In this paper, we will study compatible triples on Lie algebroids. Using a suitable decomposition for a Lie algebroid, we construct an integrable generalized distribution on the base manifold. As a result, the symplectic form on the Lie algebroid induces a symplectic form on each integral submanifold of the distribution. The induced Poisson structure on the base manifold can be represented by m...

We compute the asymptotic dimension of the rationals given with an invariant proper metric. Also we show that a countable torsion abelian group taken with an invariant proper metric has asymptotic dimension zero.

We derive and study necessary and sufficient conditions for an S-bundle to admit an invariant metric of positive or nonnegative sectional curvature. In case the total space has an invariant metric of nonnegative curvature and the base space is odd dimensional, we prove that the total space contains a flat totally geodesic immersed cylinder. We provide several examples, including a connection me...

This paper studies the extension of the Hofer metric and general Finsler metrics on the Hamiltonian symplectomorphism group Ham(M,ω) to the identity component Symp0(M,ω) of the symplectomorphism group. In particular, we prove that the Hofer metric on Ham(M,ω) does not extend to a bi-invariant metric on Symp0(M,ω) for many symplectic manifolds. We also show that for the torus T2n with the standa...

An aane invariant metric allowing one to compute aane invariant gradient descent ows is rst presented in this work. This means that given an aane invariant energy, we compute based on this metric the ow that minimizes this energy as fast as possible and in an aane invariant way. Two examples are then presented. The rst one shows that the aane ow minimizing the area enclosed by a planar curve is...

A new approach to error analysis is introduced, based on the observation that many numerical procedures can be interpreted as computations of products in a suitable Lie group. The absence of an additive error law for such procedures is intimately related to the nonexistence of bi-invariant metrics on the relevant groups. Introducing the notion of an almost Inn(G) invariant metric (a left invari...

We suggest a new method of describing invariant measures on Markov compacta and path spaces of graphs, and thus of describing characters of some groups and traces of AF-algebras. The method relies on properties of filtrations associated with the graph and, in particular, on the notion of a standard filtration. The main tool is the so-called internal metric that we introduce on simplices of meas...

AdS dynamics for massive scalar field is studied both by solving exactly the equation of motion and by constructing bulk-boundary propagator. A Robertson-Walkerlike metric is deduced from the familiar SO(2, n) invariant metric. The metric allows us to present a time-like Killing vector, which is not only invariant under space-like transformations but also invariant under the isometric transform...