a good approximation and the dominant components F ( r ) of the Dirac wave functions for the spin doublets are nearly identical, as the centrifugal barrier is much stronger than the spin-orbit potential. However, it is found that the Fock terms are dominant in the spin-orbit potential, which induce the state dependence of the effective potential and play essential roles in spin-orbit splitting. Classifying the contributions to the spin-orbit splitting into the Hartree and Fock parts, it is found that the Hartree part has an energy dependence similar to those in RMF [15], while the Fock part has an opposite tendency and cancels with the Hartree part, thus leading to good spin symmetry. The competition between the Hartree and Fock terms determines the subtle spin-orbit splitting, which explains the fluctuation of the spin-orbit splittings in DDRHF instead of monotonously decreasing with E$_{av}$ as in the RMF case. This work is partly supported by State 973 Program 2007CB815000, the NSF of China under Grant Nos. 10435010, 10775004 and 10221003, and the CNRS(France) - NSFC(China) PICS program no. 3473. One of the authors (H.L.) is grateful to the French Embassy in Beijing for the financial support for his stay in France. Acknowledgments [1] B. D. Serot, J. D. Walecka, Adv. Nucl. Phys. 16 , 1 (1986). [2] P. Ring, Prog. Part. Nucl. Phys. 37 , 193 (1996). [3] D. Vretenar, A. V. Afanasjev, G. A. Lalazissis, P. Ring, Phys. Rep. 409 , 101 (2005). [4] J. Meng, H. Toki, S. G. Zhou, S. Q. Zhang, W. H. Long, L. S. Geng, Prog. Part. Nucl. Phys. 57 , 470 (2006). [5] A. Arima, M. Harvey, K. Shimizu, Phys. Lett. B 30 , 517 (1969). [6] K. Hecht, A. Adler, Nucl. Phys. A 137 , 129 (1969). [7] A. Bohr, I. Hamamoto, B. R. Mottelson, Phys. Scripta 26 , 273 (1982). [8] C. Bahri, J. P. Draayer, and S. A. Moszkowski, Phys. Rev. Lett. 68 , 2133 (1992). 12