Four-dimensional Riemannian manifolds with two circulant structures

We consider a class (M, g, q) of four-dimensional Riemannian manifolds M, where besides the metric g there is an additional structure q, whose fourth power is the unit matrix. We use the existence of a local coordinate system such that there the coordinates of g and q are circulant matrices. In this system q has constant coordinates and q is an isometry with respect to g. By the special identity for the curvature tensor R generated by the Riemannian connection of g we define a subclass of (M, g, q). For any (M, g, q) in this subclass we get some assertions for the sectional curvatures of two-planes. We get the necessary and sufficient condition for g such that q is parallel with respect to the Riemannian connection of g

Almost Conformal Transformation in a Class of Riemannian Manifolds

We consider a 3-dimensional Riemannian manifold V with a metric g and an a±nor structure q. The local coordinates of these tensors are circulant matrices. In V we define an almost conformal transformation. Using that definition we construct an infinite series of circulant metrics which are successively almost conformaly related. In this case we get some properties

SPHERES AND CIRCLES WITH RESPECT TO AN INDEFINITE METRIC ON A RIEMANNIAN MANIFOLD WITH A SKEW-CIRCULANT STRUCTURE

We study hyper-spheres, spheres and circles, with respect to an indefinite metric, in a single tangent space on a 4-dimensional differentiable manifold. The manifold is equipped with a positive definite metric and an additional tensor structure of type (1, 1). The fourth power of the additional structure is minus identity and its components form a skew-circulant matrix in some local coordinate system. The both structures are compatible and they determine an associated indefinite metric on the manifold

Върху афинни свързаности в риманово многообразие с циркулантна метрика и две циркулантни афинорни структури

Ива Р. Докузова, Димитър Р. Разпопов - В настоящата статия е разгледан клас V оттримерни риманови многообразия M с метрика g и два афинорни тензора q и S. Дефинирана е и друга метрика ¯g в M. Локалните координати на всички тези тензори са циркулантни матрици. Намерени са: 1) зависимост между тензора на кривина R породен от g и тензора на кривина ¯R породен от ¯g; 2) тъждество за тензора на кривина R в случая, когато тензорът на кривина ¯R се анулира; 3) зависимост между секционната кривина на прозволна двумерна q-площадка {x, qx} и скаларната кривина на M.In the present paper it is considered a class V of 3-dimensional Riemannian manifolds M with a metric g and two affinor tensors q and S. It is defined another metric ¯g in M. The local coordinates of all these tensors are circulant matrices. It is found: 1) a relation between curvature tensors R and ¯R of g and ¯g, respectively; 2) an identity of the curvature tensor R of g in the case when the curvature tensor ¯R vanishes; 3) a relation between the sectional curvature of a 2-section of the type {x, qx} and the scalar curvature of M. *2000 Mathematics Subject Classification: 53C15, 53B20.This work is partially supported by project RS09 - FMI - 003 of the Scientific Research Fund, Paisii Hilendarski University of Plovdiv, Bulgaria