On transversally harmonic maps of foliated Riemannian manifolds
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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.
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.
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