Embeddings of almost Hermitian manifolds in almost hyperHermitian those. Complex and hypercomplex numbers in differential geometry

TL;DR

This paper explores embeddings of almost Hermitian manifolds into hyperHermitian structures, demonstrating that any smooth manifold can be embedded as a totally geodesic submanifold into higher-dimensional Kaehlerian and hyperKaehlerian manifolds.

Contribution

It introduces a method to embed almost Hermitian manifolds into hyperHermitian structures using tubular neighborhoods and deformations, extending embedding results to higher-dimensional complex and hypercomplex manifolds.

Findings

Any smooth manifold of dimension n can be embedded as a totally geodesic submanifold in a 2n-dimensional Kaehlerian manifold.

Similarly, it can be embedded in a 4n-dimensional hyperKaehlerian manifold.

The construction uses deformed almost hyperHermitian structures on tangent bundles.

Abstract

Tubular neighborhoods play an important role in differential topology. We have applied these constructions to geometry of almost Hermitian manifolds. At first, we consider deformations of tensor structures on a normal tubular neighborhood of a submanifold in a Riemannian manifold.Further, an almost hyperHermitian structure has been constructed on the tangent bundle TM with help of the Riemannian connection of an almost Hermitian structure on a manifold M then, we consider an embedding of the almost Hermitian manifold M in the corresponding normal tubular neighborhood of the null section in the tangent bundle TM equipped with the deformed almost hyperHermitian structure of the special form. As a result,we have obtained that any smooth manifold M of dimension n can be embedded as a totally geodesic submanifold in a Kaehlerian manifold of dimension 2n and in a hyperKaehlerian manifold of dimension 4n.