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Application of the Generalized Bochner Technique to the Study of Conformally Flat Riemannian Manifolds

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Application of the Generalized Bochner Technique to the Study of Conformally Flat Riemannian Manifolds mathematics Article Application of the Generalized Bochner Technique to the Study of Conformally Flat Riemannian Manifolds Josef Mikeš 1,* , Vladimir Rovenski 2 , Sergey Stepanov 3 and Irina Tsyganok 3 1 Department of Algebra and Geometry, Faculty of Science, Palacky University, 771 46 Olomouc, Czech Republic 2 Department of Mathematics, University of Haifa, Mount Carmel, Haifa 3498838, Israel; [email protected] 3 Department of Mathematics, Finance University, 49-55, Leningradsky Prospect, 125468 Moscow, Russia; [email protected] (S.S.); [email protected] (I.T.) * Correspondence: [email protected] Abstract: In this article, we discuss the global aspects of the geometry of locally conformally flat (complete and compact) Riemannian manifolds. In particular, the article reviews and improves some results (e.g., the conditions of compactness and degeneration into spherical or flat space forms) on the geometry “in the large" of locally conformally flat Riemannian manifolds. The results presented here were obtained using the generalized and classical Bochner technique, as well as the Ricci flow. Keywords: Riemannian manifold; locally conformally flat; curvature operator; scalar and Ricci curvatures; Ricci flow MSC: 53C20; 53C21; 53C24 Citation: Mikeš, J.; Rovenski, V.; Stepanov, S.; Tsyganok, I. Application of the Generalized Bochner Technique to the Study of Conformally Flat 1. Introduction and Main Results Riemannian Manifolds. Mathematics One of key analytical methods for differential geometry is the celebrated Bochner 2021, 9, 927. https://doi.org/ technique, which was founded by S. Bochner, K. Yano, A. Lichnerowicz and others in the 10.3390/math9090927 1950s and 1960s to study the relationship between topology and curvature of a compact Riemannian manifold (e.g., [1]; ([2] pp. 333–364)). To achieve this, the authors of the Academic Editor: Ion Mihai method used Hopf’s lemma and Green’s theorem (see also [1]). On the other hand, since the 1970s, complete (noncompact) Riemannian manifolds have been included in the range of research carried out using the Bochner technique. For this, the methods of geometric Received: 7 April 2021 analysis were developed (e.g., [3–5]). As a result, vanishing theorems of the classical Accepted: 20 April 2021 Published: 22 April 2021 Bochner technique took the form of Liouville type theorems, e.g., [5–7]. In this article, we discuss the global aspects of the geometry of locally conformally flat complete and compact Publisher’s Note: MDPI stays neutral Riemannian manifolds using the generalized and classical Bochner technique, as well as with regard to jurisdictional claims in the Ricci flow, e.g., [8,9]. Before that, we summarize and improve some known results published maps and institutional affil- (for example, the conditions of compactness and degeneration into spherical or flat space iations. forms) on the geometry in the large of locally conformally Riemannian manifolds. This article continues the series of our works [10–16] on various methods of the classical and generalized Bochner technique. In this article, (M, g) is a Riemannian manifold, namely, a connected C¥-manifold M of dimension n ≥ 2 equipped with a metric tensor g and the Levi-Civita connection r. Let Copyright: c 2021 by the authors. TM T∗ M M Nr T∗ M Licensee MDPI, Basel, Switzerland. and be the tangent and cotangent bundles of . Denote by the vector r = r( ∗ ) r = r( ∗ ) This article is an open access article bundle of r-times covariant tensors on M, and by S M S T M and L M L T M — distributed under the terms and its sub-bundles of symmetric r-tensors and differential r-forms. Riemannian metrics in the conditions of the Creative Commons fibres of these bundles are induced by the metric g and will be denoted by the same letter 2 Nr ∗ Attribution (CC BY) license (https:// g. We will set kTk = g(T, T) for arbitrary T 2 T M. creativecommons.org/licenses/by/ A locally conformally flat Riemannian manifold (M, g) is determined by the condition ¥ 4.0/). (see [17] (p. 60)) that any point x 2 M has a neighborhood W ⊂ M and a C -function f on Mathematics 2021, 9, 927. https://doi.org/10.3390/math9090927 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 927 2 of 10 2 f W such that the Riemannian manifold (W, e gj W) is flat (has zero sectional curvature). For such manifolds, the Weyl conformal curvature tensor W, see (3) in what follows, vanishes. For n ≥ 4 this condition is necessary and sufficient (see [17] (p. 60)). The Riemannian geometry “in the large” studies relations between the curvature and the topology and between local and global characteristics, of a Riemannian manifold. There are many results “in the large" on locally conformally flat Riemannian manifolds (e.g., [3,6,10,11,18,19]). The purpose of this article is to review and improve some known results in this field. The paper presents seven theorems, which provide sufficient conditions for a Riemannian manifold to admit a flat metric or metric of positive constant sectional curvature, as well as analytic compactness conditions for complete locally conformally flat Riemannian manifolds. Our theorems can be viewed as instructive applications (obtained by logical inference and noncomplicated calculations) of deep recent results on the Ricci flow under the strict assumption that the Weyl curvature tensor vanishes, thus, the article can also demonstrate to young (and experienced) researchers a visual connection between various topics of modern Riemannian geometry. S. Bochner’s theorem (see [1] (pp. 79–80)), according to which a compact locally con- formally flat Riemannian manifold of dimension n ≥ 4 with a positive definite Ricci tensor is a homological sphere—this is the first best-known result “in the large" on such manifolds. Following this, M. Tani proved (see [20]) that a compact locally conformally flat Riemannian manifold of dimension n ≥ 3 with positive constant scalar curvature and positive sectional curva- ture (or positive definite Ricci tensor) is a manifold of positive constant sectional curvature. Notice that a compact manifold of positive constant sectional curvature for even n is either a sphere Sn or a real projective space RPn with canonical metric; and for each odd n ≥ 3, there are infinitely many spherical space forms that were completely classified by group-theoretic methods (see [8] for more details). Our first theorem gives a generalization of the results of S. Bochner and M. Tani. Theorem 1. A compact locally conformally flat Riemannian manifold of dimension n ≥ 4 with positive sectional curvature admits a metric of positive constant sectional curvature and therefore is diffeomorphic to a spherical space form. Following M. Tani [20], the study was continued by S. Goldberg, who proved a number of theorems. According to one of them (see [21]), a compact locally conformally flat manifold of dimensionp n ≥ 3 with constant scalar curvature s and Ricci tensor Ric such that k Ric k < s/ n − 1 has constant sectional curvature. Note that here s > 0, and the inequality has an equivalent form k Ric k2 < s2/(n − 1). An analogue of Goldberg’s theorem for a complete locally conformally flat manifold with scalar curvature and Ricci tensor satisfying similar conditions was proved in [22]. We can prove a similar statement without the condition that the scalar curvature is constant. Theorem 2. A compact locally conformally flat Riemannian manifold of dimension n ≥ 4 with positive scalar curvature s satisfying the condition 8 ( ) s2 n = < 4/15 , 4, k Ric k2 ≤ (83/400) s2, n = 5, : n s2 n ≥ n2−1 , 6, admits a metric of positive constant sectional curvature and therefore is diffeomorphic to a spherical space form. Another S. Goldberg’s theorem (see [21,23]) states that a locally conformally flat manifold of dimension n ≥ 3 with positive definite Ricci tensor Ric and constant scalar curvature s such 2 1 2 that k Ric k ≤ n−1 s , is a manifold of constant sectional curvature. Note that the inequality 2 1 2 k Ric k ≥ n s holds for any Riemannian manifold. We formulate a similar statement without the requirement that the scalar curvature is constant. Mathematics 2021, 9, 927 3 of 10 Theorem 3. Let (M, g) be a compact locally conformally flat Riemannian manifold of dimension n ≥ 4 with s Ric > g. 2(n − 1) Then M admits a metric of positive constant sectional curvature and therefore is diffeomorphic to a spherical space form. In particular, when the dimension of M is equal to three, we have 1 Theorem 4. A three-dimensional compact conformally flat Riemannian manifold with Ric < 2 s g has constant positive sectional curvature. We consider a homogeneous Riemannian manifold (M, g), which is determined by the condition that its isometry group is transitive, i.e., for any points x, y 2 M there exists an isometry that translates x into y (see [17] (p. 178)). Recall that a homogeneous Riemannian manifold is complete (see [17] (p. 181)). S. Goldberg’s theorem in [23] states that a locally conformally flat homogeneous Riemannian manifold with Ricci tensor Ric ≥ 0 is a locally symmetric space. Since a locally symmetric space has a parallel Ricci tensor, S. Goldberg formulates yet another theorem, according to which a locally conformally flat Riemannian manifold has constant sectional curvature if its Ricci tensor is parallel and is positive or negative definite. Moreover, S. Goldberg does not include the completeness or compactness of the manifold in the conditions of the theorems. We supplement these results by the following. Theorem 5. A simply connected irreducible locally conformally flat homogeneous Riemannian manifold of dimension n ≥ 3 with non-negative definite Ricci tensor is the canonical sphere. Articles [6,7] are devoted to the search for analytic compactness conditions for com- plete locally conformally flat Riemannian manifolds.
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