Vanishing Geodesic Distance for the Riemannian Metric with Geodesic Equation the Kdv-equation
نویسندگان
چکیده
The Virasoro-Bott group endowed with the right-invariant L2metric (which is a weak Riemannian metric) has the KdV-equation as geodesic equation. We prove that this metric space has vanishing geodesic distance.
منابع مشابه
Solution of Vacuum Field Equation Based on Physics Metrics in Finsler Geometry and Kretschmann Scalar
The Lemaître-Tolman-Bondi (LTB) model represents an inhomogeneous spherically symmetric universefilledwithfreelyfallingdustlikematterwithoutpressure. First,wehaveconsideredaFinslerian anstaz of (LTB) and have found a Finslerian exact solution of vacuum field equation. We have obtained the R(t,r) and S(t,r) with considering establish a new solution of Rµν = 0. Moreover, we attempttouseFinslergeo...
متن کاملGeodesic Distance for Right Invariant Sobolev Metrics of Fractional Order on the Diffeomorphism Group
We study Sobolev-type metrics of fractional order on the group of compactly supported diffeomorphisms Diffc(M), where M is a Riemannian manifold of bounded geometry. We prove that the geodesic distance, induced by the Riemannian metric, vanishes if the order s satisfies 0 ≤ s < 1 2 . For M 6= R we show the vanishing of the geodesic distance also for s = 1 2 , and for dim(M) = 1 we show that the...
متن کاملVanishing Geodesic Distance on Spaces of Submanifolds and Diffeomorphisms
The L-metric or Fubini-Study metric on the non-linear Grassmannian of all submanifolds of type M in a Riemannian manifold (N, g) induces geodesic distance 0. We discuss another metric which involves the mean curvature and shows that its geodesic distance is a good topological metric. The vanishing phenomenon for the geodesic distance holds also for all diffeomorphism groups for the L-metric.
متن کاملar X iv : m at h . D G / 0 40 93 03 v 1 1 7 Se p 20 04 VANISHING GEODESIC DISTANCE ON SPACES OF SUBMANIFOLDS AND DIFFEOMORPHISMS
The L-metric or Fubini-Study metric on the non-linear Grassmannian of all submanifolds of type M in a Riemannian manifold (N, g) induces geodesic distance 0. We discuss another metric which involves the mean curvature and shows that its geodesic distance is a good topological metric. The vanishing phenomenon for the geodesic distance holds also for all diffeomorphism groups for the L-metric.
متن کاملTensor Glyph Warping – Visualizing Metric Tensor Fields using Riemannian Exponential Maps
The Riemannian exponential map, and its inverse the Riemannian logarithm map, can be used to visualize metric tensor fields. In this chapter we first derive the well-known metric sphere glyph from the geodesic equations, where the tensor field to be visualized is regarded as the metric of a manifold. These glyphs capture the appearance of the tensors relative to the coordinate system of the hum...
متن کامل