Complete noncompact manifolds with harmonic curvature

Abstract

Let (Mⁿ, g) be an n-dimensional complete noncompact Riemannian manifold with harmonic curvature and positive Sobolev constant. In this paper, by employing an elliptic estimation method, we show that (Mⁿ, g) is a space form if it has sufficiently small L^n/2-norms of trace-free curvature tensor and nonnegative scalar curvature. Moreover, we get a gap theorem for (Mⁿ, g) with positive scalar curvature.

Keywords

Harmonic curvature / trace-free curvature tensor / space form

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