Soldered tensor fields of normalized submanifolds

TL;DR

This paper extends the concept of soldered structures to tensor fields, demonstrating that the almost complex structure of an almost Kähler manifold is soldered to a submanifold if and only if it is invariant and totally geodesic.

Contribution

It introduces the notion of soldered tensor fields and characterizes when the almost complex structure is soldered to a submanifold in almost Kähler geometry.

Findings

Almost complex structure is soldered iff the submanifold is invariant and totally geodesic.

Generalizes soldered forms to tensor fields.

Provides conditions for soldering in almost Kähler manifolds.

Abstract

In an earlier paper we discussed soldered forms, multivector fields and Riemannian metrics. In particular, we showed that a Riemannian submanifold is totally geodesic iff the metric is soldered to the submanifold. In the present note we discuss general, soldered tensor fields. In particular, we prove that the almost complex structure of an almost K\"ahler manifold is soldered to a submanifold iff the latter is an invariant, totally geodesic submanifold.