A classification of totally geodesic and totally umbilical Legendrian submanifolds of $(\kappa,\mu)$-spaces

TL;DR

This paper classifies totally geodesic and totally umbilical Legendrian submanifolds within certain $(,)$-spaces, demonstrating they are locally isometric to explicitly constructed examples.

Contribution

It provides a complete classification of these submanifolds in $(,)$-spaces with Boeckx invariant $I \u2264 -1$, identifying all such submanifolds explicitly.

Findings

Submanifolds must be among explicitly constructed examples.

Classification holds for spaces with Boeckx invariant $I \u2264 -1$.

Results specify geometric structure of Legendrian submanifolds in these spaces.

Abstract

We present classifications of totally geodesic and totally umbilical Legendrian submanifolds of $(\kappa,\mu)$-spaces with Boeckx invariant $I \leq -1$. In particular, we prove that such submanifolds must be, up to local isometries, among the examples that we explicitly construct.