The Lie derivative of currents on Lie groups

Abstract

The aim of this work is to study the properties of the Lie derivative of currents and generalized forms on Riemann manifolds. For an application, we give some results of the Lie derivative of currents and generalized forms on Lie groups.

Reynolds transport theorem for smooth deformations of currents on manifolds

“…In addition, κ t♯ T is the image of the current in space so that for the Eulerian version of the transport theorem, or for the case where the body is identified with its current image, the transport operator is simply the dual of the Lie derivative. (See an analogous result in [20]. )…”

“…(See an analogous result in [20]. ) We finally remark that since the support of k tY T is contained in Image k t = k t {B}, the extension from v t tov t has no computational significance and the 'hat' (b) may be omitted in practice.…”

“…It is also observed that if one expresses the time derivative of k ] t v in terms of the Lie derivative as in (20), it follows that ∂ ∂t…”

“…, u rÀ1 ),ð21Þwhere in the third line we used T k t s V t =v t s k t which follows from(7). Hence,k ] t (v t y u) = V t y k ] t u:ð22ÞIt is possible now to rewrite equation(20) in terms of V t as ∂ ∂tt = t k ] t (v t ) = k ] t ( _ v t ) + d(V t y k ] t v t ) + V t y d(k ] t v t ),ð23Þwhere we used the fact that exterior derivative commutes with pullback of forms so thatk ] t (d(v t y v t )) = dk ] t (v t y v t ) = d(V t y k ] t v t ):ð24ÞConsidering the interval ½a, b & I, equation(20) implies that Falach and Segev…”

The Lie derivative of normal connections

“…In 2010, Sultanov used the Lie derivative of the linear connection to study the curvature tensor and the sorsion tensor on linear algebras [8]. In 2012, by invoking the Lie derivative of forms on the Riemannian n−dimensional manifold, the authors of [1] constructed the Lie derivative of the currents on Riemannian manifolds and given some applications on Lie groups. Recently, B. C. Van and T. T. K. Ha studied some properties of the Lie derivative of the linear connection ∇, the conjugate derivative d ∇ with the linear connection and using them for searching the curvature, the torsion of a space R n along the linear flat connection ∇ [9].…”