A new variational characterization of 𝑛-dimensional space forms

Source abstract

A Riemannian manifold ( M n , g ) (M^n,g) is associated with a Schouten ( 0 , 2 ) (0,2) -tensor C g C_g which is a naturally defined Codazzi tensor in case ( M n , g ) (M^n,g) is a locally conformally flat Riemannian manifold. In this paper, we study the Riemannian functional F k [ g ] = ∫ M σ k ( C g ) d v o l g \mathcal {F}_k[g]=\int _M\sigma _k(C_g)dvol_g defined on M 1 = { g ∈ M | V o l ( g ) = 1 } \mathcal {M}_1=\{g\in \mathcal {M}|Vol(g)=1\} , where M \mathcal {M} is the space of smooth Riemannian metrics on a compact smooth manifold M M and { σ k ( C g ) , 1 ≤ k ≤ n } \{\sigma _k(C_g),\ 1\leq k\leq n\} is the elementary symmetric functions of the eigenvalues of C g C_g with respect to g g . We prove that if n ≥ 5 n\geq 5 and a conformally flat metric g g is a critical point of F 2 | M 1 \mathcal {F}_2|_{\mathcal {M}_1} with F 2 [ g ] ≥ 0 \mathcal {F}_2[g]\geq 0 , then g g must have constant sectional curvature. This is a generalization of Gursky and Viaclovsky’s very recent theorem that the critical point of F 2 | M 1 \mathcal {F}_2|_{\mathcal {M}_1} with F 2 [ g ] ≥ 0 \mathcal {F}_2[g]\geq 0 characterized the three-dimensional space forms.