Comptes Rendus
Analyse harmonique, Géométrie et Topologie
Harmonic vector fields and the Hodge Laplacian operator on Finsler geometry
Behroz Bidabad1, 2  ; Mir Ahmad Mirshafeazadeh3
1 Department of Mathematics and Computer Sciences, Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Ave. 15914 Tehran, Iran
2 Institut de Mathematique de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France
3 Department of Mathematics, Shabestar Branch, Islamic Azad University, Shabestar, Iran
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1193-1204.

We first present the natural definitions of the horizontal differential, the divergence (as an adjoint operator) and a p-harmonic form on a Finsler manifold. Next, we prove a Hodge-type theorem for a Finsler manifold in the sense that a horizontal p-form is harmonic if and only if the horizontal Laplacian vanishes. This viewpoint provides a new appropriate natural definition of harmonic vector fields in Finsler geometry. This approach leads to a Bochner–Yano type classification theorem based on the harmonic Ricci scalar. Finally, we show that a closed orientable Finsler manifold with a positive harmonic Ricci scalar has zero Betti number.

Nous présentons d’abord les définitions naturelles de la différentielle horizontale, de la divergence (comme opérateur adjoint) et d’une forme p-harmonique sur une variété finslérienne. Ensuite, nous prouvons un théorème de type Hodge pour une variété finslérienne dans le sens où une p-forme horizontale est harmonique si et seulement si le Laplacien horizontal est nul. Ce point de vue fournit une nouvelle définition naturelle appropriée des champs de vecteurs harmoniques en géométrie finslérienne. Cette méthode conduit à un théorème de classification de type Bochner–Yano basé sur le scalaire de Ricci harmonique. Enfin, nous montrons qu’une variété finslérienne fermée et orientable, avec un scalaire de Ricci harmonique positif, a un nombre de Betti nul.

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DOI : 10.5802/crmath.287
Classification : 58B20

Behroz Bidabad 1, 2 ; Mir Ahmad Mirshafeazadeh 3

1 Department of Mathematics and Computer Sciences, Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Ave. 15914 Tehran, Iran
2 Institut de Mathematique de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France
3 Department of Mathematics, Shabestar Branch, Islamic Azad University, Shabestar, Iran
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Harmonic vector fields and the {Hodge} {Laplacian} operator on {Finsler} geometry},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1193--1204},
     publisher = {Acad\'emie des sciences, Paris},
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     year = {2022},
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Behroz Bidabad; Mir Ahmad Mirshafeazadeh. Harmonic vector fields and the Hodge Laplacian operator on Finsler geometry. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1193-1204. doi : 10.5802/crmath.287. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.287/

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