Geodesics and Submanifold Structures in Conformal Geometry

TL;DR

This paper develops the theory of extrinsic conformal geometry for submanifolds, introducing invariants to characterize geodesic submanifolds within conformal manifolds, extending classical structures to surfaces and curves.

Contribution

It extends M"obius and Laplace structures to surfaces and curves, providing new tensorial invariants for conformal embeddings and geodesic submanifold characterization.

Findings

Introduced tensorial invariants for conformal embeddings

Characterized various geodesic submanifolds using these invariants

Extended conformal structures to lower-dimensional submanifolds

Abstract

A conformal structure on a manifold $M^n$ induces natural second order conformally invariant operators, called M\"obius and Laplace structures, acting on specific weight bundles of $M$, provided that $n\ge 3$. By extending the notions of M\"obius and Laplace structures to the case of surfaces and curves, we develop here the theory of extrinsic conformal geometry for submanifolds, find tensorial invariants of a conformal embedding, and use these invariants to characterize various forms of geodesic submanifolds.