Semi-Invariant Submanifolds in Metric Geometry of Affinors

TL;DR

This paper introduces semi-invariant submanifolds within a generalized metric geometry framework based on affinors, extending concepts from complex and contact geometries, and studies their integrability properties.

Contribution

It generalizes the notion of semi-invariant submanifolds to a broad class of structured manifolds defined by affinors, providing new characterizations and simplifications.

Findings

Characterization of integrability conditions for invariant and anti-invariant distributions.

Simplified computations when the affinor is covariant constant.

Extension of CR-submanifold concepts to more general geometries.

Abstract

We introduce a generalization of structured manifolds as the most general Riemannian metric g associated to an affinor (tensor field of (1,1)-type) F and initiate a study of their semi-invariant submanifolds. These submanifolds are generalization of CR-submanifolds of almost complex geometry and semi-invariant submanifolds of several interesting geometries (almost product, almost contact and others). We characterize the integrability of both invariant and anti-invariant distribution; the special case when F is covariant constant with respect to g gives major simplifications in computations.