Geometric Configuration of Riemannian Submanifolds of arbitrary Codimension

TL;DR

This paper explores the geometric relationships between submanifolds of arbitrary codimension and their boundaries within Riemannian spaces, revealing conditions for transversality and total geodesicity based on Newton transformations.

Contribution

It establishes new relations linking the geometry of submanifolds and their boundaries, and connects ellipticity of Newton transformations to transversality and geodesic properties.

Findings

Ellipticity of generalized Newton transformations implies transversality.

Boundary geometry relates to ambient submanifold geometry.

Total geodesicity of Pn in Mn+q under certain conditions.

Abstract

In this paper we study a geometric configuration of submanifolds of arbitrary codimension in an ambient Riemannian space. We obtain relations between the geometry of a q-codimension submanifold Mn along its boundary and the geometry of the boundary of Mn as an hypersuface of a q-codimensional submanifold Pn in an ambient space Mn+q. As a consequence of these geometric ralations we get that the ellipticity of the generalized Newton transformations implies the tranversality of Mn and Pn in Pn is totally geodesic in Mn+q.