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Kaveh Eftekharinasab

Publications and source records attributed to Kaveh Eftekharinasab.

At least 19 recordsLinked to original sources
arXiv2026

A Frobenius Theorem on Fréchet Manifolds

We investigate the integrability of Fréchet tangent distributions on Fréchet manifolds. We introduce the local well-posedness Condition W for split tangent subbundles, which reduces the local integrability problem to solving initial value problems with parameters whose solutions define curves tangent to the distribution. By applying a variational approach to establish the existence and uniqueness of these solutions, we prove a Frobenius theorem stating that involutivity and Condition W are sufficient for integrability. This yields the existence of a unique maximal foliation of the manifold. Furthermore, we provide a dual formulation of the theorem using differential forms, which characterizes the algebraic conditions for integrability via the exterior derivative of the subbundle's local annihilator.

math.DG↗
arXiv2025

A Class of Functionals on the Sequence Space $s$ Satisfying the Palais-Smale Condition

We introduce a class of functionals on the space of rapidly decreasing sequences $s$, called $\mathcal{F}_s$-functionals, defined as decomposable sums of quadratic and convex terms with quadratic growth. We prove that such functionals satisfy the Palais-Smale condition and admit a unique global minimum. Furthermore, we show that the Palais-Smale condition is preserved under linear homeomorphisms. This allows us to construct corresponding functionals satisfying the Palais-Smale condition on Fréchet spaces isomorphic to $s$. We then show how this framework provides a tool for the proof of existence and uniqueness of solutions for specific operator problems, where coupled infinite-dimensional systems are transformed into diagonalized problems in the space $s$.

math.FA↗
arXiv2025

Spray-Invariant Sets in Infinite-Dimensional Manifolds

We introduce the concept of spray-invariant sets on infinite-dimensional manifolds, where any geodesic of a spray starting in the set stays within it for its entire domain. These sets, possibly including singular spaces such as stratified spaces, exhibit different geometric properties depending on their regularity: sets that are not differentiable submanifolds may show sensitive dependence, for example, on parametrization, whereas for differentiable submanifolds invariance is preserved under reparametrization. This framework offers a broader perspective on geodesic preservation than the rigid notion of totally geodesic submanifolds, with examples arising naturally even in simple settings, such as linear spaces equipped with flat sprays.

math.DG↗
arXiv2025

A Generalized Palais-Smale Condition in the Fréchet space setting

We extend the Palais-Smale condition to Keller's $C_c^1$-functionals on Fréchet spaces. Using this condition together with Ekeland's variational principle, we obtain some results regarding the existence of minima. In this setting, we prove that the Palais-Smale condition for functionals bounded below implies the coercivity.

math.DG↗
arXiv2025

Global Implicit Function Theorems and Critical Point Theory in Fréchet Spaces

We prove two versions of a global implicit function theorem, which involve no loss of derivative, for Keller's $ C_c^1 $-mappings between arbitrary Fréchet spaces. Subsequently, within this framework, we apply these theorems to establish the global existence and uniqueness of solutions to initial value problems that involve the loss of one derivative. Moreover, we prove a Lagrange multiplier theorem by employing indirect applications of the global implicit function theorems through submersions and transversality.

math.DG↗
arXiv2024

Geometry Via Sprays on Frechet Manifolds

We construct connection maps and linear symmetric connections on tangent and second-order tangent bundles for \fr manifolds using the notion of a spray. For these manifolds, we characterize linear symmetric connections on tangent bundles in terms of bilinear symmetric mappings associated with sprays. We also provide an alternative characterization of these connections using tangent structures. Furthermore, we prove that a bijective correspondence exists between linear symmetric connections on tangent bundles and sprays.

math.DG↗
arXiv2023

A Global Diffeomorphism Theorem for Fréchet spaces

We give sufficient conditions for a $ C^1_c $-local diffeomorphism between Fréchet spaces to be a global one. We extend the Clarke's theory of generalized gradients to the more general setting of Fréchet spaces. As a consequence, we define the Chang Palais-Smale condition for Lipschitz functions and show that a function which is bounded below and satisfies the Chang Palais-Smale condition at all levels is coercive. We prove a version of the mountain pass theorem for Lipschitz maps in the Fréchet setting and show that along with the Chang Palais-Smale condition we can obtain a global diffeomorphism theorem.

math.DG↗
arXiv2023

The Morse-Sard-Brown Theorem for Functionals on Bounded-Fréchet-Finsler Manifolds

In this paper, we study Lipschitz-Fredholm vector fields on Bounded-Fréchet-Finsler manifolds. In this context we generalize the Morse-Sard-Brown theorem, asserting that if $M$ is a connected smooth bounded-Fréchet-Finsler manifold endowed with a strengthened connection $\mathcal{K}$ and if $ξ$ is a smooth Lipschitz-Fredholm vector field on $M$ with respect to $\mathcal{K}$ which satisfies condition (CV). Then, for any smooth functional $l$ on $M$ which is associated to $ξ$, the set of the critical values of $l$ is of the first category in $\rr$. Therefore, the set of the regular values of $l$ is a residual Baire subset of $\mathbb{R}$.

math.DG↗
arXiv2023

Geometry of Bounded Frechet Manifolds

In this paper we develop the geometry of bounded Fréchet manifolds. We prove that a bounded Fréchet tangent bundle admits a vector bundle structure. But the second order tangent bundle $T^2M$ of a bounded Fréchet manifold $M$, becomes a vector bundle over $M$ if and only if $M$ is endowed with a linear connection. As an application, we prove the existence and uniqueness of the integral curve of a vector field on $M$.

math.DG↗
arXiv2022

Multiplicity Theorems for Frechet Manifolds

We prove multiplicity theorems for Keller $ C_c^1 $-functionals on Frechet spaces and Finsler manifolds which are invariant under the action of a discrete subgroup. For such functionals, we evaluate the minimal number of critical points by applying the Lusternik-Schnirelmann category.

math.DG↗
arXiv2022

Some critical point results for Fréchet manifolds

We prove a so-called linking theorem and some of its corollaries, namely a mountain pass theorem and a three critical points theorem for Keller $ C^1$-functional on $ C^1 $- Frechet manifolds. Our approach relies on a deformation result which is not implemented by considering the negative pseudo-gradient flows. Furthermore, for mappings between Frechet manifolds we provide a set of sufficient conditions in terms of the Palais-Smale condition that indicates when a local diffeomorphism is a global one.

math.DG↗
arXiv2021

Some applications of transversality for infinite dimensional manifolds

We present some transversality results for a category of Fréchet manifolds, the so-called $MC^k$-Fréchet manifolds. In this context, we apply the obtained transversality results to construct the degree of nonlinear Fredholm mappings by virtue of which we prove a rank theorem, an invariance of domain theorem and a Bursuk-Ulam type theorem.

math.DG↗
arXiv2020

Finslerian geodesics on Fréchet manifolds

We establish a framework, namely, nuclear bounded Fréchet manifolds endowed with Riemann-Finsler structures to study geodesic curves on certain infinite dimensional manifolds such as the manifold of Riemannian metrics on a closed manifold. We prove on these manifolds geodesics exist locally and they are length minimizing in a sense. Moreover, we show that a curve on these manifolds is geodesic if and only if it satisfies a collection of Euler-Lagrange equations. As an application, without much difficulty, we prove that the solution to the Ricci flow on an Einstein manifold is not geodesic.

math.DG↗
arXiv2015

Transversality and Lipschitz-Fredholm maps

We study transversality for Lipschitz-Fredholm maps in the context of bounded Fréchet manifolds. We show that the set of all Lipschitz-Fredholm maps of a fixed index between Fréchet spaces has the transverse stability property. We give a straightforward extension of the Smale transversality theorem by using the generalized Sard's theorem for this category of manifolds. We also provide an answer to the well known problem concerning the existence of a submanifold structure on the preimage of a transversal submanifold.

math.DG↗
arXiv2015

Fréchet Lie algebroids and their cohomology

We define Lie and Courant algebroids on Fréchet manifolds. Moreover, we construct a Dirac structure on the generalized tangent bundle of a Fréchet manifold and show that it inherits a Fréchet Lie algebroid structure. We show that the Lie algebroid cohomology of the $\bb$-cotangent bundle Lie algebroid of a weakly symplectic Fréchet manifold $M$ is the Lichnerowicz-Poisson cohomology of $M$.

math.DG↗