A new variational characterization of n-dimensional space forms

作者：Hu ZJ^*; Li HZ

来源：Transactions of the American Mathematical Society, 2004, 356(8): 3005-3023.

DOI：10.1090/S0002-9947-03-03486-X

• locally conformally flat Riemannian manifold
• Schouten tensor
• space form
• Riemannian functional.

A Riemannian manifold (M-n, g) is associated with a Schouten (0, 2)-tensor C-g which is a naturally defined Codazzi tensor in case (M-n, g) is a locally conformally flat Riemannian manifold. In this paper, we study the Riemannian functional F-k[g] = integral(M) sigma(k)(C-g) dvol(g) defined on M-1 = {g is an element ofM\V ol(g) = 1}, where M is the space of smooth Riemannian metrics on a compact smooth manifold M and {sigma(k)(C-g); 1 less than or equal to k less than or equal to n} is the elementary symmetric functions of the eigenvalues of C-g with respect to g. We prove that if n greater than or equal to 5 and a conformally flat metric g is a critical point of F-2\M-1 with F-2[g] greater than or equal to 0, then 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 with F-2[g] greater than or equal to 0 characterized the three-dimensional space forms.

• 出版日期2004
• 单位郑州大学