The geometry of product conjugate connections

TL;DR

This paper explores the properties of product conjugate connections in differential geometry, focusing on their relation to integrability of almost product structures and introducing tensor analogs from Hermitian geometry.

Contribution

It introduces structural and virtual tensors for product conjugate connections, extending concepts from Hermitian geometry to product geometry, and applies these to distributions and O'Neill-Gray tensors.

Findings

Characterization of product conjugate connections using new tensors

Expressions of tensors in terms of O'Neill-Gray tensor fields

Examples illustrating the geometric properties of distributions

Abstract

Properties of pairs of product conjugate connections are stated with a special view towards the integrability of the given almost product structure. We define the analogous in product geometry of the structural and the virtual tensors from the Hermitian geometry and express the product conjugate connections in terms of these tensors. Some examples from the geometry of a pair of complementary distributions are discussed and for this case the above structural and virtual tensors are expressed in terms of O'Neill-Gray tensor fields.