arXiv subjects

Sharief Deshmukh

Publications and source records attributed to Sharief Deshmukh.

12 recordsLinked to original sources

arXiv2026

Casorati inequalities along vertical and horizontal distributions of pointwise slant riemannian submersions

We establish optimal Casorati inequalities for pointwise slant Riemannian submersions. As such a submersion carries two distinct distributions, the vertical, tangent to the fibers, and the horizontal, governed by O'Neill's tensors T and A, we treat them separately. For submersions from generalized complex and generalized Sasakian space forms we bound the normalized scalar curvature of each distribution by its normalized Casorati curvatures. Special cases recover sharp inequalities for real, complex, Kähler, Sasakian, Kenmotsu, cosymplectic, and almost and almost space forms. Equality is characterized geometrically: invariantly quasi-umbilical fibers in the vertical case and integrability in the horizontal case. The invariant and anti-invariant limits reproduce known results, and examples illustrate sharpness.

math.DG↗

arXiv2026

Non-compact Ricci Solitons of Finite Volume with Potential Field of Constant Length

We study non-compact Ricci solitons of finite volume whose potential vector field has constant length. Under the assumptions that the scalar curvature is constant along the integral curves of the potential field and that a natural divergence term is integrable on the unit tangent bundle, we prove that such Ricci solitons are necessarily trivial. As applications, we obtain rigidity and non-existence results for Ricci solitons whose potential field is the Reeb vector field of almost contact metric and almost $α$-cosymplectic manifolds. In dimension three, we derive consequences for almost $α$-cosymplectic and contact metric manifolds, and we compare our results with the classification of homogeneous almost $α$-cosymplectic Ricci solitons due to Li and Liu. Several examples and non-examples are included to illustrate the necessity of the finite-volume and sign assumptions.

math.DG↗

arXiv2022

Spacelike Foliations on Lorentz manifolds

In this work, we study the geometric properties of spacelike foliations by hypersurfaces on a Lorentz manifold. We investigate conditions for the leaves being stable, totally geodesic or totally umbilical. We consider that $\overline{M}^{n+1}$ is equipped with a timelike closed conformal vector field $ξ$. If the foliation has constant mean curvature, we show that the leaves are stable. When the leaves are compact spacelike hypersurfaces we show that, under certain conditions, its are totally umbilic hypersurfaces. In the case of foliations by complete noncompact hypersurfaces, we using a Maximum Principle at infinity to conclude that the foliation is totally geodesic.

math.DG↗

arXiv2020

A remarkable property of concircular vector fields on a Riemannian manifold

In this paper, we show that given a nontrivial concircular vector field $\boldsymbol{u}$ on a Riemannian manifold $(M,g)$ with potential function $f$, there exists a unique smooth function $ρ$ on $M$ that connects $\boldsymbol{u}$ to the gradient of potential function $\nabla f$, which we call the connecting function of the concircular vector field $\boldsymbol{u}$. Then this connecting function is shown to be a main ingredient in obtaining characterizations of $n$-sphere $\mathbf{S}^{n}(c)$ and the Euclidean space $\mathbf{E}^{n}$. We also show that the connecting function influences topology of the Riemannian manifold.

math.DG↗

arXiv2019

On some geometric properties of quasi-product production models

In this article we obtain classification results on the quasi-product production functions in terms of the geometry of their associated graph hypersurfaces, generalizing in a new setting some recent results concerning basic production models. In particular, we obtain several results on the geometry of Spillman-Mitscherlich and transcendental production functions.

math.DG↗

arXiv2018

Conformal vector fields and Yamabe solitons

In this paper, we use less topological restrictions and more geometric and analytic conditions to obtain some sufficient conditions on Yamabe solitons such that their metrics are Yamabe metrics, that is, metrics of constant scalar curvature. More precisely, we use properties of conformal vector fields to find several sufficient conditions on the soliton vector fields of Yamabe solitons under which their metrics are of Yamabe metrics.

math.DG↗

arXiv2017

Euclidean submanifolds with conformal canonical vector field

The position vector field x is the most elementary and natural geometric object on a Euclidean submanifold $M$. The position vector field plays very important roles in mathematics as well as in physics. Similarly, the tangential component x^T of the position vector field is the most natural vector field tangent to the Euclidean submanifold $M$. We simply call the vector field x^T the \textit{canonical vector field} of the Euclidean submanifold M. In earlier articles, we investigated Euclidean submanifolds whose canonical vector fields are concurrent, concircular, or torse-forming. In this article we study Euclidean submanifolds with conformal canonical vector field. In particular, we characterize such submanifolds. Several applications are also given. In the last section we present three global results on complete Euclidean submanifolds with conformal canonical vector field.

math.DG↗

arXiv2017

Yamabe and quasi-Yamabe solitons on Euclidean submanifolds

In this paper we initiate the study of Yamabe and quasi-Yamabe solitons on Euclidean submanifolds whose soliton fields are the tangential components of their position vector fields. Several fundamental results of such solitons were proved. In particular, we classify such Yamabe and quasi-Yamabe solitons on Euclidean hypersurfaces.

math.DG↗

arXiv2014

Classification of Ricci solitons on Euclidean hypersurfaces

A Ricci soliton $(M,g,v,λ)$ on a Riemannian manifold $(M,g)$ is said to have concurrent potential field if its potential field $v$ is a concurrent vector field. Ricci solitons arisen from concurrent vector fields on Riemannian manifolds were studied recently in \cite{CD2}. The most important concurrent vector field is the position vector field on Euclidean submanifolds. In this paper we completely classify Ricci solitons on Euclidean hypersurfaces arisen from the position vector field of the hypersurfaces.

math.DG↗

arXiv2014

Ricci solitons and concurrent vector fields

A Ricci soliton $(M^n,g,v,λ)$ on a Riemannian manifold $(M^n,g)$ is said to have concurrent potential field if its potential field $v$ is a concurrent vector field. In the first part of this paper we completely classify Ricci solitons with concurrent potential fields. In the second part we derive a necessary and sufficient condition for a submanifold to be a Ricci soliton in a Riemannian manifold equipped with a concurrent vector field. In the last part, we classify shrinking Ricci solitons with $λ=1$ on Euclidean hypersurfaces. Several applications of our results are also presented.

math.DG↗

arXiv2013

Two optimal inequalities for anti-holomorphic submanifolds and their applications

The CR $δ$-invariant for CR-submanifolds was introduced in a recent article [B. Y. Chen, An optimal inequality for CR-warped products in complex space forms involving CR $δ$-invariant, Internat. J. Math. 23} (2012), no. 3, 1250045 (17 pages)]. In this paper, we prove two new optimal inequalities for anti-holomorphic submanifolds in complex space forms involving the CR $δ$-invariant. Moreover, we obtain some classification results for certain anti-holomorphic submanifolds in complex space forms which satisfy the equality case of either inequality.

math.DG↗

arXiv2012

A note on trans-Sasakian manifolds

In this paper, we obtain some sufficient conditions for a 3-dimensional compact trans-Sasakian manifold of type $(α,β)$ to be homothetic to a Sasakian manifold. A characterization of a 3-dimensional cosymplectic manifold is also obtained.

math.DG↗