Conformally Sasakian manifolds and the k-nullity condition
Abstract
We study almost contact metric manifolds that are locally conformal to Sasakian manifolds in the normalization of Olszak. We show that the local conformal factors are uniquely determined by the structure; consequently, every such manifold is globally conformal to a Sasakian manifold and its Lee form is exact. We then investigate the k-nullity condition $ R(X,Y)ξ=k\{η(Y)X-η(X)Y\}. $ We prove that a smooth nullity function is necessarily constant, including in dimension three. Moreover, the k-nullity condition is equivalent to the assertion that f=exp(σ) satisfies $ \mathrm{Hess}_g f=-(kf+C)g $ for a constant C, and hence to the fact that the conformal metrics g and g0=exp(−2σ)g have the same geodesic circles. We derive the range of the conformal factor, a local warped-product description, and a classification in the complete case: when the Lee form is nonzero, the complete metric is a simply connected space form of curvature k. Finally, we construct incomplete examples of nonconstant sectional curvature in every dimension 2n+1≥ 5.
Related
- Isoperimetric and total curvature inequalities in Cartan-Hadamard manifolds with nullity
- Hitchin sections on $3$-dimensional Sasakian manifolds
- Three-dimensional noncompact trans-Sasakian η-Einstein manifolds
- Laplace and Dolbeault operators on Sasakian manifolds
- Characterization of Sasakian Manifolds by Curvature Conditions
BibTeX
@misc{massamba2026conformally,
title = {{Conformally Sasakian manifolds and the $k$-nullity condition}},
author = {Fortuné Massamba and Pontsho Moile},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.12924},
eprint = {2611.12924},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.12924},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Conformally Sasakian manifolds and the k-nullity condition
AU - Fortuné Massamba
AU - Pontsho Moile
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.12924
UR - https://arxiv.org/abs/2611.12924
AB - We study almost contact metric manifolds that are locally conformal to Sasakian manifolds in the normalization of Olszak. We show that the local conformal factors are uniquely determined by the structure; consequently, every such manifold is globally conformal to a Sasakian manifold and its Lee form is exact. We then investigate the k-nullity condition $ R(X,Y)ξ=k\{η(Y)X-η(X)Y\}. $ We prove that a smooth nullity function is necessarily constant, including in dimension three. Moreover, the k-nullity condition is equivalent to the assertion that f=exp(σ) satisfies $ \mathrm{Hess}_g f=-(kf+C)g $ for a constant C, and hence to the fact that the conformal metrics g and g_0=exp(−2σ)g have the same geodesic circles. We derive the range of the conformal factor, a local warped-product description, and a classification in the complete case: when the Lee form is nonzero, the complete metric is a simply connected space form of curvature k. Finally, we construct incomplete examples of nonconstant sectional curvature in every dimension 2n+1≥ 5.
ER -