On a class of submanifolds in tangent bundle with g - natural metric - normal lift

TL;DR

This paper investigates a specific class of submanifolds in the tangent bundle of a Riemannian manifold, focusing on those induced by an isometric immersion and the normal bundle, with respect to a g-natural metric.

Contribution

It introduces and analyzes a new class of immersions into the tangent bundle derived from an isometric immersion and the normal bundle, expanding understanding of submanifold geometry in tangent bundles.

Findings

Characterization of the immersion induced by the original map and normal bundle.

Conditions for the immersion to be isometric or have specific geometric properties.

Examples illustrating the class of submanifolds studied.

Abstract

An isometric immersion of a Riemannian manifold $M$ into a Riemannian manifold $N$ gives rise in a natural way to variety of immersions into the tangent bundle $TN$ with a non-degenerate $g-$ natural metric $G$. In the paper we introduce and study an immersion into $TN$ defined by the immersion $f:M\longrightarrow N$ itself and the normal bundle.