‎The least action principle indicates that for the open spacetime manifolds‎, ‎there are data on the boundary‎. ‎Recently‎, ‎it has been proposed that the data for the effective actions at order $\\\\alpha\\\'$ are the values of the massless fields and their first derivatives‎. ‎These data should be respected by the T-duality transformations at order $\\\\alpha\\\'$‎. ‎Moreover‎, ‎the T-duality transformations should not change the unit vector to the boundary which in turns implies that the base space metric should be also invariant‎. ‎Assuming such restricted T-duality transformations‎, ‎we show that the transformation of the circular reduction of the parity-odd part of the effective action of the heterotic string theory at order $\\\\alpha\\\'$ under the Buscher rules is cancelled by some total derivative terms and by some restricted T-duality transformations at order $\\\\alpha\\\'$‎. ‎Using the Stokes\\\' theorem‎, ‎we then show that the boundary terms in the base space corresponding to the total derivative terms are $exactly$ cancelled by transformation of the circular reduction of the Gibbons-Hawking boundary term under the above restricted T-duality transformations‎. ‎These calculations confirm the above proposal for the data on the boundary for the effective actions at order $\\\\alpha\\\'$‎.

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