Extrinsic characterizations of biconservative surfaces in the $4$-dimensional hyperbolic space
Abstract
Biconservative submanifolds arise as a natural relaxation of the biharmonic condition and play an important role in the submanifold theory. In this paper, we study non-CMC biconservative surfaces with parallel normalized mean curvature vector field (PNMC surfaces) in the four-dimensional hyperbolic space ℍ4, for which we consider the hyperboloid model. We provide a local extrinsic description of such surfaces, showing that they are generated by a directrix curve lying in a totally geodesic hypersurface ℍ3 of ℍ4, through a certain normal flow. This extrinsic classification of non-CMC, PNMC biconservative surfaces in ℍ4 splits naturally into three cases according to the type of a certain vector field, which can be non-zero null, spacelike or timelike. We also prove that these surfaces are invariant under the action of a parabolic, elliptic, and hyperbolic one-parameter group of isometries of ℍ4, respectively. Moreover, their full groups of ambient isometries preserving the surfaces are determined. Together with the previous results, the classification of non-CMC, PNMC surfaces in four-dimensional space forms is now complete, from both intrinsic and extrinsic points of view.
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BibTeX
@misc{nistor2026extrinsic,
title = {{Extrinsic characterizations of biconservative surfaces in the $4$-dimensional hyperbolic space}},
author = {Simona Nistor and Mihaela Rusu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.22396},
eprint = {2605.22396},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.22396},
}
RIS
TY - UNPB
TI - Extrinsic characterizations of biconservative surfaces in the $4$-dimensional hyperbolic space
AU - Simona Nistor
AU - Mihaela Rusu
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2605.22396
UR - https://arxiv.org/abs/2605.22396
AB - Biconservative submanifolds arise as a natural relaxation of the biharmonic condition and play an important role in the submanifold theory. In this paper, we study non-CMC biconservative surfaces with parallel normalized mean curvature vector field (PNMC surfaces) in the four-dimensional hyperbolic space ℍ^4, for which we consider the hyperboloid model. We provide a local extrinsic description of such surfaces, showing that they are generated by a directrix curve lying in a totally geodesic hypersurface ℍ^3 of ℍ^4, through a certain normal flow. This extrinsic classification of non-CMC, PNMC biconservative surfaces in ℍ^4 splits naturally into three cases according to the type of a certain vector field, which can be non-zero null, spacelike or timelike. We also prove that these surfaces are invariant under the action of a parabolic, elliptic, and hyperbolic one-parameter group of isometries of ℍ^4, respectively. Moreover, their full groups of ambient isometries preserving the surfaces are determined. Together with the previous results, the classification of non-CMC, PNMC surfaces in four-dimensional space forms is now complete, from both intrinsic and extrinsic points of view.
ER -