Some Rigidity Results For Critical Metrics Of Quadratic Curvature Functionals On Riemannian Manifolds

In this paper,we study rigidity results on critical metrics for quadratic curvature functionals.It is well-known that Einstein metrics are critical points for the Einstein-Hilbert functional on the space of unit volume metrics M1(Mn),where R denotes the scalar curvature.Thus,many authors are natural to study rigidity results on critical metrics of the Euler-Lagrange equations for more general curvature functionals.Catino considered the following family of quadratic curvature functionals which are also defined on M1(Mn),and proved some related rigidity results,where Ric denotes the Ricci curvature.Furthermore,from the Euler-Lagrange equation(2-16)of Ft functionals:it has been observed that every Einstein metric is a critical point of for all t ? R.In the third chapter,we study some rigidity results for Einstein metrics as the critical points of a family of known quadratic curvature functionals on compact manifolds,under the conditions of Cotton tensors is zero.Using some estimates with respect to the Weyl curvature tensor and divergence theorems,we obtain that a critcal metric must be Einstein or constant sectional curvature.In the fourth chapter,we are interested in studying the functional on complete manifold and proving some related rigidity results,where t,s are real con-stants(when s?0,Ft,0 is equivalent to Ft),and Rm denote the Riemannian curvature tensor.Moreover,we also provide rigidity results by the integral inequalities involving the Weyl curvature and the Sobolev constant,accordingly.