Abstract: The compact minimal submanifold in a locally symetric and conformally flat Riemann manifold is discussed in this paper.We get the Pinching constant for scalar curvature.The result of Li^[2] is generalized,but the method is completely different. Meanwhile,we get better conclusion than that of[3].We also research the Pinching problem for sectional curvature on compact minimal submanifolds in a unit sphere, partially improving the results of S.T.Yan^[4].

Key words: Locally symetric, conformally flat, minimal submanifold, scalar curvature, sectional curvature