Geometry of Invariant Almost Semi K\"ahler Submanifolds of Flag Manifolds

TL;DR

This paper explores the geometric properties of invariant almost semi-Kähler submanifolds within flag manifolds, establishing conditions for minimality and total geodesicity.

Contribution

It proves that these submanifolds are necessarily minimal and, when also flag manifolds, are totally geodesic, advancing understanding of their geometric structure.

Findings

Invariant almost semi-Kähler submanifolds are minimal.

Such submanifolds are totally geodesic if they are also flag manifolds.

Provides new insights into the geometry of homogeneous spaces in flag manifolds.

Abstract

In this paper we discuss the geometry of homogeneous spaces witch are almost Hermitian submanifolds of flag manifolds. We prove that such spaces are necessarily minimal submanifolds and in the case where these submanifolds are also flag manifolds, they are totally geodesic.