Nazia Valiyakath | Searcharxiv

arXiv2026

Homogeneous spaces with two equivalent isotropy summands develop positive Ricci curvature under Ricci flow

We study normalized Ricci flow on simply connected homogeneous spaces G/H for which the isotropy representation splits into exactly two equivalent irreducible subrepresentations. We prove that every G-invariant metric evolves to one with positive Ricci curvature, and that the family of G-invariant metrics with positive Ricci curvature is forward-invariant under the flow. The proof relies on the fact that the phase portrait of the family of fixed-volume G-invariant metrics can be explicitly visualized.

math.DG↗

arXiv2026

On Nilpotent and Solvable Quasi-Einstein Manifolds

In this paper, we investigate nilpotent and unimodular solvable Lie groups that admit quasi-Einstein metrics $(M,g,X)$ with $X$ a left-invariant vector field, which we call \emph{totally left-invariant quasi-Einstein metrics}. We give a complete classification of nilpotent Lie groups admitting such metrics, proving that this occurs if and only if the group is isomorphic to a Heisenberg Lie group. For unimodular solvable Lie groups $S$, we show that the existence of a non-flat totally left-invariant quasi-Einstein metric forces the center of $S$ to be one-dimensional. Furthermore, under the additional assumption that the adjoint action $\operatorname{ad}_a$ of $S$ is a normal derivation, we obtain a full classification: these groups are standard and their nilradical must be a Heisenberg Lie algebra. As an application, we prove that the only near-horizon geometries on a compact nilmanifold are $Γ\backslash H_{n}$, where $ H_{n}$ is $n$-dimensional Heisenberg Lie group.

math.DG↗