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On the Topology of Singularities of the Set of Supporting Hyperplanes of a Smooth Submanifold in an Affine Space
Author(s) -
Sedykh V. D.
Publication year - 2005
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610704006003
Subject(s) - submanifold , hyperplane , mathematics , pure mathematics , gravitational singularity , catalan number , affine transformation , affine space , multiplicity (mathematics) , euler characteristic , tangent , space (punctuation) , convex cone , mathematical analysis , regular polygon , topology (electrical circuits) , convex set , combinatorics , geometry , computer science , convex optimization , operating system
Many new universal relations are obtained between the Euler numbers of manifolds of singular supporting hyperplanes of an arbitrary generic smooth closed k ‐dimensional submanifold in R n where n ⩽ 7 or k = 1. These relations are applied to Barner‐convex curves in an odd‐dimensional space R n . A universal (nontrivial) linear relation is established between the numbers of singular supporting hyperplanes of various types but of the same total multiplicity n of tangency with a given generic smooth closed connected Barner‐convex curve in R n . The coefficients of this relation are defined by Catalan numbers.