# Some Remarks on the Hyperk\"ahler Reduction

## TL;DR

This paper explores hyperk"ahler reduction using frame bundles, recovers known results, and shows fibers are totally geodesic, also describing O'Neill's tensors on principal bundles.

## Contribution

It provides a new perspective on hyperk"ahler reduction through frame bundles and establishes the geodesic property of fibers, extending previous work.

## Key findings

- Fibers of hyperk"ahler reduction are totally geodesic.
- Recovered results of Gocho and Nakajima via frame bundle approach.
- Described O'Neill's submersion tensors on principal bundles.

## Abstract

We consider a hyperk\"ahler reduction and describe it via frame bundles. Tracing the connection through the various reductions, we recover the results of Gocho and Nakajima. In addition, we show that the fibers of such a reduction are necessarily totally geodesic. As an independent result, we describe O'Neill's submersion tensors on principal bundles.