Higher Fundamental Forms and Warped Product Hypersurfaces

TL;DR

This paper explores the geometry of hypersurfaces in warped product manifolds and conformal manifolds, emphasizing the role of higher fundamental forms in their geometric characterization.

Contribution

It introduces the use of higher fundamental forms to characterize hypersurface embeddings in warped and conformal manifolds, providing new insights into their geometric structure.

Findings

Higher fundamental forms characterize warped product hypersurfaces.

Higher conformal fundamental forms are key in conformal hypersurface analysis.

The study links fundamental forms with the geometry of specific manifold classes.

Abstract

Warped products are one of the simplest families of Riemannian manifolds that can have non-trivial geometries. In this article, we characterize the geometry of hypersurface embeddings arising from warped product manifolds using the language of higher (Riemannian) fundamental forms. In a similar vein, we also study the geometry of conformal manifolds with embedded hypersurfaces that admits a trivialization of the conformal metric to a product metric, with base manifold given by the embedded hypersurface. We show that the higher conformal fundamental forms play a critical role in their characterization.