arXiv subjects

M. L. Foka

Publications and source records attributed to M. L. Foka.

2 recordsLinked to original sources

arXiv2026

Geometry of left-invariant vector fields on Lie groups

We investigate the geometry of left-invariant vector fields on simply connected nilpotent Lie groups equipped with left-invariant Riemannian metrics. Exploiting the canonical identification between the Lie algebra $\mathfrak{g}$ and the space of left-invariant vector fields, we establish complete algebraic characterizations for several fundamental classes of vector fields, including Killing, one-harmonic, harmonic, conformal, and concurrent fields. Our main results reveal a striking rigidity phenomenon: on any nilpotent Lie group, the spaces of Killing, one-harmonic, and conformal vector fields all coincide precisely with the center of the Lie algebra. Moreover, we prove that no nontrivial concurrent vector fields exist in this setting. In contrast, harmonic vector fields form a proper subspace of the Lie algebra, characterized as those central vectors orthogonal to the derived algebra.

math.DG↗

arXiv2025

Affine and Projective vector fields on five-dimensional nilpotent Lie groups

This paper presents a complete classification of left-invariant affine and projective vector fields on five-dimensional simply connected nilpotent Lie groups endowed with Riemannian metrics. Building on the classification of left-invariant metrics on five-dimensional nilpotent Lie groups made by FOKa et al., we develop an algebraic characterization of these vector fields on an arbitrary Riemannian Lie group. We then employ this framework to classify left-invariant affine and projective vector fields on simply connected, Riemannian nilpotent Lie groups of dimension five. Key results demonstrate that all projective vector fields in this context are necessarily affine, extending classical results by [Kobayashi1963] on homogeneous spaces. We provide explicit matrix representations of the relevant operators and solve the resulting systems of equations case-by-case using algebraic techniques.

math.DG↗