40 research outputs found
Last multipliers as autonomous solutions of the Liouville equation of transport
Using the characterization of last multipliers as solutions of the
Liouville's transport equation, new results are given in this approach of ODE
by providing several new characterizations, e.g. in terms of Witten and Marsden
differentials or adjoint vector field. Applications to Hamiltonian vector
fields on Poisson manifolds and vector fields on Riemannian manifolds are
presented. In Poisson case, the unimodular bracket considerably simplifies
computations while, in the Riemannian framework, a Helmholtz type decomposition
yields remarkable examples: one is the quadratic porous medium equation, the
second (the autonomous version of the previous) produces harmonic square
functions, while the third refers to the gradient of the distance function with
respect to a two dimensional rotationally symmetric metric. A final example
relates the solutions of Helmholtz (particularly Laplace) equation to provide a
last multiplier for a gradient vector field. A connection of our subject with
gas dynamics in Riemannian setting is pointed at the end.Comment: final versio
Formal Frobenius structures generated by geometric deformation algebras
Necessary and sufficient conditions for some deformation algebras to provide
formal Frobenius structures are given. Also, examples of formal Frobenius
structures with fundamental tensor that is not of the deformation type and
examples of symmetric non-metric connections are presented.Comment: 15 page
Last multipliers for multivectors with applications to Poisson geometry
The theory of the last multipliers as solutions of the Liouville's transport
equation, previously developed for vector fields, is extended here to general
multivectors. Characterizations in terms of Witten and Marsden differentials
are reobtained as well as the algebraic structure of the set of multivectors
with a common last multiplier, namely Gerstenhaber algebra. Applications to
Poisson bivectors are presented by obtaining that last multipliers count for
''how far away'' is a Poisson structure from being exact with respect to a
given volume form. The notion of exact Poisson cohomology for an unimodular
Poisson structure on is introduced.Comment: 16 page
A new approach to gradient Ricci solitons and generalizations
This short note concerns with two inequalities in the geometry of gradient
Ricci solitons on a smooth manifold . These inequalities
provide some relationships between the curvature of the Riemannian metric
and the behavior of the scalar field through two second order equations
satisfied by the scalar . We propose several generalizations of Ricci
solitons to the setting of manifolds endowed with linear connections, not
necessary of metric type.Comment: revised and expanded versio
From the Eisenhart problem to Ricci solitons in -Kenmotsu manifolds
The Eisenhart problem of finding parallel tensors is solved for the symmetric
case in the regular -Kenmotsu framework. On this way, the Olszack-Rosca
example of Einstein manifolds provided by -Kenmotsu manifolds via locally
symmetric Ricci tensors is recovered as well as a case of Killing vector
fields. Some other classes of Einstein-Kenmotsu manifolds are presented. Our
result is interpreted in terms of Ricci solitons and special quadratic first
integrals.Comment: 12 page
Weyl substructures and compatible linear connections
The aim of this paper is to study from the point of view of linear
connections the data with a smooth
dimensional real manifold, a
\textit{}\textit{\emph{dimensional semi-Riemannian distribution}}\emph{}on
the conformal structure generated by and a Weyl
substructure: a map such that
. Compatible
linear connections are introduced as a natural extension of similar notions
from Riemannian geometry and such a connection is unique if a symmetry
condition is imposed. In the foliated case the local expression of this unique
connection is obtained. The notion of Vranceanu connection is introduced for a
pair (Weyl structure, distribution) and it is computed for the tangent bundle
of Finsler spaces, particularly Riemannian, choosing as distribution the
vertical bundle of tangent bundle projection and as 1-form the Cartan form.Comment: 15 page
Ricci-Yamabe maps for Riemannian flows and their volume variation and volume entropy
The aim of this short note is to produce new examples of geometrical flows
associated to a given Riemannian flow . The considered flow in covariant
symmetric -tensor fields will be called Ricci-Yamabe map since it involves a
scalar combination of Ricci tensor and scalar curvature of . Due to the
signs of considered scalars the Ricci-Yamabe flow can be also a Riemannian or
semi-Riemannian or singular Riemannian flow. We study the associated function
of volume variation as well as the volume entropy. Finally, since the
two-dimensional case was the most handled situation we express the Ricci flow
equation in all four orthogonal separable coordinate systems of the plane
CR-structures of codimension 2 on tangent bundles in Riemann-Finsler geometry
We determine a 2-codimensional CR-structure on the slit tangent bundle
of a Finsler manifold by imposing a condition regarding the almost
complex structure associated to when restricted to the structural
distribution of a framed -structure. This condition is satisfied when is of scalar flag curvature (particularly flat) and in the Riemannian case
this last condition means that is of constant curvature. This
CR-structure is finally generalized by using one positive number but under more
difficult conditions.Comment: 10 page
Ricci solitons in manifolds with quasi-constant curvature
The Eisenhart problem of finding parallel tensors treated already in the
framework of quasi-constant curvature manifolds in \cite{x:j} is reconsidered
for the symmetric case and the result is interpreted in terms of Ricci
solitons. If the generator of the manifold provides a Ricci soliton then this
is i) expanding on para-Sasakian spaces with constant scalar curvature and
vanishing -concircular tensor field and ii) shrinking on a class of
orientable quasi-umbilical hypersurfaces of a real projective space=elliptic
space form.Comment: 10 page
(a, 1)f structures on product of spheres
Our aim in this paper is to give some examples of Riemannian
structures (a generalization of an -paracontact structure) induced on
product of spheres of codimension () in an -dimensional
Euclidean space (), endowed with an almost product structure.Comment: 10 page