Transversal Conformal fields and Killing forms on foliations

Abstract

1 Introduction Let (M,F) be a smooth manifold with a Riemannian foliation is a foliation F on a smooth manifold M such that the normal bundle Q = TM/L may be endowed with a metric gQ whose Lie derivative is zero along leaf directions ([23]). Note that we can choose a Riemannian metric gM on M such that gM∣TF = gQ; such a metric is called bundle − like. Denote by (M, gM,F). Recently, S. D. Jung and K. Richardson ([11]) proved the generalized Obata theorem which states that: (M,F) is transversally isometric to (Sq(1/c),G), where G is the discete subgroup of O(q) acting by isometries on the last q coordinates of the sphere Sq(1/c) of radius 1/c if and only if there exists a non-constant basic function f such that ∇X∇f = −c2fX for all foliated normal vectors X, where c is a positive real number and ∇ is the transverse Levi-Civita connection on the normal bundle Q.(See below) Let R∇, ρ∇ and σ∇ be the transversal curvature tensor, transverse Ricci operator and transversal scalar curvature with respect to the transversal Levi-Civita connection ∇ on Q ([23]). Let κB be the basic part of the mean curvature form of the foliation F and κ♯ B its dual vector field (see Section 2). Then we have the following well-known theorem. Theorem A. ([11]) Let (M, gM,F) be a closed, connected Riemannian manifold with a Riemannian foliation F of a nonzero constant transversal scalar curvature σ∇. If M admits a transversal nonisometric conformal field Y satisfying one of the following conditions: (1) Y = ∇h for any basic function h, or (2) θ(Y )ρ∇ = μgQ for some basic function μ, or (3) ρ∇(∇fY ) = σ∇ q ∇fY , gQ(κ♯ B,∇fY ) = 0 and gQ(Aκ♯ B ∇fY ,∇fY ) ≤ 0, when fY = 1 q div∇Y , then (M,F) is transversally isometric to (Sq(1/c),G). Now, we recall two tensor fields E∇ and Z∇ ([5], [10]) by E∇(Y ) = ρ∇(Y ) − σ∇ q Y, Y ∈ TF , (1.1) Z∇(X, Y ) = R∇(X, Y ) − R∇ σ (X, Y ), (1.2) where R∇ σ (X, Y )s = σ∇ q(q−1){gQ(π(Y ), s)π(X) − gQ(π(X), s)π(Y )} for any vector field, X, Y ∈ TM and s ∈ ΓQ. Trivially, if E∇ = 0 (resp. Z∇ = 0), then the foliation is transversally Einsteinian (resp. transversally constant sectional curvature). The tensor Z∇ is called as the transversal concircular curvature tensor, which is a generalization of the concircular curvature tensor on a Riemannian manifold. In an ordinary manifold, the concircular curvature tensor is invariant under a concircular transformation which is a conformal transformation preserving geodesic circles ([25]). Then we have the wellknown theorem. Theorem B. ([5]) Let (M, gM,F) be a closed, connected Riemannian manifold with a Riemannian foliation F of a nonzero constant transversal scalar curvature σ∇. If M admits a transversal nonisometric conformal field Y such that ∫ M gQ(E∇(∇fY ),∇fY ) ≥ 0, then (M,F) is transversally isometric to (Sq(1/c),G). Theorem C. ([7, 10]) Let (M, gM,F) be a closed, connected Riemannian manifold with a Riemannian foliation F of a nonzero constant transversal scalar curvature σ∇, and suppose that F is minimal. If M admits a transversal nonisometric conformal field Y such that (1)θ(Y )∣E∇∣2 = 0, ([7]) (2)θ(Y )∣Z∇∣2 = 0. ([10]) then (M,F) is transversally isometric to (Sq(1/c),G). Namely, we extend Theorem C as follows: There are many results about the Riemannian foliations admitting a transversal nonisometric conformal field ([5], [7], [10], [11], [21]). Main Theorem 1. Let (M, gM,F) be a closed, connected Riemannian manifold with a Riemannian foliation F of a nonzero constant transversal scalar curvature σ∇, and suppose that F is minimal. If M admits a transversal nonisometric conformal field Y such that θ(Y )∣E∇∣2 = const. θ(Y )∣Z∇∣2 = const. then (M,F) is transversally isometric to (Sq(1/c),G). Also, we study a generalization of Theorem A (2) and (3) when F is minimal. Main Theorem 2. Let (M, gM,F) be a closed, connected Riemannian manifold with a Riemannian foliation F of a nonzero constant transversal scalar curvature σ∇, and suppose that F is minimal. If M admits a transversal nonisometric conformal field Y such that θ(Y )gQ(θ(Y )E∇,E∇) ≤ 0, then (M,F) is transversally isometric to (Sq(1/c),G). Remark. See also ([26]) for the ordinary manifold. Main Theorem 3. Let (M, gM,F) be a closed, connected Riemannian manifold with a Riemannian foliation F of a nonzero constant transversal scalar curvature σ∇. If M admits a transversal conformal field ¯ Y such that Y = K + ∇h, where K is a transversal Killing field and h is a basic function, then (M,F) is transversally isometric to (Sq(1/c),G). Remark. Main Theorem 3 is a generalization of Theorem A (1). On the other hand, a transverse Killing fields and conformal fields are very important objects for studying mathematical and physical problems on foliated manifolds. As their generalizations, transverse Killing forms and conformal Killing forms were studied by many authors ([7], [24]). In 2012, S. D. Jung and K. Richardson ([13]) studied the parallelness of transverse Killing and conformal Killing forms on a compact manifold. Namely, we have the following. Theorem D. ([13]) Let F be a Riemannian foliation on a compact Riemannian manifold M. If the transversal curvature endomorphism is nonpositive, then any transverse conformal Killing r-form (1 ≤ r ≤ q − 1) is parallel, where q = codim F. When (F, J) is a transverse K¨ahler foliation, the parallelness of such forms was studied in ([6, 8]), as follows. Theorem E. Let (F, J) be a transverse K¨ahler foliation on a closed, connected Riemannian manifold. Then (1) if the mean curvature vector is transversally holomorphic, then any transverse Killing r-form (r ≥ 2) is parallel ([6]); (2) if the foliation is minimal, then for any transverse conformal Killing r-form ϕ (2 ≤ r ≤ q − 2), Jϕ is parallel ([8]); On a complete Riemannian foliation, the parallelness of L2-transverse forms was studied in ([4]) and ([12]). Namely, Theorem F. Let F be a Riemannian foliation on a complete foliated Riemannian manifold M. Assume that all leaves are compact and the mean curvature is bounded. If the transversal curvature endomorphism is nonpositive, then (1) L2-transverse Killing forms are parallel ([4]); (2) L2-transverse conformal Killing forms are parallel ([12]); The parallelness of L2-transverse conformal Killing forms on a transverse K¨ahler foliation was studied by S. D. Jung and H. Liu ([12]). That is, Theorem G. ([12]) Let (F, J) be a minimal transverse K¨ahler foliation on a complete Riemannian manifold with all leaves be compact. Then for any L2-transverse conformal Killing r-form ϕ (2 ≤ r ≤ q − 2), Jϕ is parallel. Remark. Note that any transverse Killing form is a transverse conformal Killing form. Hence from Theorem G, for any L2-transverse Killing ϕ, Jϕ is also parallel. But generally, the parallelness of Jϕ does not impty the parallelness of ϕ. Hence we study the parallelness of L2-transverse Killing forms on a transverse K¨ahler foliation. In Section 4, we study the parallelness and vanishing theorem of L2-transverse Killing forms on a transverse K¨ahler foliation. That is, Main Theorem 4. Let (F, J) be a transverse K¨ahler foliation on a complete Riemannian manifold such that all leaves be compact. Assume that the mean curvature vector field is transversally holomorphic, coclosed and bounded. Then L2-transverse Killing rforms (r ≥ 2) are parallel. In addition, if vol(M) is infinite, then L2-transverse Killing r-forms (r ≥ 2) are trivial.