By Yorozu S.
We provide a generalization of the end result acquired by way of C. Currais-Bosch. Weconsider the -operator linked to a transverse Killing box v on acomplete foliated Riemannian manifold . lower than a definite assumption,we turn out that, for every belongs to the Lie algebra of the linearholonomy crew . a distinct case of our consequence, the model of the foliationby issues, implies the implications given via B. Kostant (compact case) andC. Currfis-Bosch (non-compact case).
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Additional info for A-operator on complete foliated Riemannian manifolds
Indeed, the harmonicity of f is equivalent to α Fkk = 0. k Note that α Fkk εα . 33) and we use the structure equations to get β α α α j α j dFkt − Fjt ϕk − Fkj ϕt + Fkt ωβ ∧ ϕt β α α α j α j + dFkt − Fjt ϕk − Fkj ϕt + Fkt ωβ ∧ ϕ t 1 1 i = − Fiα Hklt ϕl ∧ ϕt + Fkβ 2 2 N γ δ Riemα βγδ ω ∧ ω . We deﬁne β α α l α l α α j α j Fktl ϕ + Fktl ϕ = dFkt − Fjk ϕk − Fkj ϕt + Fkt ωβ , j j β l α l α α α α Fktl ϕ + Fktl ϕ = dFkt − Fjt ϕk − Fkj ϕt + Fkt ωβα . 30) and the above equations we deduce the commutation relations α α = Fklt − Fkβ Ftγ Flδ Fktl N Riemα βγδ , α α Fktl = Fklt − Fkβ Ftγ Flδ N Riemα βγδ , 1 1 α α i Fktl = Fklt − Fiα Hklt − Fkβ Ftγ Flδ − Ftδ Flγ 2 2 N Riemα βδγ .
As for part (ii) , let G (t) be non-decreasing. Fix x0 ∈ M , set rx0 = dist (x0 , x) , and observe that, by the triangle inequality, Ric (x) ≥ − (m − 1) G (r (x)) ≥ − (m − 1) Gx0 (rx0 (x)) . 14 are met with respect to the origin x0 . 44 Chapter 2. 39) can be explicitly estimated. 14, these estimates can now be applied to get upper volume bounds. 13 above). 17. Let (M, , ) be a complete, m-dimensional manifold satisfying Ric (x) ≥ − (m − 1) G (r (x)) on M for some non-negative function G ∈ C 0 ([0, +∞)) , where r (x) = dist (x, o) is the distance from a ﬁxed reference origin o ∈ M .
Next, we compute the Laplacian of u. We observe that du = uk ϕk + uk ϕk with, according to the previous formulas, α α B i Bik . 5. Weitzenb¨ ock-type formulas 25 Hence, with the aid of the above calculations, ukt ϕt + ukt ϕt = duk − ut ϕtk with α α α α ukt = Bik B it + B i Bikt . By the deﬁnition of the Laplacian on the K¨ ahler manifold M , we have ukk ∆u = 4 k α α α α Bik B ik + B i Bikk =4 α,i,k α α α,i,k α j B i Bjα Hikk −4 α Bik B ik + 4 =4 α,i,k δ α B i Biβ Btγ B t Kβγδ . 27. Let (M, , , JM ), (N, (, ) , JN ) be K¨ ahler manifolds and let f : M → N be a holomorphic map.
A-operator on complete foliated Riemannian manifolds by Yorozu S.