Moving finite unit tight frames for $S^n$

TL;DR

This paper explores the existence of moving finite unit tight frames on spheres, showing they exist on odd-dimensional spheres and providing a method to construct such frames using vector fields.

Contribution

It establishes that odd-dimensional spheres admit moving finite unit tight frames for their tangent bundles and offers a construction method for these frames.

Findings

Moving finite unit tight frames exist on all odd-dimensional spheres.

A characterization of when sets of vector fields form such frames.

A procedure for constructing vector fields on spheres.

Abstract

Frames for $\R^n$ can be thought of as redundant or linearly dependent coordinate systems, and have important applications in such areas as signal processing, data compression, and sampling theory. The word "frame" has a different meaning in the context of differential geometry and topology. A moving frame for the tangent bundle of a smooth manifold is a basis for the tangent space at each point which varies smoothly over the manifold. It is well known that the only spheres with a moving basis for their tangent bundle are $S^1$, $S^3$, and $S^7$. On the other hand, after combining the two separate meanings of the word "frame", we show that the $n$-dimensional sphere, $S^n$, has a moving finite unit tight frame for its tangent bundle if and only if $n$ is odd. We give a procedure for creating vector fields on $S^{2n-1}$ for all $n\in\N$, and we characterize exactly when sets of such vector fields form a moving finite unit tight frame.