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Zohreh Fathi

Publications and source records attributed to Zohreh Fathi.

6 recordsLinked to original sources
arXiv2026

On classification of Finslerian spaces with nontrivial concircular transformations

We prove that in a non-trivial conformal circle-preserving transformation (CPT for short), $e^{\sigma}F$ on a (forward or backward) complete Finslerian manifold $(M,F)$, the conformal factor has at most two critical points. Then a diffeomorphism classification based on the number of critical points as well as some curvature rigidity properties - of Finslerian manifolds that admit nontrivial conformal CPTs - is presented.

math.DG↗
arXiv2026

On the rigidity of Finslerian conformal circle-preserving transformations

We prove that if a forward or backward complete Berwaldian or reversible Finslerian manifold $(M,F)$ admits a non-trivial (non-homothety) conformal concircular transformation ({\bf c}ircle-{\bf p}reserving {\bf t}ransformation or {\cpt} for short), $e^{\sigma} F$, where $\sigma$ has at least one critical point, then, $(M,F)$ is Riemannian. Consequently, $(M,g)$ is conformally diffeomorphic to either 1) the standard sphere, 2) the Euclidean space, or 3) the hyperbolic space. In particular, a compact Berwaldian or reversible Finslerian manifold does not admit any non-trivial conformal {\cpt}s, unless it is conformally diffeomorphic to the standard sphere.

math.DG↗
arXiv2023

On Rigidity of ALE vector bundles

We discuss topological rigidity of vector bundles with asymptotically conical (AC) total spaces of rank greater than 1 with a sufficiently connected link; our focus will mainly be on ALE (asymptotically locally Euclidean) bundles. Within the smooth category, we topologically classify all ALE tangent bundles by showing only 2-sphere, projective plane and open contractible manifolds admit ALE tangent bundles. We also discuss other interesting topological and geometric rigidities of ALE vector bundles.

math.DG↗
arXiv2022

Discrete Ollivier-Ricci curvature

We analyze both continuous and discrete-time Ollivier-Ricci curvatures of locally-finite weighted graphs $\G$ equipped with a given distance "$\dist$" (w.r.t. which $\G$ is metrically complete) and for general random walks. We show the continuous-time Ollivier-Ricci curvature is well-defined for a large class of Markovian and non-Markovian random walks and provide a criterion for existence of continuous-time Ollivier-Ricci curvature; the said results generalize the previous rather limited constructions in the literature. In addition, important properties of both discrete-time and continuous-time Ollivier-Ricci curvatures are obtained including -- to name a few -- Lipschitz continuity, concavity properties, piece-wise regularity (piece-wise linearity in the case of linear walks) for the discrete-time Ollivier-Ricci as well as Lipschitz continuity and limit-free formulation for the continuous-time Ollivier-Ricci. these properties were previously known only for very specific distances and very specific random walks. As an application of Lipschitz continuity, we obtain existence and uniqueness of generalized continuous-time Ollivier-Ricci curvature flows. Along the way, we obtain -- by optimizing McMullen's upper bounds -- a sharp upper bound estimate on the number of vertices of a convex polytope in terms of number of its facets and the ambient dimension, which might be of independent interest in convex geometry. The said upper bound allows us to bound the number of polynomial pieces of the discrete-time Ollivier-Ricci curvature as a function of time in the time-polynomial random walk. The limit-free formulation we establish allows us to define an operator theoretic Ollivier-Ricci curvature which is a non-linear concave functional on suitable operator spaces.

math.MG↗
arXiv2021

Bakry-Émery Ricci curvature of doubly warped product of weighted spaces

We introduce a notion of doubly warped product of weighted graphs that is consistent with the doubly warped product in the Riemannian setting. We establish various discrete Bakry-Émery Ricci curvature-dimension bounds for such warped products in terms of the curvature of the constituent graphs. This requires deliberate analysis of the quadratic forms involved, prompting the introduction of some crucial notions such as curvature saturation at a vertex. In the spirit of being thorough and to provide a frame of reference, we also introduce the $\left(R_1,R_2\right)$-doubly warped products of smooth measure spaces and establish $\N$-Bakry-Émery Ricci curvature (lower) bounds thereof in terms of those of the factors. At the end of these notes, we present examples and demonstrate applications of warped products with some toy models.

math.DG↗