Abstract: The invariances under the parameter transformation in the parameter space and under the mapping to a sufficient statistic in the sample space for the basic geometrical quantities, especially for the curvatures, in the statistical manifold are discussed in the present paper. It is shown that if the geometrical structures of a family of the complete distributions are invariant under any mapping to a sufficient statistio in the sample space, then the Fisher information is the only metric and the a-connection defined by Amari is the only one-parameter affine connections in the statistical manifold with a ignorable constant factor The invariances studied are connected with the sufficiency, so they might be more preferable in the statistical problems.