Local Rigidity Theorem In Positive Constant Curvature Space

The positive mass theorem is one of the most important result in both generalrelativity and differential geometry, and with the related research, it has been grown tobe a very attractive area in mathematics.Before the proof accomplished by R.Scheon and S.T.Yau, A.Fisher and J.Marsdenhad proved the local case of Positive mass theorem in 1975. That is, when deforminga ?at metric along the directions of increasing positive scalar curvature on a compactsubset of a ?at manifold, what we would obtain is still the ?at metric. In another way,the ?at metric is local rigid with positive scalar curvature. While in 1996, M.Min-Ooproved that if we assumed the Manifold is spin, then the constant positive curvaturespace is also local rigid. i.e. if deforming the metric along the direction, the manifoldis isometric to the hemisphere.In this paper, first with the aid of decomposition of elliptic operators, we will provethat the scalar curvature is weak negative definite around the constant negative curvaturemetric, and then using Morse's lemma, we have proved local rigidity theorem for ahemisphere. Together with the result of Fisher and Marsden, we obtain the local rigidityfor the nonnegative constant curvature manifold, when deforming in the directions ofincreasing the scalar curvature.Compared with the proof of M.Min-Oo, here we do not presumed that the mani-fold is spin, thus it is more general. The result here can be viewed as a weak form ofpositive mass theorem on the positive curvature manifold, so it will be helpful for us tounderstand the positive mass theorem better. Meanwhile, it will be helpful in furtherunderstanding of some basic properties of Einstein manifolds, also some applicationin general relativity.