The geodesic distance vanishes on the group Diffc(M) of compactly supported diffeomorphisms of a Riemannian manifold M of bounded geometry, for the right invariant weak Riemannian metric which is induced by the Sobolev metric Hs of order 0 ≤ s < 1 2 on the Lie algebra Xc(M) of vector fields with compact support.

We construct a Polish group with an invariant metric in which Lie sums and Lie brackets do not exist. The construction of the group and the proof of the main theorem use some facts of combinatorial nature about the free group with two generators equipped with a Graev metric.

Using the Ponce de Leon background metric, which describes a 5D universe in an apparent vacuum: ¯ G AB = 0, we study the effective 4D evolution of both, the inflaton and gauge-invariant scalar metric fluctuations, in the recently introduced model of space time matter inflation.

We study the anti-self-dual equation for non-diagonal SU(2)-invariant metrics and give an equivalent ninth-order system. This system reduce to a sixth-order system if the metric is in the conformal class of scalar-flatKähler metric.

Using the Ponce de Leon background metric, which describes a 5D universe in an apparent vacuum: ¯ G AB = 0, we study the effective 4D evolution of both, the inflaton and gauge-invariant scalar metric fluctuations, in the recently introduced model of space time matter inflation.

In [KO2] we developed a general classification scheme for metric Lie algebras, i.e. for finite-dimensional Lie algebras equipped with a non-degenerate invariant inner product. Here we determine all nilpotent Lie algebras l with dim l = 2 which are used in this scheme. Furthermore, we classify all nilpotent metric Lie algebras of dimension at most 10.

We study the indefinite metric G in the contact phase space (P, θ) of a homogeneous thermodynamical system introduced by R. Mrugala. We calculate the curvature tensor, Killing vector fields, second fundamental form of Legendre submanifolds of P constitutive surfaces of different homogeneous thermodynamical systems. We established an isomorphism of the space (P, θ,G) with the Heisenberg Lie grou...

Dynamical equations describing evolution of state functions in space-time of a given metric are important components of physical theories of particles. A method based on a group of the metric is used to obtain an infinite set of general dynamical equations for a scalar and analytical function representing free and spinless particles. It is shown that this set of equations is the same for any gr...

This brief paper investigates the consequences for the metric tensor of space-time when the Weyl tensor (in its conformally invariant form) and the energy-momentum tensor is specified. It is shown that, unless rather special conditions hold, the metric is uniquely determined up to a constant conformal factor.

We derive the (g − 2)(g − 3)/2 linearly independent relations among the products of pairs in a basis of holomorphic abelian differentials in the case of compact non-hyperelliptic Riemann surfaces of genus g ≥ 4. By the Kodaira-Spencer map this leads to the modular invariant metric on the moduli space induced by the Siegel metric.