$, where here $\mathbb{R}^n$ denotes Euclidean space;

(2) when the quadric is a paraboloid the relation between Lagrangian and Legendrian singularities is the most natural one, and the calculations, formulas and proofs are simpler;

(3) using the classical theory of poles, polars and polar duality, we construct the natural isomorphism between the front of the Lagrange submanifold of the normal map (considered as a subvariety in $J^0(\mathbb{R}^n) = \mathbb{R}^n \times \mathbb{R}$) and the front of the Legendre submanifold of the tangential map (considered as a subvariety in the space $(\mathbb{R}^{n + 1})^\vee$ of affine n-dimensional subspaces in $\mathbb{R}^{n + 1}$).

As a consequence of our results we obtain a formula to calculate the vertices of smooth curves in $\mathbb{R}^n$. Our results may be applied to calculate and study umbilic points of surfaces in $\mathbb{R}^3$, or more generally to study the contact of submanifolds in Euclidean space $\mathbb{R}^n$ with k-spheres, for $k = 1,\ldots,n - 1$, in terms of the contact of submanifolds in $\mathbb{R}^{n + 1}$ with $(k + 1)$-planes.

We remark that it is possible to obtain additional geometric information when stereographic projection is replaced by an inversion.