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Milnor fibrations on rho-tubes and rho-spheres for real analytic map germs
Abstract
For a real analytic map germ G:(ℝm,0)→(ℝp,0), we study Milnor fibrations obtained by replacing the squared Euclidean distance with an analytic control function ρ defining the origin. We formulate the corresponding Milnor set and condition (b), derive criteria for tube and sphere fibrations, and examine their dependence on ρ. The family Gρ=(xz, yzρ) provides explicit changes of regularity under elliptic controls. In particular, the germ G(x,y,z)=(xz, yz(10x2+y2+3z2)) admits neither the induced tube nor sphere fibration for the Euclidean control, while both fibrations exist for the adapted function ρ=10x2+y2+3z2. Thus the control function can be essential to the local fibration structure.
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BibTeX
@misc{ribeiro2026milnor,
title = {{Milnor fibrations on rho-tubes and rho-spheres for real analytic map germs}},
author = {Maico Ribeiro and Mateus de Melo and Gabriel Santos},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17374},
eprint = {2609.17374},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17374},
}
RIS
TY - UNPB
TI - Milnor fibrations on rho-tubes and rho-spheres for real analytic map germs
AU - Maico Ribeiro
AU - Mateus de Melo
AU - Gabriel Santos
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17374
UR - https://arxiv.org/abs/2609.17374
AB - For a real analytic map germ G:(ℝ^m,0)→(ℝ^p,0), we study Milnor fibrations obtained by replacing the squared Euclidean distance with an analytic control function ρ defining the origin. We formulate the corresponding Milnor set and condition (b), derive criteria for tube and sphere fibrations, and examine their dependence on ρ. The family G_ρ=(xz, yzρ) provides explicit changes of regularity under elliptic controls. In particular, the germ G(x,y,z)=(xz, yz(10x^2+y^2+3z^2)) admits neither the induced tube nor sphere fibration for the Euclidean control, while both fibrations exist for the adapted function ρ=10x^2+y^2+3z^2. Thus the control function can be essential to the local fibration structure.
ER -