PreprintCC0arXiv
The equal norm Grassmannian: topology and applications to frame fields
Abstract
Given an orientable rank-d vector bundle over a d-manifold, the Euler class is the unique obstruction to finding a nowhere-zero section. An equal norm Parseval frame is a certain collection of sections, one of whose properties is that the lengths of each vector at each point are equal. This makes the Euler class an obvious obstruction to finding such a frame. In this work we show that this is the only obstruction so long as the number of sections is at least 2d+2. We also identify precise obstructions when the number of sections is 2d and produce examples showing that for smaller numbers of sections it is sometimes impossible to find Parseval frame fields.
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BibTeX
@misc{ballas2026equal,
title = {{The equal norm Grassmannian: topology and applications to frame fields}},
author = {Samuel A. Ballas and Ferhat Karabatman and Tom Needham and Clayton Shonkwiler},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38669},
eprint = {2610.38669},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38669},
note = {CC0},
}
RIS
TY - UNPB
TI - The equal norm Grassmannian: topology and applications to frame fields
AU - Samuel A. Ballas
AU - Ferhat Karabatman
AU - Tom Needham
AU - Clayton Shonkwiler
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38669
UR - https://arxiv.org/abs/2610.38669
AB - Given an orientable rank-d vector bundle over a d-manifold, the Euler class is the unique obstruction to finding a nowhere-zero section. An equal norm Parseval frame is a certain collection of sections, one of whose properties is that the lengths of each vector at each point are equal. This makes the Euler class an obvious obstruction to finding such a frame. In this work we show that this is the only obstruction so long as the number of sections is at least 2d+2. We also identify precise obstructions when the number of sections is 2d and produce examples showing that for smaller numbers of sections it is sometimes impossible to find Parseval frame fields.
ER -