On the frame bundle adapted to a submanifold

TL;DR

This paper explores the geometry of the adapted frame bundle over a submanifold within a Riemannian manifold, linking horizontal distributions, extrinsic minimality, and intrinsic curvature computations.

Contribution

It characterizes the horizontal distribution of the adapted frame bundle and relates minimality to harmonicity of the Gauss map under deformed metrics.

Findings

Horizontal distribution characterized and related to Levi-Civita connection.

Minimality of submanifold equivalent to harmonic Gauss map.

Curvature formulas computed for intrinsic geometry.

Abstract

Let $M$ be a submanifold of a Riemannian manifold $(N,g)$. $M$ induces a subbundle $O(M,N)$ of adapted frames over $M$ of the bundle of orthonormal frames $O(N)$. Riemannian metric $g$ induces natural metric on $O(N)$. We study the geometry of a submanifold $O(M,N)$ in $O(N)$. We characterize the horizontal distribution of $O(M,N)$ and state its correspondence with the horizontal lift in $O(N)$ induced by the Levi--Civita connection on $N$. In the case of extrinsic geometry, we show that minimality is equivalent to harmonicity of the Gauss map of the submanifold $M$ with deformed Riemannian metric. In the case of intrinsic geometry we compute the curvatures.