On Properties of the Intrinsic Geometry of Submanifolds in a Riemannian Manifold

TL;DR

This paper investigates the intrinsic metric properties of submanifolds in Riemannian manifolds, extending classical surface geometry to include boundary behavior through a continuity-based metric extension.

Contribution

It develops a new approach to intrinsic geometry by analyzing the metric extension to boundaries, building on Alexandrov's classical surface theory.

Findings

Extended the intrinsic metric to submanifold boundaries.

Provided new insights into boundary behavior of intrinsic metrics.

Connected classical surface geometry with modern boundary analysis.

Abstract

In the paper we study the properties of a metric function which is the extension by continuity of the intrinsic metric of the interior of a submanifold to its boundary. This approach is the development of the classical intrinsic geometry of surfaces studied by A.D. Alexandrov and his school.