Abstract We classify 3-dimensional compact Riemannian manifolds ( M 3 , g ) that admit a non-constant solution to the equation − Δfg +Hess f Ric = μ + λg for some special constants λ ), under assumption manifold has cyclic parallel Ricci tensor. Namely, structures we study here are: positive static triples, critical metrics of volume functional, and total scalar curvature functional. also n -di...

We study a well-known scalar quantity in Riemannian geometry, the Ricci scalar, in the context of diffusion tensor imaging (DTI), which is an emerging non-invasive medical imaging modality. We derive a physical interpretation for the Ricci scalar and explore experimentally its significance in DTI. We also extend the definition of the Ricci scalar to the case of high angular resolution diffusion...

In the present study, we considered 3-dimensional generalized (κ, μ)-contact metric manifolds. We proved that a 3-dimensional generalized (κ, μ)-contact metric manifold is not locally φ-symmetric in the sense of Takahashi. However such a manifold is locally φ-symmetric provided that κ and μ are constants. Also it is shown that if a 3-dimensional generalized (κ, μ) -contact metric manifold is Ri...

We obtain the evolution equations for the Riemann tensor, the Ricci tensor and the scalar curvature induced by the mean curvature flow. The evolution for the scalar curvature is similar to the Ricci flow, however, negative, rather than positive, curvature is preserved. Our results are valid in any dimension.

We prove that for a solution (M, g(t)), t ∈ [0, T ), where T < ∞, to the Ricci flow on a complete non-compact Riemannian manifold with the Ricci curvature tensor uniformly bounded by some constant C on M × [0, T ), the curvature tensor stays uniformly bounded on M × [0, T ).

A generalization of Ricci-like solitons with torse-forming potential, which is a constant multiple the Reeb vector field, studied. The conditions under these are equivalent to almost Einstein-like metrics given. Some results obtained for parallel symmetric second-order covariant tensor. Finally, an explicit example arbitrary dimension given and some illustrated.