Edmond Bonan (born January 27, 1937, Haifa) is a French mathematician, known particularly for his work on special holonomy

${\displaystyle \sum _{0\leq i\leq n}\ \sum _{0\leq j\leq i}\ \sum _{{1 \over 2}(j-i)\leq k\leq n-i}\ \sum _{0\leq g\leq n-i-k_{+}}\ \sum _{0\leq h\leq i-j-2k_{-}}K_{1}^{h}K_{0}^{g}\zeta _{i,j,i-j+2k}}$

In fact Edmond Bonan proved ^[1] that on a compacte hyperkähler manifold

${\displaystyle p\leq n,\ b_{2p}\geq (p+1)(p+2)/2}$

${\displaystyle p\leq n-1,\ b_{2p-1}\equiv 0,\ mod\ 4}$

${\displaystyle p\leq n,\ \phi _{p}=\sum _{0\leq a+b+c\leq [p/2]}F_{i}^{a}\wedge F_{j}^{b}\wedge F_{k}^{c}\wedge \psi _{a,b,c},\ \ F_{i}\wedge *\psi _{a,b,c}=F_{j}\wedge *\psi _{a,b,c}=F_{k}\wedge *\psi _{a,b,c}=0}$

${\displaystyle \sum _{0\leq i\leq n}\ \sum _{0\leq j\leq i}\ \sum _{{1 \over 2}(j-i)\leq k\leq n-i}\ \sum _{0\leq g\leq n-i-k_{+}}\ \sum _{0\leq h\leq i-j-2k_{-}}K_{1}^{h}K_{0}^{g}\zeta _{i,j,i-j+2k}}$

In fact Edmond Bonan proved ^[2] that on a compacte hyperkähler manifold

${\displaystyle p\leq n,\ b_{2p}\geq (p+1)(p+2)/2}$

${\displaystyle p\leq n-1,\ b_{2p-1}\equiv 0,\ mod\ 4}$

Publications of Edmond Bonan at the Comptes Rendus de l'Académie des Sciences Paris - Séries I - Mathematics

• Structures presque quaternioniennes, vol. 258, 1964, pp. 792–795.
• Connexions presque quaternioniennes, vol. 258, 1964, pp. 1696–1699.
• Structures presque hermitiennes quaternioniennes, vol. 258, 1964, pp. 1988–1991.
• Tenseur de structure d'une variété presque quaternionienne, vol. 259, 1964, pp. 45–48.
• Structure presque quaternale sur une variété différentiable, vol. 261, 1965, pp. 5445–5448.
• Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7), vol. 320, 1966, pp. 127–129.
• Sur un lemme adapté au théorème de Tietze-Uryshon, vol. 270, 1970, pp. 1226–1228.
• Relèvements-Prolongements à valeurs dans les espaces de Fréchet, vol. 272, 1971, pp. 714–717.
• Sur les relèvements-Prolongements à valeurs dans les espaces de Fréchet, vol. 274, 1972, pp. 448–450.
• Sur l'algèbre extérieure d'une variété presque hermitienne quaternionique, vol. 295, 1982, pp. 115–118.
• Sur l'algèbre extérieure d'une variété presque hermitienne quaternionique, vol. 296, 1983, pp. 601–602.
• Comparaison d'un corps convexe avec ses deux ellipsoïdes optimaux, vol. 315, 1992, pp. 557–560.
• Décomposition de l'algèbre extérieure d'une variété hyperkählérienne, vol. 320, 1995, pp. 457–462.
• Sur certaines variétés presque hermitiennes quaternioniques, vol. 320, 1995, pp. 981–984.
• Sur certaines variétés presque hyperhermitiennes, vol. 321, 1995, pp. 95–96.
• Connexions pour les variétés riemanniennes avec une structure de type G2 ou Spin(7), vol. 343, 2006, pp. 755–758.

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Alumnus of the École polytechnique, Bonan completed in 1967 his doctoral dissertation ^[4] in Differential geometry at the University of Paris under the supervision of André Lichnerowicz. From 1968 to 1997, he held the post of lecturer and then professor at the University of Picardie Jules Verne in Amiens, currently professor emeritus. At the same time, from 1969 to 1981, he lectured at École polytechnique.

• G₂ manifold
• Spin(7) manifold.
• Holonomy
• Quaternion-Kähler manifold
• Calibrated geometry
• Hypercomplex manifold
• Hyperkähler manifold

If the forms ${\displaystyle F_{2}}$ and ${\displaystyle F_{3}}$ are associated to ${\displaystyle J}$ and ${\displaystyle K}$ then the complex symplectic form ${\displaystyle \mu =F_{2}+iF_{3}}$ can be written ${\displaystyle \mu =\sum _{\alpha =1}^{n}\theta ^{\alpha }\wedge \theta ^{\alpha +n}}$ where ${\displaystyle \theta ^{r}=0\ (1\leq r\leq 2n)\ }$ define the set of ${\displaystyle 2n}$-plane ${\displaystyle S_{2n}}$ field of the tangent eigenvector for ${\displaystyle I}$ with eigenvalue say i. We calculate ${\displaystyle d\mu :=\sum _{\alpha =1}^{n}(d\theta ^{\alpha }\wedge \theta ^{\alpha +n}-\theta ^{\alpha }\wedge d\theta ^{\alpha +n})}$. If ${\displaystyle F_{2}}$ and ${\displaystyle F_{3}}$ are closed then from ${\displaystyle d\mu =0\ }$, wedging by all but one ${\displaystyle \theta ^{s}(1\leq s\leq 2n)}$ we get ${\displaystyle (\wedge _{s=1}^{2n}\theta ^{s})\wedge d\theta ^{r}=0\ (1\leq r\leq 2n)\ }$. By Frobenius theorem the field ${\displaystyle S_{2n}}$ is completely integrable and the almost complex structure ${\displaystyle I}$ is then analytic.