284 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
Generalized Secure Distributed Source Coding with Side Information
In this paper, new inner and outer bounds on the achievable
compression-equivocation rate region for generalized secure data compression
with side information are given that do not match in general. In this setup,
two senders, Alice and Charlie intend to transmit information to Bob via
channels with limited capacity so that he can reliably reconstruct their
observations. The eavesdropper, Eve, has access to one of the channels at each
instant and is interested in the source of the same channel at the time. Bob
and Eve also have their own observations which are correlated with Alice's and
Charlie's observations. In this model, two equivocation and compression rates
are defined with respect to the sources of Alice and Charlie. Furthermore,
different special cases are discussed where the inner and outer bounds match.
Our model covers the previously obtained results as well
A New Secret key Agreement Scheme in a Four-Terminal Network
A new scenario for generating a secret key and two private keys among three
Terminals in the presence of an external eavesdropper is considered. Terminals
1, 2 and 3 intend to share a common secret key concealed from the external
eavesdropper (Terminal 4) and simultaneously, each of Terminals 1 and 2 intends
to share a private key with Terminal 3 while keeping it concealed from each
other and from Terminal 4. All four Terminals observe i.i.d. outputs of
correlated sources and there is a public channel from Terminal 3 to Terminals 1
and 2. An inner bound of the "secret key-private keys capacity region" is
derived and the single letter capacity regions are obtained for some special
cases.Comment: 6 pages, 3 figure
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
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