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Singularities of non-degenerate n-ruled (n + 1)-manifolds in Euclidean space

Kentaro Saji (2004)

Banach Center Publications

The objective of this paper is to study singularities of n-ruled (n + 1)-manifolds in Euclidean space. They are one-parameter families of n-dimensional affine subspaces in Euclidean space. After defining a non-degenerate n-ruled (n + 1)-manifold we will give a necessary and sufficient condition for such a map germ to be right-left equivalent to the cross cap × interval. The behavior of a generic n-ruled (n + 1)-manifold is also discussed.

Smoothability of proper foliations

John Cantwell, Lawrence Conlon (1988)

Annales de l'institut Fourier

Compact, C 2 -foliated manifolds of codimension one, having all leaves proper, are shown to be C ∞ -smoothable. More precisely, such a foliated manifold is homeomorphic to one of class C ∞ . The corresponding statement is false for foliations with nonproper leaves. In that case, there are topological distinctions between smoothness of class C r and of class C r + 1 for every nonnegative integer r .

Smoothing of real algebraic hypersurfaces by rigid isotopies

Alexander Nabutovsky (1991)

Annales de l'institut Fourier

Define for a smooth compact hypersurface M n of R n + 1 its crumpleness κ ( M n ) as the ratio diam R n + 1 ( M n ) / r ( M n ) , where r ( M n ) is the distance from M n to its central set. (In other words, r ( M n ) is the maximal radius of an open non-selfintersecting tube around M n in R n + 1 . ) We prove that any n -dimensional non-singular compact algebraic hypersurface of degree d is rigidly isotopic to an algebraic hypersurface of degree d and of crumpleness ≤ exp ( c ( n ) d α ( n ) d n + 1 ) . Here c ( n ) , α ( n ) depend only on n , and rigid isotopy means an isotopy passing only through hypersurfaces of degree...

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