Gao, Zhiyong2009-06-042009-06-041990Chang, Shun-Cheng. "Critical Riemannian metrics." (1990) Diss., Rice University. <a href="https://hdl.handle.net/1911/16328">https://hdl.handle.net/1911/16328</a>.https://hdl.handle.net/1911/16328Let $(M,g)$ be a compact oriented n-dimensional smooth Riemannian manifold. Consider the following quadratic Riemannian functional$$SR(g) = \int\sb{M}\ \vert R\sb{ijkl}(g)\vert \sp{2}d\mu$$which is homogeneous of degree ${n\over2}-2,$ where $R\sb{ijkl}$(g) is the curvature tensors of $(M,g)$ and $d\mu$ is the volume element measured by g. A critical point of $SR(g)$ is called a critical metric on M, that is, the Ricci tensor satisfies the critical equations grad$SR\sb{g}$ = 0. In particular, for a compact 4-manifold M, every Einstein metric is a critical metric for SR on M. In this thesis, we propose an extension of the compactness property for Einstein metrics to critical metrics on a compact smooth Riemannian 4-manifold M. More precisely, first we consider the subspace $G(M)$ of all critical metrics on M with the injectivity radius bounded from below by a constant $i\sb{0} >$ 0 and diameter bounded from above by d. Then we are able to prove that $G(M)$ is compact as a subset of moduli space of critical metrics in the $C\sp{\infty}$-topology (Theorem 6.1). Second, we replaced the injectivity radius lower bound by the local volume bound, then we get a compact 4-dimensional critical orbifold (Theorem 7.1). Furthermore, by using the fundamental equations of Riemannian submersions with totally geodesic fibers, we construct some critical Riemannian 4-manifolds.96 p.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.MathematicsThesisThesis Math. 1990 Chang