Illinois Journal of Mathematics

The foliated structure of contact metric $(\kappa,\mu)$-spaces

Beniamino Cappelletti Montano

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In this note, we study the foliated structure of a contact metric $(\kappa,\mu)$-space. In particular, using the theory of Legendre foliations, we give a geometric interpretation of the Boeckx's classification of contact metric $(\kappa,\mu)$-spaces and we find necessary conditions for a contact manifold to admit a compatible contact metric $(\kappa,\mu)$-structure. Finally, we prove that any contact metric $(\kappa,\mu)$-space $M$ whose Boeckx invariant $I_M$ is different from $\pm1$ admits a compatible Sasakian or Tanaka–Webster parallel structure according to the circumstance that $|I_M|>1$ or $|I_M| \lt 1$, respectively.

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Illinois J. Math., Volume 53, Number 4 (2009), 1157-1172.

First available in Project Euclid: 22 November 2010

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Primary: 53C12: Foliations (differential geometric aspects) [See also 57R30, 57R32] 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C26: Hyper-Kähler and quaternionic Kähler geometry, "special" geometry 57R30: Foliations; geometric theory


Cappelletti Montano, Beniamino. The foliated structure of contact metric $(\kappa,\mu)$-spaces. Illinois J. Math. 53 (2009), no. 4, 1157--1172. doi:10.1215/ijm/1290435344.

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