Abstract
We obtain a list of simple classes of singularities of function germs with respect to the quasi m-boundary equivalence relation, with m ≥ 2 . The results obtained in this paper are a natural extension of Zakalyukin’s work on the new non-standard equivalent relation. In spite of the rather artificial nature of the definitions, the quasi relations have very natural applications in symplectic geometry. In particular, they are used to classify singularities of Lagrangian projections equipped with a submanifold. The main method that is used in the classification is the standard Moser’s homotopy technique. In addition, we adopt the version of Arnold’s spectral sequence method, which is described in Lemma 2. Our main results are Theorem 4 on the classification of simple quasi classes, and Theorem 5 on the classification of Lagrangian submanifolds with smooth varieties. The brief description of the main results is given in the next section.
Highlights
In [1], Vladimir Zakalyukin classified function germs with respect to a new non-standard equivalence relation, which he named quasi boundary equivalence
Using Zakalyukin’s idea, a similar quasi equivalence relation was introduced in which case the smooth hypersurface was replaced by an arbitrary one, called a border, in particular, when the border is a cylinder over a corner or a cuspidal edge, as considered in [2,3], respectively
The behavior of critical points of a function can be shown by their discriminants inside and on a certain domain with a semi-border
Summary
In [1], Vladimir Zakalyukin classified function germs with respect to a new non-standard equivalence relation, which he named quasi boundary equivalence. We will discuss in the current paper the connection between the obtained simple classes and the singularities of Lagrangian projections in the presence of a submanifold of codimension m via the quasi semi-border miniversal deformation of a function germ f : (Rn , 0) → (R, 0), which can be constructed in the standard way. This lemma completes Arnold’s description, which is given for power series only, and it can be proven by induction on function germs of the Newton order of at least γ via similar methods to the standard ones, given in Sections 12.5–12.17 of [5] It is useful in cases not covered by the technique of Chapter 12 of [5], such as the case when the principal part f 0 is quasi homogeneous of specific degree d with indices corresponding to fixed coordinates, while basic fields, which are tangent to the semi-border, are not quasi homogeneous.
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