Harmonic-Killing vector fields

Infinitesimal Deformations of Harmonic Maps and Morphisms

“…Here h c denotes the complete lift metric associated to h (see [26]). When ϕ : (M, g) → (M, g) is the identity map, any vector field v can be interpreted as a section v : (M, g) → (T M, g c ) of the tangent bundle and we deduce [15,48] that v is a Jacobi field if and only if it is a harmonic section (for information on harmonic sections, see, for example, Wood [65]).…”

“…This formula was used by Yano and Nagano [70] who studied Jacobi fields along the identity map under the name geodesic vector fields. Jacobi fields along the identity map have also been studied by Dodson et al [15], under the name 1-harmonic-Killing fields, and by Stepanov and Shandra [56], under the name infinitesimal harmonic transformations. b When M is a complex manifold, see [16,56].…”

“…2.7 and 2.8. For more results on Jacobi fields and related types of vector fields, see [15,16,47,56,70]; for composition properties, see Sec. 3.3 and [6].…”

“…Harmonic-Killing fields. Recall that Dodson et al [15] call a Jacobi field v along the identity a 1-harmonic-Killing field. They call v harmonic-Killing if it integrates to give a local 1-parameter group of harmonic diffeomorphisms.…”

“…b There is an error in [56, Definition 2.1]; "harmonic" should say "1-harmonic" as in[15, Sec. 3], see Sec.…”