A Riemannian manifold (M, g) is semi-symmetric if (R(X,Y ) ◦ R)(U, V,W ) = 0. It is called pseudo-symmetric if R ◦ R = F, F being a given function of X, . . . ,W and g. It is called partially pseudosymmetric if this last relation is fulfilled by not all values of X, . . . ,W . Such manifolds were investigated by several mathematicians: I.Z. Szabó, S. Tanno, K. Nomizu, R. Deszcz and others. In t...

The equation also appears in investigation of geodesically equivalent metrics. Recall that two metrics on one manifold are geodesically equivalent, if every geodesic of one metric is a reparametrized geodesic of the second metric. Solodovnikov [9] has shown that Riemannian metrics on (n > 3)−dimensional manifolds admitting nontrivial 3-parameter family of geodesically equivalent metrics allow n...

The general relativity theory is redefined equivalently in almost Kähler variables: symplectic form, θ[g], and canonical symplectic connection, D̂[g] (distorted from the Levi–Civita connection by a tensor constructed only from metric coefficients and their derivatives). The fundamental geometric and physical objects are uniquely determined in metric compatible form by a (pseudo) Riemannian metri...

A new framework to perturbative quantum gravity is proposed following the geometry of nonholonomic distributions on (pseudo) Riemannian manifolds. There are considered such distributions and adapted connections, also completely defined by a metric structure, when gravitational models with infinite many couplings reduce to two–loop renormalizable effective actions. We use a key result from our p...

Lie groups appear in many fields fromMedical Imaging to Robotics. In Medical Imaging and particularly in Computational Anatomy, an organ’s shape is often modeled as the deformation of a reference shape, in other words: as an element of a Lie group. In this framework, if one wants to model the variability of the human anatomy, e.g. in order to help diagnosis of diseases, one needs to perform sta...

The geodesic motion of pseudo-classical spinning particles in Euclidean Taub-NUT space is analysed. The constants of motion are expressed in terms of Killing-Yano tensors. Some previous results from the literature are corrected. PACS number(s): 04.20.Jb, 02.40.-K The con guration space of spinning particles (spinning space) is an supersymmetric extension of an ordinary Riemannian manifold, para...

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 evol...

We investigate connections between pairs of (pseudo-)Riemannian metrics whose sum is a (tensor) product of a covector field with itself. A bijective mapping between the classes of Euclidean and Lorentzian metrics is constructed as a special result. The existence of such maps on a differentiable manifold is discussed. Similar relations for metrics of arbitrary signature on a manifold are conside...

The pseudo-Riemannian manifold M = (M, g), n ≥ 2 is paraquaternionic Kähler if hol(M) ⊂ sp(n,R)⊕sp(1, R). If hol(M) ⊂ sp(n, R), than the manifold M is called para-hyperKähler. The other possible definitions of these manifolds use certain parallel para-quaternionic structures in End(TM), similarly to the quaternionic case. In order to relate these different definitions we study para-quaternionic...

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