The extrinsic holonomy Lie algebra of a parallel submanifold

TL;DR

This paper studies the extrinsic holonomy Lie algebra of parallel submanifolds in Riemannian symmetric spaces, providing explicit descriptions especially for symmetric submanifolds, and introduces the concept of 1-fullness related to the second fundamental form.

Contribution

It introduces the intrinsic property of 1-fullness to distinguish symmetric submanifolds and explicitly describes the holonomy Lie algebra of the second osculating bundle for complete parallel submanifolds.

Findings

Holonomy Lie algebra of the second osculating bundle can be described via the second fundamental form and curvature tensor.

Explicit formulas for the holonomy Lie algebra are obtained for full symmetric submanifolds.

The concept of 1-fullness characterizes symmetric submanifolds among parallel submanifolds.

Abstract

We investigate parallel submanifolds of a Riemannian symmetric space $N$. The special case of a symmetric submanifold has been investigated by many authors before and is well understood. We observe that there is an intrinsic property of the second fundamental form which distinguishes full symmetric submanifolds from arbitrary full parallel submanifolds of $N$, usually called "1-fullness of $M$". Furthermore, for every parallel submanifold $M$ of $N$ we consider the pullback bundle $TN|M$ with its induced connection, which admits a distinguished parallel subbundle $osc M$, usually called the "second osculating bundle of $M$". If $M$ is a complete parallel submanifold of $N$, then we can describe the corresponding holonomy Lie algebra of $osc M$ by means of the second fundamental form of $M$ and the curvature tensor of $N$ at the origin. If moreover $N$ is simply connected and $M$ is even a full symmetric submanifold of $N$, then we will calculate the holonomy Lie algebra of $TN|M$ in an explicit form.