Abstract
This is a survey on our recent works on bi-harmonic maps on CR-manifolds and foliated Riemannian manifolds, and also a research paper on bi-harmonic maps principal G-bundles. We will show, (1) for a complete strictly pseudoconvex CR manifold , every pseudo bi-harmonic isometric immersion into a Riemannian manifold of non-positive curvature, with finite energy and finite bienergy, must be pseudo harmonic; (2) for a smooth foliated map of a complete, possibly non-compact, foliated Riemannian manifold into another foliated Riemannian manifold, of which transversal sectional curvature is non-positive, we will show that if it is transversally bi-harmonic map with the finite energy and finite bienergy, then it is transversally harmonic; (3) we will claim that the similar result holds for principal G-bundle over a Riemannian manifold of negative Ricci curvature.
Highlights
The theory of harmonic maps has been extensively developed and applied in many problems in topology and differential geometry
This is a survey on our recent works on bi-harmonic maps on CR-manifolds and foliated Riemannian manifolds, and a research paper on bi-harmonic maps principal G-bundles
We will show, (1) for a complete strictly pseudoconvex CR manifold ( M, gθ ), every pseudo bi-harmonic isometric immersion φ :(M, gθ ) → ( N, h) into a Riemannian manifold of non-positive curvature, with finite energy and finite bienergy, must be pseudo harmonic; (2) for a smooth foliated map of a complete, possibly non-compact, foliated Riemannian manifold into another foliated Riemannian manifold, of which transversal sectional curvature is non-positive, we will show that if it is transversally bi-harmonic map with the finite energy and finite bienergy, it is transversally harmonic; (3) we will claim that the similar result holds for principal G-bundle over a Riemannian manifold of negative Ricci curvature
Summary
The theory of harmonic maps has been extensively developed and applied in many problems in topology and differential geometry (cf. [1] [2] [3], etc.). Y. Chen’s conjecture: Every biharmonic isometric immersion of a Riemannian manifold (M , g ) into a Riemannian manifold ( N, h) of non-positive curvature must be harmonic (minimal). The CR analogue of the generalized Chen’s conjecture: Let ( M , gθ ) be a complete strictly pseudoconvex CR manifold, and ( N, h) , a Riemannian manifold of non-positive curvature. (cf Theorem 2.1) Let φ be a pseudo biharmonic map of a strictly pseudoconvex complete CR manifold (M , gθ ) into another Riemannian manifold ( N, h) of non positive curvature. Riemannian manifolds were introduced by Chiang and Wolak (cf [21]) and see [22] [23] [24] [25] [26] They are generalizations of transversally harmonic maps introduced by Konderak and Wolak (cf [27] [28]). If π :( P, g ) → (M , h) is biharmonic, it is harmonic
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