Balanced Hermitian metrics from SU(2)-structures

TL;DR

This paper explores the geometric structures of hypersurfaces in 6-manifolds with balanced Hermitian SU(3)-structures, providing conditions for embeddings, constructing new examples, and analyzing holonomy of Bismut connections.

Contribution

It introduces balanced SU(2)-structures on 5-manifolds, establishes embedding conditions into balanced SU(3)-manifolds, and constructs new examples with specific holonomy properties.

Findings

Any 5-dimensional compact nilmanifold admits an invariant balanced SU(2)-structure.

New balanced Hermitian SU(3)-metrics are constructed from balanced SU(2)-structures.

Examples of compact manifolds with balanced SU(n)-structures and SU(n) holonomy Bismut connections are provided.

Abstract

We study the intrinsic geometrical structure of hypersurfaces in 6-manifolds carrying a balanced Hermitian SU(3)-structure, which we call {\em balanced} SU(2)-{\em structures}. We provide conditions which imply that such a 5-manifold can be isometrically embedded as a hypersurface in a manifold with a balanced SU(3)-structure. We show that any 5-dimensional compact nilmanifold has an invariant balanced SU(2)-structure as well as new examples of balanced Hermitian SU(3)-metrics constructed from balanced SU(2)-structures. Moreover, for $n=3,4$, we present examples of compact manifolds, endowed with a balanced SU(n)-structure, such that the corresponding Bismut connection has holonomy equal to SU(n).