In this paper, we study the flow of closed, starshaped hypersurfaces in $\mathbb{R}^{n+1}$ with speed $r^\alpha\sigma_2^{1/2},$ where $\sigma_2^{1/2}$ is normalized square root scalar curvature, $\alpha\geq 2,$ and $r$ distance from points on hypersurface to origin. We prove that exists for all time starshapedness preserved. Moreover, after normalization, show converges exponentially fast a sph...

We show that a complete [Formula: see text]-dimensional Riemannian manifold text] with finitely generated first homology has macroscopic dimension if it satisfies the following “macroscopic curvature” assumptions: every ball of radius in volume at most text], and loop is null-homologous concentric text].

Every Finsler metric naturally induces a spray but not so for the converse. The notion sprays of scalar (resp. isotropic) curvature has been known as generalization metrics flag curvature. In this paper, new notion, constant curvature, is introduced and especially it shows that isotropic necessarily even in dimension $$n\ge 3$$ . Further, complete conditions are given constant) to be metrizable...