Metallic conjugate connections

Adara M. Blaga,Cristina-Elena Hrețcanu

doi:10.33044/revuma.v59n1a09

Abstract

Properties of metallic conjugate connections are stated by pointing out their relation to product conjugate connections. We define the analogous in metallic geometry of the structural and the virtual tensors from the almost product geometry and express the metallic conjugate connections in terms of these tensors. From an applied point of view we consider invariant distributions with respect to the metallic structure and for a natural pair of complementary distributions, the above structural and virtual tensors are expressed in terms of O'Neill–Gray tensor fields.

Highlights

Besides the very well known almost complex, almost tangent, and almost product structures on a differentiable manifold M, some other polynomial structures naturally arise as C∞-tensor fields J of type (1, 1) which are roots of the algebraic equation
In order to find a measure of how far away a linear connection is from being in CJ (M ) we introduce the notion of metallic conjugate connection
An important tool in our work is provided by the pair defined for the almost product geometry in [5] and considered here in the last part of Section 1

Summary

Introduction

Besides the very well known almost complex, almost tangent, and almost product structures on a differentiable manifold M , some other polynomial structures naturally arise as C∞-tensor fields J of type (1, 1) which are roots of the algebraic equation. Metallic structure; (conjugate) linear connection; metallic Riemannian manifold; structural and virtual tensor field; invariant distributions. Fix J a metallic structure on M and define the associated linear connections as follows. We remark that Bejan and Crasmareanu [1] studied the conjugate connections with respect to a quadratic endomorphism as a generalization of almost complex and almost product cases. An important tool in our work is provided by the pair (structural tensor, virtual tensor) defined for the almost product geometry in [5] and considered here in the last part of Section 1.

Metallic conjugate connection

The duality of metallic and product conjugate connections

Invariant distributions