56 research outputs found
Ricci Solitons conformally equivalent to left invariant metrics
In this paper we study the geometry of Riemannian metrics conformally
equivalent to invariant metrics on Lie groups. Then we give a necessary and
sufficient condition for these metrics to be Ricci solitons. Using this
condition, many explicit examples of shrinking, steady and expanding Ricci
solitons are given. Finally, we give an example of Ricci solitons which is not
conformally equivalent to a left invariant Riemannian metric
Randers Ricci soliton homogeneous nilmanifolds
Let be a left invariant Randers metric on a simply connected nilpotent
Lie group , induced by a left invariant Riemannian metric
and a vector field which is
-invariant. If the Ricci flow equation has a
unique solution then, is a Ricci soliton if and only if is a
semialgebraic Ricci soliton
On the left invariant Randers and Matsumoto metrics of Berwald type on 3-dimensional Lie groups
In this paper we identify all simply connected 3-dimensional real Lie groups
which admit Randers or Matsumoto metrics of Berwald type with a certain
underlying left invariant Riemannian metric. Then we give their flag curvatures
formulas explicitly
On the Existence of Homogeneous Geodesics in Homogeneous Kropina Spaces
Recently, it is shown that each regular homogeneous Finsler space admits
at least one homogeneous geodesic through any point . The purpose of
this article is to study the existence of homogeneous geodesics on singular
homogeneous -spaces, specially, homogeneous Kropina spaces. We
show that any homogeneous Kropina space admits at least one homogeneous
geodesic through any point. It is shown that, under some conditions, the same
result is true for any -homogeneous space. Also, in the case of
homogeneous Kropina space of Douglas type, a necessary and sufficient condition
for a vector to be a geodesic vector is given. Finally, as an example,
homogeneous geodesics of -dimensional non-unimodular real Lie groups
equipped with a left invariant Randers metric of Douglas type are investigated
Symplectic Connections Induced by the Chern Connection
Let be a symplectic manifold and be a Finsler structure on
. In the present paper we define a lift of the symplectic two-form
on the manifold , and find the conditions that the Chern
connection of the Finsler structure preserves this lift of . In
this situation if admits a nowhere zero vector field then we have a
non-empty family of Fedosov structures on
Left invariant lifted -metrics of Douglas type on tangent Lie groups
In this paper we study lifted left invariant -metrics of
Douglas type on tangent Lie groups. Let be a Lie group equipped with a left
invariant -metric of Douglas type , induced by a left
invariant Riemannian metric . Using vertical and complete lifts, we
construct the vertical and complete lifted -metrics and
on the tangent Lie group and give necessary and sufficient
conditions for them to be of Douglas type. Then, the flag curvature of these
metrics are studied. Finally, as some special cases, the flag curvatures of
and in the cases of Randers metrics of Douglas type, and Kropina
and Matsumoto metrics of Berwald type are given
Riemannian Geometry of Two Families of Tangent Lie Groups
Using vertical and complete lifts, any left invariant Riemannian metric on a
Lie group induces a left invariant Riemannian metric on the tangent Lie group.
In the present article we study the Riemannian geometry of tangent bundle of
two families of Lie groups. The first one is the family of special Lie groups
considered by J. Milnor and the second one is the class of Lie groups with
one-dimensional commutator groups. The Levi-Civita connection, sectional and
Ricci curvatures have been investigated
Left invariant Ricci solitons on five-dimensional nilmanifolds
In 2002, using a variational method, Lauret classified five-dimensional
nilsolitons. In this work, using the algebraic Ricci soliton equation, we
obtain the same classification. We show that, among ten classes of
five-dimensional nilmanifolds, seven classes admit Ricci soliton structure. In
any case, the derivation which satisfies the algebraic Ricci soliton equation
is computed
Naturally reductive homogeneous -metric spaces
In the present paper we study naturally reductive homogeneous
-metric spaces. Under some conditions, we give some necessary
and sufficient conditions for a homogeneous -metric space to be
naturally reductive. Then we show that for such spaces the two definitions of
naturally reductive homogeneous Finsler space, given in literature, are
equivalent. Finally we compute the flag curvature of naturally reductive
homogeneous -metric spaces
The Relation Between Automorphism Group and Isometry Group of Left Invariant -metrics
This work generalizes the results of an earlier paper by the second author,
from Randers metrics to -metrics. Let be an
-metric which is defined by a left invariant vector field and a
left invariant Riemannian metric on a simply connected real Lie group . We
consider the automorphism and isometry groups of the Finsler manifold
and their intersection. We prove that for an arbitrary left invariant vector
field and any compact subgroup of automorphisms which is invariant
under them, there exists an -metric such that is a subgroup
of its isometry group
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