Singularity obstructions to immersions.

Intrinsic Obstructions to the Existence of Isometric Minimal Immersions

S.S. Chern raised the problem to find necessary and sufficient conditions for a given Riemannian manifold to be realizable on a minimal submanifold of a Euclidean space. The aim of this paper is to provide new necessary conditions. For minimal submanifolds in a Euclidean space we consider the negative of the Ricci tensor as defining a new metric, which is nothing but the third fundamental form,...

Geometric and Topological Obstructions to Various Immersions in Submanifold Theory and Some Related Open Problems

In this article we survey known results on geometric and topological obstructions to various important classes of isometric immersions in submanifold theory. In addition we present several related open problems. The main purpose of this article is an invitation for further research on the topics that it deals with.

Minimal Obstructions for 1-Immersions and Hardness of 1-Planarity Testing

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by no more than one other edge. A non-1-planar graph G is minimal if the graph G − e is 1-planar for every edge e of G. We construct two infinite families of minimal non-1-planar graphs and show that for every integer n ≥ 63, there are at least 2(n−54)/4 nonisomorphic minimal non-1-planar graphs of order n. It is a...

Obstructions to Shiftedness

In this paper we show results on the combinatorial properties of shifted simplicial complexes. We prove two intrinsic characterization theorems for this class. The first theorem is in terms of a generalized vicinal preorder. It is shown that a complex is shifted if and only if the preorder is total. Building on this we characterize obstructions to shiftedness and prove there are finitely many i...

Obstructions to Homotopy Invariance