Formally integrable complex structures on higher dimensional knot spaces

TL;DR

This paper proves that certain higher-dimensional knot spaces become formally Kähler manifolds when the ambient manifold admits a parallel vector cross product and the manifold's dimension matches the cross product's order minus one.

Contribution

It generalizes previous results by showing that spaces of free immersions modulo diffeomorphisms are formally Kähler under specific geometric conditions.

Findings

Spaces of free immersions are formally Kähler when conditions are met.

Generalizes known results for codimension 2 submanifolds and special manifolds.

Extends the understanding of complex structures on higher-dimensional knot spaces.

Abstract

Let $S$ be a compact oriented finite dimensional manifold and $M$ a finite dimensional Riemannian manifold, let ${\rm Imm}_f(S,M)$ the space of all free immersions $\varphi:S \to M$ and let $B^+_{i,f}(S,M)$ the quotient space ${\rm Imm}_f(S,M)/{\rm Diff}^+(S)$, where ${\rm Diff}^+(S)$ denotes the group of orientation preserving diffeomorphisms of $S$. In this paper we prove that if $M$ admits a parallel $r$-fold vector cross product $\varphi \in \Omega ^r(M, TM)$ and $\dim S = r-1$ then $B^+_{i,f}(S,M)$ is a formally K\"ahler manifold. This generalizes Brylinski's, LeBrun's and Verbitsky's results for the case that $S$ is a codimension 2 submanifold in $M$, and $S = S^1$ or $M$ is a torsion-free $G_2$-manifold respectively.