We introduce a Riemannin metric of diagonal type on the cotangent bundle of a Riemannian manifold and show that T*M with this metric is locally symmetric Einstein manifold. Also, we obtain a locally symmetric Kähler Einstein structure on the cotangent bundle of a Riemannian manifold of negative constant sectional curvature. Similar results are obtained on a tube around zero section in the cotangent bundle, in the case of a Riemannian manifold of positive constant sectional curvature.

We classify the paracontact Riemannian manifolds that their Riemannian curvature satis es in the certain condition and we show that this classi cation holds for the special cases semi-symmetric and locally symmetric spaces. Finally we study paracontact Riemannian manifolds satisfying R(X;  )
S = 0, where S is the Ricci tensor.

Let M be an n-dimensional Riemannian manifold and TM its tangent bundle. The conformal and fiber preserving vector fields on TM have well-known physical interpretations and have been studied by physicists and geometricians. Here we define a Riemannian or pseudo-Riemannian lift metric ḡ on TM , which is in some senses more general than other lift metrics previously defined on TM , and seems to complete these works. Next we study the lift conformal vector fields ds on (TM, ḡ) and prove among the others that, every complete lift conformal vector field on TM is homothetic, and moreover, every horizontal or vertical lift conformal vector field on TM is a Killing vector.

We consider the unit tangent sphere bundle of Riemannian manifold (M, g) with g-natural metric ˜ G and we equip it to an almost contact B-metric structure. Considering this structure, we show that there is a direct correlation between the Riemannian curvature tensor of (M, g) and local symmetry property of ˜ G. More precisely, we prove that the flatness of metric g is necessary and sufficient for the g-natural metric ˜ G to be locally symmetric.

Among the eigenvalue problems of the Laplacian, the biharmonic operator eigenvalue problems are interesting projects because these problems root in physics and geometric analysis. The buckling problem is one of the most important problems in physics, and many studies have been done by the researchers about the solution and the estimate of its eigenvalue. In this paper, first, we obtain the evolution equation of the first nonzero eigenvalue of the buckling problem on closed Riemannian manifold (compact and without boundary Riemannian manifold) along the unnormalized Ricci flow and normalized Ricci flow and by using them, we prove that the first nonzero eigenvalue and some quantities dependent to this eigenvalue are monotonic along the Ricci flow, under the some geometric conditions. Then, on special manifold such as homogeneous, 3-dimensional, 2-dimensional manifolds, we study the evolutionary behavior of this eigenvalue. Especially in the 2-dimensional state, depending on the value of the scalar curvature along the normalized Ricci flow, we find the quantities dependent on the first eigenvalue that are monotonic under the normalized Ricci flow. Finally, we give examples of soliton states and Einstein manifolds, and we obtain the evolution of the first eigenvalue of the buckling problem under the Ricci flow on these examples.

The symmetric curvature and associated curvatures of a vector bundle E with connection Ñ on a manifold M with connection Ñ were introduced. It is well-known that a total space of semi-Riemannian vector bundle over a semi-Riemannian manifold can be made into a semi-Riemannian manifold. In this case, the relation between curvatures of the Levi-Civita connections of E and M was studied. Here, the relation between symmetric curvatures of the Levi-Civita connections and their associated curvatures of E and M is studied.

The notion of a weakly symmetric and weakly projective symmetric Riemannian manifolds have been introduced by Tamassy and Binh [11], [12] and then after studied by so many authors such as De, Shaikh and Jana, Shaikh and Hui, Shaikh, Jana and Eyasmin ([1], [3], [4], [5], [6], [7], [8]). Recently, Singh and Khan [10] introduced the notion of Special weakly symmetric Riemannian manifolds and denoted such manifold by (SW S)n. A. U. Khan and Q. Khan found some results On Special Weakly Projective Symmetric Manifolds [13]. And P. Verma, P. Kanaujia and S. Kishor found some results on M-Projective Curvature Tensor on (k, µ)-Contact Space Forms and Sasakian-Space-Forms ([16], [17]). Motivated from the above, we have studied the nature of Ricci tensor R of type (1, 1) in a special weakly M-projective symmetric Riemannian manifold (SWMS)n and also explored some interesting results on (SWMS)n.