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Conformally equivariant quantization : existence and uniqueness

Christian Duval, Pierre Lecomte, Valentin Ovsienko (1999)

Annales de l'institut Fourier

We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudo-riemannian manifold ( M , g ) . In other words, we establish a canonical isomorphism between the spaces of polynomials on T * M and of differential operators on tensor densities over M , both viewed as modules over the Lie algebra o ( p + 1 , q + 1 ) where p + q = dim ( M ) . This quantization exists for generic values of the weights of the tensor densities and we compute the critical values of the weights yielding...

Conformally geodesic mappings satisfying a certain initial condition

Hana Chudá, Josef Mikeš (2011)

Archivum Mathematicum

In this paper we study conformally geodesic mappings between pseudo-Riemannian manifolds ( M , g ) and ( M ¯ , g ¯ ) , i.e. mappings f : M M ¯ satisfying f = f 1 f 2 f 3 , where f 1 , f 3 are conformal mappings and f 2 is a geodesic mapping. Suppose that the initial condition f * g ¯ = k g is satisfied at a point x 0 M and that at this point the conformal Weyl tensor does not vanish. We prove that then f is necessarily conformal.

Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane

Yuri L. Sachkov (2010)

ESAIM: Control, Optimisation and Calculus of Variations

The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is studied. Local and global optimality of extremal trajectories is characterized. Lower and upper bounds on the first conjugate time are proved. The cut time is shown to be equal to the first Maxwell time corresponding to the group of discrete symmetries of the exponential mapping. Optimal synthesis on an open dense subset of the state space is described.

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