[Paper Review] Existence and non-existence of skew branes

Serge Tabachnikov, Yu. Tyurina|ArXiv.org|Apr 23, 2005

Geometric and Algebraic Topology7 references4 citations

TL;DR

This paper establishes topological obstructions to the existence of skew branes—immersions of manifolds into Euclidean space where no two tangent spaces are parallel—by using characteristic classes and homology intersection theory. It proves that closed, oriented manifolds with non-zero Euler characteristic in codimension 2 must have pairs of negatively parallel tangent spaces, and constructs explicit perturbations of spheres and tori in R^4 and R^{2n+1} that avoid all parallel tangent planes, demonstrating existence in cases where non-existence theorems would otherwise apply.

ABSTRACT

Following recent work by Ghomi, Solomon and Tabachnikov, we study geometry and topology of skew branes. A skew brane is a codimension 2 submanifold in affine space such that the tangent spaces at any pair of distinct points are not parallel. We prove that if an oriented closed manifold has a non-zero Euler characteristic $ξ$ then it is not a skew brane; generically, the number of oppositely oriented pairs of parallel tangent spaces is not less than $(ξ^2)/4$. We also construct examples of skew odd-dimensional spheres and skew two-dimensional tori.

Motivation & Objective

• To establish topological obstructions to the existence of skew branes—immersions where no two tangent spaces are parallel—using characteristic classes and homology intersection theory.
• To prove that any closed, oriented, embedded submanifold of dimension 2n in R^{2n+2} with non-zero Euler characteristic must have at least one pair of negatively parallel tangent spaces.
• To construct explicit examples of skew branes in R^{2n+1} and R^4, including perturbations of the standard sphere and torus, that avoid all pairs of parallel tangent planes.
• To show that the bounds in the non-existence theorems are sharp by constructing an immersed sphere in R^4 with one double point and no negatively parallel tangent planes.

Proposed method

• Use the tangent Gauss map to associate a fundamental class in the oriented Grassmannian G_{2n}^+(2n+2), and compute its homology class and intersection with the orientation-reversing involution.
• Apply characteristic classes of tautological bundles over the Grassmannian to compute the homology intersection number, which yields a lower bound on the number of negatively parallel tangent pairs.
• Use a decomposition of functions on the torus into Z_2^2-invariant and anti-invariant components to analyze the system of equations governing the tangent planes under perturbation.
• Construct explicit perturbations of the standard sphere S^{2n-1} in R^{2n+1} and the standard torus T^2 in R^4 by modifying their parametrizations with small, carefully chosen functions of angular coordinates.
• Compute Plücker coordinates of tangent planes in the linear approximation in ε to analyze when pairs of tangent planes become parallel or negatively parallel.
• Verify that the perturbed manifolds avoid all pairs of parallel tangent planes by showing that the resulting systems of equations for parallelism have no solutions for generic choices of perturbation functions.

Experimental results

Research questions

• RQ1 Under what topological conditions does a closed, oriented submanifold of dimension 2n in R^{2n+2} necessarily have a pair of negatively parallel tangent spaces?
• RQ2 Can the standard sphere S^{2n-1} in R^{2n+1} be perturbed to avoid all pairs of parallel tangent planes, despite the non-existence result for quadratic hypersurfaces?
• RQ3 Is it possible to perturb the standard torus T^2 in R^4 to eliminate all pairs of parallel tangent planes, including negatively parallel ones?
• RQ4 What is the sharpness of the lower bound on the number of negatively parallel tangent pairs for immersed surfaces in R^4, as given by the algebraic count of double points?
• RQ5 Can an immersed sphere in R^4 with a single double point be constructed without any pair of negatively parallel tangent planes?

Key findings

• Any closed, oriented, embedded submanifold M^{2n} ⊂ R^{2n+2} with non-zero Euler characteristic χ must have at least one pair of distinct points with negatively parallel tangent spaces.
• For a generic such manifold, the number of unordered pairs of negatively parallel tangent spaces is at least χ²/4.
• There exists a small C¹-perturbation of the standard sphere S^{2n-1} ⊂ R^{2n+1} that is free from all pairs of parallel tangent planes, demonstrating existence of skew branes on non-quadratic hypersurfaces.
• A small perturbation of the standard torus T^2 ⊂ R^4 exists that is free from all pairs of parallel tangent planes, providing a new construction of a skew torus.
• An immersed sphere in R^4 with exactly one double point and no pair of negatively parallel tangent planes exists, showing that the bound in Theorem 2 is sharp for the spherical case.
• The perturbation of the sphere in R^4 with a single double point is constructed by modifying the parametrization with a function g(α) = sin 2α + sin 4α, which breaks the symmetry that would otherwise force negatively parallel tangent planes.