Critical Metrics Of Some Riemannian Functionals

This paper includes two chapters.In chapter one, we study critical metrics of the Riemannian functional (?) on a compact,connected orientable smooth n-manifold M~n, n≥3, denned by the L~2-norm of the trace-free Ricci tensor, normalized by an appropriate power of the volume of M~n with respectto Riemannian metric g, i.e.,where E_g=Ric_g-(?)g denotes the trace-free Ricci tensor, Ric_g and R_g denote the Riccitensor and the scalar curvature of the Riemannian metric g. It is shown that critical metricsof this functional have remarkable properties, and characterizations of locally conformallyflat critical metrics are also obtained.In chapter two, we study a class of special Riemannian functionals F, defined bywhere a,b,c are constants with respect to the dimension n of the Riemannian manifoldand not equal to zero at the same time, Riem_g denotes the Riemannian curvature tensorof the Riemannian metric g. We compute the Euler-Lagrange equations of F, and findthat the locally conformally flat critical metric of F is also an Einstein metric under someconditions.