Regularity Theorems in Riemannian Geometry. II. Harmonic Curvature and the Weyl Tensor

TLDR: DeTurck and Goldschmidt as discussed by the authors showed that a metric with a Ricci curvature is real-analytic in harmonic coordinates if its Weyl tensor vanishes at a point at which the Ricci curve has only simple eigenvalues.

Abstract: We study the regularity of metrics satisfying geometric conditions imposed on the Ricci or Weyl curvatures. In particular, we show that a metric with harmonic curvature is realanalytic in harmonic coordinates and that such a metric on a manifold of dimension > 4 is locally conformally flat if its Weyl tensor vanishes at a point at which the Ricci curvature has only simple eigenvalues. 1980 Mathematics Subject Classification (1985 Revision): 58G30, 53B20, 53C25. Introduction In the study of the differentiability properties of tensor fields, it is important to take into account the profound effect which the choice of coordinate System has on the regularity of tensors expressed in the System. In [8] it is shown that the optimal regularity of Riemannian metrics is attained in harmonic coordinates: if a metric is of Holder class C' in some coordinate System, then it is at least of class C*' in harmonic coordinates. This paper is a sequel to [8]; here, we are mainly concerned with the regularity of metrics possessing specific properties and the consequences of this regularity. We are also interested in obtaining optimal regularity within a given conformal class of metrics; it turns out that it is achieved by the metric of constant scalar curvature, belonging to this class, expressed in harmonic coordinates (see Proposition 2). Most of our regularity results are obtained by showing that certain overdetermined Systems of partial differential equations are elliptic. 1 Supported by the Sloan Foundation, NSF Grant MGS 85-03302 and NATO Subvention 0153/87. 2 Supported in part by NSF Grant DMS 87-04209. 378 D. DeTurck, H. Goldschmidt We consider metrics satisfying various geometric conditions involving the Ricci curvature or the Weyl tensor. The covariant derivative of the Ricci curvature of a Riemannian manifold (X, g) of dimension n is a tensor of rank three with a certain symmetry. As such, it decomposes according to the action of GL(n, R) on the tangent spaces of the manifold into two components. Theorem l asserts that the metric g is real-analytic in harmonic coordinates whenever one of these two components vanishes. The hypothesis of this theorem holds if either the covariant derivative of the Ricci curvature is totally Symmetrie, in which case we say that g has harmonic curvature, or if the Symmetrie part of the covariant derivative vanishes. Note that in both cases, g has constant scalar curvature. Moreover, Theorem l gives us the realanalyticity of Einstein metrics proved in [8]. We examine how the regularity of the Weyl tensor implies that of the metric in Theorems 2, 4 and Proposition 4. Conformally flat metrics with constant scalar curvature are real-analytic in harmonic coordinates. The final section is devoted to the study of metrics with harmonic curvature and to the consequences of the analyticity of such metrics. In particular, we prove that a metric with harmonic curvature on a manifold of dimension > 4 is locally conformally flat if its Weyl tensor vanishes at a point at which the Ricci curvature has only simple eigenvalues (Theorem 5). We also extend results of Derdzinski and Shen [7] and derive global topological conditions on a Riemannian manifold with harmonic curvature, which depend on the number of distinct simple eigenvalues of the Ricci curvature (Theorem 6). 1. Preliminary results Let Xbz a C^-manifold of dimension n > 3. We denote by Γ the tangent b ndle of X, and by Γ* its cotangent b ndle. By ST*9 /\\T* and (X) Γ* we shall mean the &-th Symmetrie power of Γ*, the /-th exterior power of Γ* and the tensor product of m copies of Γ*, respectively. The Symmetrie product ^ 2, with ^ 2£ T*, is equal to i ® β2 + 02 ® β u the £-th Symmetrie power of β e Γ* will be denoted by . A tensor or an object on X will be supposed to be of class C°° unless i t is explicitly stated that it is of class C or C', with 0 < α < 1. We say that an object is of class € if it is real-analytic. Let E and E be vector bundles over X. We denote by Jk(E) the b ndle of fc-jets of sections of E and by jk(s) the fc-jet of a section s of E of class C. For / > 0, let nk: Jk+l(E) -> Jk(E) and π: Jk(E) -+ Xbe the natural projections. We now consider a quasi-linear differential operator from sections of E to sections of E'. In other words, let F be an open fibered submanifold of Jk(E) and φ: F-* E' be a quasi-linear morphism of fibered manifolds over Xin the sense of [5], Chapter IX, Section 2. The differential operator corresponding to φ sends a section s of E of class C, for which jk(s) is a section ofF, into the section φ Ok (s)) of E'. The symbol of φ is a morphism of vector bundles Regularity Theorems in Riemannian Geometry. II 379 over %_! F. If χ Ε Χ, β e T* and p E πλ_! F, with n (p) = x, we consider the linear mapping