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Behroz Bidabad

Publications and source records attributed to Behroz Bidabad.

18 recordsLinked to original sources
arXiv2024

Ricci flow-based brain surface covariance descriptors for diagnosing Alzheimer's disease

Automated feature extraction from MRI brain scans and diagnosis of Alzheimer's disease are ongoing challenges. With advances in 3D imaging technology, 3D data acquisition is becoming more viable and efficient than its 2D counterpart. Rather than using feature-based vectors, in this paper, for the first time, we suggest a pipeline to extract novel covariance-based descriptors from the cortical surface using the Ricci energy optimization. The covariance descriptors are components of the nonlinear manifold of symmetric positive-definite matrices, thus we focus on using the Gaussian radial basis function to apply manifold-based classification to the 3D shape problem. Applying this novel signature to the analysis of abnormal cortical brain morphometry allows for diagnosing Alzheimer's disease. Experimental studies performed on about two hundred 3D MRI brain models, gathered from Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset demonstrate the effectiveness of our descriptors in achieving remarkable classification accuracy.

eess.IV↗
arXiv2023

Ricci flow on Finsler manifolds

This paper investigates the short-time existence and uniqueness of Ricci flow solutions on Finsler manifolds. The main results of this paper are theorems demonstrating the short-time existence of the flow solution for $n$-dimensional Finsler manifolds and the uniqueness of the solution for isotropic Finsler manifolds. Two examples are also presented to illustrate the results.

math.DG↗
arXiv2023

The Schwarzian derivative on Finsler manifolds of constant curvature

Lagrange introduced the notion of Schwarzian derivative and Thurston discovered its mysterious properties playing a role similar to that of curvature on Riemannian manifolds. Here we continue our studies on the development of the Schwarzian derivative on Finsler manifolds. First, we obtain an integrability condition for the Möbius equations. Then we obtain a rigidity result as follows; Let $(M, F)$ be a connected complete Finsler manifold of positive constant Ricci curvature. If it admits non-trivial Möbius mapping, then $M$ is homeomorphic to the $n$-sphere. Finally, we reconfirm Thurston's hypothesis for complete Finsler manifolds and show that the Schwarzian derivative of a projective parameter plays the same role as the Ricci curvature on theses manifolds and could characterize a Bonnet-Mayer-type theorem.

math.DG↗
arXiv2023

On projective symmetries on Finsler Spaces

There are two definitions of Einstein-Finsler spaces introduced by Akbar-Zadeh, which we will show is equal along the integral curves of $I$-invariant projective vector fields. The sub-algebra of the $C$-projective vector fields, leaving the $H$-curvature invariant, has been studied extensively. Here we show on a closed Finsler space with negative definite Ricci curvature reduces to that of Killing vector fields. Moreover, if an Einstein-Finsler space admits such a projective vector field then the flag curvature is constant. Finally, a classification of compact isotropic mean Landsberg manifolds admitting certain projective vector fields is obtained with respect to the sign of Ricci curvature.

math.DG↗
arXiv2023

On the geometry of Zermelo's optimal control trajectories

In the present work, we study the optimal control paths in the Zermelo navigation problem from the geometric and differential equations point of view rather than the optimal control point of view, where the latter has been carried out in our recent work. Here, we obtain the precise form of the system of ODE where the solutions are optimal trajectories of Zermelo's navigation problem. Having a precise equation allows optimizing a cost function more accurately and efficiently. The advantage of these equations is to approximate optimal trajectories in the general case by the first-order approximation of external fields $w$. The latter could be solved numerically since we have retrieved simpler equations for these paths.

math.OC↗
arXiv2020

The Schwarzian derivative and conformal transformation on Finsler manifolds

Thurston, in 1986, discovered that the Schwarzian derivative has mysterious properties similar to the curvature on a manifold. After his work, there are several approaches to develop this notion on Riemannian manifolds. Here, a tensor field is identified in the study of global conformal diffeomorphisms on Finsler manifolds as a natural generalization of the Schwarzian derivative. Then, a natural definition of a Mobius mapping on Finsler manifolds is given and its properties are studied. In particular, it is shown that Mobius mappings are mappings that preserve circles and vice versa. Therefore, if a forward geodesically complete Finsler manifold admits a Mobius mapping, then the indicatrix is conformally diffeomorphic to the Euclidean sphere $ S^{n-1}$ in $ \mathbb{R}^n $. In addition, if a forward geodesically complete absolutely homogeneous Finsler manifold of scalar flag curvature admits a non-trivial change of Mobius mapping, then it is a Riemannian manifold of constant sectional curvature.

math.DG↗
arXiv2020

Complete Finsler spaces of constant negative Ricci curvature

Here, using the projectively invariant pseudo-distance and Schwarzian derivative, it is shown that every connected complete Finsler space of the constant negative Ricci scalar is reversible. In particular, every complete Randers metric of constant negative Ricci (or flag) curvature is Riemannian.

math.DG↗
arXiv2020

On Sobolev spaces and density theorems on Finsler manifolds

Here, a natural extension of Sobolev spaces is defined for a Finsler structure $F$ and it is shown that the set of all real $C^{\infty}$ functions with compact support on a forward geodesically complete Finsler manifold $(M, F)$, is dense in the extended Sobolev space $H_1^p (M)$. As a consequence, the weak solutions $u$ of the Dirichlet equation $Δu=f$ can be approximated by $C^\infty$ functions with compact support on $M$. Moreover, let $W \subset M$ be a regular domain with the $C^r$ boundary $\partial W$, then the set of all real functions in $C^r (W) \cap C^0 (\overline W)$ is dense in $H_k^p (W)$, where $k\leq r$. Finally, several examples are illustrated and sharpness of the inequality $k\leq r$ is shown.

math.DG↗
arXiv2018

Complete Ricci solitons on Finsler manifolds

The geometric flow theory and its applications turned into one of the most intensively developing branches of modern geometry. Here, a brief introduction to Finslerian Ricci flow and their self-similar solutions known as Ricci solitons are given and some recent results are presented. They are a generalization of Einstein metrics and are previously developed by the present authors for Finsler manifolds. In the present work, it is shown that a complete shrinking Ricci soliton Finsler manifold has a finite fundamental group.

math.DG↗
arXiv2018

Ricci flow on Finsler surfaces

Here, we study the existence and uniqueness of solutions to the Ricci flow on Finsler surfaces and show short time existence of solutions for such flows. To this purpose, we first study the Finslerian Ricci-DeTurck flow on Finsler surfaces and find a unique short time solution to this flow. Then, we find a solution to the original Ricci flow by pulling back the solution of the Ricci-DeTurck flow using appropriate diffeomorphisms. In the end, we illustrate this argument with some examples.

math.DG↗
arXiv2015

On compact Ricci solitons in Finsler geometry

Ricci solitons on Finsler spaces, previously developed by the present authors, are a generalization of Einstein spaces, which can be considered as a solution to the Ricci flow on compact Finsler manifolds. In the present work it is shown that on a Finslerian space, a forward complete shrinking Ricci soliton is compact if and only if it is bounded. Moreover, it is proved that a compact shrinking Finslerian Ricci soliton has finite fundamental group and hence the first de Rham cohomology group vanishes.

math.DG↗
arXiv2015

On a Projectively Invariant Pseudo-distance in Finsler geometry

Here, a non-linear analysis method is applied rather than classical one to study projective changes of Finsler metrics. More intuitively, a projectively invariant pseudo-distance is introduced and characterized with respect to the Ricci tensor and its covariant derivatives.

math.DG↗
arXiv2014

On quasi-Eienstein Finsler spaces

The notion of quasi-Einstein metric in physics is equivalent to the notion of Ricci soliton in Riemannian spaces. Quasi-Einstein metrics serve also as solution to the Ricci flow equation. Here, the Riemannian metric is replaced by a Hessian matrix derived from a Finsler structure and a quasi-Einstein Finsler metric is defined. In compact case, it is proved that the quasi-Einstein metrics are solution to the Finslerian Ricci flow and conversely, certain form of solutions to the Finslerian Ricci flow are quasi-Einstein Finsler metrics.

math.DG↗
arXiv2013

On Sobolev spaces and density theorems on Finsler manifolds

Let $(M,F)$ be a $C^\infty$ Finsler manifold, $p\geq 1$ a real number, $k$ a positive integer and $H_k^p (M)$ a certain Sobolev space determined by a Finsler structure $F$. Here, it is shown that the set of all real $C^{\infty}$ functions with compact support on $M$ is dense in the Sobolev space $H_1^p (M)$. This result permits to approximate certain solution of Dirichlet problem living on $H_1^p (M)$ by $C^ \infty$ functions with compact support on $(M,F)$. Moreover, let $W \subset M$ be a regular domain with the $C^r$ boundary $\partial W$, then the set of all real functions in $C^r (W) \cap C^0 (\overline W)$ is dense in $H_k^p (W)$, where $k\leq r$. This work is an extension of some density theorems of T. Aubin on Riemannian manifolds.

math.DG↗
arXiv2012

3D-Algorithms of Composed Pursuit Navigation

The problem of pursuing a moving target is always one of the main topics in navigation. In the literatures, there are two well-known algorithms called Pure Pursuit and Pure Rendezvous navigation in the 3-dimensional space $\mathbb{R}^3$. In this paper, these two methods are combined to introduce a novel family of pursuing algorithms called Composed Pursuit Navigation. The Kinematic and geometric properties of this navigation is studied. The trajectories of this new family of algorithms benefit the advantages of two known methods and its prominence is demonstrated in two real examples. Moreover, it is shown that the metric related to the algorithms are given by Matsumoto metrics.

math.DG↗
arXiv2011

Circle-preserving transformations in Finsler spaces

Here, by extending the definition of circle to Finsler geometry, we show that, every circle-preserving local diffeomorphism is conformal. This result implies that in Finsler geometry, the definition of concircular change of metrics, a priori, does not require the conformal assumption.

math.DG↗
arXiv2006

Invariant conformal geometry on Finsler manifolds

The electric capacity of a conductor in the 3-dimensional Euclidean space $R^3 $ is defined as a ratio of a given positive charge on the conductor to the value of potential on the surface. This definition of the capacity is independent of the given charge. The capacity of a set as a mathematical notion was defined first by N. Wiener (1924) and was developed by O. Forstman, C. J. de La Vallee Poussin, and several other French mathematicians in connection with potential theory. This paper develops the theory of conformal invariants for Finsler manifolds. More precisely we prove: The capacity of a compact set and the capacity of the condenser of two closed sets are conformally invariant. By mean of the notion of capacity, we construct and study four conformal invariant functions $ ρ$, $ ν$, $μ$ and $λ$ which have similarities with the classical invariants on $S^n$, $ R^n $ or $H^n$. Their properties and especially their continuity are efficient tools for solving some problems of conformal geometry in the large.

math.DG↗