In this article we have showed that a gradient $\rho$-Einstein soliton with vector field of bounded norm and satisfying some other conditions is isometric to the Euclidean sphere. Later, proved non-trivial complete finite weighted Dirichlet integral certain restriction on Ricci curvature must be constant scalar steady flat. Finally, non-shrinking or non-expanding traceless possessing steady.

An important component of Hamilton’s program for the Ricci flow on compact 3-manifolds is the classification of singularities which form under the flow for certain initial metrics. In particular, Type I singularities, where the evolving metrics have curvatures whose maximums are inversely proportional to the time to blow-up, are modelled on the 3-sphere and the cylinder S × R and their quotient...

Let (N, γ) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their ‘almost’ versions). We define a left invariant Riemannian metric on N compatible with γ to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. We prove that minimal metrics...

Let (N, γ) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their ‘almost’ versions). We define a left invariant Riemannian metric on N compatible with γ to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. We prove that minimal metrics...

We derive matter collineations for some static spherically symmetric spacetimes and compare the results with Killing, Ricci and Curvature symmetries. We conclude that matter and Ricci collineations are not, in general, the same.

In this paper, we initiate the study of a generalized soliton on Riemannian manifold, find characterization for Euclidean space, and in compact case, sufficient condition under which it reduces to quasi-Einstein manifold. We also conditions an Einstein Note that Ricci solitons being self-similar solutions heat flow, topic is related symmetry geometry manifolds. Moreover, generalizations are nat...

We propose fundamental inequalities for contact pseudo-slant submanifolds of (ϵ)-para Sasakian space form employing generalized normalized δ-Casorati curvature. characterize which equality cases hold and illustrate the main result with some applications. Further, we have considered a certain type submanifold Ricci soliton after computing its scalar curvature, developed an inequality to find cor...

In this paper, we prove a volume growth estimate for steady gradient Ricci solitons with bounded Nash entropy. We show that such soliton has rate no smaller than $$r^{\frac{n+1}{2}}.$$ This result not only improves the in (Chan et al., arXiv:2107.01419 , 2021, Theorem 1.3), but also is optimal since Bryant and Appleton’s (Appleton, arXiv:1708.00161 2017) have exactly rate.

Assuming that there exists a translating soliton $u_\infty$ with speed $C$ in domain $\Omega$ and prescribed contact angle on $\partial\Omega$, we prove graphical solution to the mean curvature flow same converges $u_\infty +Ct$ as $t\to\infty$. We also generalize recent existence result of Gao, Ma, Wang Weng non-Euclidean settings under suitable bounds convexity Ricci $\Omega$.

We introduce $$\mu $$ -scalar curvature for a Kähler metric with moment map and start up study on constant as generalization of both cscK Kähler–Ricci soliton continuity path to extremal metric. some fundamental constraints the existence by investigating volume functional Tian-Zhu’s work, which is closely related Perelman’s W-functional. A new K-energy studied an approach uniqueness problem pre...