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...

Abstract We investigate the pluriclosed flow on Oeljeklaus–Toma manifolds. parameterize left-invariant metrics manifolds, and we classify ones which lift to an algebraic soliton of universal covering. further show that starting from a metric has long-time solution $\omega _t$ once normalized collapses torus in Gromov–Hausdorff sense. Moreover, $\tfrac {1}{1+t}\omega covering manifold converges ...

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...

In Kirchheim-Magnani [7] the authors construct a left invariant distance ρ on the Heisenberg group such that the identity map id is 1-Lipschitz but it is not metrically differentiable anywhere. In this short note we give an interpretation of the Kirchheim-Magnani counterexample to metric differentiability. In fact we show that they construct something which fails shortly from being a dilatation...

We construct left invariant special Kähler structures on the cotangent bundle of a flat pseudo-Riemannian Lie group. introduce twisted cartesian product two algebras according to linear representations by infinitesimal transformations. also exhibit double extension process algebra which allows us get all simply connected groups with bi-invariant symplectic connections. All constructed performin...

Abstract In the infinite dimensional Heisenberg group, we construct a left invariant weak Riemannian metric that gives degenerate geodesic distance. The same construction yields sub-Riemannian We show how standard notion of sectional curvature adapts to our framework, but it cannot be defined everywhere and is unbounded on suitable sequences planes. vanishing distance precisely occurs along thi...

We study the interplay between following types of special non-Kähler Hermitian metrics on compact complex manifolds (locally conformally Kähler, k-Gauduchon, balanced, and locally balanced) prove that a Kähler nilmanifold carrying balanced or left-invariant k-Gauduchon metric is necessarily torus. Combined with main result in [FV16], this leads to fact 2-step endowed whichever two metrics—balan...

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...