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Ahmed Mohammed Cherif

Publications and source records attributed to Ahmed Mohammed Cherif.

At least 19 recordsLinked to original sources
arXiv2026

Ricci Solitons on the Lie Group $Sol^3_{m,n}$ and Applications

In this paper, we study Ricci solitons on the three-dimensional solvable Lie group $\mathrm{Sol}^{3}_{m,n}$ equipped with a left-invariant Riemannian metric, viewed as a generalization of the classical $\mathrm{Sol}^{3}$ geometry. We investigate harmonic maps, harmonic sections, and geodesic curves, including the geodesic properties of the integral curves of the Ricci soliton vector field. We also characterize harmonic linear maps from $\mathrm{Sol}^{3}_{m,n}$ into Euclidean spaces.

math.DG↗
arXiv2026

On generalized $ξ$-Parallel Maps

In this paper, we introduce and study the notion of generalized $ξ$-parallel maps between Riemannian manifolds, where $ξ$ is a vector field on the target manifold. We define the associated energy functional and derive its first variation formula, leading to the Euler--Lagrange equation characterizing generalized $ξ$-parallel maps. We establish fundamental properties of this new class of maps and investigate its relationship with harmonic and biharmonic maps. Furthermore, we study generalized $ξ$-parallel curves in space forms and examine generalized $ξ$-parallel submanifolds of space forms admitting concircular vector fields.

math.DG↗
arXiv2026

Dirichlet Energy of the Scalar Curvature

In this article, we derive the first variation formula for the Dirichlet scalar curvature energy functional on Riemannian manifolds. Motivated by this variational framework, we introduce a natural generalization of Einstein metrics, called Dirichlet--Einstein metrics, and we investigate some of their fundamental properties.

math.DG↗
arXiv2026

Second Variation of F-Einstein-Hilbert Functional

This article describes a formula for second variation of generalized Einstein-Hilbert functional on Riemannian manifolds. This work extends the definition of stable Einstein manifolds, and we present some properties.

math.DG↗
arXiv2026

Harmonic basis vector fields on surfaces

In this paper, we study local parameterizations of surfaces in the Euclidean space $\mathbb{R}^3$ whose coordinate vector fields are harmonic sections with respect to the induced Riemannian metric. After introducing the notion of harmonic basis vector fields on a surface, we derive necessary and sufficient conditions under which the coordinate vector fields $\frac{\partial}{\partial u}$ and $\frac{\partial}{\partial v}$ are harmonic. We then focus on two important classes of surfaces: surfaces of revolution and translation surfaces. For these families, we obtain a complete classification of all parameterizations admitting harmonic basis vector fields.

math.DG↗
arXiv2026

On the non-existence of symphonic maps

In this paper, we study the existence of symphonic maps on compact or complet non-compact Riemannian manifold into Riemannian manifolds admitting a conformal vector field or a non-trivial Ricci solitons.

math.DG↗
arXiv2026

Construction of normally biharmonic submanifolds

We examine biharmonic submanifolds within warped product structures. For a submanifold $(M,g)\subset (N,h)$ and a positive smooth function $f:I\to\mathbb{R}^+$, we study the inclusion $φ:(I\times M,\widetilde{g})\to (I\times N,\overline{h})$, where $\widetilde{g}=dt^2+f^2g$ and $\overline{h}=dt^2+f^2h$. We relate the tension and bitension fields of $φ$ to the warping function and the geometry of $M$. We further characterize tangentially and normally biharmonic cases via differential conditions on $f$, and interpret these conditions in terms of the Ricci curvature of $(M,g)$ and $(I\times M,\widetilde{g})$.

math.DG↗
arXiv2026

On the generalization of biharmonic hypersurfaces and biharmonic curves

In this work, we extend the concepts of $p$-biharmonic maps and $p$-biharmonic hypersurfaces to provide a broader characterization of $(p,q)$-harmonic hypersurfaces and $(p,q)$-harmonic curves in Riemannian manifolds, including Einstein spaces. Moreover, we present new explicit examples of proper $(p,q)$-harmonic hypersurfaces and $(p,q)$-harmonic curves in space forms.

math.DG↗
arXiv2026

Bi-symphonic maps between Riemannian manifolds

This note introduces an extension to the definition of symphonic maps, denoted as $φ:(M,g)\longrightarrow(N,h)$, by exploring variations in the bi-energy functional associated with the pullback metric $φ^*h$ between two Riemannian manifolds.

math.DG↗
arXiv2026

$H$-tensional hypersurfaces in $4$-dimensional space forms

In this paper, we investigate the classification of $H$-tensional hypersurfaces $M$ in a $4$-dimensional space form $N^4(c)$ of constant sectional curvature $c$. Our results show that minimal hypersurfaces are the only $H$-tensional hypersurfaces in $4$-dimensional space forms, thereby providing an affirmative partial answer to Conjecture 3 proposed in \cite{kacimi}.

math.DG↗
arXiv2026

On Ricci Solitons and Harmonic Vector Fields in the Thurston Geometry $F^4$

In this paper, we consider a left-invariant Riemannian metric $g$ on the Lie group $F^4$. We classify Ricci solitons on $(F^4,g)$ and show that all such solitons are expanding and non-gradient. Moreover, we study the existence of harmonic maps from compact Riemannian manifolds into $(F^4,g)$. Finally, we characterize a class of harmonic vector fields on $(F^4,g)$.

math.DG↗
arXiv2026

$HS$-tensional maps and $HM$-tensional maps

Let $ψ: (M,g)\longrightarrow (N,h)$ be a smooth map between Riemannian manifolds. The tension field of $ψ$ can be regarded as a map from $(M,g)$ into the Riemannian vector bundle $ψ^{-1}TN$, equipped with the Sasaki metric $G_{S}$. In this paper, we study certain aspects of two types of maps: those whose tension fields are harmonic maps (called $HM$-tensional maps) and those whose tension fields are harmonic sections (called $HS$-tensional maps).

math.DG↗
arXiv2025

Geometry of Harmonic Identity Maps

An identity map $(M,g)\longrightarrow(M,g)$ is a harmonic from a Riemannian manifold $(M,g)$ onto itself. In this paper, we study the harmonicity of identity maps $(M,g)\longrightarrow(M,g-df\otimes df)$ and $(M,g-df\otimes df)\longrightarrow(M,g)$ where $f$ is a smooth function with gradient norm $<1$ on $(M,g)$. We construct new examples of identity harmonic maps. We define a symmetric tensor field on $M$ whose properties are related to the harmonicity of these identity maps.

math.DG↗