Notes on a Three-Dimensional Riemannian Manifold with an Additional Structure (original) (raw)
Related papers
Riemannian manifolds with two circulant structures
2013
We consider a three-dimensional Riemannian manifold equipped with two circulant structures - a metric g and a structure q, which is an isometry with respect to g and the third power of q is minus identity. We discuss some curvature properties of this manifold, we give an example of such a manifold and find a condition for q to be parallel with respect to the Riemannian connection of g.
On a Three-Dimensional Riemannian Manifold with an Additional Structure
arXiv: Differential Geometry, 2011
We consider a 3-dimensional Riemannian manifold M with a metric tensor g, and a-nors q and S. We note that the local coordinates of these three tensors are circulant matrices. We have that the third degree of q is the identity and q is compatible with g. We discuss the sectional curvatures in case when q is parallel with respect to the connection of g.
Three-dimensional Riemannian manifolds with circulant structures
We consider a class of three-dimensional Riemannian manifolds M with a circulant metric g and a circulant affinor q. We have that the third degree of q is the identity and q is accordant with g. In a manifold in such class we get some curvature properties.
Some curvatures of Riemannian manifolds with circulant structures
We consider a four-dimensional Riemannian manifold equipped with two circulant structures -- a metric ggg and a structure qqq, which is an isometry with respect to ggg and the forth power of qqq is the identity. We discuss some curvature properties of this manifold.
On a Riemannian manifold with a circulant structure whose third power is the identity
Filomat
It is studied a 3-dimensional Riemannian manifold equipped with a tensor structure of type (1,1), whose third power is the identity. This structure has a circulant matrix with respect to some basis, i.e. the structure is circulant. On such a manifold a fundamental tensor by the metric and by the covariant derivative of the circulant structure is defined. An important characteristic identity for this tensor is obtained. It is established that the image of the fundamental tensor with respect to the usual conformal transformation satisfies the same identity. A Lie group as a manifold of the considered type is constructed and some of its geometrical characteristics are found.
Curvature properties of Riemannian manifolds with skew-circulant structures
2021
We consider a 4-dimensional Riemannian manifold M endowed with a right skew-circulant tensor structure S, which is an isometry with respect to the metric g and the fourth power of S is minus identity. We determine a class of manifolds (M, g, S), whose curvature tensors are invariant under S. For such manifolds we obtain properties of the Ricci tensor. Also we get expressions of the sectional curvatures of some special 2-planes in a tangent space of (M, g, S). Mathematics Subject Classification (2020): 53B20, 53C25, 53C15, 53C55.
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 \bar{g} in M. The local coordinates of all these tensors are circulant matrices. It is found: 1)\ a relation between curvature tensors R and \bar{R} of g and \bar{g}, respectively; 2)\ an identity of the curvature tensor R of g in the case when the curvature tensor \bar{R} vanishes; 3)\ a relation between the sectional curvature of a 2-section of the type \{x, qx\} and the scalar curvature of M.
On 3-dimensional almost Einstein manifolds with circulant structures
2020
Abstract: A 3-dimensional Riemannian manifold equipped with a tensor structure of type (1, 1), whose third power 4 is the identity, is considered. This structure and the metric have circulant matrices with respect to some basis, i.e., 5 these structures are circulant. An associated manifold, whose metric is expressed by both structures, is studied. Three 6 classes of such manifolds are considered. Two of them are determined by special properties of the curvature tensor of the 7 manifold. The third class is composed by manifolds whose structure is parallel with respect to the Levi-Civita connection 8 of the metric. Some geometric characteristics of these manifolds are obtained. Examples of such manifolds are given. 9
Local properties of almost-Riemannian structures in dimension 3
Discrete & Continuous Dynamical Systems - A, 2015
A 3D almost-Riemannian manifold is a generalized Riemannian manifold defined locally by 3 vector fields that play the role of an orthonormal frame, but could become collinear on some set Z called the singular set. Under the Hormander condition, a 3D almost-Riemannian structure still has a metric space structure, whose topology is compatible with the original topology of the manifold. Almost-Riemannian manifolds were deeply studied in dimension 2. In this paper we start the study of the 3D case which appear to be reacher with respect to the 2D case, due to the presence of abnormal extremals which define a field of directions on the singular set. We study the type of singularities of the metric that could appear generically, we construct local normal forms and we study abnormal extremals. We then study the nilpotent approximation and the structure of the corresponding small spheres. We finally give some preliminary results about heat diffusion on such manifolds.