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Mehdi Rafie-Rad

Mehdi Rafie-Rad

Academic rank: Associate Professor
ORCID: 0000-0002-8214-3835
Education: PhD.
ScopusId: 23493274700
Faculty: Faculty of Mathematical Sciences
Address:
Phone: 01135302464

Research

Title
ON THE RIEMANN CURVATURE OPERATORS IN RANDERS SPACES
Type
JournalPaper
Keywords
Riemann curvature operator; Randers metric; principal curvature; S-curvature
Year
2013
Journal INTERNATIONAL JOURNAL OF GEOMETRIC METHODS IN MODERN PHYSICS
DOI
Researchers Mehdi Rafie-Rad

Abstract

The Riemann curvature in Riemann–Finsler geometry can be regarded as a collection of linear operators on the tangent spaces. The algebraic properties of these operators may be linked to the geometry and the topology of the underlying space. The principal curvatures of a Finsler space (M,F) at a point x are the eigenvalues of the Riemann curvature operator at x. They are real functions κ on the slit tangent manifold TM0. A principal curvature κ(x, y) is said to be isotropic respectively, quadratic) if κ(x, y)/F (x, y) is a function of x only (respectively, κ(x, y) is quadratic with respect to y). On the other hand, the Randers metrics are the most popular and prominent metrics in pure and applied disciplines. Here, it is proved that if a Randers metric admits an isotropic principal curvature, then F is of isotropic S-curvature. The same result is also established for F to admit a quadratic principal curvature. These results extend Shen’s verbal results about Randers metrics of scalar flag curvature K = K(x) as well as those Randers metrics with quadratic Riemann curvature operator. The Riemann curvature Ri k may be broken into two operators Ri k and J i k. The isotropic and quadratic principal curvature are characterized in terms of the eigenvalues of R and J .