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Constructing generic smooth maps of a manifold into a surface with prescribed singular loci

Osamu Saeki (1995)

Annales de l'institut Fourier

It is known that the singular set S ( f ) of a generic smooth map f : M → N of an n -dimensional manifold into a surface is a closed 1-dimensional submanifold of M and that it has a natural stratification induced by the absolute index. In this paper, we give a complete characterization of those 1-dimensional (stratified) submanifolds which arise as the singular set of a generic map in terms of the homology class they represent.

Constructions on second order connections

J. Kurek, W. M. Mikulski (2007)

Annales Polonici Mathematici

We classify all ℱ ℳ m , n -natural operators : J ² ⟿ J ² V A transforming second order connections Γ: Y → J²Y on a fibred manifold Y → M into second order connections ( Γ ) : V A Y → J ² V A Y on the vertical Weil bundle V A Y → M corresponding to a Weil algebra A.

Contact elements on fibered manifolds

Ivan Kolář, Włodzimierz M. Mikulski (2003)

Czechoslovak Mathematical Journal

For every product preserving bundle functor T μ on fibered manifolds, we describe the underlying functor of any order ( r , s , q ) , s ≥ r ≤ q . We define the bundle K k , l r , s , q Y of ( k , l ) -dimensional contact elements of the order ( r , s , q ) on a fibered manifold Y and we characterize its elements geometrically. Then we study the bundle of general contact elements of type μ . We also determine all natural transformations of K k , l r , s , q Y into itself and of T ( K k , l r , s , q Y ) into itself and we find all natural operators lifting projectable vector fields and horizontal one-forms...

Contact geometry of multidimensional Monge-Ampère equations: characteristics, intermediate integrals and solutions

Dmitri V. Alekseevsky, Ricardo Alonso-Blanco, Gianni Manno, Fabrizio Pugliese (2012)

Annales de l’institut Fourier

We study the geometry of multidimensional scalar 2 n d order PDEs (i.e. PDEs with n independent variables), viewed as hypersurfaces ℰ in the Lagrangian Grassmann bundle M ( 1 ) over a ( 2 n + 1 ) -dimensional contact manifold ( M , 𝒞 ) . We develop the theory of characteristics of ℰ in terms of contact geometry and of the geometry of Lagrangian Grassmannian and study their relationship with intermediate integrals of ℰ . After specializing such results to general Monge-Ampère equations (MAEs), we focus our attention to MAEs of...

Contact hamiltonians distinguishing locally certain Goursat systems

Piotr Mormul (2000)

Banach Center Publications

For the first time in dimension 9, the Goursat distributions are not locally smoothly classified by their small growth vector at a point. As shown in [M1], in dimension 9 of the underlying manifold 93 different local behaviours are possible and four irregular pairs of them have coinciding small growth vectors. In the present paper we distinguish geometrically objects in three of those pairs. Smooth functions in three variables - contact hamiltonians in the terminology of Arnold, [A] - help to do...

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