| This paper mainly studies the geometric and topological effects of scalar curvature on Riemannian manifold.It includes the extremal/rigidity metric problem about the scalar curvature,the geometric and topological embodiment of scalar curvature in general relativity and the scalar curvature of a two dimensional manifold after Tl expansion.The results are as follows(1)In 2019,Gromov proposed whether the bi-invariant metric on the compact Lie group is extremal or not.We show that bi-invariant metric are rigid in the class of left invariant metric in compact semi-simple Lie groups.Some positive answers to Gromov’s conjectures are given,and some topological conditions are excluded.If the metric is rigid,then Lie groups do not contain tori as a subgroup.This differs from the results of M.Laurll and S.Goette and U.Semmelmann.(2)We consider the Riemannian manifold(Mn,g)whose asymptotic end is Rk×Xn-k,where(X,gX)is required to be scalar curvature flat.By studying the compactness problem on(Mn,g),the positive mass theorem is proved under the condition of a general connected sum has no positive scalar curvature metric.It is proved that when the mass is zero,M is isomorphic to(Rk×Xn-k,gRk+gX).That is,for the case where the asymptotic end is Rk×Xn-k,under the condition of compactification,we make the positive mass theorem to be the case where X is scalar curvature flat and prove the rigid part.(3)Study the T1 extension of a two dimensional Riemannian manifold S2,if the T1 extension of S2 is(T1×S2,e-2ηdt+gS2),where η is a smooth function on S2,and its scalar curvature has a positive lower bounded,then we give the estimate of the first eigenvalue of the Δη=Δ-▽η.This is different from the condition that Bakry Emery Ricci tensor has a positive lower bound. |
