Journal
JOURNAL OF GEOMETRY
Volume 113, Issue 1, Pages -Publisher
SPRINGER BASEL AG
DOI: 10.1007/s00022-021-00617-z
Keywords
Transversal tension field; transversally harmonic map; Liouville type theorem
Categories
Funding
- Jeju National University
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This paper investigates the relationship between complete foliated Riemannian manifolds and foliated Riemannian manifolds with non-positive transversal sectional curvature. The main result states that under certain conditions on the transversal Ricci curvature, every transversally harmonic map of finite transversal energy is transversally constant.
Let (M, F) be a complete foliated Riemannian manifold and all leaves be compact. Let (M', F') be a foliated Riemannian manifold of non-positive transversal sectional curvature. Assume that the transversal Ricci curvature Ric(Q) of M satisfies Ric(Q) >= -lambda(0) at all point x is an element of M and Ric(Q) > -lambda(0) at some point x(0), where lambda(0) is the infimum of the spectrum of the basic Laplacian acting on L-2-basic functions on M. Then every transversally harmonic map phi: M -> M' of finite transversal energy is transversally constant.
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