On Transversally Harmonic Maps of Foliated Riemannian Manifolds

Abstract

Abstract. Let (M, F ) and (M ′ , F ′ ) be two foliated Riemannian manifolds with M compact. If the transversal Ricci curvature of F is nonnegative and the transversal sectional curvature of F ′ is nonpositive, then any transversally harmonic map ϕ : (M, F) → (M ′ , F ′ ) is transversally totally geodesic. In addition, if the transversal Ricci curvature is positive at some point, then ϕ is transversally constant.

Liouville type theorem for (F;F')p-harmonic maps on foliations

“…B (E) if and only if i(X)Ψ = 0 and θ(X)Ψ = 0 for all X ∈ ΓL. Then the generalized Weitzenböck type formula (2.9) is extended to Ω * B (E) as follows [10]:…”

“…Transversally harmonic maps on foliated Riemannian manifolds have been studied by many authors [3,12,13,18]. However, a transversally harmonic map is not a critical point of the transversal energy [10]…”

Liouville type theorems for transversally harmonic and biharmonic maps

“…where {E a }(a = 1, • • • , q) is a local orthonormal basic frame on Q. Trivially, the transversal tension field τ b (φ) is a section of φ −1 Q ′ . A foliated map φ : (M, g, F ) → (M ′ , g ′ , F ′ ) is said to be transversally harmonic if the transversal tension field vanishes, i.e., τ b (φ) = 0 [6]. And the transversal energy of φ on a compact domain Ω is defined by…”

“…Let (M, F ) and (M ′ , F ′ ) be foliated Riemannian manifolds and let φ : M → M ′ be a smooth foliated map, i.e., φ is a smooth leaf-preserving map. Then φ is said to be transversally harmonic if the transversal tension field τ b (φ) = tr Q ∇tr d T φ vanishes, where d T φ = dφ| Q and Q is the normal bundle of F (see [6,14,15] for details). When F is minimal, a transversally harmonic map is a critical point of the transversal energy E B (φ) [6], which is given by…”

“…Transversally harmonic and biharmonic maps are generalizations of harmonic and biharmonic maps because transversally harmonic and biharmonic maps are just harmonic and biharmonic maps on the point foliation, respectively. For more information about transversally harmonic and biharmonic maps, see [4,6,9,14,15]. For harmonic maps, the classical Liouville theorem is well-known.…”