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Using canonical method we investigate $Dp-$brane world-volumenoncommutativity in weakly curved background. The term weakly curved means thatin the leading order, the source of non-flatness is infinitesimally smallKalb-Ramond field $B_{\mu\nu}$ linear in coordinate, while the Ricci tensordoes not contribute being the infinitesimal of the second order. On thesolution of boundary conditions, we find simple expression for the space-timecoordinates in terms of the effective coordinates and momenta. This basicrelation helped us to prove that noncommutativity appears only on theworld-sheet boundary. The noncommutativity parameter has a standard form butwith infinitesimally small and coordinate dependent antisymmetric tensor$B_{\mu\nu}$. This result coincides with that obtained on the group manifoldsin the limit of the large level $n$ of current algebra. After quantization thealgebra of functions on Dp-brane world-volume is represented with theKontsevich star product instead of the Moyal one in the flat background.

Noncommutativity in weakly curved background by canonical methods