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In hep-th/9903210 (curvature)$^2$ terms of the effective D-brane action werederived to lowest order in the string coupling. Their results are correct up toambiguous terms which involve the second fundamental form of the D-brane. Wecompute five point string amplitudes on the disk. We compare the subleadingorder in $\alpha'$ of the string amplitudes with the proposed lagrangian ofhep-th/9903210 supplemented by the ambiguous terms. The comparison determinesthe complete form of the gravitational terms in the effective D-brane action toorder ${\calO}(\alpha^{' 2})$. Our results are valid for arbitrary ambientgeometries and world-volume embeddings.Comment: 34 pages, LaTeX, 4 Postscript figures, expanded introduction and  conclusions, typos corrected, references added, final version to appear in  JHEP09(2001)00

On α′2 corrections to the D-brane action for non-geodesic world-volume embeddings

On α′² corrections to the D-brane action for non-geodesic world-volume embeddings