# On the Finsler Geometry of Four-Dimensional Einstein Lie Groups

On the Finsler Geometry of Four-Dimensional Einstein Lie Groups In this paper, we study left invariant $$(\alpha ,\beta )$$ ( α , β ) -metrics on four-dimensional real Lie groups equipped with left invariant Einstein Riemannian metrics. We classify all left invariant $$(\alpha ,\beta )$$ ( α , β ) -metrics of Berwald type induced by a left invariant Einstein Riemannian metric and a left invariant vector field and show that all of them are locally Minkowskian. All left invariant Randers metrics of Douglas type, and all Einstein Kropina metrics induced by a left invariant Riemannian metric and a left invariant vector field, are classified. Finally, the flag curvatures of these spaces are investigated and in a special case the geodesics are computed. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Iranian Journal of Science and Technology, Transactions A: Science Springer Journals

# On the Finsler Geometry of Four-Dimensional Einstein Lie Groups

6 pages

/lp/springer_journal/on-the-finsler-geometry-of-four-dimensional-einstein-lie-groups-rMpgY9iFE7
Publisher
Springer Journals
Subject
Engineering; Engineering, general; Chemistry/Food Science, general; Earth Sciences, general; Life Sciences, general; Materials Science, general; Physics, general
ISSN
1028-6276
eISSN
2364-1819
D.O.I.
10.1007/s40995-018-0583-z
Publisher site
See Article on Publisher Site

### Abstract

In this paper, we study left invariant $$(\alpha ,\beta )$$ ( α , β ) -metrics on four-dimensional real Lie groups equipped with left invariant Einstein Riemannian metrics. We classify all left invariant $$(\alpha ,\beta )$$ ( α , β ) -metrics of Berwald type induced by a left invariant Einstein Riemannian metric and a left invariant vector field and show that all of them are locally Minkowskian. All left invariant Randers metrics of Douglas type, and all Einstein Kropina metrics induced by a left invariant Riemannian metric and a left invariant vector field, are classified. Finally, the flag curvatures of these spaces are investigated and in a special case the geodesics are computed.

### Journal

Iranian Journal of Science and Technology, Transactions A: ScienceSpringer Journals

Published: Jun 4, 2018

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